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A-Z From Wikipedia, the free encyclopedia Chapter 1

Antepredicament

Antepredicaments, in logic, are certain previous matters requisite to a more easy and clear apprehension of the doctrine of predicaments or . Such are definitions of common terms, as equivocals, univocals, etc., with divisions of things, their differences, etc. They are thus called because treated them before the predicaments, hoping that the thread of discourse might not afterwards be interrupted.

1.1

• This article incorporates text from a publication now in the public domain: Chambers, Ephraim, ed. (1728). "*article needed". Cyclopædia, or an Universal Dictionary of Arts and Sciences (first ed.). James and John Knapton, et al.

2 Chapter 2

Apophasis

Not to be confused with Apophysis (disambiguation), Apoptosis, or Apophis (disambiguation).

Apophasis is a rhetorical device wherein the speaker or writer brings up a by either denying it, or denying that it should be brought up.*[1] Accordingly, it can be seen as a rhetorical relative of irony. Also called paralipsis (παράλειψις) – also spelled paraleipsis or paralepsis –, or occupatio,*[2]*[3]*[4]*[5] and known also as prae- teritio, preterition, antiphrasis (ἀντίφρασις), or parasiopesis (παρασιώπησις), apophasis is usually employed to make a subversive attack, which makes it a frequently used tactic in political speeches to make an attack on one's opponent. Using apophasis in this way is often considered to be bad form. The device is typically used to distance the speaker from unfair claims, while still bringing them up. For instance, a politician might say, “I don't even want to talk about the allegations that my opponent is a drunk.”It can also be used in denying such claims entirely, for example by saying “I'm sure that my opponent is not lying; however, his grasp of the seems to be shaky.”*[6]

2.1 Name

The is Late Latin, from Greek ἀπόφασις from ἀπόφημι apophemi,*[7] “to say no”.*[8]

2.2 Proslepsis

When paralipsis is taken to its extreme, proslepsis occurs, and the speaker provides full details stating and/or drawing attention to something in the very act of pretending to pass it over: “I will not stoop to mentioning the occasion last winter when our esteemed opponent was found asleep in an alleyway with an empty bottle of vodka still pressed to his lips.”*[9] Paralipsis was often used by Cicero in his orations:

“Obliviscor iam iniurias tuas, Clodia, depono memoriam doloris mei”(“I now forget your wrongs, Clodia, I aside the memory of my [that you caused].”) —Cicero,"Pro Caelio", Chapter 50

“It would be superfluous in me to point out to your lordship that this is war.” —Charles Francis Adams, U.S. Ambassador to Britain, dispatch to Earl Russell, 5 September 1863, concerning Britain's relations with the Confederacy.

“Ssh,”said Grace Makutsi, putting a finger to her lips. “It's not polite to talk about it. SO I won't mention the Double Comfort Furniture Shop, which is one of the businesses my fiance owns, you know.

3 4 CHAPTER 2. APOPHASIS

I must not talk about that. But do you know the store, Mma? If you save up, you should come in some day and buy a chair.” —Alexander McCall Smith, Blue Shoes and , Chapter 4

As a rhetorical device, it can serve various purposes, often dependent on the relationship of the speaker to the ad- dressee and the extent of their shared knowledge. Apophasis is rarely literal; instead, it conveys through implications that may depend on this context. As an example of how meaning shifts, the English phrase“needless to say”invokes shared understanding, but its actual meaning depends on whether that understanding was really shared. The speaker is alleging that it is not necessary to say something because the addressee already knows it, but is it so? If it is, it may merely emphasize a pertinent . If the knowledge is weighted with , it may be an indirect way of levying an accusation (“needless to say, because you are responsible"). If the addressee does not actually already possess the knowledge, it may be a way to condescend: the speaker suspected as much but wanted to call attention to the addressee's ignorance. Conversely, it could be a sincere and polite way to share necessary that the addressee may or may not know without implying that the addressee is ignorant. An example of the last type of paralipsis/paralepsis, where it serves to politely avoid suggestion of ignorance, is found in the narrative style of Adso of Melk in Umberto Eco's The Name of the Rose, where the character fills in details of early fourteenth-century history for the reader by stating it is unnecessary to speak of them.*[10]

2.3 With proper

When it is taboo to speak of an entity by name, an epithet or sobriquet can be used in place of the name. For example, when it was forbidden in Myanmar to speak the name of political prisoner Aung San Suu Kyi, she was commonly referred to as “The Lady”. Various names of in Judaism are used to avoid writing or speaking sacred names. The name of the fictional Lord Voldemort in the popular Harry Potter is taboo, and he is commonly referred with epithets such as “He-Who-Must-Not-Be-Named”and “You-Know-Who”.

2.4 Examples

2.4.1 Political

In 1988, President Ronald Reagan said of Michael Dukakis, a presidential candidate who was rumored to have received psychological treatment, “Look, I'm not going to pick on an invalid.”*[11] In 2015, Donald Trump said of fellow Republican presidential candidate and former Hewlett-Packard CEO Carly Fiorina, “I promised I would not say that she ran Hewlett-Packard into the ground.”*[12] In 2016, Trump tweeted of journalist Megyn Kelly, “I refuse to call Megyn Kelly a bimbo, because that would not be politically correct.”*[13]

2.5 See also

• from ignorance • • Elephant in the room • The lady doth protest too much, methinks • Glossary of rhetorical terms • • Apophatic • Ironic process theory • Unsaid 2.6. NOTES 5

2.6 Notes

[1] Baird, A. Craig; Thonssen, Lester (1948). “Chapter 15 The Style of Public Address”. Speech Criticism, the Development of Standards for Rhetorical Appraisal. Ronald Press Co. p. 432.

[2] Kathryn L. Lynch (2000). Chaucer's Philosophical Visions. Boydell & Brewer Ltd. pp. 144–. ISBN 978-0-85991-600-4. Retrieved 22 May 2013.

[3] Anthony David Nuttall (1980). Overheard by God: fiction and prayer in Herbert, Milton, Dante and St. John. Methuen. p. 96. Retrieved 22 May 2013.

[4] Fārūq Shūshah; Muḥammad Muḥammad ʻInānī (al-Duktūr.) (2003). Beauty bathing in the river: poems. Egyptian State Pub. House (GEBO). p. 19. Retrieved 22 May 2013.

[5] K. V. Tirumalesh (1999). Matters: Essays on Language, Literature, and Translation. Allied Publishers. p. 113. ISBN 978-81-7023-947-5. Retrieved 22 May 2013.

[6] Safire, William (October 9, 1988). “ON LANGUAGE; Debatemanship”. The New York Times.

[7] “Henry George Liddell, Robert Scott, A Greek-English Lexicon”. Perseus Digital Library. Tufts University. Retrieved 7 April 2013.

[8] “apophasis”. Dictionary.com. Retrieved 1 June 2011.

[9] Burton, Gideon O. “paralipsis”. Silva Rhetoricae: The Forest of . Brigham Young University. Archived from the original on 25 May 2011. Retrieved 1 June 2011.

[10] Eco, Umberto (1984). “Postscript to the Name of the Rose”. The Name of the Rose. Translated by William Weaver. San Diego: Harcourt Brace Jovanovich. p. 39. Eco and Weaver use the spelling paralepsis or “passing over”for the phenomenon.

[11] Lamar Jr., Jacob V. (15 August 1988). “Reagan: Part Fixer, Part Hatchet Man”. Time Magazine. Retrieved 16 August 2015.

[12] Strauss, Daniel (14 August 2015). “Donald Trump bad-mouths his rivals”. Politico. Retrieved 16 August 2015.

[13] “Donald J. Trump on Twitter”. Twitter. Retrieved 2016-01-27.

2.7 References

• Smyth, Herbert Weir (1984) [1920]. Greek Grammar. Cambridge, MA: Harvard University Press. p. 680. ISBN 0-674-36250-0.

2.8 External links

• Figures of rhetoric: Apophasis

• A Handbook of Rhetorical Devices: Apophasis • Wordsmith: Paralipsis Chapter 3

Argument map

An argument map

In and philosophy, an argument map or argument diagram is a visual representation of the structure of an argument. An argument map typically includes the key components of the argument, traditionally called the conclusion and the , also called contention and .*[1] Argument maps can also show co-premises, objections, counterarguments, rebuttals, and lemmas. There are different styles of argument map but they are often functionally equivalent and represent an argument's individual claims and the relationships between them. Argument maps are commonly used in the context of teaching and applying .*[2] The purpose of mapping is to uncover the logical structure of , identify unstated assumptions, evaluate the support an argument offers for a conclusion, and understanding of debates. Argument maps are often designed to support deliberation of issues, and arguments in wicked problems.*[3] An argument map is not to be confused with a map or a mind map, which are less strict in relating claims.

3.1 Key features of an argument map

A number of different kinds of argument map have been proposed but the most common, which Chris Reed and Glenn Rowe called the standard diagram,*[4] consists of a tree structure with each of the reasons leading to the conclusion.

6 3.2. REPRESENTING AN ARGUMENT AS AN ARGUMENT MAP 7

There is no consensus as to whether the conclusion should be at the top of the tree with the reasons leading up to it or whether it should be at the bottom with the reasons leading down to it.*[4] Another variation diagrams an argument from left to right.*[5] According to Doug Walton and colleagues, an argument map has two basic components: “One component is a set of circled numbers arrayed as points. Each number represents a ( or conclusion) in the argument diagrammed. The other component is a set of lines or arrows joining the points. Each line () represents an . The whole network of points and lines represents a kind of overview of the reasoning in the given argu- ment...”*[6] With the introduction of software for producing argument maps, it has become common for argument maps to consist of boxes containing the actual rather than numbers referencing those propositions. There is disagreement on the terminology to be used when describing argument maps,*[7] but the standard diagram contains the following structures: Dependent premises or co-premises, where at least one of the joined premises requires another premise before it can give support to the conclusion: An argument with this structure has been called a linked argument.*[8] Independent premises, where the premise can support the conclusion on its own: Although independent premises may jointly make the conclusion more convincing, this is to be distinguished from situations where a premise gives no support unless it is joined to another premise. Where several premises or groups of premises lead to a final conclusion the argument might be described as convergent. This is distinguished from a divergent argument where a single premise might be used to support two separate conclusions.*[9] Intermediate conclusions or sub-conclusions, where a claim is supported by another claim that is used in turn to support some further claim, i.e. the final conclusion or another intermediate conclusion: In the following diagram, 4 is an intermediate conclusion in that it is a conclusion in relation to statement 5 but is a premise in relation to the final conclusion, i.e. statement 1. An argument with this structure is sometimes called a complex argument. If there is a single chain of claims containing at least one intermediate conclusion, the argument is sometimes described as a serial argument or a chain argument.*[10] Each of these structures can be represented by the equivalent “box and line”approach to argument maps. In the following diagram, the conclusion is shown at the top, and the boxes linked to it represent supporting reasons, which comprise one or more premises. The , 1A, comprising two premises 1A-a and 1A-b, support the conclusion: Argument maps can also represent counterarguments. In the following diagram, the objection 1A weakens the con- clusion, while the reason 2A supports premise 1A-b of the objection:

3.2 Representing an argument as an argument map

A written text can be transformed into an argument map by following a sequence of steps. Monroe Beardsley's 1950 book Practical Logic recommended the following procedure:*[11]

1. Separate statements by brackets and number them.

2. Put circles around the logical indicators.

3. Supply, in parenthesis, any logical indicators that are left out.

4. Set out the statements in a diagram in which arrows show the relationships between statements.

Beardsley gave the first example of a text being analysed in this way:

Though ① [people who talk about the “social significance”of the arts don’t like to admit it], ② [music and painting are bound to suffer when they are turned into mere vehicles for propaganda]. For ③ [propaganda appeals to the crudest and most vulgar feelings]: (for) ④ [look at the academic monstrosities produced by the official Nazi painters]. What is more important, ⑤ [art must be an end in itself for the artist], because ⑥ [the artist can do the best work only in an atmosphere of complete freedom].

Beardsley said that the conclusion in this example is statement ②. Statement ④ needs to be rewritten as a declarative , e.g. “Academic monstrosities [were] produced by the official Nazi painters.”Statement ① points out that the conclusion isn't accepted by everyone, but statement ① is omitted from the diagram because it doesn't support the 8 CHAPTER 3. ARGUMENT MAP

Statements 1 and 2 are dependent premises or co-premises 3.3. HISTORY 9

Statements 2, 3, 4 are independent premises

conclusion. Beardsley said that the logical relation between statement ③ and statement ④ is unclear, but he proposed to diagram statement ④ as supporting statement ③. More recently, philosophy professor Maralee Harrell recommended the following procedure:*[12]

1. Identify all the claims being made by the author.

2. Rewrite them as independent statements, eliminating non-essential .

3. Identify which statements are premises, sub-conclusions, and the main conclusion.

4. Provide missing, implied conclusions and implied premises. (This is optional depending on the purpose of the argument map.)

5. Put the statements into boxes and draw a line between any boxes that are linked.

6. Indicate support from premise(s) to (sub)conclusion with arrows.

Argument maps are useful not only for representing and analyzing existing writings, but also for thinking through issues as part of a problem-structuring process or writing process, which has been called “reflective argumentation” .*[13]

3.3 History 10 CHAPTER 3. ARGUMENT MAP

Statement 4 is an intermediate conclusion or sub-conclusion

3.3.1 The philosophical origins and tradition of argument mapping

In the Elements of Logic, which was published in 1826 and issued in many subsequent editions,*[14] Archbishop Richard Whately gave probably the first form of an argument map, introducing it with the suggestion that “many students probably will find it a very clear and convenient mode of exhibiting the logical analysis of the course of argument, to draw it out in the form of a Tree, or Logical Division”. However, the technique did not become widely used, possibly because for complex arguments, it involved much writing and rewriting of the premises. Legal and theorist John Henry Wigmore produced maps of legal arguments using numbered premises in the early 20th century,*[15] based in part on the ideas of 19th century philosopher who used lines 3.3. HISTORY 11

A box and line diagram

to indicate relations between terms.*[16]

3.3.2 Anglophone argument diagramming in the 20th century

Dealing with the failure of formal reduction of informal argumentation, English speaking de- veloped diagrammatic approaches to informal reasoning over a period of fifty years. Monroe Beardsley proposed a form of argument diagram in 1950.*[11] His method of marking up an argument and representing its components with linked numbers became a standard and is still widely used. He also introduced terminology that is still current describing convergent, divergent and serial arguments. , in his groundbreaking and influential 1958 book The Uses of Argument,*[17] identified several elements to an argument which have been generalized. The Toulmin diagram is widely used in educational critical teaching.*[18]*[19] Whilst Toulmin eventually had a significant impact on the development of informal logic he had little initial impact and the Beardsley approach to diagramming arguments along with its later developments became the standard approach in this field. Toulmin introduced something that was missing from Beardsley's approach. In Beardsley, “arrows link reasons and conclusions (but) no support is given to the implication itself between them. There is no theory, in other words, of inference distinguished from logical deduction, the passage is always deemed not controversial and not subject to support and evaluation”.*[20] Toulmin introduced the concept of warrant which “can be considered as representing the reasons behind the inference, the backing that authorizes the link”.*[21] Beardsley's approach was refined by Stephen N. Thomas, whose 1973 book Practical Reasoning In Natural Lan- guage*[22] introduced the term linked to describe arguments where the premises necessarily worked together to support the conclusion.*[23] However, the actual distinction between dependent and independent premises had been made prior to this.*[23] The introduction of the linked structure made it possible for argument maps to represent missing or “hidden”premises. In addition, Thomas suggested showing reasons both for and against a conclusion with the reasons against being represented by dotted arrows. Thomas introduced the term argument diagram and defined basic reasons as those that were not supported by any others in the argument and the final conclusion as that which was not used to support any further conclusion. Michael Scriven further developed the Beardsley-Thomas approach in his 1976 book Reasoning.*[24] Whereas Beardsley had said “At first, write out the statements...after a little practice, refer to the statements by number alone”*[25] Scriven advocated clarifying the meaning of the statements, listing them and then using a tree diagram with numbers to display the structure. Missing premises (unstated assumptions) were to be included and indicated with an alphabetical letter instead of a number to mark them off from the explicit statements. Scriven introduced 12 CHAPTER 3. ARGUMENT MAP

A sample argument using objections counterarguments in his diagrams, which Toulmin had defined as rebuttal.*[26] This also enabled the diagramming of “balance of consideration”arguments.*[27] In the 1990s, Tim van Gelder and colleagues developed a series of computer software applications that permitted the premises to be fully stated and edited in the diagram, rather than in a legend.*[28] Van Gelder's first program, Reason!Able, was superseded by two subsequent programs, bCisive and Rationale.*[29] Throughout the 1990s and 2000s, many other software applications were developed for argument visualization. By 2013, more than 60 such software systems existed.*[30] One of the differences between these software systems is whether collaboration is supported.*[31] Single-user argumentation systems include Convince Me, iLogos, LARGO, Athena, Araucaria, and Carneades; small group argumentation systems include Digalo, QuestMap, Compendium, 3.3. HISTORY 13

A diagram of the example from Beardsley's Practical Logic

Belvedere, and AcademicTalk; community argumentation systems include Debategraph and Collaboratorium.*[31] For more software examples, see: § External links. In 1998 a series of large-scale argument maps released by Robert E. Horn stimulated widespread interest in argument mapping.*[32] 14 CHAPTER 3. ARGUMENT MAP

A box and line diagram of Beardsley's example, produced using Harrell's procedure

From Whately's Elements of Logic p467, 1852 edition

3.4 Applications

Argument maps have been applied in many areas, but foremost in educational, academic and business settings, includ- ing design rationale.*[33] Argument maps are also used in forensic science,*[34] , and artificial intelligence.*[35] It has also been proposed that argument mapping has a great potential to improve how we understand and execute , in to the ongoing evolution of e-democracy.*[36] 3.4. APPLICATIONS 15

Wigmore evidence chart, from 1905

A Toulmin argument diagram, redrawn from his 1959 Uses of Argument

3.4.1 Difficulties with the philosophical tradition

It has traditionally been hard to separate teaching critical thinking from the philosophical tradition of teaching logic and method, and most critical thinking textbooks have been written by . Informal logic textbooks are re- plete with philosophical examples, but it is unclear whether the approach in such textbooks transfers to non-philosophy students.*[18] There appears to be little statistical effect after such classes. Argument mapping, however, has a mea- surable effect according to many studies.*[37] For example, instruction in argument mapping has been shown to improve the critical thinking skills of business students.*[38] 16 CHAPTER 3. ARGUMENT MAP

A generalised Toulmin diagram

Scriven's argument diagram. The explicit premise 1 is conjoined with additional unstated premises a and b to imply 2.

3.4.2 Evidence that argument mapping improves critical thinking ability

There is that the skills developed in argument-mapping-based critical thinking courses substan- tially transfer to critical thinking done without argument maps. Alvarez's meta-analysis found that such critical thinking courses produced gains of around 0.70 SD, about twice as much as standard critical-thinking courses.*[39] The tests used in the reviewed studies were standard critical-thinking tests.

3.4.3 How argument mapping helps with critical thinking

The use of argument mapping has occurred within a number of disciplines, such as philosophy, management report- ing, military and intelligence analysis, and public debates.*[33] Logical structure: Argument maps display an argument's logical structure more clearly than does the standard linear way of presenting arguments. Critical thinking : In learning to argument map, students master such key critical thinking concepts as“reason” , “objection”, “premise”, “conclusion”, “inference”, “rebuttal”, “unstated assumption”, “co-premise” , “strength of evidence”, “logical structure”, “independent evidence”, etc. Mastering such concepts is not just a matter of memorizing their definitions or even being able to apply them correctly; it is also understanding why the distinctions these words mark are important and using that understanding to guide one's reasoning. 3.5. STANDARDS 17

Visualization: Humans are highly visual and argument mapping may provide students with a basic set of visual schemas with which to understand argument structures. More careful reading and listening: Learning to argument map teaches people to read and listen more carefully, and highlights for them the key “What is the logical structure of this argument?" and“How does this sentence fit into the larger structure?" In-depth cognitive processing is thus more likely. More careful writing and speaking: Argument mapping helps people to state their reasoning and evidence more precisely, because the reasoning and evidence must fit explicitly into the map's logical structure. Literal and intended meaning: Often, many statements in an argument do not precisely assert what the author meant. Learning to argument map enhances the complex skill of distinguishing literal from intended meaning. Externalization: Writing something down and reviewing what one has written often helps reveal gaps and clarify one's thinking. Because the logical structure of argument maps is clearer than that of linear prose, the benefits of mapping will exceed those or ordinary writing. Anticipating replies: Important to critical thinking is anticipating objections and considering the plausibility of different rebuttals. Mapping develops this anticipation skill, and so improves analysis.

3.5 Standards

3.5.1 Argument Interchange Format

The Argument Interchange Format, AIF, is an international effort to develop a representational mechanism for ex- changing argument resources between research groups, tools, and domains using a semantically rich language.*[40] AIF-RDF is the extended represented in the Resource Framework (RDFS) semantic language. Though AIF is still something of a moving target, it is settling down.*[41]

3.5.2 Legal Knowledge Interchange Format

The Legal Knowledge Interchange Format (LKIF),*[42] developed in the European ESTRELLA project,*[43] is an XML schema for rules and arguments, designed with the goal of becoming a standard for representing and inter- changing policy, legislation and cases, including their justificatory arguments, in the legal domain. LKIF builds on and uses the Web (OWL) for representing concepts and includes a reusable basic ontology of legal concepts.

3.6 See also

• Flow (policy debate) • Informal • Information graphics

3.7 Notes

[1] Freeman 1991, pp. 49–90

[2] For example: Davies 2012; Facione 2013, p. 86; Fisher 2004; Kelley 2014, p. 73; Kunsch, Schnarr & van Tyle 2014; Walton 2013, p. 10

[3] For example: Culmsee & Awati 2013; Hoffmann & Borenstein 2013; Metcalfe & Sastrowardoyo 2013; Ricky Ohl, “Computer supported argument visualisation: modelling in consultative democracy around wicked problems”, in Okada, Buckingham Shum & Sherborne 2014, pp. 361–380

[4] Reed & Rowe 2007, p. 64

[5] For example: Walton 2013, pp. 18–20 18 CHAPTER 3. ARGUMENT MAP

[6] Reed, Walton & Macagno 2007, p. 2 [7] Freeman 1991, pp. 49–90; Reed & Rowe 2007 [8] Harrell 2010, p. 19 [9] Freeman 1991, pp. 91–110; Harrell 2010, p. 20 [10] Beardsley 1950, pp. 18–19; Reed, Walton & Macagno 2007, pp. 3–8; Harrell 2010, pp. 19–21 [11] Beardsley 1950 [12] Harrell 2010, p. 28 [13] For example: Hoffmann & Borenstein 2013; Hoffmann 2015 [14] Whately 1834 (first published 1826) [15] Wigmore 1913 [16] Goodwin 2000 [17] Toulmin 2003 (first published 1958) [18] Simon, Erduran & Osborne 2006 [19] Böttcher & Meisert 2011; Macagno & Konstantinidou 2013 [20] Reed, Walton & Macagno 2007, p. 8 [21] Reed, Walton & Macagno 2007, p. 9 [22] Thomas 1997 (first published 1973) [23] Snoeck Henkemans 2000, p. 453 [24] Scriven 1976 [25] Beardsley 1950, p. 21 [26] Reed, Walton & Macagno 2007, p. 10–11 [27] van Eemeren et al. 1996, p. 175 [28] van Gelder 2007 [29] Berg et al. 2009 [30] Walton 2013, p. 11 [31] Scheuer et al. 2010 [32] Holmes 1999; Horn 1998 and Robert E. Horn, “Infrastructure for navigating interdisciplinary debates: critical decisions for representing argumentation”, in Kirschner, Buckingham Shum & Carr 2003, pp. 165–184 [33] Kirschner, Buckingham Shum & Carr 2003; Okada, Buckingham Shum & Sherborne 2014 [34] For example: Bex 2011 [35] For example: Verheij 2005; Reed, Walton & Macagno 2007; Walton 2013 [36] Hilbert 2009 [37] Twardy 2004; Álvarez Ortiz 2007; Harrell 2008; Yanna Rider and Neil Thomason, “Cognitive and pedagogical benefits of argument mapping: LAMP guides the way to better thinking”, in Okada, Buckingham Shum & Sherborne 2014, pp. 113–134; Dwyer 2011; Davies 2012 [38] Carrington et al. 2011; Kunsch, Schnarr & van Tyle 2014 [39] Álvarez Ortiz 2007, pp. 69–70 et seq [40] See the AIF original draft description (2006) and the full AIF-RDF ontology specifications in RDFS format. [41] Bex et al. 2013 [42] Boer, Winkels & Vitali 2008 [43] “Estrella project website”. estrellaproject.org. Archived from the original on 2016-02-12. Retrieved 2016-02-24. 3.8. REFERENCES 19

3.8 References

• Álvarez Ortiz, Claudia María (2007). Does philosophy improve critical thinking skills? (PDF) (M.A.). Depart- ment of Philosophy, . OCLC 271475715.

• Beardsley, Monroe C. (1950). Practical logic. New York: Prentice-Hall. OCLC 4318971.

• Berg, Timo ter; van Gelder, Tim; Patterson, Fiona; Teppema, Sytske (2009). Critical thinking: reasoning and communicating with Rationale. : Pearson Benelux. ISBN 9043018015. OCLC 301884530.

• Bex, Floris J. (2011). Arguments, stories and criminal evidence: a formal hybrid theory. Law and philosophy library 92. Dordrecht; New York: Springer. doi:10.1007/978-94-007-0140-3. ISBN 9789400701397. OCLC 663950184.

• Bex, Floris J.; Modgil, Sanjay; Prakken, Henry; Reed, Chris (2013). “On logical specifications of the Argu- ment Interchange Format”(PDF). Journal of Logic and Computation 23 (5): 951–989. doi:10.1093/logcom/exs033.

• Boer, Alexander; Winkels, Radboud; Vitali, Fabio (2008). “MetaLex XML and the Legal Knowledge Inter- change Format” (PDF). In Casanovas, Pompeu; Sartor, Giovanni; Casellas, Núria; Rubino, Rossella. Com- putable models of the law: , dialogues, games, . Lecture notes in computer science 4884. Berlin; New York: Springer. pp. 21–41. doi:10.1007/978-3-540-85569-9_2. ISBN 9783540855682. OCLC 244765580. Retrieved 2016-02-24.

• Böttcher, Florian; Meisert, Anke (February 2011). “Argumentation in science education: a model-based framework”. Science & Education 20 (2): 103–140. doi:10.1007/s11191-010-9304-5.

• Carrington, Michal; Chen, Richard; Davies, Martin; Kaur, Jagjit; Neville, Benjamin (June 2011).“The effec- tiveness of a single intervention of computer‐aided argument mapping in a marketing and a financial accounting subject”(PDF). Higher Education Research & Development 30 (3): 387–403. doi:10.1080/07294360.2011.559197. Retrieved 2016-02-24.

• Culmsee, Paul; Awati, Kailash (2013) [2011].“Chapter 7: Visualising reasoning, and Chapter 8: Argumentation- based rationale”. The heretic's guide to best practices: the of managing complex problems in organisations. Bloomington, IN: iUniverse, Inc. pp. 153–211. ISBN 9781462058549. OCLC 767703320.

• Davies, Martin (Summer 2012). “Computer-aided argument mapping as a tool for teaching critical thinking” . International Journal of Learning and Media 4 (3-4): 79–84. doi:10.1162/IJLM_a_00106.

• Dwyer, Christopher Peter (2011). The evaluation of argument mapping as a learning tool (PDF) (Ph.D.). School of , National University of Ireland, Galway. OCLC 812818648. Retrieved 2016-02-24.

• van Eemeren, Frans H.; Grootendorst, Rob; Snoeck Henkemans, A. Francisca; Blair, J. Anthony; Johnson, Ralph H.; Krabbe, Erik C. W.; Plantin, Christian; Walton, Douglas N.; Willard, Charles A.; Woods, John (1996). Fundamentals of argumentation theory: a handbook of historical backgrounds and contemporary de- velopments. Mahwah, NJ: Lawrence Erlbaum Associates. doi:10.4324/9780203811306. ISBN 0805818618. OCLC 33970847.

• Facione, Peter A. (2013) [2011]. THINK critically (2nd ed.). Boston: Pearson. ISBN 0205490980. OCLC 770694200.

• Fisher, Alec (2004) [1988]. The logic of real arguments (2nd ed.). Cambridge; New York: Cambridge Univer- sity Press. doi:10.1017/CBO9780511818455. ISBN 0521654815. OCLC 54400059. Retrieved 2016-02-24.

• Freeman, James B. (1991). and the macrostructure of arguments: a theory of argument structure. Berlin; New York: Foris Publications. ISBN 3110133903. OCLC 24429943. Retrieved 2016-02-24.

• van Gelder, Tim (2007). “The rationale for Rationale” (PDF). Law, and Risk 6 (1-4): 23–42. doi:10.1093/lpr/mgm032.

• Goodwin, Jean (2000). “Wigmore's chart method”. Informal Logic 20 (3): 223–243.

• Harrell, Maralee (December 2008). “No computer program required: even pencil-and-paper argument map- ping improves critical-thinking skills”(PDF). Teaching Philosophy 31 (4): 351–374. doi:10.5840/teachphil200831437. 20 CHAPTER 3. ARGUMENT MAP

• Harrell, Maralee (August 2010). “Creating argument diagrams”. academia.edu.

• Hilbert, Martin (April 2009).“The maturing concept of e-democracy: from e-voting and online consultations to democratic value out of jumbled online chatter” (PDF). Journal of Information Technology and Politics 6 (2): 87–110. doi:10.1080/19331680802715242.

• Hoffmann, Michael H. G. (November 2015). “Reflective argumentation: a cognitive function of arguing”. Argumentation. doi:10.1007/s10503-015-9388-9.

• Hoffmann, Michael H. G.; Borenstein, Jason (February 2013).“Understanding ill-structured engineering problems through a collaborative learning and argument visualization approach”. Science and Engineering Ethics 20 (1): 261–276. doi:10.1007/s11948-013-9430-y. PMID 23420467.

• Holmes, Bob (10 July 1999). “Beyond words”. New Scientist (2194). Archived from the original on 28 September 2008.

• Horn, Robert E. (1998). Visual language: global for the 21st century. Bainbridge Island, WA: MacroVU, Inc. ISBN 189263709X. OCLC 41138655.

• Kelley, David (2014) [1988]. The art of reasoning: an introduction to logic and critical thinking (4th ed.). New York: W. W. Norton & Company. ISBN 0393930785. OCLC 849801096.

• Kirschner, Paul Arthur; Buckingham Shum, Simon J; Carr, Chad S, eds. (2003). Visualizing argumentation: software tools for collaborative and educational sense-making. Computer supported work. New York: Springer. doi:10.1007/978-1-4471-0037-9. ISBN 1852336641. OCLC 50676911. Retrieved 2016- 02-24.

• Kunsch, David W.; Schnarr, Karin; van Tyle, Russell (November 2014). “The use of argument mapping to enhance critical thinking skills in business education”. Journal of Education for Business 89 (8): 403–410. doi:10.1080/08832323.2014.925416.

• Macagno, Fabrizio; Konstantinidou, Aikaterini (August 2013). “What students' arguments can tell us: using argumentation schemes in science education”. Argumentation 27 (3): 225–243. doi:10.1007/s10503-012- 9284-5.

• Metcalfe, Mike; Sastrowardoyo, Saras (November 2013). “Complex project conceptualisation and argument mapping”. International Journal of Project Management 31 (8): 1129–1138. doi:10.1016/j.ijproman.2013.01.004.

• Okada, Alexandra; Buckingham Shum, Simon J; Sherborne, Tony, eds. (2014) [2008]. Knowledge cartogra- phy: software tools and mapping techniques. Advanced information and knowledge processing (2nd ed.). New York: Springer. doi:10.1007/978-1-4471-6470-8. ISBN 9781447164692. OCLC 890438015. Retrieved 2016-02-24.

• Reed, Chris; Rowe, Glenn (2007).“A pluralist approach to argument diagramming”(PDF). Law, Probability and Risk 6 (1-4): 59–85. doi:10.1093/lpr/mgm030. Retrieved 2016-02-24.

• Reed, Chris; Walton, Douglas; Macagno, Fabrizio (March 2007). “Argument diagramming in logic, law and artificial intelligence”. The Knowledge Engineering Review 22 (1): 1–22. doi:10.1017/S0269888907001051.

• Scheuer, Oliver; Loll, Frank; Pinkwart, Niels; McLaren, Bruce M. (2010). “Computer-supported argumen- tation: a review of the state of the art” (PDF). International Journal of Computer-Supported Collaborative Learning 5 (1): 43–102. doi:10.1007/s11412-009-9080-x.

• Scriven, Michael (1976). Reasoning. New York: McGraw-Hill. ISBN 0070558825. OCLC 2800373.

• Simon, Shirley; Erduran, Sibel; Osborne, Jonathan (2006). “Learning to teach argumentation: research and development in the science classroom”(PDF). International Journal of Science Education 28 (2-3): 235–260. doi:10.1080/09500690500336957.

• Snoeck Henkemans, A. Francisca (November 2000). “State-of-the-art: the structure of argumentation”. Argumentation 14 (4): 447–473. doi:10.1023/A:1007800305762.

• Thomas, Stephen N. (1997) [1973]. Practical reasoning in natural language (4th ed.). Upper Saddle River, NJ: Prentice-Hall. ISBN 0136782698. OCLC 34745923. 3.9. FURTHER READING 21

• Toulmin, Stephen E. (2003) [1958]. The uses of argument (Updated ed.). Cambridge; New York: Cambridge University Press. doi:10.1017/CBO9780511840005. ISBN 0521534836. OCLC 57253830. Retrieved 2016- 02-24.

• Twardy, Charles R. (June 2004). “Argument maps improve critical thinking” (PDF). Teaching Philosophy 27 (2): 95–116. doi:10.5840/teachphil200427213.

• Verheij, Bart (2005). Virtual arguments: on the design of argument assistants for lawyers and other arguers. Information technology & law series 6. : T.M.C. Asser Press. ISBN 9789067041904. OCLC 59617214.

• Walton, Douglas N. (2013). Methods of argumentation. Cambridge; New York: Cambridge University Press. doi:10.1017/CBO9781139600187. ISBN 1107677335. OCLC 830523850. Retrieved 2016-02-24.

• Whately, Richard (1834) [1826]. Elements of logic: comprising the substance of the article in the Encyclopædia metropolitana: with additions, &c. (5th ed.). : B. Fellowes. OCLC 1739330. Retrieved 2016-02-24.

• Wigmore, John Henry (1913). The of judicial : as given by logic, psychology, and general experience, and illustrated in judicial trials. Boston: Little Brown. OCLC 1938596. Retrieved 2016-02-24.

3.9 Further reading

• van Eemeren, Frans H.; Garssen, Bart; Krabbe, Erik C. W.; Snoeck Henkemans, A. Francisca; Verheij, Bart; Wagemans, Jean H. M. (2014). Handbook of argumentation theory. New York: Springer. doi:10.1007/978- 90-481-9473-5. ISBN 9789048194728. OCLC 871004444.

• Facione, Peter A.; Facione, Noreen C. (2007). Thinking and reasoning in human decision making: the method of argument and heuristic analysis. Milbrae, CA: California Academic Press. ISBN 1891557580. OCLC 182039452.

• van Gelder, Tim (17 February 2009).“What is argument mapping?". timvangelder.com. Retrieved 12 January 2015.

• Harrell, Maralee (June 2005). “Using argument diagramming software in the classroom” (PDF). Teaching Philosophy 28 (2): 163–177. doi:10.5840/teachphil200528222.

3.10 External links

3.10.1 Argument mapping software

• Araucaria (open source, cross platform/Java)

• Argumentative (open source, Windows); supports single-user, graphical argumentation

• Argunet (open source, cross platform)

• Compendium (open source, cross platform/Java)

• iLogos (cross platform/Java)

• OVA (Web based, Online Visualisation of Argument)

• PIRIKA (PIlot for the RIght Knowledge and Argument) (open source, Linux, Windows) 22 CHAPTER 3. ARGUMENT MAP

3.10.2 Online, collaborative software

• AGORA-net (user interface in English, German, Spanish, Chinese, and Russian) • bCisiveOnline

• Carneades (open source, argument (re)construction, evaluation, mapping and interchange) • Collam (JavaScript library for visualizing argument maps)

• Debategraph • TruthMapping

• Arguman (Open source, English, Turkish, and Chinese) Chapter 4

Argumentation theory

Argumentation theory, or argumentation, is the interdisciplinary study of how conclusions can be reached through logical reasoning; that is, claims based, soundly or not, on premises. It includes the arts and sciences of civil debate, dialogue, conversation, and . It studies rules of inference, logic, and procedural rules in both artificial and real world settings. Argumentation includes debate and negotiation which are concerned with reaching mutually acceptable conclusions. It also encompasses eristic dialog, the branch of social debate in which victory over an opponent is the primary goal. This art and science is often the means by which people protect their beliefs or self-interests in rational dialogue, in common parlance, and during the process of arguing. Argumentation is used in law, for example in trials, in preparing an argument to be presented to a court, and in testing the of certain kinds of evidence. Also, argumentation scholars study the post hoc rationalizations by which organizational actors try to justify decisions they have made irrationally.

4.1 Key components of argumentation

• Understanding and identifying arguments, either explicit or implied, and the goals of the participants in the different types of dialogue.

• Identifying the premises from which conclusions are derived

• Establishing the "burden of proof" – determining who made the initial claim and is thus responsible for pro- viding evidence why his/her position merits acceptance.

• For the one carrying the “burden of proof”, the advocate, to marshal evidence for his/her position in order to convince or force the opponent's acceptance. The method by which this is accomplished is producing valid, sound, and cogent arguments, devoid of weaknesses, and not easily attacked.

• In a debate, fulfillment of the burden of proof creates a burden of rejoinder. One must try to identify faulty reasoning in the opponent's argument, to attack the reasons/premises of the argument, to provide counterex- amples if possible, to identify any , and to show why a valid conclusion cannot be derived from the reasons provided for his/her argument.

4.2 Internal structure of arguments

Typically an argument has an internal structure, comprising the following

1. a set of assumptions or premises

2. a method of reasoning or deduction and

3. a conclusion or point.

23 24 CHAPTER 4. ARGUMENTATION THEORY

An argument has one or more premises and one conclusion. Often is used as the method of reasoning so that the conclusion follows logically from the assumptions or support. One challenge is that if the set of assumptions is inconsistent then anything can follow logically from inconsistency. Therefore, it is common to insist that the set of assumptions be consistent. It is also good practice to require the set of assumptions to be the minimal set, with respect to set inclusion, necessary to infer the consequent. Such arguments are called MINCON arguments, short for minimal consistent. Such argumentation has been applied to the fields of law and medicine. A second school of argumentation investigates abstract arguments, where 'argument' is considered a primitive term, so no internal structure of arguments is taken on account. In its most common form, argumentation involves an individual and an interlocutor/or opponent engaged in dialogue, each contending differing positions and trying to persuade each other. Other types of dialogue in addition to persua- sion are eristic, information seeking, , negotiation, deliberation, and the dialectical method (Douglas Walton). The dialectical method was made famous by and his use of critically questioning various characters and historical figures.

4.3 Argumentation and the grounds of knowledge

Argumentation theory had its origins in , a theory of knowledge () in the field of philosophy. It sought to find the grounds for claims in the forms (logic) and materials (factual ) of a universal system of knowledge. But argument scholars gradually rejected Aristotle's systematic philosophy and the in Plato and Kant. They questioned and ultimately discarded the that argument premises take their soundness from formal philosophical systems. The field thus broadened.*[1] Karl R. Wallace's seminal essay, “The Substance of Rhetoric: Good Reasons”in the Quarterly Journal of Speech (1963) 44, led many scholars to study“marketplace argumentation”– the ordinary arguments of ordinary people. The seminal essay on marketplace argumentation is Ray Lynn Anderson and C. David Mortensen,"Logic and Marketplace Argumentation”Quarterly Journal of Speech 53 (1967): 143-150.*[2]*[3] This line of thinking led to a natural alliance with late developments in the of knowledge.*[4] Some scholars drew connections with recent developments in philosophy, namely the of and . Rorty has called this shift in emphasis“the ". In this new hybrid approach argumentation is used with or without empirical evidence to establish convincing con- clusions about issues which are moral, scientific, epistemic, or of a in which science alone cannot answer. Out of pragmatism and many intellectual developments in the and social sciences, “non-philosophical” argumentation theories grew which located the formal and material grounds of arguments in particular intellectual fields. These theories include informal logic, social epistemology, ethnomethodology, speech acts, the sociology of knowledge, the sociology of science, and . These new theories are not non-logical or anti-logical. They find logical coherence in most communities of discourse. These theories are thus often labeled “sociological” in that they focus on the social grounds of knowledge.

4.4 Approaches to argumentation in communication and informal logic

In general, the label “argumentation”is used by communication scholars such as (to name only a few) Wayne E. Brockriede, Douglas Ehninger, Joseph W. Wenzel, Richard Rieke, Gordon Mitchell, Carol Winkler, Eric Gander, Dennis S. Gouran, Daniel J. O'Keefe, Mark Aakhus, Bruce Gronbeck, James Klumpp, G. Thomas Goodnight, Robin Rowland, Dale Hample, C. Scott Jacobs, Sally Jackson, David Zarefsky, and Charles Arthur Willard, while the term "informal logic" is preferred by philosophers, stemming from University of Windsor philosophers Ralph H. Johnson and J. Anthony Blair. Harald Wohlrapp developed a criterion for validness (Geltung, Gültigkeit) as freedom of objections. Trudy Govier, Douglas Walton, Michael Gilbert, Harvey Seigal, Michael Scriven, and John Woods (to name only a few) are other prominent authors in this tradition. Over the past thirty years, however, scholars from several disciplines have co-mingled at international conferences such as that hosted by the University of Amsterdam (the ) and the International Society for the Study of Argumentation (ISSA). Other international conferences are the biannual conference held at Alta, Utah sponsored by the (US) National Communication Association and American Forensics Association and conferences sponsored by the Ontario Society for the Study of Argumentation (OSSA). Some scholars (such as Ralph H. Johnson) construe the term“argument”narrowly, as exclusively written discourse 4.5. KINDS OF ARGUMENTATION 25

or even discourse in which all premises are explicit. Others (such as Michael Gilbert) construe the term“argument” broadly, to include spoken and even nonverbal discourse, for instance the degree to which a war memorial or pro- paganda poster can be said to argue or “make arguments.”The philosopher Stephen E. Toulmin has said that an argument is a claim on our attention and , a that would seem to authorize treating, say, propaganda posters as arguments. The dispute between broad and narrow theorists is of long standing and is unlikely to be settled. The views of the majority of argumentation theorists and analysts fall somewhere between these two extremes.

4.5 Kinds of argumentation

4.5.1 Conversational argumentation

Main articles: Conversation Analysis and Discourse Analysis

The study of naturally-occurring conversation arose from the field of sociolinguistics. It is usually called conver- sation analysis. Inspired by ethnomethodology, it was developed in the late 1960s and early 1970s principally by the sociologist Harvey Sacks and, among others, his close associates Emanuel Schegloff and Gail Jefferson. Sacks died early in his career, but his work was championed by others in his field, and CA has now become an established force in sociology, , , speech-communication and psychology.*[5] It is particularly influential in interactional sociolinguistics, discourse analysis and discursive psychology, as well as being a coherent discipline in its own right. Recently CA techniques of sequential analysis have been employed by phoneticians to explore the fine phonetic details of speech. Empirical studies and theoretical formulations by Sally Jackson and Scott Jacobs, and several generations of their students, have described argumentation as a form of managing conversational disagreement within communication contexts and systems that naturally prefer agreement.

4.5.2 Mathematical argumentation

Main article: Philosophy of

The basis of mathematical has been the subject of long debate. Frege in particular sought to demonstrate (see , The Foundations of Arithmetic, 1884, and Logicism in Philosophy of mathematics) that arithmetical can be derived from purely logical and therefore are, in the end, logical truths. The project was devel- oped by Russell and Whitehead in their . If an argument can be cast in the form of sentences in Symbolic Logic, then it can be tested by the application of accepted proof procedures. This has been carried out for Arithmetic using . Be that as it may, an argument in Mathematics, as in any other discipline, can be considered valid only if it can be shown that it cannot have true premises and a conclusion.

4.5.3 Scientific argumentation

Main article:

Perhaps the most radical statement of the social grounds of scientific knowledge appears in Alan G.Gross's The (Cambridge: Harvard University Press, 1990). Gross holds that science is rhetorical “without remainder,”meaning that scientific knowledge itself cannot be seen as an idealized ground of knowledge. Scientific knowledge is produced rhetorically, meaning that it has special epistemic authority only insofar as its communal methods of verification are trustworthy. This thinking represents an almost complete rejection of the foundationalism on which argumentation was first based.

4.5.4 Interpretive argumentation

Main article: Interpretive discussion 26 CHAPTER 4. ARGUMENTATION THEORY

Interpretive argumentation is a dialogical process in which participants explore and/or resolve interpretations often of a text of any medium containing significant in meaning. Interpretive argumentation is pertinent to the humanities, , literary theory, linguistics, , , , and . Topics in conceptual include aesthetic, judicial, logical and religious interpretation. Topics in scientific interpretation include scientific modeling.

4.5.5 Legal argumentation

Main articles: Oral argument and Closing argument

Legal arguments are spoken presentations to a judge or appellate court by a lawyer, or parties when representing themselves of the legal reasons why they should prevail. Oral argument at the appellate level accompanies written briefs, which also advance the argument of each party in the legal dispute. A closing argument, or summation, is the concluding statement of each party's counsel reiterating the important arguments for the trier of fact, often the jury, in a court case. A closing argument occurs after the presentation of evidence.

4.5.6 Political argumentation

Main article: Political argument

Political arguments are used by academics, media pundits, candidates for political office and officials. Political arguments are also used by citizens in ordinary interactions to comment about and understand political events.*[6] The of the public is a major in this line of research. Political scientist Samuel L. Popkin coined the expression "low information voters" to describe most voters who know very little about politics or the world in general. In practice, a "low information voter" may not be aware of legislation that their representative has sponsored in Congress. A low-information voter may base their ballot box decision on a media sound-bite, or a flier received in the mail. It is possible for a media sound-bite or campaign flier to present a political position for the incumbent candidate that completely contradicts the legislative action taken in Washington D.C. on behalf of the constituents. It may only take a small percentage of the overall voting group who base their decision on the inaccurate information, a voter block of 10 to 12%, to swing an overall election result. When this happens, the constituency at large may have been duped or fooled. Nevertheless, the election result is legal and confirmed. Savvy Political consultants will take advantage of low-information voters and sway their votes with disinformation because it can be easier and sufficiently effective. Fact checkers have come about in recent years to help counter the effects of such campaign tactics.

4.6 Psychological aspects

Psychology has long studied the non-logical aspects of argumentation. For example, studies have shown that simple repetition of an idea is often a more effective method of argumentation than appeals to reason. Propaganda often utilizes repetition.*[7] Nazi rhetoric has been studied extensively as, inter alia, a repetition campaign. Empirical studies of communicator credibility and attractiveness, sometimes labeled charisma, have also been tied closely to empirically-occurring arguments. Such studies bring argumentation within the ambit of persuasion theory and practice. Some psychologists such as William J. McGuire believe that the is the basic unit of human reasoning. They have produced a large body of empirical work around McGuire's famous title “A Syllogistic Analysis of Cognitive Relationships.”A central line of this way of thinking is that logic is contaminated by psychological variables such as “,”in which subjects confound the likelihood of predictions with the desirability of the predictions. People hear what they want to hear and see what they expect to see. If planners want something to happen they see it as likely to happen. If they hope something will not happen, they see it as unlikely to happen. Thus smokers think that they personally will avoid cancer. Promiscuous people practice unsafe sex. Teenagers drive recklessly. 4.7. THEORIES 27

4.7 Theories

4.7.1 Argument fields

Stephen E. Toulmin and Charles Arthur Willard have championed the idea of argument fields, the former drawing upon 's notion of language games, (Sprachspiel) the latter drawing from communication and argumentation theory, sociology, , and social epistemology. For Toulmin, the term “field”des- ignates discourses within which arguments and factual claims are grounded.*[8] For Willard, the term “field”is interchangeable with“community,”“audience,”or“readership.”*[9] Along similar lines, G. Thomas Goodnight has studied“spheres”of argument and sparked a large literature created by younger scholars responding to or using his ideas.*[10] The general tenor of these field theories is that the premises of arguments take their meaning from social communities.*[11] Field studies might focus on social movements, issue-centered publics (for instance, pro-life versus pro-choice in the abortion dispute), small activist groups, corporate public relations campaigns and issue management, scientific com- munities and disputes, political campaigns, and intellectual traditions.*[12] In the manner of a sociologist, ethnogra- pher, anthropologist, participant-observer, and journalist, the field theorist gathers and reports on real-world human discourses, gathering case studies that might eventually be combined to produce high-order explanations of argu- mentation processes. This is not a quest for some master language or master theory covering all specifics of human activity. Field theorists are agnostic about the possibility of a single grand theory and skeptical about the usefulness of such a theory. Theirs is a more modest quest for “mid-range”theories that might permit generalizations about families of discourses.

4.7.2 Stephen E. Toulmin's contributions

By far, the most influential theorist has been Stephen Toulmin, the Cambridge educated philosopher and student of Wittgenstein.*[13] What follows below is a sketch of his ideas.

An alternative to absolutism and

Toulmin has argued that absolutism (represented by theoretical or analytic arguments) has limited practical value. Absolutism is derived from Plato's idealized formal logic, which advocates universal truth; thus absolutists believe that moral issues can be resolved by adhering to a standard set of moral principles, regardless of context. By contrast, Toulmin asserts that many of these so-called standard principles are irrelevant to real situations encountered by human in daily life. To describe his vision of daily life, Toulmin introduced the concept of argument fields; in The Uses of Argument (1958), Toulmin states that some aspects of arguments vary from field to field, and are hence called“field-dependent,” while other aspects of argument are the same throughout all fields, and are hence called“field-invariant.”The flaw of absolutism, Toulmin , lies in its unawareness of the field-dependent aspect of argument; absolutism assumes that all aspects of argument are field invariant. Toulmin's theories resolve to avoid the defects of absolutism without resorting to relativism: relativism, Toulmin asserted, provides no basis for distinguishing between a moral or immoral argument. In Human Understanding (1972), Toulmin suggests that anthropologists have been tempted to side with relativists because they have noticed the influence of cultural variations on rational arguments; in other words, the anthropologist or relativist overemphasizes the importance of the“field-dependent”aspect of arguments, and becomes unaware of the“field-invariant”elements. In an attempt to provide solutions to the problems of absolutism and relativism, Toulmin attempts throughout his work to develop standards that are neither absolutist nor relativist for assessing the worth of ideas. Toulmin believes that a good argument can succeed in providing good justification to a claim, which will stand up to criticism and earn a favourable verdict.

Components of argument

In The Uses of Argument (1958), Toulmin proposed a layout containing six interrelated components for analyzing arguments: 28 CHAPTER 4. ARGUMENTATION THEORY

1. Claim: Conclusions whose must be established. For example, if a person tries to convince a listener that he is a British citizen, the claim would be “I am a British citizen.”(1) 2. Data: The facts we appeal to as a foundation for the claim. For example, the person introduced in 1 can support his claim with the supporting data “I was born in Bermuda.”(2) 3. Warrant: The statement authorizing our movement from the data to the claim. In order to move from the data established in 2,“I was born in Bermuda,”to the claim in 1,“I am a British citizen,”the person must supply a warrant to bridge the gap between 1 & 2 with the statement“A man born in Bermuda will legally be a British Citizen.”(3) 4. Backing: Credentials designed to certify the statement expressed in the warrant; backing must be introduced when the warrant itself is not convincing enough to the readers or the listeners. For example, if the listener does not deem the warrant in 3 as credible, the speaker will supply the legal provisions as backing statement to show that it is true that “A man born in Bermuda will legally be a British Citizen.” 5. Rebuttal: Statements recognizing the restrictions to which the claim may legitimately be applied. The rebuttal is exemplified as follows, “A man born in Bermuda will legally be a British citizen, unless he has betrayed Britain and has become a spy of another country.” 6. Qualifier: Words or phrases expressing the speaker's degree of force or certainty concerning the claim. Such words or phrases include “possible,”“probably,”“impossible,”“certainly,”“presumably,”“as far as the evidence goes,”or “necessarily.”The claim “I am definitely a British citizen”has a greater degree of force than the claim “I am a British citizen, presumably.”

The first three elements “claim,”“data,”and “warrant”are considered as the essential components of practical arguments, while the second triad“qualifier,”“backing,”and“rebuttal”may not be needed in some arguments. When first proposed, this layout of argumentation is based on legal arguments and intended to be used to analyze the rationality of arguments typically found in the courtroom; in fact, Toulmin did not realize that this layout would be applicable to the field of rhetoric and communication until his works were introduced to rhetoricians by Wayne Brockriede and Douglas Ehninger. Only after he published Introduction to Reasoning (1979) were the rhetorical applications of this layout mentioned in his works.

The evolution of knowledge

Toulmin's Human Understanding (1972) asserts that conceptual change is evolutionary. This book attacks 's explanation of conceptual change in The Structure of Scientific Revolutions. Kuhn held that conceptual change is a (as opposed to an evolutionary) process in which mutually exclusive compete to replace one another. Toulmin criticizes the relativist elements in Kuhn's thesis, as he points out that the mutually exclusive paradigms provide no ground for comparison; in other words, Kuhn's thesis has made the relativists' error of overem- phasizing the “field variant”while ignoring the “field invariant,”or commonality shared by all argumentation or scientific paradigms. Toulmin proposes an evolutionary model of conceptual change comparable to Darwin's model of biological evolution. On this reasoning, conceptual change involves innovation and selection. Innovation accounts for the appearance of conceptual variations, while selection accounts for the survival and perpetuation of the soundest conceptions. Innova- tion occurs when the professionals of a particular discipline come to view things differently from their predecessors; selection subjects the innovative concepts to a process of debate and inquiry in what Toulmin considers as a “forum of competitions.”The soundest concepts will survive the forum of competition as replacements or revisions of the traditional conceptions. From the absolutists' point of view, concepts are either valid or invalid regardless of contexts; from a relativists' perspective, one concept is neither better nor worse than a rival concept from a different cultural context. From Toulmin's perspective, the evaluation depends on a process of comparison, which determines whether or not one concept will provide improvement to our more so than its rival concepts.

Rejection of certainty

In Cosmopolis (1990), Toulmin traces the quest for certainty back to Descartes and Hobbes, and lauds Dewey, Wittgenstein, Heidegger and Rorty for abandoning that tradition. 4.8. ARTIFICIAL INTELLIGENCE 29

4.7.3 Pragma-dialectics

Main article: Pragma-dialectics

Scholars at the University of Amsterdam in the Netherlands have pioneered a rigorous modern version of under the name pragma-dialectics. The intuitive idea is to formulate clearcut rules that, if followed, will yield rational discussion and sound conclusions. Frans H. van Eemeren, the late Rob Grootendorst, and many of their students have produced a large body of work expounding this idea. The dialectical conception of reasonableness is given by ten rules for critical discussion, all being instrumental for achieving a resolution of the difference of opinion (from Van Eemeren, Grootendorst, & Snoeck Henkemans, 2002, p. 182-183). The theory postulates this as an ideal model, and not something one expects to find as an empirical fact. The model can however serve as an important heuristic and critical tool for testing how reality approximates this ideal and point to where discourse goes wrong, that is, when the rules are violated. Any such violation will constitute a fallacy. Albeit not primarily focused on fallacies, pragma-dialectics provides a systematic approach to deal with them in a coherent way.

4.7.4 Walton's logical argumentation method

Doug Walton developed a distinctive of logical argumentation built around a set of practi- cal methods to help a user identify, analyze and evaluate arguments in everyday conversational discourse and in more structured areas such as debate, law and scientific fields.*[14] There are four main components: argumentation schemes,*[15] dialogue structures, argument mapping tools, and formal argumentation systems. The method uses the notion of commitment in dialogue as the fundamental tool for the analysis and evaluation of argumentation rather than the notion of belief.*[16] Commitments are statements that the agent has expressed or formulated, and has pledged to carry out, or has publicly asserted. According to the commitment model, agents interact with each other in a dialogue in which each takes its turn to contribute speech acts. The dialogue framework uses critical questioning as a way of testing plausible explanations and finding weak points in an argument that raise doubt concerning the acceptability of the argument. Walton's logical argumentation model takes a different view of proof and justification from that taken in the dominant epistemology in analytical philosophy, which is based on a true belief framework. On the logical argumentation approach, knowledge is seen as form of belief commitment firmly fixed by an argumentation procedure that tests the evidence on both sides, and use standards of proof to determine whether a proposition qualifies as knowledge. On this evidence-based approach, scientific knowledge must be seen as defeasible.

4.8 Artificial intelligence

See also: Argument mapping and Argumentation framework

Efforts have been made within the field of artificial intelligence to perform and analyze the act of argumentation with computers. Argumentation has been used to provide a proof-theoretic semantics for non-monotonic logic, starting with the influential work of Dung (1995). Computational argumentation systems have found particular application in domains where formal logic and classical are unable to capture the richness of reasoning, domains such as law and medicine. In Elements of Argumentation, Philippe Besnard and Anthony Hunter introduce techniques for formalizing deductive argumentation in artificial intelligence, emphasizing emerging formalizations for practical argumentation.*[17] Within Computer Science, the ArgMAS workshop series (Argumentation in Multi-Agent Systems), the CMNA workshop series,*[18] and now the COMMA Conference,*[19] are regular annual events attracting participants from every continent. The journal Argument & Computation*[20] is dedicated to exploring the intersection between argumentation and computer science.

4.9 See also

• A fortiori argument 30 CHAPTER 4. ARGUMENTATION THEORY

• Argument (logic)

• Argumentation ethics

• Critical thinking

• Criticism

• Defeasible reasoning

• Dialectic

• Discourse ethics

• Essentially contested concepts

• Forensics

• Legal theory

• Logical argument

• Logic of Argumentation

• Negotiation theory

• Pars destruens/pars construens

• Public Sphere

• Rationality

• Rhetoric

• Social engineering (political science)

• Social psychology (psychology)

• Sophistry

• Source criticism

• Straight and Crooked Thinking (book)

4.10 Notes

[1] Bruce Gronbeck. “From Argument to Argumentation: Fifteen Years of Identity Crisis.”Jack Rhodes and Sara Newell, ed.s Proceedings of the Summer Conference on Argumentation. 1980.

[2] See Joseph W. Wenzel “Perspectives on Argument.”Jack Rhodes and Sara Newell, ed.s Proceedings of the Summer Conference on Argumentation. 1980.

[3] David Zarefsky. “Product, Process, or Point of View? Jack Rhodes and Sara Newell, ed.s Proceedings of the Summer Conference on Argumentation. 1980.

[4] See Ray E. McKerrow. “Argument Communities: A Quest for Distinctions.”

[5] Psathas, George (1995): Conversation Analysis, Thousand Oaks: Sacks, Harvey. (1995). Lectures on Conversation. Blackwell Publishing. ISBN 1-55786-705-4. Sacks, Harvey, Schegloff, Emanuel A., & Jefferson, Gail (1974). A simple systematic for the organization of turn-taking for conversation. Language, 50, 696-735. Schegloff, Emanuel A. (2007). Sequence Organization in Interaction: A Primer in Conversation Analysis, Volume 1, Cambridge: Cambridge University Press. Ten Have, Paul (1999): Doing Conversation Analysis. A Practical Guide, Thousand Oaks: Sage.

[6] Michael McGee. “The 'Ideograph' as a Unit of Analysis in Political Argument.”Jack Rhodes and Sara Newell, eds. Proceedings of the Summer Conference on Argumentation. 1980.

[7] Jacques Ellul, Propaganda, Vintage, 1973, ISBN 0-394-71874-7 ISBN 978-0394718743. 4.11. SOURCES 31

[8] Stephen E. Toulmin. The uses of argument. 1959.

[9] Charles Arthur Willard. “Some Questions About Toulmin's View of Argument Fields.”Jack Rhodes and Sara Newell, eds. Proceedings of the Summer Conference on Argumentation. 1980. “Field Theory: A Cartesian Meditation.”George Ziegelmueller and Jack Rhodes, eds. Dimensions of Argument: Proceedings of the Second Summer Conference on Argu- mentation.

[10] G. T. Goodnight, “The Personal, Technical, and Public Spheres of Argument.”Journal of the American Forensics Asso- ciation. (1982) 18:214-227.

[11] Bruce E. Gronbeck. “Sociocultural Notions of Argument Fields: A Primer.”George Ziegelmueller and Jack Rhodes, eds. Dimensions of Argument: Proceedings of the Second Summer Conference on Argumentation. (1981) 1-20.

[12] Robert Rowland, “Purpose, Argument Fields, and Theoretical Justification.”Argumentation. vol. 22 Number 2 (2008) 235-250.

[13] Loui, Ronald P. (2006). “A Citation-Based Reflection on Toulmin and Argument”. In Hitchcock, David; Verheij, Bart. Arguing on the Toulmin Model: New Essays in Argument Analysis and Evaluation. Springer Netherlands. pp. 31–38. doi:10.1007/978-1-4020-4938-5_3. ISBN 978-1-4020-4937-8. Retrieved 2010-06-25. Toulmin's 1958 work is essential in the field of argumentation

[14] Walton, Douglas (2013). Methods of Argumentation. Cambridge: Cambridge University Press.

[15] Walton, Douglas; Reed, Chris; Macagno, Fabrizio (2008). Argumentation Schemes. New York: Cambridge University Press.

[16] Walton, Douglas; Krabbe, E. C. W. (1995). Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning. Albany: SUNY Press.

[17] P. Besnard & A. Hunter, “Elements of Argumentation.”MIT Press, 2008. See also: http://mitpress.mit.edu/catalog/ item/default.asp?ttype=2&tid=11482

[18] Computational Models of Natural Argument

[19] Computational Models of Argument

[20] Journal of Argument & Computation

4.11 Sources

• J. Robert Cox and Charles Arthur Willard, eds. Advances in Argumentation Theory and Research 1982.

• Dung, P. M. “On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games.”Artificial Intelligence, 77: 321-357 (1995).

• Bondarenko, A., Dung, P. M., Kowalski, R., and Toni, F.,“An abstract, argumentation-theoretic approach to default reasoning”, Artificial Intelligence 93(1-2) 63-101 (1997).

• Dung, P. M., Kowalski, R., and Toni, F. “Dialectic proof procedures for assumption-based, admissible argu- mentation.”Artificial Intelligence. 170(2), 114-159 (2006).

• Frans van Eemeren, Rob Grootendorst, Sally Jackson, and Scott Jacobs, Reconstructing Argumentative Dis- course 1993.

• Frans Van Eemeren & Rob Grootendorst. A systematic theory of argumentation. The pragma-dialected - proach. 2004.

• Eemeren, F.H. van, Grootendorst, R. & Snoeck Henkemans, F. et al. (1996). Fundamentals of Argumentation Theory. A Handbook of Historical Backgrounds and Contemporary Developments. Mahwah, NJ: Erlbaum.

• Richard H. Gaskins Burdens of Proof in Modern Discourse. Yale University Press. 1993.

• Michael A. Gilbert Coalescent Argumentation 1997.

• Trudy Govier, Problems in Argument Analysis and Evaluation. 1987. 32 CHAPTER 4. ARGUMENTATION THEORY

• Dale Hample. (1979).“Predicting belief and belief change using a cognitive theory of argument and evidence.” Communication Monographs. 46, 142-146.

• Dale Hample. (1978). “Are attitudes arguable?" Journal of Value Inquiry. 12, 311-312.

• Dale Hample. (1978). “Predicting immediate belief change and adherence to argument claims.”Communi- cation Monographs, 45, 219-228.

• Dale Hample & Judy Hample. (1978). “Evidence credibility.”Debate Issues. 12, 4-5.

• Dale Hample. (1977). “Testing a model of value argument and evidence.”Communication Monographs. 14, 106-120.

• Dale Hample. (1977).“The Toulmin model and the syllogism.”Journal of the American Forensic Association. 14, 1-9.

• Trudy Govier, A Practical Study of Argument2nd ed. 1988.

• Sally Jackson and Scott Jacobs,“Structure of Conversational Argument: Pragmatic Bases for the Enthymeme.” The Quarterly Journal of Speech. LXVI, 251-265.

• Ralph H. Johnson. Manifest Rationality: A Pragmatic Theory of Argument. Lawrence Erlbaum, 2000.

• Ralph H. Johnson and J. Anthony Blair. “Logical Self-Defense”, IDEA, 2006. First published, McGraw Hill Ryerson, , ON, 1997, 1983, 1993. Reprinted, McGraw Hill, New York, NY, 1994.

• Ralph Johnson. and Blair, J. Anthony (1987),“The Current State of Informal Logic”, Informal Logic, 9(2–3), 147–151.

• Ralph H. Johnson. H. (1996). The rise of informal logic. Newport News, VA: Vale Press

• Ralph H. Johnson. (1999). The relation between formal and informal logic. Argumentation, 13(3) 265-74.

• Ralph H. Johnson. & Blair, J. A. (1977). Logical self-defense. Toronto: McGraw-Hill Ryerson. US Edition. (2006). New York: Idebate Press.

• Ralph H. Johnson. & Blair, J. Anthony. (1987). The current state of informal logic. Informal Logic 9, 147-51.

• Ralph H. Johnson. & Blair, J. Anthony. (1996). Informal logic and critical thinking. In F. van Eemeren, R. Grootendorst, & F. Snoeck Henkemans (Eds.), Fundamentals of Argumentation Theory. (pp. 383–86). Mahwah, NJ: Lawrence Erlbaum Associates

• Ralph H. Johnson, Ralph. H. & Blair, J. Anthony. (2000). “Informal logic: An overview.”Informal Logic. 20(2): 93-99.

• Ralph H. Johnson, Ralph. H. & Blair, J. Anthony. (2002). Informal logic and the reconfiguration of logic. In D. Gabbay, R. H. Johnson, H.-J. Ohlbach and J. Woods (Eds.). Handbook of the logic of argument and inference: The turn towards the practical. (pp. 339–396). Elsivier: North Holland.

• Chaim Perelman and Lucie Olbrechts-Tyteca, The New Rhetoric, Notre Dame, 1970.

• Stephen Toulmin. The uses of argument. 1959.

• Stephen Toulmin. The Place of Reason in Ethics. 1964.

• Stephen Toulmin. Human Understanding: The Use and Evolution of Concepts. 1972.

• Stephen Toulmin. Cosmopolis. 1993.

• Douglas N. Walton, The Place of Emotion in Argument. 1992.

• Joseph W. Wenzel 1990 Three perspectives on argumentation. In R Trapp and J Scheutz, (Eds.), Perspectives on argumentation: Essays in honour of Wayne Brockreide. 9-26 Waveland Press: Prospect Heights, IL

• John Woods. (1980). What is informal logic? In J.A. Blair & R. H. Johnson (Eds.), Informal Logic: The First International Symposium .(pp. 57–68). Point Reyes, CA: Edgepress. 4.12. FURTHER READING 33

• John Woods. (2000). How Philosophical is Informal Logic? Informal Logic. 20(2): 139-167. 2000

• Charles Arthur Willard Liberalism and the Problem of Knowledge: A New Rhetoric for Modern Democracy. University of Chicago Press. 1996.

• Charles Arthur Willard, A Theory of Argumentation. University of Alabama Press. 1989. • Charles Arthur Willard, Argumentation and the Social Grounds of KnowledgeUniversity of Alabama Press. 1982. • Harald Wohlrapp. Der Begriff des Arguments. Über die Beziehungen zwischen Wissen, Forschen, Glaube, Subjektivität und Vernunft. Würzburg: Königshausen u. Neumann, 2008 ISBN 978-3-8260-3820-4

4.12 Further reading

Flagship journals:

• Argumentation

• Informal Logic • Argumentation and Advocacy (formerly Journal of the American Forensic Association)

• Social Epistemology • Episteme: A Journal of Social Epistemology

• Journal of Argument and Computation

4.13 External links

• Universiteit (Dutch)

• Universiteit Twente (Dutch)

• L'Argumentation: Introduction à l'étude du discours (French) Free on-line book by Mariana Tutescu previously published in 1998 as ISBN 973-575-248-4

• Argumentum.ch, E-course of Argumentation Theory for the Human and Social Sciences Chapter 5

Ariadne's thread (logic)

Ariadne's thread, named for the legend of Ariadne, is the solving of a problem with multiple apparent means of proceeding - such as a physical maze, a logic puzzle, or an ethical dilemma - through an exhaustive application of logic to all available routes. It is the particular method used that is able to follow completely through to trace steps or take point by point a series of found truths in a contingent, ordered search that reaches an end position. This process can take the form of a mental record, a physical marking, or even a philosophical debate; it is the process itself that assumes the name.

5.1 Implementation

The key element to applying Ariadne's thread to a problem is the creation and maintenance of a record - physical or otherwise - of the problem's available and exhausted options at all times. This record is referred to as the “thread” , regardless of its actual medium. The purpose the record serves is to permit backtracking - that is, reversing earlier decisions and trying alternatives. Given the record, applying the algorithm is straightforward:

• At any moment that there is a choice to be made, make one arbitrarily from those not already marked as failures, and follow it logically as far as possible. • If a results, back up to the last decision made, mark it as a failure, and try another decision at the same point. If no other options exist there, back up to the last place in the record that does, mark the failure at that level, and proceed onward.

This algorithm will terminate upon either finding a solution or marking all initial choices as failures; in the latter case, there is no solution. If a thorough examination is desired even though a solution has been found, one can revert to the previous decision, mark the success, and continue on as if a solution were never found; the algorithm will exhaust all decisions and find all solutions.

5.2 Distinction from trial and error

The terms“Ariadne's thread”and "trial and error" are often used interchangeably, which is not necessarily correct. They have two distinctive differences:

•“Trial and error”implies that each “trial”yields some particular value to be studied and improved upon, removing “errors”from each iteration to enhance the quality of future trials. Ariadne's thread has no such mechanic, making all decisions arbitrarily. For example, the scientific method is trial and error; puzzle-solving is Ariadne's thread.

• Trial-and-error approaches are rarely concerned with how many solutions may exist to a problem, and indeed often assume only one correct solution exists. Ariadne's thread makes no such assumption, and is capable of locating all possible solutions to a purely logical problem.

34 5.3. APPLICATIONS 35

In short, trial and error approaches a desired solution; Ariadne's thread blindly exhausts the search space completely, finding any and all solutions. Each has its appropriate distinct uses. They can be employed in tandem - for example, although the editing of a Wikipedia article is arguably a trial-and-error process (given how in theory it approaches an ideal state), article provide the record for which Ariadne's thread may be applied, reverting detrimental edits and restoring the article back to the most recent error-free version, from which other options may be attempted.

5.3 Applications

Obviously, Ariadne's thread may be applied to the solving of mazes in the same manner as the legend; an actual thread can be used as the record, or chalk or a similar marker can be applied to label passages. If the maze is on paper, the thread may well be a pencil. Logic problems of all natures may be resolved via Ariadne's thread, the maze being but an example. At present, it is most prominently applied to Sudoku puzzles, used to attempt values for as-yet-unsolved cells. The medium of the thread for puzzle-solving can vary widely, from a pencil to numbered chits to a computer program, but all accomplish the same task. Note that as the compilation of Ariadne's thread is an inductive process, and due to its exhaustiveness leaves no room for actual study, it is largely frowned upon as a solving method, to be employed only as a last resort when deductive methods fail. Artificial intelligence is heavily dependent upon Ariadne's thread when it comes to game-playing, most notably in programs which play chess; the possible moves are the decisions, game-winning states the solutions, and game-losing states failures. Due to the massive depth of many games, most algorithms cannot afford to apply Ariadne's thread entirely on every move due to time constraints, and therefore work in tandem with a heuristic that evaluates game states and limits a breadth-first search only to those that are most likely to be beneficial, a trial-and-error process. Even circumstances where the concept of “solution”is not so well defined have had Ariadne's thread applied to them, such as navigating the World Wide Web, making sense of patent law, and in philosophy; “Ariadne's Thread” is a popular name for websites of many purposes, but primarily for those that feature philosophical or ethical debate.

5.4 See also

• Depth First Search

• Labyrinth •

• Computer chess • J. Hillis Miller

• Gordian Knot

5.5 References

• Solving Sudoku Step-by-step guide by Michael Mepham; includes history of Ariadne's thread and demonstra- tion of application • Constructing Sudoku A flow chart shows how to and solve Sudoku by using Ariadne's thread (back- tracking technique) • Ariadne and the Minotaur: The Cultural Role of a Philosophy of Rhetoric Article by Andrea Battistini detailing Ariadne's thread as a philosophical metaphor • Philosophy in Labyrinths A study of the logic behind and meaning of labyrinths; includes rather literal inter- pretations of Ariadne's thread. Chapter 6

Austrian Ludwig Wittgenstein Society

The Austrian Ludwig Wittgenstein Society was first established in 1974 to promote philosophical conferences, workshops, summer schools, and research that are inspired by the work of Ludwig Wittgenstein and the Circle.*[4]*[5] It is an international society, which also has a publication series.*[6]

6.1 History

In 1974, , Werner Leinfellner, Rudolf Haller, Paul Weingartner, and Adolf Hübner founded the Austrian Ludwig Wittgenstein Society (ALWS), but it was the impetuous of the first International Wittgenstein Symposium in Kirchberg am Wechsel, Lower that led to its development as an international society. The location of the ALWS is in Kirchberg am Wechsel and the location was selected because Ludwig Wittgenstein taught at elementary schools close Kirchberg am Wechsel in the 1920s.*[5]

6.2 Goals

The ALWS has two primary goals.

• To promote the analysis, tradition, and dissemination of Wittgenstein’s philosophy. • To promote, more generally, philosophy that has its roots in Austria and other countries with emphasis on analytic philosophy inspired by Wittgenstein and philosophy of science inspired by, but not limited to, the philosophy of the .*[7]

6.3 Membership

They currently report 120 members both national and international. Individuals can download an application form their website. Membership is decided by the Executive Committee of the ALWS based on academic criteria. Students pay half the fee. Members receive a 30% discount on fees for the International Wittgenstein Symposium and for ALWS publications.*[8]

6.4 Publications

Since the first International Wittgenstein Symposium, the ALWS has published its proceedings. In 2006, they started a new series (published by ontos verlag). The new series contains:*[6]

• The official proceedings of the International Wittgenstein Symposium • Special workshops

36 6.5. INTERNATIONAL LUDWIG WITTGENSTEIN INSTITUTE (ILWI) 37

• High-quality publications submitted to and reviewed by the ALWS.

6.5 International Ludwig Wittgenstein Institute (ILWI)

The ILWI was recently founded by the ALWS and supports scientific and cultural projects that are related to Wittgen- stein’s philosophy and the goals of the ALWS. The ILWI presents:*[9]

• Workshops

• Summer schools

• Wittgenstein lectures

• Research opportunities at the ILWI

6.6 Sponsorship

The Austrian Ludwig Wittgenstein Society is funded by the government of Lower Austria, the Austrian Federal Ministry for Science and Research, Austrian National , Raiffeisenbank NÖ-Süd Alpin, Kirchberg am Wechsel, membership fees, and donations.*[10]

6.7 See also

• Ludwig Wittgenstein

• International Wittgenstein Symposium

•

• Tractatus Logico-Philosophicus

• Philosophical Investigations

•

6.8 References

[1] “Institute Vienna Circle”.

[2] “Die Internationale Ludwig Wittgenstein Gesellschaft”.

[3] "Österreichische Gesellschaft für Philosophie”.

[4] “Graz: Austrian Ludwig Wittgenstein Society”.

[5] “Austrian Ludwig Wittgenstein Society”.

[6] “Publications of the Austrian Ludwig Wittgenstein Society”.

[7] “Goals”.

[8] “Membership”.

[9] “ILWI”.

[10] “Sponsors”. 38 CHAPTER 6. AUSTRIAN LUDWIG WITTGENSTEIN SOCIETY

6.9 External links

• Austrian Ludwig Wittgenstein Society

• British Wittgenstein Society • Nordic Wittgenstein Society Chapter 7

Belief revision

Belief revision is the process of changing beliefs to take into account a new piece of information. The logical formalization of belief revision is researched in philosophy, in databases, and in artificial intelligence for the design of rational agents. What makes belief revision non-trivial is that several different ways for performing this operation may be possible. For example, if the current knowledge includes the three facts " A is true”," B is true”and “if A and B are true then C is true”, the introduction of the new information " C is false”can be done preserving consistency only by removing at least one of the three facts. In this case, there are at least three different ways for performing revision. In general, there may be several different ways for changing knowledge.

7.1 Revision and update

Two kinds of changes are usually distinguished: update the new information is about the situation at present, while the old beliefs refer to the past; update is the operation of changing the old beliefs to take into account the change; revision both the old beliefs and the new information refer to the same situation; an inconsistency between the new and old information is explained by the possibility of old information being less reliable than the new one; revision is the process of inserting the new information into the set of old beliefs without generating an inconsistency.

The main assumption of belief revision is that of minimal change: the knowledge before and after the change should be as similar as possible. In the case of update, this formalizes the assumption of inertia. In the case of revision, this principle enforces as much information as possible to be preserved by the change.

7.1.1 Example

The following classical example shows that the operations to perform in the two settings of update and revision are not the same. The example is based on two different interpretations of the set of beliefs {a ∨ b} and the new piece of information ¬a : update in this scenario, two satellites, Unit A and Unit B, orbit around Mars; the satellites are programmed to land while transmitting their status to Earth; Earth has received a transmission from one of the satellites, communicating that it is still in orbit; however, due to interference, it is not known which satellite sent the signal; subsequently, Earth receives the communication that Unit A has landed; this scenario can be modeled in the following way; two propositional variables a and b indicate that Unit A and Unit B, respectively, are still in orbit; the initial set of beliefs is {a ∨ b} (either one of the two satellites is still in orbit) and the new piece of information is ¬a (Unit A has landed, and is therefore not in orbit); the only rational result of the update is ¬a ; since the initial information that one of the two satellites had not landed yet was possibly coming from the Unit A, the position of the Unit B is not known;

39 40 CHAPTER 7. BELIEF REVISION revision the play “Six Characters in Search of an Author”will be performed in one of the two local theatres; this information can be denoted by {a ∨ b} , where a and b indicates that the play will be performed at the first or at the second theatre, respectively; a further information that “Jesus Christ Superstar”will be performed at the first theatre indicates that ¬a holds; in this case, the obvious conclusion is that “Six Characters in Search of an Author”will be performed at the second but not the first theatre, which is represented in logic by ¬a ∧ b .

This example shows that revising the belief a ∨ b with the new information ¬a produces two different results ¬a and ¬a ∧ b depending on whether the setting is that of update or revision.

7.2 Contraction, expansion, revision, consolidation, and merging

In the setting in which all beliefs refer to the same situation, a distinction between various operations that can be performed is made: contraction removal of a belief; expansion addition of a belief without checking consistency; revision addition of a belief while maintaining consistency; consolidation restoring consistency of a set of beliefs; merging fusion of two or more sets of beliefs while maintaining consistency.

Revision and merging differ in that the first operation is done when the new belief to incorporate is considered more reliable than the old ones; therefore, consistency is maintained by removing some of the old beliefs. Merging is a more general operation, in that the priority among the belief sets may or may not be the same. Revision can be performed by first incorporating the new fact and then restoring consistency via consolidation. This is actually a form of merging rather than revision, as the new information is not always treated as more reliable than the old knowledge.

7.3 The AGM postulates

The AGM postulates (named after the names of their proponents, Alchourrón, Gärdenfors, and Makinson) are prop- erties that an operator that performs revision should satisfy in order for that operator to be considered rational. The considered setting is that of revision, that is, different pieces of information referring to the same situation. Three operations are considered: expansion (addition of a belief without a consistency check), revision (addition of a belief while maintaining consistency), and contraction (removal of a belief). The first six postulates are called“the basic AGM postulates”. In the settings considered by Alchourrón, Gärdenfors, and Makinson, the current set of beliefs is represented by a deductively closed set of logical formulae K called belief base, the new piece of information is a logical formula P , and revision is performed by a binary operator ∗ that takes as its operands the current beliefs and the new information and produces as a result a belief base representing the result of the revision. The + operator denoted expansion: K + P is the deductive closure of K ∪ {P } . The AGM postulates for revision are:

1. K ∗ P is a belief base (i.e., a deductively closed set of formulae); 2. P ∈ K ∗ P 3. K ∗ P ⊆ K + P 4. If(¬P ) ̸∈ K, then K ∗ P = K + P 5. K ∗ P is inconsistent only if P is inconsistent or K is inconsistent 7.4. CONDITIONS EQUIVALENT TO THE AGM POSTULATES 41

6. IfP and Q then equivalent, logically are K ∗ P = K ∗ Q (see )

7. K ∗ (P ∧ Q) ⊆ (K ∗ P ) + Q

8. If(¬Q) ̸∈ K ∗ P then (K ∗ P ) + Q ⊆ K ∗ (P ∧ Q)

A revision operator that satisfies all eight postulates is the full meet revision, in which K ∗ P is equal to K + P if consistent, and to the deductive closure of P otherwise. While satisfying all AGM postulates, this revision operator has been considered to be too conservative, in that no information from the old knowledge base is maintained if the revising formula is inconsistent with it.

7.4 Conditions equivalent to the AGM postulates

The AGM postulates are equivalent to several different conditions on the revision operator; in particular, they are equivalent to the revision operator being definable in terms of structures known as selection functions, epistemic entrenchments, systems of spheres, and preference relations. The latter are reflexive, transitive, and total relations over the set of models.

Each revision operator ∗ satisfying the AGM postulates is associated to a set of preference relations ≤K , one for each possible belief base K , such that the models of K are exactly the minimal of all models according to ≤K . The revision operator and its associated family of orderings are related by the fact that K ∗ P is the set of formulae whose set of models contains all the minimal models of P according to ≤K . This condition is equivalent to the set of models of K ∗ P being exactly the set of the minimal models of P according to the ordering ≤K .

A preference ordering ≤K represents an order of implausibility among all situations, including those that are conceiv- able but yet currently considered false. The minimal models according to such an ordering are exactly the models of the knowledge base, which are the models that are currently considered the most likely. All other models are greater than these ones, and are indeed considered less plausible. In general, I

7.5 Contraction

Contraction is the operation of removing a belief P from a knowledge base K ; the result of this operation is denoted by K − P . The operators of revision and contractions are related by the Levi and Harper identities:

K ∗ P = (K − ¬P ) + P

K − P = K ∩ (K ∗ ¬P ) Eight postulates have been defined for contraction. Whenever a revision operator satisfies the eight postulates for revision, its corresponding contraction operator satisfies the eight postulates for contraction, and vice versa. If a contraction operator satisfies at least the first six postulates for contraction, translating it into a revision operator and then back into a contraction operator using the two identities above leads to the original contraction operator. The same holds starting from a revision operator. One of the postulates for contraction has been longly discussed: the recovery postulate:

K = (K − P ) + P

According to this postulate, the removal of a belief P followed by the reintroduction of the same belief in the belief base should lead to the original belief base. There are some examples showing that such behavior is not always reasonable: in particular, the contraction by a general condition such as a ∨ b leads to the removal of more specific conditions such as a from the belief base; it is then unclear why the reintroduction of a ∨ b should also lead to the reintroduction of the more specific condition a . For example, if George was previously believed to have German 42 CHAPTER 7. BELIEF REVISION

citizenship, it was also believed to be European. Contracting this latter belief amounts to stop believing that George is European; therefore, that George has German citizenship is also retracted from the belief base. If George is later discovered to have Austrian citizenship, then the fact that he is European is also reintroduced. According to the recovery postulate, however, the belief that he also has German citizenship should also be reintroduced. The correspondence between revision and contraction induced by the Levi and Harper identities is such that a con- traction not satisfying the recovery postulate is translated into a revision satisfying all eight postulates, and that a revision satisfying all eight postulates is translated into a contraction satisfying all eight postulates, including recov- ery. As a result, if recovery is excluded from consideration, a number of contraction operators are translated into a single revision operator, which can be then translated back into exactly one contraction operator. This operator is the only one of the initial group of contraction operators that satisfies recovery; among this group, it is the operator that preserves as much information as possible.

7.6 The Ramsey test

The evaluation of a counterfactual conditional a > b can be done, according to the Ramsey test (named for Frank P. Ramsey), to the hypothetical addition of a to the set of current beliefs followed by a check for the truth of b . If K is the set of beliefs currently held, the Ramsey test is formalized by the following correspondence:

a > b b ∈ K ∗ a

If the considered language of the formulae representing beliefs is propositional, the Ramsey test gives a consistent definition for counterfactual conditionals in terms of a belief revision operator. However, if the language of formulae representing beliefs itself includes the counterfactual conditional connective > , the Ramsey test leads to the Garden- fors triviality result: there is no non-trivial revision operator that satisfies both the AGM postulates for revision and the condition of the Ramsey test. This result holds in the assumption that counterfactual formulae like a > b can be present in belief bases and revising formulae. Several solutions to this problem have been proposed.

7.7 Non-monotonic inference relation

Given a fixed knowledge base K and a revision operator ∗ , one can define a non-monotonic inference relation using the following definition: P ⊢ Q if and only if K ∗ P |= Q . In other words, a formula P entails another formula Q if the addition of the first formula to the current knowledge base leads to the derivation of Q . This inference relation is non-monotonic. The AGM postulates can be translated into a set of postulates for this inference relation. Each of these postulates is entailed by some previously considered set of postulates for non-monotonic inference relations. Vice versa, condi- tions that have been considered for non-monotonic inference relations can be translated into postulates for a revision operator. All these postulates are entailed by the AGM postulates.

7.8 Foundational revision

In the AGM framework, a belief set is represented by a deductively closed set of propositional formulae. While such sets are infinite, they can always be finitely representable. However, working with deductively closed sets of formulae leads to the implicit assumption that equivalent belief bases should be considered equal when revising. This is called the principle of irrelevance of . This principle has been and is currently debated: while {a, b} and {a ∧ b} are two equivalent sets, revising by ¬a should produce different results. In the first case, a and b are two separate beliefs; therefore, revising by ¬a should not produce any effect on b , and the result of revision is {¬a, b} . In the second case, a ∧ b is taken a single belief. The fact that a is false contradicts this belief, which should therefore be removed from the belief base. The result of revision is therefore {¬a} in this case. The problem of using deductively closed knowledge bases is that no distinction is made between pieces of knowledge that are known by themselves and pieces of knowledge that are merely consequences of them. This distinction is instead done by the foundational approach to belief revision, which is related to foundationalism in philosophy. 7.9. MODEL-BASED REVISION AND UPDATE 43

According to this approach, retracting a non-derived piece of knowledge should lead to retracting all its consequences that are not otherwise supported (by other non-derived pieces of knowledge). This approach can be realized by using knowledge bases that are not deductively closed and assuming that all formulae in the knowledge base represent self-standing beliefs, that is, they are not derived beliefs. In order to distinguish the foundational approach to belief revision to that based on deductively closed knowledge bases, the latter is called the coherentist approach. This name has been chosen because the coherentist approach aims at restoring the coherence (consistency) among all beliefs, both self-standing and derived ones. This approach is related to in philosophy. Foundationalist revision operators working on non-deductively closed belief bases typically select some of K that are consistent with P , combined them in some way, and then conjoined them with P . The following are two non-deductively closed base revision operators.

WIDTIO (When in Doubt, Throw it Out) the maximal subsets of K that are consistent with P are intersected, and P is added to the resulting set; in other words, the result of revision is composed by P and of all formulae of K that are in all maximal subsets of K that are consistent with P ;

Ginsberg-Fagin-Ullman-Vardi the maximal subsets of K ∪ {P } that are consistent and contain P are combined by disjunction;

Nebel similar to the above, but a priority among formulae can be given, so that formulae with higher priority are less likely to being retracted than formulae with lower priority.

A different realization of the foundational approach to belief revision is based on explicitly declaring the dependences among beliefs. In the truth maintenance systems, dependence links among beliefs can be specified. In other worlds, one can explicitly declare that a given fact is believed because of one or more other facts; such a dependency is called a justification. Beliefs not having any justifications play the role of non-derived beliefs in the non-deductively closed knowledge base approach.

7.9 Model-based revision and update

A number of proposals for revision and update based on the set of models of the involved formulae were developed independently of the AGM framework. The principle behind this approach is that a knowledge base is equivalent to a set of possible worlds, that is, to a set of scenarios that are considered possible according to that knowledge base. Revision can therefore be performed on the sets of possible worlds rather than on the corresponding knowledge bases. The revision and update operators based on models are usually identified by the name of their authors: Winslett, Forbus, Satoh, Dalal, Hegner, and Weber. According to the first four of these proposal, the result of revising/updating a formula K by another formula P is characterized by the set of models of P that are the closest to the models of K . Different notions of closeness can be defined, leading to the difference among these proposals.

Dalal the models of P having a minimal Hamming distance to models of K are selected to be the models that result from the change;

Satoh similar to Dalal, but distance between two models is defined as the set of literals that are given different values by them; similarity between models is defined as set containment of these differences;

Winslett for each model of K , the closest models of P are selected; comparison is done using set containment of the difference;

Borgida equal to Winslett's if K and P are inconsistent; otherwise, the result of revision is K ∧ P ;

Forbus similar to Winslett, but the Hamming distance is used.

The revision operator defined by Hegner makes K not to affect the value of the variables that are mentioned in P . What results from this operation is a formula K′ that is consistent with P , and can therefore be conjoined with it. The revision operator by Weber is similar, but the literals that are removed from K are not all literals of P , but only the literals that are evaluated differently by a pair of closest models of K and P according to the Satoh measure of closeness. 44 CHAPTER 7. BELIEF REVISION

7.10 Iterated revision

The AGM postulates are equivalent to a preference ordering (an ordering over models) to be associated to every knowledge base K . However, they do not relate the orderings corresponding to two non-equivalent knowledge bases. In particular, the orderings associated to a knowledge base K and its revised version K ∗ P can be completely different. This is a problem for performing a second revision, as the ordering associated with K ∗ P is necessary to calculate K ∗ P ∗ Q . Establishing a relation between the ordering associated with K and K ∗ P has been however recognized not to be the right solution to this problem. Indeed, the preference relation should depend on the previous history of revisions, rather than on the resulting knowledge base only. More generally, a preference relation gives more information about the state of mind of an agent than a simple knowledge base. Indeed, two states of mind might represent the same piece of knowledge K while at the same time being different in the way a new piece of knowledge would be incorporated. For example, two people might have the same idea as to where to go on holiday, but yet they differ on how they would change this idea if they win a million-dollar lottery. Since the basic condition of the preference ordering is that their minimal models are exactly the models of their associated knowledge base, a knowledge base can be considered implicitly represented by a preference ordering (but not vice versa). Given that a preference ordering allows deriving its associated knowledge base but also allows performing a single step of revision, studies on iterated revision have been concentrated on how a preference ordering should be changed in response of a revision. While single-step revision is about how a knowledge base K has to be changed into a new knowledge base K ∗P , iterated revision is about how a preference ordering (representing both the current knowledge and how much situations believed to be false are considered possible) should be turned into a new preference relation when P is learned. A single step of iterated revision produces a new ordering that allows for further revisions. Two kinds of preference ordering are usually considered: numerical and non-numerical. In the first case, the level of plausibility of a model is representing by a non-negative integer number; the lower the rank, the more plausible the situation corresponding to the model. Non-numerical preference orderings correspond to the preference relations used in the AGM framework: a possibly total ordering over models. The non-numerical preference relation were initially considered unsuitable for iterated revision because of the impossibility of reverting a revision by a number of other revisions, which is instead possible in the numerical case. Darwiche and Pearl*[1] formulated the following postulates for iterated revision.

1. if α |= µ then (ψ ∗ µ) ∗ α ≡ ψ ∗ α ;

2. if α |= ¬µ , then (ψ ∗ µ) ∗ α ≡ ψ ∗ α ;

3. if ψ ∗ α |= µ , then (ψ ∗ µ) ∗ α |= µ ;

4. if ψ ∗ α ̸|= ¬µ , then (ψ ∗ µ) ∗ α ̸|= ¬µ .

Specific iterated revision operators have been proposed by Spohn, Boutilier, Williams, Lehmann, and others.

Spohn rejected revision this non-numerical proposal has been first considered by Spohn, who rejected it based on the fact that revisions can change some orderings in such a way the original ordering cannot be restored with a sequence of other revisions; this operator change a preference ordering in view of new information P by making all models of P being preferred over all other models; the original preference ordering is maintained when comparing two models that are both models of P or both non-models of P ;

Natural revision while revising a preference ordering by a formula P , all minimal models (according to the pref- erence ordering) of P are made more preferred by all other ones; the original ordering of models is preserved when comparing two models that are not minimal models of P ; this operator changes the ordering among models minimally while preserving the that the models of the knowledge base after revising by P are the minimal models of P according to the preference ordering;

Transmutations these are two forms of revision, conditionalization and adjustment, which work on numerical preference orderings; revision requires not only a formula but also a number indicating its degree of plausi- bility; while the preference ordering is still inverted (the lower a model, the most plausible it is) the degree of plausibility of a revising formula is direct (the higher the degree, the most believed the formula is); 7.11. MERGING 45

Ranked revision a ranked model, which is an assignment of non-negative integers to models, has to be specified at the beginning; this rank is similar to a preference ordering, but is not changed by revision; what is changed by a sequence of revisions are a current set of models (representing the current knowledge base) and a number called the rank of the sequence; since this number can only monotonically non-decrease, some sequences of revision lead to situations in which every further revision is performed as a full meet revision.

7.11 Merging

The assumption implicit in the revision operator is that the new piece of information P is always to be considered more reliable than the old knowledge base K . This is formalized by the second of the AGM postulates: P is always believed after revising K with P . More generally, one can consider the process of merging several pieces of information (rather than just two) that might or might not have the same reliability. Revision becomes the particular instance of this process when a less reliable piece of information K is merged with a more reliable P . While the input to the revision process is a pair of formulae K and P , the input to merging is a multiset of formulae K , T , etc. The use of multisets is necessary as two sources to the merging process might be identical. When merging a number of knowledge bases with the same degree of plausibility, a distinction is made between arbitration and majority. This distinction depends on the assumption that is made about the information and how it has to be put together.

arbitration the result of arbitrating two knowledge bases K and T entails K ∨ T ; this condition formalizes the assumption of maintaining as much as the old information as possible, as it is equivalent to imposing that every formula entailed by both knowledge bases is also entailed by the result of their arbitration; in a view, the “real”world is assumed one of the worlds considered possible according to at least one of the two knowledge bases;

majority the result of merging a knowledge base K with other knowledge bases can be forced to entail K by adding a sufficient number of other knowledge bases equivalent to K ; this condition corresponds to a kind of vote- by-majority: a sufficiently large number of knowledge bases can always overcome the“opinion”of any other fixed set of knowledge bases.

The above is the original definition of arbitration. According to a newer definition, an arbitration operator is a merging operator that is insensitive to the number of equivalent knowledge bases to merge. This definition makes arbitration the exact opposite of majority. Postulates for both arbitration and merging have been proposed. An example of an arbitration operator satisfying all postulates is the classical disjunction. An example of a majority operator satisfying all postulates is that selecting all models that have a minimal total Hamming distance to models of the knowledge bases to merge. A merging operator can be expressed as a family of orderings over models, one for each possible multiset of knowledge bases to merge: the models of the result of merging a multiset of knowledge bases are the minimal models of the ordering associated to the multiset. A merging operator defined in this way satisfies the postulates for merging if and only if the family of orderings meets a given set of conditions. For the old definition of arbitration, the orderings are not on models but on pairs (or, in general, tuples) of models.

7.12

Many revision proposals involve orderings over models representing the relative plausibility of the possible alterna- tives. The problem of merging amounts to combine a set of orderings into a single one expressing the combined plausibility of the alternatives. This is similar with what is done in social choice theory, which is the study of how the preferences of a group of agents can be combined in a rational way. Belief revision and social choice theory are similar in that they combine a set of orderings into one. They differ on how these orderings are interpreted: prefer- ences in social choice theory; plausibility in belief revision. Another difference is that the alternatives are explicitly enumerated in social choice theory, while they are the propositional models over a given alphabet in belief revision. 46 CHAPTER 7. BELIEF REVISION

7.13 Complexity

The problem about belief revision that is the most studied from the point of view of computational complexity is that of query answering in the propositional case. This is the problem of establishing whether a formula follows from the result of a revision, that is, K ∗ P |= Q , where K , P , and Q are propositional formulae. More generally, query answering is the problem of telling whether a formula is entailed by the result of a belief revision, which could be update, merging, revision, iterated revision, etc. Another problem that has received some attention is that of model checking, that is, checking whether a model satisfies the result of a belief revision. A related question is whether such result can be represented in space polynomial in that of its arguments. Since a deductively closed knowledge base is infinite, complexity studies on belief revision operators working on deductively closed knowledge bases are done in the assumption that such deductively closed knowledge base are given in the form of an equivalent finite knowledge base. A distinction is made among belief revision operators and belief revision schemes. While the former are simple mathematical operators mapping a pair of formulae into another formula, the latter depend on further information such as a preference relation. For example, the Dalal revision is an operator because, once two formulae K and P are given, no other information is needed to compute K ∗ P . On the other hand, revision based on a preference relation is a revision scheme, because K and P do not allow determining the result of revision if the family of preference orderings between models is not given. The complexity for revision schemes is determined in the assumption that the extra information needed to compute revision is given in some compact form. For example, a preference relation can be represented by a sequence of formulae whose models are increasingly preferred. Explicitly storing the relation as a set of pairs of models is instead not a compact representation of preference because the space required is exponential in the number of propositional letters. The complexity of query answering and model checking in the propositional case is in the second level of the polynomial hierarchy for most belief revision operators and schemas. Most revision operators suffer from the problem of representational blow up: the result of revising two formulae is not necessarily representable in space polynomial in that of the two original formulae. In other words, revision may exponentially increase the size of the knowledge base.

7.14 Implementations

Systems specifically implementing belief revision are: Immortal, SATEN, and BReLS. Two systems including a belief revision feature are SNePS and Cyc. Truth maintenance systems are used in Artificial Intelligence to implement belief revision.

7.15 See also

• Artificial intelligence

• Inquiry

• Knowledge representation

• Belief propagation

• Reason maintenance

• Epistemic closure

• Non-monotonic logic

• Defeasible reasoning

• Reasoning

• Philosophy of science

• Discursive dilemma 7.16. REFERENCES 47

7.16 References

[1] Darwiche, A. and Pearl, J. (1997) On the logic of iterated belief revision. Artificial Intelligence 89(1-2): 1-29.

• C. E. Alchourròn, P. Gärdenfors, and D. Makinson (1985). On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic, 50:510–530.

• C. Boutilier (1993). Revision sequences and nested conditionals. In Proceedings of the Thirteenth International Joint Conference on Artificial Intelligence (IJCAI'93), pages 519–525.

• C. Boutilier (1995). Generalized update: belief change in dynamic settings. In Proceedings of the Fourteenth International Joint Conference on Artificial Intelligence (IJCAI'95), pages 1550–1556.

• C. Boutilier (1996). Abduction to plausible causes: an -based model of belief update. Artificial Intelli- gence, 83:143–166.

• M. Cadoli, F. M. Donini, P. Liberatore, and M. Schaerf (1999). The size of a revised knowledge base. Artificial Intelligence, 115(1):25–64.

• T. Chou and M. Winslett (1991). Immortal: A model-based belief revision system. In Proceedings of the Conference on the Principles of Knowledge Representation and Reasoning (KR'91), pages 99–110. Morgan Kaufmann Publishers.

• M. Dalal (1988). Investigations into a theory of knowledge base revision: Preliminary report. In Proceedings of the Seventh National Conference on Artificial Intelligence (AAAI'88), pages 475–479.

• T. Eiter and G. Gottlob (1992). On the complexity of propositional knowledge base revision, updates and counterfactuals. Artificial Intelligence, 57:227–270.

• T. Eiter and G. Gottlob (1996). The complexity of nested counterfactuals and iterated knowledge base revi- sions. Journal of Computer and System Sciences, 53(3):497–512.

• R. Fagin, J. D. Ullman, and M. Y. Vardi (1983). On the semantics of updates in databases. In Proceedings of the Second ACM SIGACT SIGMOD Symposium on Principles of Database Systems (PODS'83), pages 352–365.

• M. A. Falappa, G. Kern-Isberner, G. R. Simari (2002): Explanations, belief revision and defeasible reasoning. Artificial Intelligence, 141(1–2): 1–28.

• M. Freund and D. Lehmann (2002). Belief Revision and Rational Inference. Arxiv preprint cs.AI/0204032.

• N. Friedman and J. Y. Halpern (1994). A knowledge-based framework for belief change, part II: Revision and update. In Proceedings of the Conference on the Principles of Knowledge Representation and Reasoning (KR'94), pages 190–200.

• A. Fuhrmann (1991). Theory contraction through base contraction. Journal of , 20:175– 203.

• D. Gabbay, G. Pigozzi, and J. Woods (2003). Controlled Revision – An algorithmic approach for belief revi- sion, Journal of Logic and Computation, 13(1): 15–35.

• P. Gärdenfors and D. Makinson (1988). Revision of knowledge systems using epistemic entrenchment. In Proceedings of the Second Conference on Theoretical Aspects of Reasoning about Knowledge (TARK'88), pages 83–95.

• P. Gärdenfors and H. Rott (1995). Belief revision. In Handbook of Logic in Artificial Intelligence and Logic Programming, Volume 4, pages 35–132. . 48 CHAPTER 7. BELIEF REVISION

• G. Grahne and Alberto O. Mendelzon (1995). Updates and subjunctive queries. Information and Computation, 2(116):241–252.

• G. Grahne, Alberto O. Mendelzon, and P. Revesz (1992). Knowledge transformations. In Proceedings of the Eleventh ACM SIGACT SIGMOD SIGART Symposium on Principles of Database Systems (PODS'92), pages 246–260.

• S. O. Hansson (1999). A Textbook of Belief Dynamics. Dordrecht: Kluwer Academic Publishers.

• A. Herzig (1996). The PMA revised. In Proceedings of the Conference on the Principles of Knowledge Representation and Reasoning (KR'96), pages 40–50.

• A. Herzig (1998). for belief base updating. In D. Dubois, D. Gabbay, H. Prade, and P. Smets, editors, Handbook of defeasible reasoning and uncertainty management, volume 3 – Belief Change, pages 189–231. Kluwer Academic Publishers.

• H. Katsuno and A. O. Mendelzon (1991). On the difference between updating a knowledge base and revising it. In Proceedings of the Second International Conference on the Principles of Knowledge Representation and Reasoning (KR'91), pages 387–394.

• H. Katsuno and A. O. Mendelzon (1991). Propositional knowledge base revision and minimal change. Artificial Intelligence, 52:263–294.

• S. Konieczny and R. Pino Perez (1998). On the logic of merging. In Proceedings of the Sixth International Conference on Principles of Knowledge Representation and Reasoning (KR'98), pages 488–498.

• D. Lehmann (1995). Belief revision, revised. In Proceedings of the Fourteenth International Joint Conference on Artificial Intelligence (IJCAI'95), pages 1534–1540.

• P. Liberatore (1997). The complexity of iterated belief revision. In Proceedings of the Sixth International Conference on Database Theory (ICDT'97), pages 276–290.

• P. Liberatore and M. Schaerf (1998). Arbitration (or how to merge knowledge bases). IEEE Transactions on Knowledge and Data Engineering, 10(1):76–90.

• P. Liberatore and M. Schaerf (2000). BReLS: A system for the integration of knowledge bases. In Proceedings of the Seventh International Conference on Principles of Knowledge Representation and Reasoning (KR 2000), pages 145–152.

• D. Makinson (1985). How to give up: A survey of some formal aspects of the logic of theory change. , 62:347–363.

• A. Perea (2003). Proper Rationalizability and Belief Revision in Dynamic Games. Research Memoranda 048: METEOR, Maastricht Research School of of Technology and Organization.

• B. Nebel (1991). Belief revision and default reasoning: Syntax-based approaches. In Proceedings of the Second International Conference on the Principles of Knowledge Representation and Reasoning (KR'91), pages 417–428.

• B. Nebel (1994). Base revision operations and schemes: Semantics, representation and complexity. In Pro- ceedings of the Eleventh European Conference on Artificial Intelligence (ECAI'94), pages 341–345.

• B. Nebel (1996). How hard is it to revise a knowledge base? Technical Report 83, Albert-Ludwigs-Universität Freiburg, Institut für Informatik.

• G. Pigozzi (2005). Two aggregation in social decision making: the Ostrogorski and the discursive dilemma, Episteme: A Journal of Social Epistemology, 2(2): 33–42. 7.17. EXTERNAL LINKS 49

• G. Pigozzi (2006). Belief merging and the discursive dilemma: an argument-based account to paradoxes of judgment aggregation. Synthese 152(2): 285–298.

• P. Z. Revesz (1993). On the semantics of theory change: Arbitration between old and new information. In Proceedings of the Twelfth ACM SIGACT SIGMOD SIGART Symposium on Principles of Database Systems (PODS'93), pages 71–82.

• K. Satoh (1988). Nonmonotonic reasoning by minimal belief revision. In Proceedings of the International Conference on Fifth Generation Computer Systems (FGCS'88), pages 455–462.

• Shoham, Yoav; Leyton-Brown, Kevin (2009). Multiagent Systems: Algorithmic, Game-Theoretic, and Log- ical Foundations. New York: Cambridge University Press. ISBN 978-0-521-89943-7. See Section 14.2; downloadable free online.

• V. S. Subrahmanian (1994). Amalgamating knowledge bases. ACM Transactions on Database Systems, 19(2):291– 331.

• A. Weber (1986). Updating propositional formulas. In Proc. of First Conf. on Expert Database Systems, pages 487–500.

• M. Williams (1994). Transmutations of knowledge systems. In Proceedings of the Fourth International Con- ference on the Principles of Knowledge Representation and Reasoning (KR'94), pages 619–629.

• M. Winslett (1989). Sometimes updates are circumscription. In Proceedings of the Eleventh International Joint Conference on Artificial Intelligence (IJCAI'89), pages 859–863.

• M. Winslett (1990). Updating Logical Databases. Cambridge University Press.

• Y. Zhang and N. Foo (1996). Updating knowledge bases with disjunctive information. In Proceedings of the Thirteenth National Conference on Artificial Intelligence (AAAI'96), pages 562–568.

7.17 External links

• Belief revision at PhilPapers • Logic of Belief Revision at the Indiana Philosophy Ontology Project

• Logic of Belief Revision entry in the Stanford Encyclopedia of Philosophy • Beliefrevision.org

• Defeasible Reasoning: 4.3 Belief Revision Theory at Stanford Encyclopedia of Philosophy Chapter 8

Boolean network

State space of a Boolean Network with N=4 nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the which is a simple“copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.

A Boolean network consists of a discrete set of Boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on

50 8.1. CLASSICAL MODEL 51

the state of the network at time t. This may be done synchronously or asynchronously. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.*[1]*[2] The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.*[3]

8.1 Classical model

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Boolean networks are related to cellular automata. A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed. The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks*[4] but their mathematically understanding only started in the 2000's.*[5]

8.1.1 Attractors

Since a Boolean network has only 2*N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a cycle. In the literature in this field, each cycle is also called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states*[6] and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.*[3] With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.*[7]

8.2 Stability

The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes ( Kc ), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In the critical regime, the Hamming distance is small compared with the number of nodes ( N ) in the network. * For N-K-model [13] the network is stable if K < Kc , critical if K = Kc , and unstable if K > Kc .

The state of a given node ni is updated according to its , whose outputs are randomly populated. pi denotes the probability of assigning an off output to a given series of input signals.

If pi = p = const. for every node, the transition between the stable and chaotic range depends on p . The critical * value of the average number of connections is Kc = 1/[2p(1 − p)] . [14] If K is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is in * * * in in determined by ⟨K ⟩ [15] [16] [17] The network is stable if ⟨K ⟩ < Kc , critical if ⟨K ⟩ = Kc , and unstable in if ⟨K ⟩ > Kc . The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: P (K) ∝ K−γ , and ⟨Kin⟩ = ⟨Kout⟩ , since every out-link from a node is an in-link to another.*[18] Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. 52 CHAPTER 8. BOOLEAN NETWORK

For random Boolean networks, qi = 2pi(1 − pi) . In the general case, stability of the network is governed by the * largest eigenvalue λQ of matrix Q , where Qij = qiAij , and A is the adjacency matrix of the network. [19] The network is stable if λQ < 1 , critical if λQ = 1 , unstable if λQ > 1 .

8.3 Varations of the model

8.3.1 Other topologies

One theme is to study different underlying graph topologies.

• The homogeneous case simply refers to a grid which is simply the reduction to the famous Ising model. • Scale-free topologies may be chosen for Boolean networks.*[20] One can distinguish the case where only in- degree distribution in power-law distributed,*[21] or only the out-degree-distribution or both.

8.3.2 Other updating schemes

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes usually not simultaneously changing their state,*[22] different alternatives have been introduced. A common classification*[23] is the following:

• Deterministic asynchronous updated Boolean networks (DRBNs) are not synchronously updated but a * deterministic still exists. A node i will be updated when t ≡ Qi (mod Pi) where t is the time step. [24] • The most general case is full stochastic updating (GARBN, general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.

8.4 See also

• NK model

8.5 References

[1] Albert, Réka; Othmer, Hans G (July 2003).“The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster”. Journal of Theoretical Biology 223 (1): 1–18. doi:10.1016/S0022- 5193(03)00035-3.

[2] Li, J.; Bench, A. J.; Vassiliou, G. S.; Fourouclas, N.; Ferguson-Smith, A. C.; Green, A. R. (30 April 2004). “Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies”. Proceedings of the National Academy of Sciences 101 (19): 7341–7346. Bibcode:2004PNAS..101.7341L. doi:10.1073/pnas.0308195101. PMC 409920. PMID 15123827. Retrieved 25 November 2014.

[3] Drossel, Barbara (December 2009). Schuster, Heinz Georg, ed. “Chapter 3. Random Boolean Networks”. Reviews of Nonlinear Dynamics and Complexity. Reviews of Nonlinear Dynamics and Complexity (Wiley): 69–110. doi:10.1002/9783527626359.ch3. Retrieved 25 November 2014.

[4] Kauffman, Stuart (11 October 1969). “Homeostasis and Differentiation in Random Genetic Control Networks”. Nature 224 (5215): 177–178. Bibcode:1969Natur.224..177K. doi:10.1038/224177a0.

[5] Aldana, Maximo; Coppersmith, Susan; Kadanoff, Leo P. (2003). “Boolean Dynamics with Random Couplings”. Per- spectives and Problems in Nonlinear Sciences: 23–89. doi:10.1007/978-0-387-21789-5_2. Retrieved 8 January 2016.

[6] Wuensche, Andrew (2011). Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]. Frome, England: Luniver Press. p. 16. ISBN 9781905986316. Retrieved 12 January 2016.

[7] Greil, Florian (2012).“Boolean Networks as Modeling Framework”. Frontiers in Plant Science 3. doi:10.3389/fpls.2012.00178. 8.6. EXTERNAL LINKS 53

[8] Bastolla, U.; Parisi, G. (May 1998). “The modular structure of Kauffman networks”. Physica D: Nonlinear Phenomena 115 (3-4): 219–233. arXiv:cond-mat/9708214. Bibcode:1998PhyD..115..219B. doi:10.1016/S0167-2789(97)00242-X.

[9] Bilke, Sven; Sjunnesson, Fredrik (December 2001). “Stability of the Kauffman model”. Physical Review E 65 (1). arXiv:cond-mat/0107035. Bibcode:2002PhRvE..65a6129B. doi:10.1103/PhysRevE.65.016129.

[10] Socolar, J.; Kauffman, S. (February 2003). “Scaling in Ordered and Critical Random Boolean Networks”. Physical Review Letters 90 (6). arXiv:cond-mat/0212306. Bibcode:2003PhRvL..90f8702S. doi:10.1103/PhysRevLett.90.068702. Retrieved 26 November 2014.

[11] Samuelsson, Björn; Troein, Carl (March 2003). “Superpolynomial Growth in the Number of Attractors in Kauffman Networks”. Physical Review Letters 90 (9). Bibcode:2003PhRvL..90i8701S. doi:10.1103/PhysRevLett.90.098701.

[12] Mihaljev, Tamara; Drossel, Barbara (October 2006). “Scaling in a general of critical random Boolean networks”. Physical Review E 74 (4). arXiv:cond-mat/0606612. Bibcode:2006PhRvE..74d6101M. doi:10.1103/PhysRevE.74.046101.

[13] Kauffman, S. A. (1969).“Metabolic stability and epigenesis in randomly constructed genetic nets”. Journal of Theoretical Biology 22 (3): 437–467. doi:10.1016/0022-5193(69)90015-0. PMID 5803332.

[14] Derrida, B; Pomeau, Y (1986-01-15). “Random Networks of Automata: A Simple Annealed Approximation”. Euro- physics Letters (EPL) 1 (2): 45–49. doi:10.1209/0295-5075/1/2/001.

[15] Solé, Ricard V.; Luque, Bartolo (1995-01-02). “Phase transitions and antichaos in generalized Kauffman networks”. Physics Letters A 196 (5–6): 331–334. doi:10.1016/0375-9601(94)00876-Q.

[16] Luque, Bartolo; Solé, Ricard V. (1997-01-01). “Phase transitions in random networks: Simple analytic determination of critical points”. Physical Review E 55 (1): 257–260. doi:10.1103/PhysRevE.55.257.

[17] Fox, Jeffrey J.; Hill, Colin C. (2001-12-01). “From topology to dynamics in biochemical networks”. Chaos: An Inter- disciplinary Journal of Nonlinear Science 11 (4): 809–815. doi:10.1063/1.1414882. ISSN 1054-1500.

[18] Aldana, Maximino; Cluzel, Philippe (2003-07-22). “A natural class of robust networks”. Proceedings of the National Academy of Sciences 100 (15): 8710–8714. doi:10.1073/pnas.1536783100. ISSN 0027-8424. PMC 166377. PMID 12853565.

[19] Pomerance, Andrew; Ott, Edward; Girvan, Michelle; Losert, Wolfgang (2009-05-19). “The effect of network topology on the stability of discrete state models of genetic control”. Proceedings of the National Academy of Sciences 106 (20): 8209–8214. doi:10.1073/pnas.0900142106. ISSN 0027-8424. PMC 2688895. PMID 19416903.

[20] Aldana, Maximino (October 2003). “Boolean dynamics of networks with scale-free topology”. Physica D: Nonlinear Phenomena (Elsevier) 185 (1): 45–66. doi:10.1016/s0167-2789(03)00174-x.

[21] Drossel, Barbara; Greil, Florian (4 August 2009). “Critical Boolean networks with scale-free in-degree distribution”. Physical Review E 80 (2). doi:10.1103/PhysRevE.80.026102.

[22] Harvey, Imman; Bossomaier, Terry (1997). Husbands, Phil; Harvey, Imman, eds. “Time out of joint: Attractors in asynchronous random Boolean networks”. Proceedings of the Fourth European Conference on Artificial Life (ECAL97) (MIT Press): 67–75.

[23] Gershenson, Carlos (2002). Standish, Russell K; Bedau, Mark A, eds. “Classification of Random Boolean Networks” . Proceedings of the eighth international conference on Artificial life. Artificial Life (Cambridge, Massachusetts, USA) 8: 1–8. Retrieved 12 January 2016.

[24] Gershenson, Carlos; Broekaert, Jan; Aerts, Diederik (14 September 2003). “Contextual Random Boolean Networks” [7th European Conference, ECAL 2003]. Advances in Artificial Life. Lecture Notes in Computer Science (Dortmund, ) 2801: 615–624. doi:10.1007/978-3-540-39432-7_66. ISBN 978-3-540-39432-7. Retrieved 12 January 2016.

• Dubrova, E., Teslenko, M., Martinelli, A., (2005). *Kauffman Networks: Analysis and Applications, in“Pro- ceedings of International Conference on Computer-Aided Design”, pages 479-484.

8.6 External links

• DDLab

• Analysis of Dynamic Algebraic Models (ADAM) v1.1 54 CHAPTER 8. BOOLEAN NETWORK

• RBNLab

• NetBuilder Boolean Networks Simulator • Open Source Boolean Network Simulator

• JavaScript Kauffman Network • Probabilistic Boolean Networks (PBN)

• A SAT-based tool for computing attractors in Boolean Networks Chapter 9

Canon (basic principle)

The concept of canon is very broad; in a general sense it refers to being a rule or a body of rules. There are definitions that state it as: “the body of rules, principles, or standards accepted as axiomatic and universally binding in a field of study or art”.*[1] This can be related to such topics as literary canons or the canons of rhetoric, which is a topic within itself that describes the rules of giving a speech. There are five key principles, and when grouped together, are the principles set for giving speeches as seen with regard to Rhetoric. This is one such example of how the term canon is used in regard to rhetoric.*[2]*[3]*[4]*[5]*[6]

9.1 See also

• • Canon

• Canon law (Catholic Church) • (philosophy)

• Principle •

• Rhetoric

9.2 References

[1] “Define Canon”. Dictionary.reference.com. Retrieved 2015-04-07.

[2] WordNet 3.1. retrieved 2011-12-03 from: Canon Search Word.

[3] W.C Sayers (1915–1916) established a system of canons of classification Sayers, W.C. (1915-1916). Canons of classifi- cation applied to “The subject”, “The expansive”, “The decimal”and “The Library of Congress”classifications: A study in bibliographical classification method. Lindon: Grafton.

[4] S. R. Ranganathan developed a theory of facet analysis which he presented as a detailed series of 46 canons, 13 postulates and 22 principles. in Prolegomena to library classification. New York: Asia Publishing House. Spiteri, Louise (1998). A Simplified Model for Facet Analysis: Ranganathan 101. Canadian Journal of Information and Library Science—Revue Canadienne des Sciences de l'Information et de Bibliotheconomie, 23(1-2), 1-30., Retrieved from: http://iainstitute.org/ en/learn/research/a_simplified_model_for_facet_analysis.php

[5] Toye, Richard (2013). Rhetoric A Very Short Introduction. Oxford, : Oxford University Press. ISBN 978-0-19-965136-8.

[6] “Canon”. Dictionary.com. Random House, Inc. Retrieved September 29, 2014.

55 Chapter 10

Canonical normal form

In , any Boolean function can be put into the canonical (CDNF) or minterm canonical form and its dual canonical (CCNF) or maxterm canonical form. Other canonical forms include the complete sum of prime implicants or Blake canonical form (and its dual), and the (also called Zhegalkin or Reed–Muller). Minterms are called products because they are the logical AND of a set of variables, and maxterms are called sums because they are the logical OR of a set of variables. These concepts are dual because of their complementary- symmetry relationship as expressed by De Morgan's laws. Two dual canonical forms of any Boolean function are a “sum of minterms”and a “product of maxterms.”The term "Sum of Products" or "SoP" is widely used for the canonical form that is a disjunction (OR) of minterms. Its De Morgan dual is a "Product of Sums" or "PoS" for the canonical form that is a conjunction (AND) of maxterms. These forms can be useful for the simplification of these functions, which is of great importance in the optimization of Boolean formulas in general and digital circuits in particular.

10.1 Summary

One application of Boolean algebra is digital circuit design. The goal may be to minimize the number of gates, to minimize the settling time, etc. There are sixteen possible functions of two variables, but in digital logic hardware, the simplest gate circuits implement only four of them: conjunction (AND), disjunction (inclusive OR), and the respective complements of those (NAND and NOR). Most gate circuits accept more than 2 input variables; for example, the spaceborne , which pioneered the application of integrated circuits in the 1960s, was built with only one type of gate, a 3-input NOR, whose output is true only when all 3 inputs are false.*[1]

10.2 Minterms

For a boolean function of n variables x1, . . . , xn , a product term in which each of the n variables appears once (in either its complemented or uncomplemented form) is called a minterm. Thus, a minterm is a logical expression of n variables that employs only the complement operator and the conjunction operator. For example, abc , ab′c and abc′ are 3 examples of the 8 minterms for a Boolean function of the three variables a , b , and c . The customary reading of the last of these is a AND b AND NOT-c. There are 2*n minterms of n variables, since a variable in the minterm expression can be in either its direct or its complemented form—two choices per variable.

56 10.3. MAXTERMS 57

10.2.1 Indexing minterms

Minterms are often numbered by a binary encoding of the complementation pattern of the variables, where the variables are written in a standard order, usually alphabetical. This assigns the value 1 to the direct form ∑n ′ i ′ ( xi ) and 0 to the complemented form ( xi ); the minterm is then 2 value(xi) . For example, minterm abc is i=1 numbered 1102 = 610 and denoted m6 .

10.2.2 Functional equivalence

A given minterm n gives a true value (i.e., 1) for just one combination of the input variables. For example, minterm 5, a b' c, is true only when a and c both are true and b is false—the input arrangement where a = 1, b = 0, c = 1 results in 1. Given the truth table of a logical function, it is possible to write the function as a “sum of products”. This is a special form of disjunctive normal form. For example, if given the truth table for the arithmetic sum bit u of one bit position's logic of an adder circuit, as a function of x and y from the addends and the carry in, ci: Observing that the rows that have an output of 1 are the 2nd, 3rd, 5th, and 8th, we can write u as a sum of minterms ′ ′ ′ ′ m1, m2, m4, and m7 . If we wish to verify this: u(ci, x, y) = m1 + m2 + m4 + m7 = (ci , x , y) + (ci , x, y ) + (ci, x′, y′) + (ci, x, y) evaluated for all 8 combinations of the three variables will match the table.

10.3 Maxterms

For a boolean function of n variables x1, . . . , xn , a sum term in which each of the n variables appears once (in either its complemented or uncomplemented form) is called a maxterm. Thus, a maxterm is a logical expression of n variables that employs only the complement operator and the disjunction operator. Maxterms are a dual of the minterm idea (i.e., exhibiting a complementary symmetry in all respects). Instead of using ANDs and complements, we use ORs and complements and proceed similarly. For example, the following are two of the eight maxterms of three variables:

a + b' + c a' + b + c

There are again 2*n maxterms of n variables, since a variable in the maxterm expression can also be in either its direct or its complemented form—two choices per variable.

10.3.1 Indexing maxterms

Each maxterm is assigned an index based on the opposite conventional binary encoding used for minterms. The ′ maxterm convention assigns the value 0 to the direct form (xi) and 1 to the complemented form (xi) . For example, ′ ′ we assign the index 6 to the maxterm a + b + c (110) and denote that maxterm as M6. Similarly M0 of these three ′ ′ ′ variables is a + b + c (000) and M7 is a + b + c (111).

10.3.2 Functional equivalence

It is apparent that maxterm n gives a false value (i.e., 0) for just one combination of the input variables. For example, maxterm 5, a' + b + c', is false only when a and c both are true and b is false—the input arrangement where a = 1, b = 0, c = 1 results in 0. If one is given a truth table of a logical function, it is possible to write the function as a “product of sums”. This is a special form of conjunctive normal form. For example, if given the truth table for the carry-out bit co of one bit position's logic of an adder circuit, as a function of x and y from the addends and the carry in, ci: Observing that the rows that have an output of 0 are the 1st, 2nd, 3rd, and 5th, we can write co as a product of maxterms M0,M1,M2 and M4 . If we wish to verify this: co(ci, x, y) = M0M1M2M4 = (ci + x + y)(ci + x + y') (ci + x' + y)(ci + x' + y') evaluated for all 8 combinations of the three variables will match the table. 58 CHAPTER 10. CANONICAL NORMAL FORM

10.4 Dualization

The complement of a minterm is the respective maxterm. This can be easily verified by using de Morgan's law. For ′ ′ ′ ′ ′ example: M5 = a + b + c = (ab c) = m5

10.5 Non-canonical PoS and SoP forms

It is often the case that the canonical minterm form can be simplified to an equivalent SoP form. This simplified form would still consist of a sum of product terms. However, in the simplified form, it is possible to have fewer product terms and/or product terms that contain fewer variables. For example, the following 3-variable function: has the canonical minterm representation: f = a′bc + abc , but it has an equivalent simplified form: f = bc . In this trivial example, it is obvious that bc = a′bc + abc , but the simplified form has both fewer product terms, and the term has fewer variables. The most simplified SoP representation of a function is referred to as a minimal SoP form. In a similar manner, a canonical maxterm form can have a simplified PoS form. While this example was easily simplified by applying normal algebraic methods [ f = (a′ + a)bc ], in less obvious cases a convenient method for finding the minimal PoS/SoP form of a function with up to four variables is using a . The minimal PoS and SoP forms are very important for finding optimal implementations of boolean functions and minimizing logic circuits.

10.6 Application example

The sample truth tables for minterms and maxterms above are sufficient to establish the canonical form for a single bit position in the addition of binary numbers, but are not sufficient to design the digital logic unless your inventory of gates includes AND and OR. Where performance is an issue (as in the Apollo Guidance Computer), the available parts are more likely to be NAND and NOR because of the complementing action inherent in logic. The values are defined as voltage states, one near ground and one near the DC supply voltage Vcc, e.g. +5 VDC. If the higher voltage is defined as the 1 “true”value, a NOR gate is the simplest possible useful logical element. Specifically, a 3-input NOR gate may consist of 3 bipolar junction with their emitters all grounded, their collectors tied together and linked to Vcc through a load impedance. Each base is connected to an input signal, and the common collector point presents the output signal. Any input that is a 1 (high voltage) to its base shorts its transistor's emitter to its collector, causing current to flow through the load impedance, which brings the collector voltage (the output) very near to ground. That result is independent of the other inputs. Only when all 3 input signals are 0 (low voltage) do the emitter-collector impedances of all 3 transistors remain very high. Then very little current flows, and the voltage-divider effect with the load impedance imposes on the collector point a high voltage very near to Vcc. The complementing property of these gate circuits may seem like a drawback when trying to implement a function in canonical form, but there is a compensating bonus: such a gate with only one input implements the complementing function, which is required frequently in digital logic. This example assumes the Apollo parts inventory: 3-input NOR gates only, but the discussion is simplified by sup- posing that 4-input NOR gates are also available (in Apollo, those were compounded out of pairs of 3-input NORs).

10.6.1 Canonical and non-canonical consequences of NOR gates

Fact #1: a set of 8 NOR gates, if their inputs are all combinations of the direct and complement forms of the 3 input variables ci, x, and y, always produce minterms, never maxterms—that is, of the 8 gates required to process all combinations of 3 input variables, only one has the output value 1. That's because a NOR gate, despite its name, could better be viewed (using De Morgan's law) as the AND of the complements of its input signals. Fact #2: the reason Fact #1 is not a problem is the duality of minterms and maxterms, i.e. each maxterm is the complement of the like-indexed minterm, and vice versa.

In the minterm example above, we wrote u(ci, x, y) = m1 + m2 + m4 + m7 but to perform this with a 4-input 10.6. APPLICATION EXAMPLE 59

NOR gate we need to restate it as a product of sums (PoS), where the sums are the opposite maxterms. That is,

u(ci, x, y) = AND( M0,M3,M5,M6 ) = NOR( m0, m3, m5, m6 ). Truth tables:

In the maxterm example above, we wrote co(ci, x, y) = M0M1M2M4 but to perform this with a 4-input NOR gate we need to notice the to the NOR of the same minterms. That is,

co(ci, x, y) = AND( M0,M1,M2,M4 ) = NOR( m0, m1, m2, m4 ). Truth tables:

10.6.2 Design -offs considered in addition to canonical forms

One might suppose that the work of designing an adder stage is now complete, but we haven't addressed the fact that all 3 of the input variables have to appear in both their direct and complement forms. There's no difficulty about the addends x and y in this respect, because they are static throughout the addition and thus are normally held in latch circuits that routinely have both direct and complement outputs. (The simplest latch circuit made of NOR gates is a pair of gates cross-coupled to make a flip-flop: the output of each is wired as one of the inputs to the other.) There is also no need to create the complement form of the sum u. However, the carry out of one bit position must be passed as the carry into the next bit position in both direct and complement forms. The most straightforward way to do this is to pass co through a 1-input NOR gate and label the output co', but that would add a gate delay in the worst possible place, slowing down the rippling of carries from right to left. An additional 4-input NOR gate building the canonical form of co' (out of the opposite minterms as co) solves this problem.

′ co (ci, x, y) = AND(M3,M5,M6,M7) = NOR(m3, m5, m6, m7).

Truth tables: The trade-off to maintain full speed in this way includes an unexpected cost (in addition to having to use a bigger gate). If we'd just used that 1-input gate to complement co, there would have been no use for the minterm m7 , and the gate that generated it could have been eliminated. Nevertheless, it's still a good trade. Now we could have implemented those functions exactly according to their SoP and PoS canonical forms, by turning NOR gates into the functions specified. A NOR gate is made into an OR gate by passing its output through a 1-input NOR gate; and it is made into an AND gate by passing each of its inputs through a 1-input NOR gate. However, this approach not only increases the number of gates used, but also doubles the number of gate delays processing the signals, cutting the processing speed in half. Consequently, whenever performance is vital, going beyond canonical forms and doing the Boolean algebra to make the unenhanced NOR gates do the job is well worthwhile.

10.6.3 Top-down vs. bottom-up design

We have now seen how the minterm/maxterm tools can be used to design an adder stage in canonical form with the addition of some Boolean algebra, costing just 2 gate delays for each of the outputs. That's the “top-down”way to design the digital circuit for this function, but is it the best way? The discussion has focused on identifying“fastest”as “best,”and the augmented canonical form meets that criterion flawlessly, but sometimes other factors predominate. The designer may have a primary goal of minimizing the number of gates, and/or of minimizing the fanouts of signals to other gates since big fanouts reduce resilience to a degraded power supply or other environmental factors. In such a case, a designer may develop the canonical-form design as a baseline, then try a bottom-up development, and finally compare the results. The bottom-up development involves noticing that u = ci XOR (x XOR y), where XOR means [true when either input is true but not when both are true], and that co = ci x + x y + y ci. One such development takes twelve NOR gates in all: six 2-input gates and two 1-input gates to produce u in 5 gate delays, plus three 2-input gates and one 3-input gate to produce co' in 2 gate delays. The canonical baseline took eight 3-input NOR gates plus three 4-input NOR gates to produce u, co and co' in 2 gate delays. If the circuit inventory actually includes 4-input NOR gates, the top-down canonical design looks like a winner in both gate count and speed. But if (contrary to our convenient supposition) the circuits are actually 3-input NOR gates, of which two are required for each 4-input NOR function, then the canonical design takes 14 gates compared to 12 for the bottom-up approach, but still produces the sum digit u considerably faster. The fanout comparison is tabulated as: What's a decision-maker to do? An observant one will have noticed that the description of the bottom-up development mentions co' as an output but not co. Does that design simply never need the direct form of the carry out? Well, yes 60 CHAPTER 10. CANONICAL NORMAL FORM and no. At each stage, the calculation of co' depends only on ci', x' and y', which means that the carry propagation ripples along the bit positions just as fast as in the canonical design without ever developing co. The calculation of u, which does require ci to be made from ci' by a 1-input NOR, is slower but for any word length the design only pays that penalty once (when the leftmost sum digit is developed). That's because those calculations overlap, each in what amounts to its own little pipeline without affecting when the next bit position's sum bit can be calculated. And, to be sure, the co' out of the leftmost bit position will probably have to be complemented as part of the logic determining whether the addition overflowed. But using 3-input NOR gates, the bottom-up design is very nearly as fast for doing parallel addition on a non-trivial word length, cuts down on the gate count, and uses lower fanouts ... so it wins if gate count and/or fanout are paramount! We'll leave the exact circuitry of the bottom-up design of which all these statements are true as an exercise for the interested reader, assisted by one more algebraic formula: u = ci(x XOR y) + ci'(x XOR y)']'. Decoupling the carry propagation from the sum formation in this way is what elevates the performance of a carry-lookahead adder over that of a ripple carry adder. To see how NOR gate logic was used in the Apollo Guidance Computer's ALU, visit http://klabs.org/history/ech/ agc_schematics/index.htm, select any of the 4-BIT MODULE entries in the Index to Drawings, and expand images as desired.

10.7 See also

• Algebraic normal form • Canonical form

• Blake canonical form

• List of Boolean algebra topics

10.8 Footnotes

[1] Hall, Eldon C. (1996). Journey to the Moon: The History of the Apollo Guidance Computer. AIAA. ISBN 1-56347-185-X.

10.9 References

• Bender, Edward A.; Williamson, S. Gill (2005). A Short Course in Discrete Mathematics. Mineola, NY: Dover Publications, Inc. ISBN 0-486-43946-1. The authors demonstrate a proof that any Boolean (logic) function can be expressed in either disjunctive or conjunctive normal form (cf pages 5–6); the proof simply proceeds by creating all 2*N rows of N Boolean variables and demonstrates that each row (“minterm”or“maxterm”) has a unique Boolean expression. Any Boolean function of the N variables can be derived from a composite of the rows whose minterm or maxterm are logical 1s (“trues”)

• McCluskey, E. J. (1965). Introduction to the Theory of Switching Circuits. NY: McGraw–Hill Book Company. p. 78. LCCN 65-17394. Canonical expressions are defined and described

• Hill, Fredrick J.; Peterson, Gerald R. (1974). Introduction to Switching Theory and Logical Design (2nd ed.). NY: John Wiley & Sons. p. 101. ISBN 0-471-39882-9. Minterm and maxterm designation of functions

10.10 External links

• Boole, George (1848). Translated by Wilkins, David R.. “The of Logic”. Cambridge and Dublin Mathematical Journal III: 183—198. Chapter 11

Charles Sanders Peirce's type–token distinction

There are only 26 letters in the English alphabet and yet there are more than 26 letters in this sentence. Moreover, every time a child writes the alphabet 26 new letters have been created.

The word 'letter' was used three times in the above paragraph, each time in a different meaning. The word 'letter' is one of many words having“type–token ambiguity”. This article disambiguates 'letter' by separating the three senses using terminology standard in logic today. The key distinctions were first made by the American logician-philosopher in 1906 using terminology that he established.*[1] The letters that are created by writing are physical objects that can be destroyed by various means: these are letter TOKENS or letter INSCRIPTIONS. The 26 letters of the alphabet are letter TYPES or letter FORMS. Peirce's type–token distinction, also applies to words, sentences, paragraphs, and so on: to anything in a universe of discourse of character-string theory, or concatenation theory.*[lower-alpha 1] There is only one word type spelled el-ee--tee-ee-ar,*[2] namely, 'letter'; but every time that word type is written, a new word token has been created. Some logicians consider a word type to be the class of its tokens. Other logicians counter that the word type has a permanence and constancy not found in the class of its tokens. The type remains the same while the class of its tokens is continually gaining new members and losing old members. The word type 'letter' uses only four letter types: el, ee, tee, and ar. Nevertheless, it uses ee twice and tee twice. In standard terminology, the word type 'letter' has six letter OCCURRENCES and the letter type ee OCCURS twice in the word type 'letter'. Whenever a word type is inscribed, the number of letter tokens created equals the number of letter occurrences in the word type. Peirce's original words are the following.“A common mode of estimating the amount of matter in a …printed book is to count the number of words. There will ordinarily be about twenty 'thes' on a page, and, of course, they count as twenty words. In another sense of the word 'word,' however, there is but one word 'the' in the English language; and it is impossible that this word should lie visibly on a page, or be heard in any voice …. Such a …Form, I propose to term a Type. A Single … …such as this or that word on a single line of a single page of a single copy of a book, I will venture to call a Token. …. In order that a Type may be used, it has to be embodied in a Token which shall be a of the Type, and thereby of the object the Type signifies.”– Peirce 1906, Ogden-Richards, 1923, 280-1. These distinctions are subtle but solid and easy to master. Reflection on the simple case of occurrences of numerals is often helpful.*[lower-alpha 2] This article ends using the new terminology to disambiguate the first paragraph.

There are 26 letter types in the English alphabet and yet there are more than 26 letter occurrences in this sentence type. Moreover, every time a child writes the alphabet 26 new letter tokens have been created.

11.1 See also

•

61 62 CHAPTER 11. CHARLES SANDERS PEIRCE'S TYPE–TOKEN DISTINCTION

• Map–territory relation

• Problem of universals #Peirce

11.2 Notes

[1] Concatenation theory

[2] Occurrences of numerals

11.3 References

[1] Charles Sanders Peirce, Prolegomena to an apology for , Monist, vol.16 (1906), pp. 492–546.

[2] Using a variant of 's structural-descriptive naming found in John Corcoran , Schemata: the Concept of Schema in the , Bulletin of Symbolic Logic, vol. 12 (2006), pp. 219–40. Chapter 12

Circumscription (taxonomy)

In biological taxonomy, circumscription is the definition of a taxon, that is, a group of organisms. For every taxon, the circumscription is based on a set of attributes that characterise every member of the taxon, and exclude every other organism. One goal of biological taxonomy is to achieve a stable circumscription for every taxon. Achieving stability is not yet a certainty in most taxa, and many that had been regarded as stable for decades are in upheaval in the light of rapid developments in molecular phylogenetics. In , new discoveries may invalidate the application of irrelevant attributes used in established or obsolete circumscriptions, or present new attributes useful in cladistic taxonomy. An example of a taxonomic group with unstable circumscription is Anacardiaceae, a family of flowering plants. Some experts favor a circumscription*[1] in which this family includes the Blepharocaryaceae, Julianaceae, and Podoaceae, which are sometimes considered to be separate families.*[2]

12.1 See also

• Glossary of scientific naming • Circumscription (logic)

• Circumscriptional name • Circumscribed circle

•

12.2 References

[1] Anacardiaceae in L. Watson and M.J. Dallwitz (1992 onwards). The families of flowering plants.

[2] Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 [and more or less continuously updated since].

63 Chapter 13

Classical logic

Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well.*[1]*[2] They are characterised by a number of properties:*[3]

1. and double negative elimination

2. , and the principle of explosion

3. Monotonicity of entailment and idempotency of entailment

4. Commutativity of conjunction

5. De Morgan duality: every logical operator is dual to another

While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics.*[4]*[5] The intended semantics of classical logic is bivalent. With the advent of it became apparent however that classical admits other semantics. In Boolean-valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra;“true”corresponds to the maximal element of the algebra, and “false”corresponds to the minimal element. Intermediate elements of the algebra correspond to truth values other than “true”and “false”. The holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements.

13.1 Examples of classical logics

• Aristotle's Organon introduces his theory of , which is a logic with a restricted form of judgments: assertions take one of four forms, All Ps are Q, Some Ps are Q, No Ps are Q, and Some Ps are not Q. These judgments find themselves if two pairs of two dual operators, and each operator is the of another, relationships that Aristotle summarised with his square of oppositions. Aristotle explicitly formulated the law of the excluded middle and law of non-contradiction in justifying his system, although these laws cannot be expressed as judgments within the syllogistic framework.

• 's algebraic reformulation of logic, his system of Boolean logic;

• The first-order logic found in Gottlob Frege's Begriffsschrift.

13.2 Non-classical logics

Main article: Non-classical logic

64 13.3. REFERENCES 65

• is a semantically constructed formal theory of computability—as opposed to classical logic, which is a formal theory of truth—integrates and extends classical, linear and intuitionistic logics. • Many-valued logic, including , which rejects the law of the excluded middle and allows as a any real number between 0 and 1. • rejects the law of the excluded middle, double negative elimination, and the De Morgan's laws; • rejects idempotency of entailment as well;

• extends classical logic with non-truth-functional (“modal”) operators. • (e.g., and ) rejects the law of noncontradiction;

• Relevance logic, linear logic, and non-monotonic logic reject monotonicity of entailment;

In Deviant Logic, Fuzzy Logic: Beyond the , Susan Haack divided non-classical logics into deviant, quasi- deviant, and extended logics.*[5]

13.3 References

[1] Nicholas Bunnin; Jiyuan Yu (2004). The Blackwell dictionary of . Wiley-Blackwell. p. 266. ISBN 978-1-4051-0679-5.

[2] L. T. F. Gamut (1991). Logic, language, and meaning, Volume 1: Introduction to Logic. University of Chicago Press. pp. 156–157. ISBN 978-0-226-28085-1.

[3] Gabbay, Dov, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), Handbook of Logic in Artificial Intelligence and Logic Programming, volume 2, chapter 2.6. Oxford University Press.

[4] Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: The Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/

[5] Haack, Susan, (1996). Deviant Logic, Fuzzy Logic: Beyond the Formalism. Chicago: The University of Chicago Press.

13.4 Further reading

• Graham Priest, An Introduction to Non-Classical Logic: From If to Is, 2nd Edition, CUP, 2008, ISBN 978-0- 521-67026-5 • Warren Goldfard, “Deductive Logic”, 1st edition, 2003, ISBN 0-87220-660-2 Chapter 14

Colorless green ideas sleep furiously

S

NP VP

N' V'

AdjP N' V' AdvP

Adj' AdjP N' V Adv'

Adj Adj' N sleep Adv Colorless Adj ideas furiously green

Approximate X-Bar representation of “Colorless green ideas sleep furiously.”See phrase structure rules.

"Colorless green ideas sleep furiously" is a sentence composed by in his 1957 book Syntactic Structures as an example of a sentence that is grammatically correct, but semantically nonsensical. The sentence was originally used in his 1955 thesis “Logical Structures of Linguistic Theory”and in his 1956 paper “Three Models for the Description of Language”.*[1] Although the sentence is grammatically correct, no obvious understandable meaning can be derived from it, and thus it demonstrates the distinction between syntax and semantics. As an example of a category mistake, it was used to show inadequacy of the then-popular probabilistic models of grammar, and the need for more structured models.

66 14.1. DETAILS 67

14.1 Details

The full passage says:

1. Colorless green ideas sleep furiously. 2. *Furiously sleep ideas green colorless. It is fair to assume that neither sentence (1) nor (2) (nor indeed any part of these sentences) has ever occurred in an English discourse. Hence, in any statistical model for grammaticalness, these sentences will be ruled out on identical grounds as equally “remote”from English. Yet (1), though nonsensical, is grammatical, while (2) is not grammatical.*[2]

While the meaninglessness of the sentence is often considered fundamental to Chomsky's point, Chomsky was only relying on the sentences having never been spoken before. Thus, even if one were to prescribe a likely and reasonable meaning to the sentence, the grammaticality of the sentence is concrete despite being the first time a person had ever uttered the statement, or any part thereof in such a combination. This was used then as a counter-example to the idea that the human speech engine was based upon statistical models, such as a Markov chain, or simple statistics of words following others.

14.2 Attempts at meaningful interpretations

The sentence can be partially interpreted through polysemy. Both green and colorless have figurative meanings, which allow colorless to be interpreted as “nondescript”and green as either “immature”or pertaining to environmental consciousness. The sentence can therefore be construed as“nondescript immature ideas have violent nightmares”, a phrase with less oblique semantics. In particular, the phrase can have legitimate meaning too, if green is understood to mean “newly formed”and sleep can be used to figuratively express mental or verbal dormancy. “Furiously” remains problematic when applied to the verb “sleep”, since “furiously”denotes “angrily”, “violently”, and “intensely energetically”, meanings which are generally incompatible with sleep, dormancy, and unconscious agents typically construed as conscious ones, e.g. animals or humans, which truly “sleep”.*[3] Another possible interpretation is that green is often associated with jealousy, but since that's not really green, it is actually a 'colorless green'. Jealousy is an 'idea'. Jealousy can fall to the subconscious mind or 'sleep', but even unconscious jealousy can build up 'furiously' until you cannot contain it anymore and then you feel hatred towards someone. Writers have attempted to provide the sentence meaning through context, the first of which was written by Chinese linguist Yuen Ren Chao.*[4] In 1985, a literary competition was held at Stanford University in which the contestants were invited to make Chomsky's sentence meaningful using not more than 100 words of prose or 14 lines of verse.*[5] An example entry from the competition, from C.M. Street, is:

It can only be the thought of verdure to come, which prompts us in the autumn to buy these dormant white lumps of vegetable matter covered by a brown papery skin, and lovingly to plant them and care for them. It is a marvel to me that under this cover they are labouring unseen at such a rate within to give us the sudden awesome beauty of spring flowering bulbs. While winter reigns the earth reposes but these colourless green ideas sleep furiously.

14.3 Statistical challenges

Fernando Pereira of the University of Pennsylvania has fitted a simple statistical Markov model to a body of newspaper text, and shown that under this model, “Furiously sleep ideas green colorless”is about 200,000 times less probable than “Colorless green ideas sleep furiously”.*[6] This statistical model defines a similarity metric, whereby sentences which are more like those within a corpus in certain respects are assigned higher values than sentences less alike. Pereira's model assigns an ungrammatical version of the same sentence a lower probability than the syntactically correct form demonstrating that statistical models can learn grammaticality distinctions with minimal linguistic assumptions. However, it is not clear that the model assigns every ungrammatical sentence a lower probability than every grammatical sentence. That is, “colorless green ideas sleep furiously”may still be statistically more“remote”from English than some ungrammatical sentences. To this, 68 CHAPTER 14. COLORLESS GREEN IDEAS SLEEP FURIOUSLY

it may be argued that no current theory of grammar is capable of distinguishing all grammatical English sentences from ungrammatical ones.

14.4 Related and similar examples

There is at least one earlier example of such a sentence, and probably many more. The pioneering French syntactician Lucien Tesnière came up with the French sentence "Le silence vertébral indispose la voile licite"(“The vertebral silence indisposes the licit sail”). The game of cadavre exquis (1925) is a method for generating nonsense sentences. It was named after the first sentence generated, Le cadavre exquis boira le vin nouveau (the exquisite corpse will drink the new wine). In the popular game of "Mad Libs", a chosen player asks each other player to provide parts of speech without providing any contextual information (e.g., “Give me a proper noun”, or “Give me an adjective”), and these words are inserted into pre-composed sentences with a correct grammatical structure, but in which certain words have been omitted. The humor of the game is in the generation of sentences which are grammatical but which are meaningless or have absurd or ambiguous meanings (such as 'loud sharks'). The game also tends to generate humorous double entendres. There are likely earlier examples of such sentences, possibly from the literature, but not necessarily uncontroversial ones, given that the focus has been mostly on borderline cases. For example, followers of logical held that “metaphysical”(i.e. not empirically verifiable) statements are simply meaningless; e.g. wrote an article where he argued that almost every sentence from Heidegger was grammatically correct, yet meaningless. Of course, some philosophers who were not logical positivists disagreed with this. The philosopher used the sentence“Quadruplicity drinks procrastination”to make a similar point; W.V. Quine took issue with him on the grounds that for a sentence to be false is nothing more than for it not to be true; and since quadruplicity doesn't drink anything, the sentence is simply false, not meaningless. In a sketch about linguistics, British comedy duo Fry and Laurie used the similarly nonsensical sentence “Hold the newsreader's nose squarely, waiter, or friendly milk will countermand my trousers.” Examples like Tesnière's and Chomsky's are the least controversially nonsensical, and Chomsky's example remains by far the most famous. John Hollander wrote a poem titled “Coiled Alizarine” in his book The Night Mirror. It ends with Chomsky's sentence. Clive James wrote a poem titled "A Line and a Theme from Noam Chomsky" in his book Other Passports: Poems 1958-1985. It opens with Chomsky's second meaningless sentence and discusses the Vietnam War. Another approach is to create a syntactically-correct, easily parsable sentence using nonsense words; a famous such example is "The gostak distims the doshes". 's Jabberwocky is also famous for using this technique, although in this case for literary purposes; similar sentences used in neuroscience experiments are called Jabberwocky sentences. In Russian schools of linguistics, the glokaya kuzdra example has similar characteristics. Other arguably “meaningless”utterances are ones that make sense, are grammatical, but have no reference to the present state of the world, such as“The King of is bald”, since there is no King of France today (see definite description).

14.5 See also

• List of linguistic example sentences

• Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo

• James while John had...had a better effect on the teacher

• Gostak

• Moore's paradox

• Poverty of the stimulus 14.6. REFERENCES 69

• Universal grammar

• Philosophy of language • Linguistics

• Pseudoword • Semantics

• Jabberwocky • Glokaya kuzdra

14.6 References

[1] Chomsky, Noam (September 1956). “Three Models for the Description of Language”. IRE Transactions on Information Theory 2 (3): 113–124. doi:10.1109/TIT.1956.1056813.

[2] Chomsky, Noam (1957). Syntactic Structures. The Hague/Paris: Mouton. p. 15. ISBN 3-11-017279-8.

[3]“Furiously”American Heritage Dictionary, 2014. http://dictionary.reference.com/browse/furiously?s=t

[4] Chao, Yuen Ren. “Making Sense Out of Nonsense”. The Sesquipedalian, vol. VII, no. 32 (June 12, 1997). Archived from the original on 2006-08-30. Retrieved 2006-08-30.

[5] “LINGUIST List 2.457”. 1991-09-03. Retrieved 2007-03-14.

[6] Pereira, Fernando (2000). “Formal grammar and information theory: together again?" (PDF). Philosophical Transactions of the Royal Society 358 (1769): 1239–1253. doi:10.1098/rsta.2000.0583.. See also this post at Language Log. Chapter 15

Composition of Causes

The Composition of Causes was a set of philosophical laws advanced by in his watershed essay A System of Logic. These laws outlined Mill's view of the epistemological components of , a school of philosophical laws that posited a decidedly opportunistic approach to the classic dilemma of causation nullification. Mill was determined to prove that the intrinsic properties of all things relied on three primary tenets, which he called the Composition of Causes. These were: 1. The Cause of Inherent Efficiency, a methodological understanding of deterministic forces engaged in the perpetual axes of the , as it pertained to its own self-awareness. 2. The so-called Sixth Cause, a conceptual notion embodied by the system of inter-related segments of social and elemental vitra. This was a hotly debated matter in early 17th-century philosophical circles, especially in the halls of the Reichtaven in Meins, where the of Geudl still lingered. 3. The Cause of Multitude, an evolutionary step taken from Hemmlich's Plurality of a Dysfunctional Enterprise, detailing the necessary linkage between both sets of perception-based self-awareness. Furthermore, the Composition of Causes elevated Mill's standing in ontological circles, lauded by his contemporaries for applying a conceptual vision of an often-argued discipline.

15.1 External links

• “Of the Composition of Causes”(1859) full text

70 Chapter 16

Concatenation theory

For the theory of strings in physics, see String theory.

Concatenation theory, also called string theory, character-string theory, or theoretical syntax, studies character strings over finite alphabets of characters, , , or marks. String theory is foundational for formal linguistics, computer science, logic, and especially .*[1] A generative grammar can be seen as a recursive definition in string theory. The most basic operation on strings is concatenation, connecting two strings to form a longer string whose length is the sum of the lengths of the operands: abcde is the concatenation of ab with cde, in symbols abcde = ab ^ cde. Strings, and concatenation of strings can be treated as an algebraic system with some properties resembling those of the addition of integers; in modern mathematics, this system is called a free . In 1956 Alonzo Church wrote: “Like any branch of mathematics, theoretical syntax may, and ultimately must, be studied by the axiomatic method”.*[2] Church was evidently unaware that string theory already had two axiomatiza- tions from the 1930s: one by Hans Hermes and one by Alfred Tarski.*[3] Coincidentally, the first English presentation of Tarski’s 1933 axiomatic foundations of string theory appeared in 1956 – the same year that Church called for such axiomatizations.*[4] As Tarski himself noted using other terminology, serious difficulties arise if strings are construed as tokens rather than types in the sense of Pierce's type-token distinction, not to be confused with similar distinctions underlying other type-token distinctions.

16.1 References

[1] John Corcoran and Matt Lavine, “Discovering string theory”. Bulletin of Symbolic Logic. 19 (2013) 253–4.

[2] Alonzo Church, Introduction to , Princeton UP, Princeton, 1956

[3] John Corcoran, William Frank and Michael Maloney, “String theory”, Journal of Symbolic Logic, vol. 39 (1974) pp. 625– 637

[4] Pages 173–4 of Alfred Tarski, The concept of truth in formalized languages, reprinted in Logic, Semantics, Metamathematics, Hackett, Indianapolis, 1983, pp. 152–278

71 Chapter 17

Condensed detachment

Condensed detachment (Rule D) is a method of finding the most general possible conclusion given two formal logical statements. It was developed by the Irish logician Carew Meredith in the 1950s and inspired by the work of Łukasiewicz.

17.1 Informal description

A rule of detachment (often referred to as ) says: “Given that p implies q , and given p , infer q .” The condensed detachment goes a step further and says: “Given that p implies q , and given an r , use a unifier of p and r to make p and r the same, then use a standard rule of detachment.” A A that when applied to p produces t , and substitution B that when applied to r produces t , are called unifiers of p and r . Various unifiers may produce expressions with varying numbers of free variables. Some possible unifying expressions are substitution instances of others. If one expression is a substitution instance of another (and not just a variable renaming), then that other is called “more general”. If the most general unifier is used in condensed detachment, then the logical result is the most general conclusion that can be made in the given inference with the given second expression. (And since any weaker inference you can get is a substitution instance of the most general one, nothing less than the most general unifier is ever used in practice.) In some logics (such as standard PC) have a set of defining axioms with the “D-completeness”property. If a set of axioms is D-Complete, then any valid theorems of the system can be generated by condensed detachment alone. Note that “D-completeness”is a property of an axiomatic basis for a system, not an intrinsic property of a logic system itself. J.A.Kalman proved that any conclusion that can be generated by a sequence of uniform substitution and modus ponens steps can either be generated by condensed detachment alone, or is a substitution instance of something that can be generated by condensed detachment alone. This makes condensed detachment useful for any logic system that has modus ponens and substitution, regardless of whether or not it is D-complete.

17.2 D-notation

Since a given major premise and a given minor premise uniquely determine the conclusion (up to variable renaming), Meredith observed that it was only necessary to note which two statements were involved and that the condensed detachment can be used without any other notation required. This led to the“D-notation”for proofs. This notation uses the“D”operator to mean condensed detachment, and takes 2 arguments, in a standard prefix notation string. For example, if you have four axioms a typical proof in D-notation might look like: DD12D34 which shows a condensed detachment step using the result of two prior condensed detachment steps, the first of which used axioms 1 and 2, and the second of which used axioms 3 and 4.

72 17.3. ADVANTAGES 73

This notation, besides being used in some automated theorem provers, sometimes appears in catalogs of proofs. Condensed detachment's use of unification predates the resolution techniques of automated theorem proving.

17.3 Advantages

For automated theorem proving condensed detachment has a number of advantages over raw modus ponens and uniform substitution. At a Modus Ponens and substitution proof you have an infinite number of choices for what you can substitute for variables. This means that you have an infinite number of possible next steps. With condensed detachment there are only a finite number of possible next steps in a proof. The D-notation for complete condensed detachment proofs allows easy description of proofs for cataloging and search. A typical complete 30 step proof is less than 60 characters long in D-notation (excluding the statement of the axioms.)

17.4 References

• J.A. Kalman (Dec 1983). “Condensed Detachment as a Rule of Inference”. Studia Logica 42 (4): 443— 451. • Hindley, J. Roger (1993), “BCK and BCI logics, condensed detachment and the 2-property”, Notre Dame Journal of Formal Logic 34 (2): 231–250, doi:10.1305/ndjfl/1093634655, MR 1231287 • William McCune and Larry Wos (1992). “Experiments in Automated Deduction with Condensed Detach- ment” (PDF). In D. Kapur. Proc. 11th International Conference on Automated Deduction (CADE). LNCS 607. Springer. pp. 209—223. Chapter 18

Condition (philosophy)

Comprehensive treatment of the word "condition" requires emphasizing that it is ambiguous in the sense of having multiple normal meanings and that its meanings are often vague in the sense of admitting borderline cases. According to the 2007 : an Encyclopedia, in one widely used sense, conditions are or resemble qualities, properties, features, characteristics, or attributes.*[1] In these senses, a condition is often denoted by a nominalization of a grammatical predicate: 'being equilateral' is a nominalization of the predicate 'is equilateral'. Being equilateral is a necessary condition for being square. Being equilateral and being equiangular are two necessary conditions for being a square. In order for a polygon to be a square, it is necessary for it to be equilateral—and it is necessary for it to be equiangular. Being a quadrangle that is both equilateral and equiangular is a sufficient condition for being a square. In order for a quadrangle to be a square, it is sufficient for it to be both equilateral and equiangular. Being equilateral and being equiangular are separately necessary and jointly sufficient conditions for a quadrangle to be a square. Every condition is both necessary and sufficient for itself. The relational phrases 'is necessary for' and 'is sufficient for' are often elliptical for 'is a necessary condition for' and 'is a sufficient condition for'. These senses may be called attributive; other senses that may be called instrumental, causal, and situational are discussed below. Every condition applies to everything that satisfies it. Every individual satisfies every condition that applies to it. The condition of being equilateral applies to every square, and every square satisfies the condition of being equilateral. The satisfaction relation relates individuals to conditions, and the application relation relates conditions to individuals. The satisfaction and application relations are converses of each other. Necessity and sufficiency, the relations expressed by 'is a necessary condition for' and 'is a sufficient condition for', relate conditions to conditions, and they are converses of each other. Every condition necessary for a given condition is one that the given condition is sufficient for, and conversely. As a result of a chain of developments tracing back to George Boole and Augustus De Morgan, it has become somewhat standard to limit the individuals pertinent to a given discussion. The collection of pertinent individuals is usually called the universe of discourse, an expression coined by Boole in 1854. In discussions of ordinary Euclidean plane , for example, the universe of discourse can be taken to be the class of plane figures. Thus, squares are pertinent [individuals], but conditions, propositions, proofs, and geometers are not. Moreover, the collection of pertinent conditions is automatically limited to those coherently applicable to individuals in the universe of discourse. Thus, triangularity and circularity are pertinent [conditions], but truth, validity, rationality, bravery, and sincerity are not. Some philosophers posit universal and null conditions. A universal condition applies to or is satisfied by every pertinent individual. A null condition applies to or is satisfied by no pertinent individual. In ordinary Euclidean plane geometry, the condition of being planar is universal and the condition of being both round and square is null. Every figure satisfies the condition of being planar. No figure satisfies the condition of being round and square. Some philosophers posit for each given condition a complementary condition that applies to a pertinent individual if and only if the individual does not satisfy the given condition. In some of several senses, consequence is a relation between conditions. Being equilateral and being equiangular are two consequences of being square. In the sense used here, given any two conditions, the first is a consequence of the second if and only if the second is a sufficient condition for the first. Equivalently, being a consequence of a given condition is coextensive with being a necessary condition for it. The relational verb 'implies' is frequently used for the of the relational verb phrase 'is a consequence of'. Given any two conditions, the first implies the second if and only if the second is a consequence of the first. In the attributive senses under discussion, a consequence of a

74 75 condition cannot be said to be a result of the condition nor can the condition be said to be a cause of its consequences. It would be incoherent to say that being equilateral is caused by being square.*[1] There are reflexive and non-reflexive senses of 'consequence' applicable to conditions. Both are useful. In the reflexive senses, which are used in this article, every condition is a consequence of itself. In the non-reflexive senses, which are not used in this article, no condition is a consequence of itself. There are material, intensional, and logical senses of 'consequence' applicable to conditions. All are useful. Because of space limitations, in this article, only material consequence is used although the other two are also described. Given any two conditions, the first is a material consequence of (is materially implied by) the second if and only if every individual that satisfies the second satisfies the first. Being equilateral is a material consequence of being an equiangular , but not of being an equiangular quadrangle. As is evident, material consequence is entirely extensional in the sense that whether one given condition is a material consequence of another is determined by their two extensions, the collections of individuals that satisfy them. Given any two conditions, the first is an intensional consequence of (is intensionally implied by) the second if and only if the proposition that every individual that satisfies the second satisfies the first is analytic or intensionally true. Being equal-sided is an intensional consequence of being an equilateral triangle. Given any two conditions, the first is a of (is logically implied by) the second if and only if the proposition that every individual that satisfies the second satisfies the first is tautological or logically true. Being equilateral is a logical consequence of being an equilateral triangle.*[1] Besides the one-place conditions – such as being three-sided or being equilateral – that are satisfied or not by a given individual, there are two-place conditions – such as being equal-to or being part-of – that relate or do not relate one given individual to another. There are three-place conditions such as numerical betweenness as in “two is between one and three”. Given any three numbers, in order for the first to satisfy the betweenness condition with respect to the second and third, it is necessary and sufficient for either the second to precede the first and the third the second or the second to precede the third and the first the second. There are four-place conditions such as numerical proportionality as in “one is to two as three is to six”. Given any four numbers, in order for the first to satisfy the proportionality condition with respect to the second, third, and fourth, it is necessary and sufficient that the first be to the second as the third is to the fourth. Charles Sanders Peirce discussed polyadic or multi-place conditions as early as 1885.*[2] There are many debated philosophical issues concerning conditions and consequences. Traditional philosophers ask ontological and epistemological questions about conditions. What are conditions? Do they change? Do they exist apart from the entities satisfying them? How do we know of them? How are propositions about them known to be true or to be false? In view of modern focus on identity criteria, philosophers now want to ask the questions involving them. One such ontological question asks for an identity criterion for conditions: what is a necessary and sufficient condition for “two”conditions to be identical? The widely accepted identity criterion for extensions of conditions is that given any two conditions, in order for the extension of the first to be [identical to] that of the second, it is necessary and sufficient for the two conditions to be satisfied by the same entities. There are questions concerning the ontological status of conditions. Are conditions mental, material, ideal, linguistic, or social, or do they have some other character? What is the relation of conditions to properties? A given individual satisfies (or fulfills) a given condition if and only if the condition applies to the individual. A given individual has (or possesses) a given property if and only if the property belongs to the individual. Are the last two sentences simply translations of each other?*[1] Philosophical terminology is not uniform. Before any of the above questions can be fully meaningful, it is necessary to interpret them or to locate them in the context of the work of an individual philosopher. We should never ask an abstract question such as what it means to say that something satisfies a condition. Rather we should ask a more specific question such as what Peirce meant by saying that accuracy of speech is an important condition of accurate thinking. John Dewey's voluminous writings provide a rich source of different senses for the words 'condition' and 'conse- quence'. Except where explicitly noted, all references to Dewey are by volume number and page in the Southern Illinois UP critical edition. It would be useful to catalogue the various senses Dewey attaches to 'condition' and 'consequence' the way that A.O. Lovejoy famously catalogued senses of 'pragmatism'. In several passages, Dewey links a sense of 'condition' with a corresponding sense of 'consequence' just as senses of these words were linked above. Two corresponding usages occur repeatedly in his writings and, it should be said, in most writings concerned with human activity including government and technology. In one, condition/consequence is somewhat analogous to means/end. In fact, Dewey sometimes uses the words 'condition' and 'means' almost interchangeably as in his famous pronouncement: “Every intelligent act involves selection of certain things as means to other things as their consequences”.*[3] A little later, he adds:“…in all in which there is an end in view (consequences to be brought into ) there is a selective ordering of existing conditions as means …”.*[4] In the other sense, condition/consequence is 76 CHAPTER 18. CONDITION (PHILOSOPHY)

similar to cause/effect – although identification is probably not warranted in either case. Dewey studiously avoids sharp distinctions, dualisms, , and other artificialities. There are passages where both contrasts are relevant, but as far as I know, Dewey never explicitly notes that 'condition/consequence' was used for both. The means/end sense occurs, for example, in his 1945 Journal of Philosophy article, “Ethical Subject-Matter and Language”(15, 139),*[5] where he suggested that the inquiry into “conditions and consequences”should draw upon the whole knowledge of relevant fact. The cause/effect sense occurs on page 543 in his response to critics in the 1939 Library of Living Philosophers volume.*[6] Here he wrote: “Correlation between changes that form conditions of desires, etc., and changes that form their consequences when acted upon have the same standing and function …that physical objects have …”There are scattered passages suggesting that Dewey regarded the means/end relation as one kind of cause/effect relation. In fact he regards a causal proposition as one “whose content is a relation of conditions that are means to other conditions that are consequences”.*[4] In some of the senses Dewey uses, conditions are or resemble qualities, properties, features, characteristics, or at- tributes. These senses were referred to above as attributive. However, in the two of senses in question, the instrumen- tal sense and the causal sense, let us say, conditions are or resemble states or events more than qualities, properties, features, characteristics, or attributes. After all, the attributive condition of being equiangular, which is a conse- quence of the condition of being an equilateral triangle, could hardly be said to be brought about through use of the latter as means or said to be caused by the latter. Accordingly, an attributive condition is neither earlier nor later than its consequences, whereas an instrumental or causal condition necessarily precedes its consequences. As Dewey himself puts it, “The import of the causal relation as one of means-consequences is thus prospective”.*[3] From a practical point of view, Dewey's causal and instrumental senses of 'condition' and 'consequence' are at least as important as the attributive senses. In the causal sense, fuel, oxygen, and ignition are conditions for combustion as a consequence. In the instrumental sense, understanding, evidence, and judgment are conditions for knowledge as a consequence. For another important example, the Cambridge Dictionary of Philosophy *[7] defines 'condition' in an important sense not explained above: a condition is a state of affairs, “way things are”or situation—most commonly referred to by a nominalization of a sentence. The expression 'Snow's being white', which refers to the condition snow's being white, is a nominalization of the sentence 'Snow is white'.*[7] 'The truth of the proposition that snow is white' is a nominalization of the sentence 'the proposition that snow is white is true'. Snow's being white is a necessary and sufficient condition for the truth of the proposition that snow is white. Conditions in this sense may be called situational. Usually, necessity and sufficiency relate conditions of the same kind. Being an animal is a necessary attributive condition for being a dog. Fido's being an animal is a necessary situational condition for Fido's being a dog.

18.1 References

[1] John Corcoran . Conditions and Consequences. American Philosophy: an Encyclopedia. 2007. Eds. John Lachs and Robert Talisse. New York: Routledge. Pages 124–7.

[2] Peirce, C. S. 1992. The Essential Peirce: Selected Philosophical Writings (1867–1893). Vol. I. Eds. N. Houser and C. Kloesel. Bloomington: Indiana UP. Pages 225ff

[3] Dewey, John. 1986. John Dewey: The Later Works, 1925–1953. Volume 12: 1938. Ed. Jo Ann Boydston. Carbondale and Edwardsville: Southern Illinois UP.Page 454.

[4] Dewey, John. 1986. John Dewey: The Later Works, 1925–1953. Volume 12: 1938. Ed. Jo Ann Boydston. Carbondale and Edwardsville: Southern Illinois UP. Page 455

[5] Dewey, John. 1986. John Dewey: The Later Works, 1925–1953. Volume 15: 1942–1953. Ed. Jo Ann Boydston. Carbondale and Edwardsville: Southern Illinois UP.Page 139.

[6] Schilpp, Paul A., Ed. 1939. The Philosophy of John Dewey. Library of Living Philosophers. LaSalle, IL: Open Court.

[7] , 1999. “Condition”. Cambridge Dictionary of Philosophy. R.Audi, Ed. Cambridge: Cambridge UP. p. 171. Chapter 19

Conditional proof

A is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent. The assumed antecedent of a conditional proof is called the conditional proof assumption (CPA). Thus, the goal of a conditional proof is to demonstrate that if the CPA were true, then the desired conclusion necessarily follows. The validity of a conditional proof does not require that the CPA be actually true, only that if it were true it would lead to the consequent. Conditional proofs are of great importance in mathematics. Conditional proofs exist linking several otherwise un- proven conjectures, so that a proof of one conjecture may immediately imply the validity of several others. It can be much easier to show a proposition's truth to follow from another proposition than to prove it independently. A famous network of conditional proofs is the NP-complete class of complexity theory. There are a large number of interesting tasks, and while it is not known if a polynomial-time solution exists for any of them, it is known that if such a solution exists for any of them, one exists for all of them. Similarly, the Riemann hypothesis has a large number of consequences already proven.

19.1 Symbolic logic

As an example of a conditional proof in symbolic logic, suppose we want to prove A → C (if A, then C) from the first two premises below:

19.2 See also

• Deduction theorem

• Logical consequence • Propositional calculus

19.3 References

• Robert L. Causey, Logic, sets, and recursion, Jones and Barlett, 2006.

• Dov M. Gabbay, Franz Guenthner (eds.), Handbook of philosophical logic, Volume 8, Springer, 2002.

77 Chapter 20

Conditional quantifier

In logic, a conditional quantifier is a kind of Lindström quantifier (or generalized quantifier) QA that, relative to a classical model A, satisfies some or all of the following conditions ("X" and "Y" range over arbitrary formulas in one free variable): (The implication arrow denotes material implication in the metalanguage.) The minimal conditional logic M is char- acterized by the first six properties, and stronger conditional logics include some of the other ones. For example, the quantifier ∀A, which can be viewed as set-theoretic inclusion, satisfies all of the above except [symmetry]. Clearly [symmetry] holds for ∃A while e.g. [] fails. A semantic interpretation of conditional quantifiers involves a relation between sets of subsets of a given structure— i.e. a relation between properties defined on the structure. Some of the details can be found in the article Lindström quantifier. Conditional quantifiers are meant to capture certain properties concerning conditional reasoning at an abstract level. Generally, it is intended to clarify the role of conditionals in a first-order language as they relate to other connectives, such as conjunction or disjunction. While they can cover nested conditionals, the greater complexity of the formula, specifically the greater the number of conditional nesting, the less helpful they are as a methodological tool for un- derstanding conditionals, at least in some sense. Compare this methodological strategy for conditionals with that of first-degree entailment logics.

20.1 References

Serge Lapierre. Conditionals and Quantifiers, in Quantifiers, Logic, and Language, Stanford University, pp. 237– 253, 1995.

78 Chapter 21

Conflation

For other uses, see Conflation (disambiguation). Conflation happens when the identities of two or more individuals, concepts, or places, sharing some characteristics

Flag of the Lord Warden of the Cinque Ports, a heraldic emblem which displays conflated or “con-joined”images.

of one another, seem to be a single identity —the differences appear to become lost.*[1] In logic, it is the practice of treating two distinct concepts as if they were one, which produces errors or misunderstandings as a fusion of distinct subjects tends to obscure analysis of relationships which are emphasized by contrasts.*[2] However, if the distinctions between the two concepts appear to be superficial, intentional conflation may be desirable for the sake of conciseness and recall.

21.1 Communication and reasoning

The result of conflating concepts may give rise to fallacies of ambiguity, including the in a categorical syllogism. For example, the word “bat”has at least two distinct meanings: a flying animal, and a piece of sporting equipment (such as a baseball bat or cricket bat). If these meanings are not distinguished, the result may

79 80 CHAPTER 21. CONFLATION

be the following categorical syllogism, which may be seen as a joke (pun):

1. All bats are animals. 2. Some wooden objects are bats. 3. Therefore, some wooden objects are animals.

21.1.1 Conventional conflation

The international press features recurring news stories about the G8, which refers to a“”composed of nine members.*[3] The initial“Group of Six”has been conflated with the subsequent "" and today's "Group of Eight".*[4]

The 34th G8 at Toyako, (2008).

While it may be obvious that the G7 and the G6 were explicitly distinguishable in 1976, and the G8 and the G7 were readily differentiated in 1998, something unforeseen happened in the years since then. The 2008 summit of G8 leaders held in Hokkaido, Japan was identified as the "" and the "" in as well as the "" in were also widely reported in the international press and elsewhere. These ordinal numbers imply a process of counting backwards through the years, which requires conflating the G6 and the G7 and the G8, deliberately ignoring that each of the terms refer to distinct and different amalgamations.*[5]

*‡ European Communities (EC) were reformed into the (EU) as the Maastricht Treaty came into effect in November 1993.

21.1.2 Logical conflation

Using words with different meanings can help to clarify or it can cause real confusion. English words with multiple verb meanings can be illustrated by instances in which a motion is merged with or a causation with manner,*[6] e.g. The bride floated towards her future. In this example, the bride may: Be married on a boat, an airplane, or a hot-air balloon, etc.—not all marriages occur in a church*[7] She could gracefully walk the aisle towards matrimony.*[8] The verb of “float”has multiple meanings and both verb meanings in the example may be proper uses of a bride “floating”to a future. The “manner”of the scene (further context) would further explain the true meaning of the sentence. 21.2. TYPES 81

In an alternate illustrative example, respect is used both in the sense of “recognise a right”and “have high regard for”. We can recognise someone's right to the opinion that the is secretly controlled by alien lizards on the moon, without holding this idea in high regard. But conflation of these two different concepts leads to the notion that all ideological ideas, for example, should be treated with respect, rather than just the right to hold these ideas. Conflation in logical terms is very similar to, if not identical to, . Deliberate Idiom conflation is the amalgamation of two different expressions. In most cases, the combination results in a new expression that makes little sense literally, but clearly expresses an idea because it references well-known idioms.

21.2 Types

All conflations fit into one of two major categories: “congruent”conflations and “incongruent”conflations.

21.2.1 Congruent conflations

Congruent conflations are the more ideal, and more sought-after, examples of the concept. These occur when the two root expressions reflect similar thoughts. For example, “look who's calling the kettle black”can be formed using the root expressions“look who's talking”and "the pot calling the kettle black". These root expressions really mean the same thing: they are both a friendly way to point out hypocritical behavior. Of course, “look who's calling the kettle black”does not directly imply anything, yet the implication is understood because the conflation clearly refers to two known idioms. An illustrative conflation brings together two Roman Catholic saints named Lazarus. One, a lame beggar covered with sores which dogs are licking, appears in a New Testament story at Luke 16:19–31.*[9] The other, Lazarus of Bethany, is identified in John 11:41–44*[10] as the man whom Jesus raised from the dead. The beggar's Feast Day is June 21, and Lazarus of Bethany's day is December 17.*[11] However, both saints are depicted with crutches, and the blessing of dogs (associated with the beggar saint) usually takes place on December 17, the date associated with the resurrected Lazarus. The two characters' identities have become conflated in most cultural contexts, including the iconography of both saints.*[12]

21.2.2 Incongruent conflations

Incongruent conflation occurs when the root expressions do not mean the same thing, but share a common word or theme. For example,“a bull in a candy store”can be formed from the root expressions“a kid in a candy store”and “a bull in a shop”. The former root expression paints a picture of someone who is extraordinarily happy and excited, whereas the latter root brings to mind the image of a person who is extremely clumsy, indelicate, not suited to a certain environment, prone to act recklessly, or easily provoked. The conflation expresses both of these ideas at the same time. Without context, the speaker's intention is not entirely clear. An illustrative conflation seems to merge disparate figures as in Santería. St. Lazarus is conflated with the Yoruba Babalu Aye, and celebrated on December 17,*[11] despite Santería's reliance on the iconography associated with the begging saint whose Feast Day is June 21.*[12] By blending the identity of the two conflated St. Lazarus individuals with the identity of the Babalu Aye, Santería has gone one step further than the conflation within Catholicism, to become the kind of religious conflation known as syncretism, in which or concepts from two different are conflated to form a third.

21.2.3 Humorous conflations

Idiom conflation has been used as a source of humor in certain situations. For example, the Mexican character El Chapulín Colorado once said

"Mas vale pájaro en mano que dios lo ayudará...no, no...Dios ayuda al que vuela como pájaro...no... bueno, la idea es esa."

meaning 82 CHAPTER 21. CONFLATION

“A bird in the hand will get the worm...no, wait...The early bird is worth two in the bush...no... well, that's the idea.” by combining two popular expressions:

• "Más vale pájaro en mano que cientos volando"(“A bird in the hand is worth two in the bush.”) • "Al que madruga Dios lo ayuda"(“The early bird gets the worm.”)

This was typical of the character, and he did it with several other expressions over the course of his comedy routine. In popular culture, identities are sometimes intentionally conflated. In the early 2000s, the popular American ac- tors Ben Affleck and Jennifer Lopez were dating, and the tabloid press referred to them playfully as a third entity, Bennifer.*[13]

21.3 Taxonomic conflation

In taxonomies, a conflative term is always a polyseme.*[14]

21.4 See also

• Amalgamation (names) • Essentialism • Portmanteau • Skunked term • Stemming algorithm • Syncretism

21.5 Notes

[1] Haught, John F. (1995). Science and : From Conflict to Conversation, p. 13. [2] Haught, Science and Religion: From Conflict to Conversation, p. 14. [3] European Union: EU and the G8 [4] Brown, Stephen et al. “Italy revamps G8 as UK makes financial crisis focus,” . 15 January 2009. [5] Japanese Ministry of Foreign Affairs (MOFA): List of summits, 1–25; G7/8 summits. [6] Mateu, Jaume and Gemma Eigeu. (2002). “A Minimalist Account of Conflation Processes,”in Theoretical Approaches to Universals, pp. 211–212. [7] “Float”. dictionary.reference.com. Verb, item 3. Retrieved 25 September 2015. to rest or move in a liquid, the air, etc. [8] “Float”. dictionary.reference.com. Verb, item 4. Retrieved 25 September 2015. to move lightly and gracefully [9] Luke 16:19–31 in Roman Catholic New Advent Bible. [10] John 11:41–44 in Roman Catholic New Advent Bible. [11] With sackcloth and rum, Cubans hail Saint Lazarus, December 17, 1998. Reuters news story. [12] Money talks: folklore in the public sphere December 2005, Folklore magazine. [13] Sigman, Michael (September 10, 2010). “Inflation May Be Under Control, But Watch Out for Conflation”. Huffington Post. Retrieved 25 April 2011. [14] Malone, Joseph L. (1988). The Science of Linguistics in the Art of Translation: Some Tools from Linguistics for the Analysis and Practice of Translation, p. 112. 21.6. REFERENCES 83

21.6 References

• Alexiadou, Artemus. (2002). Theoretical Approaches to Universals. Amsterdam: John Benjamins Publishing Company. ISBN 978-90-272-2770-6; OCLC 49386229 • Haught, John F. (1995). Science and Religion: From Conflict to Conversation. New York: Paulist Press. ISBN 978-0-8091-3606-3; OCLC 32779780

• Malone, Joseph L. (1988). The Science of Linguistics in the Art of Translation: Some Tools from Linguistics for the Analysis and Practice of Translation. Albany, New York: State University of New York Press. ISBN 978-0-88706-653-5; OCLC 15856738

21.7 External links

• Conflations Chapter 22

Counterexample

In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule or law. For example, consider the proposition“all students are lazy”. Because this statement makes the claim that a certain property (laziness) holds for all students, even a single example of a diligent student will prove it false. Thus, any hard-working student is a counterexample to “all students are lazy”. More precisely, a counterexample is a specific instance of the falsity of a universal quantification (a “for all”statement). In mathematics, this term is (by a slight abuse) also sometimes used for examples illustrating the necessity of the full hypothesis of a theorem, by considering a case where a part of the hypothesis is not verified, and where one can show that the conclusion does not hold.

22.1 In mathematics

In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterex- amples to show that certain conjectures are false, mathematical researchers avoid going down blind alleys and learn how to modify conjectures to produce provable theorems.

22.1.1 Rectangle example

Suppose that a mathematician is studying geometry and shapes, and she wishes to prove certain theorems about them. She conjectures that “All rectangles are squares". She can either attempt to prove the truth of this statement using deductive reasoning, or if she suspects that her conjecture is false, she might attempt to find a counterexample. In this case, a counterexample would be a rectangle that is not a square, like a rectangle with two sides of length 5 and two sides of length 7. However, despite having found rectangles that were not squares, all the rectangles she did find had four sides. She then makes the new conjecture “All rectangles have four sides”. This is weaker than her original conjecture, since every square has four sides, even though not every four-sided shape is a square. The previous paragraph explained how a mathematician might weaken her conjecture in the face of counterexamples, but counterexamples can also be used to show that the assumptions and hypothesis are needed. Suppose that after a while the mathematician in question settled on the new conjecture“All shapes that are rectangles and have four sides of equal length are squares”. This conjecture has two parts to the hypothesis: the shape must be 'a rectangle' and 'have four sides of equal length' and the mathematician would like to know if she can remove either assumption and still maintain the truth of her conjecture. So she needs to check the truth of the statements: (1) “All shapes that are rectangles are squares”and (2)“All shapes that have four sides of equal length are squares”. A counterexample to (1) was already given, and a counterexample to (2) is a non-square rhombus. Thus the mathematician sees that both assumptions were necessary.

22.1.2 Other mathematical examples

See also: Counterexamples in topology and Minimal counterexample

84 22.2. IN PHILOSOPHY 85

A counterexample to the statement“all prime numbers are odd numbers" is the number 2, as it is a prime number but is not an odd number. Neither of the numbers 7 or 10 is a counterexample, as neither contradicts the statement. In this example, 2 is the only possible counterexample to the statement, but only a single example is needed to contradict "All prime numbers are odd numbers”. Similarly the statement“All natural numbers are either prime or composite" has the number 1 as a counterexample as 1 is neither prime nor composite. Euler's sum of powers conjecture was disproved by counterexample. It asserted that at least n n*th powers were necessary to sum to another n*th power. The conjecture was disproven in 1966*[1] with a counterexample involving n = 5; other n = 5 counterexamples are now known, as are some n = 4 counterexamples. Witsenhausen's counterexample shows that it is not always true for control problems that a quadratic loss function and a linear equation of evolution of the state variable imply optimal control laws that are linear. Other examples include the disproofs of the Seifert conjecture, the Pólya conjecture, the conjecture of Hilbert's fourteenth problem, Tait's conjecture, and the Ganea conjecture.

22.2 In philosophy

In philosophy, counterexamples are usually used to argue that a certain philosophical position is wrong by showing that it does not apply in certain cases. Unlike mathematicians, philosophers cannot prove their claims beyond any doubt, so other philosophers are free to disagree and try to find counterexamples in response. Of course, now the first philosopher can argue that the alleged counterexample does not really apply. Alternatively, the first philosopher can modify their claim so that the counterexample no longer applies; this is anal- ogous to when a mathematician modifies a conjecture because of a counterexample. For example, in Plato's , Callicles, trying to define what it means to say that some people are “better”than others, claims that those who are stronger are better. But Socrates replies that, because of their strength of numbers, the class of common rabble is stronger than the propertied class of nobles, even though the masses are prima facie of worse character. Thus Socrates has proposed a counterexample to Callicles' claim, by looking in an area that Callicles perhaps did not expect —groups of people rather than individual persons. Callicles might challenge Socrates' counterexample, arguing perhaps that the common rabble really are better than the nobles, or that even in their large numbers, they still are not stronger. But if Callicles accepts the counterexample, then he must either withdraw his claim or modify it so that the counterexample no longer applies. For example, he might modify his claim to refer only to individual persons, requiring him to think of the common people as a collection of individuals rather than as a mob. As it happens, he modifies his claim to say “wiser”instead of “stronger”, arguing that no amount of numerical superiority can make people wiser.

22.3 See also

• Exception that proves the rule

• Contradiction

22.4 References

[1] http://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf

22.5 Further reading

Using counterexamples in this way proved to be so useful that there are several books collecting them: 86 CHAPTER 22. COUNTEREXAMPLE

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology, Springer, New York 1978, ISBN 0-486-68735-X. • Joseph P. Romano and Andrew F. Siegel: Counterexamples in Probability and Statistics, Chapman & Hall, New York, London 1986, ISBN 0-412-98901-8. • Gary L. Wise and Eric B. Hall: Counterexamples in Probability and Real Analysis. Oxford University Press, New York 1993. ISBN 0-19-507068-2. • Bernard R. Gelbaum, John M. H. Olmsted: Counterexamples in Analysis. Corrected reprint of the second (1965) edition, Dover Publications, Mineola, NY 2003, ISBN 0-486-42875-3. • Jordan M. Stoyanov: Counterexamples in Probability. Second edition, Wiley, Chichester 1997, ISBN 0-471- 96538-3. • Michael Copobianco & John Mulluzzo (1978) Examples and Counterexamples in , Elsevier North-Holland ISBN 0-444-00255-3. Chapter 23

Counterintuitive

A counterintuitive proposition is one that does not seem likely to be true when assessed using intuition, common sense, or gut feelings.*[1] Scientifically discovered, objective truths are often called counterintuitive when intuition, emotions, and other cognitive processes outside of deductive rationality interpret them to be wrong. However, the subjective nature of intuition limits the of what to call counterintuitive because what is counter-intuitive for one may be intuitive for an- other. This might occur in instances where intuition changes with knowledge. For instance, many aspects of or may sound counterintuitive to a layman, while they may be intuitive to a particle physi- cist. Nevertheless, counter-intuitive concepts are psychologically more preferred than intuitive concepts*[2] like in Von Restorff effect. Flawed intuitive understanding of a problem may lead to counter-productive behavior with undesirable outcomes. In some such cases, counterintuitive policies may then produce a more desirable outcome.*[3] This can lead to conflicts between those who hold deontological and consequentialist ethical perspectives on those issues.

23.1 Counterintuition in science

Many scientific ideas that are generally accepted by people today were formerly considered to be contrary to intuition and common sense. For example, most everyday experience suggests that the Earth is flat; actually, this view turns out to be a remarkably good approximation to the true state of affairs, which is that the Earth is a very big (relative to the day-to-day scale familiar to humans) oblate spheroid. Furthermore, prior to the Copernican revolution, heliocentrism, the belief that the Earth goes around the Sun, rather than vice versa, was considered to be contrary to common sense. Another counterintuitive scientific idea concerns space travel: it was initially believed that highly streamlined shapes would be best for re-entering the earth's atmosphere. In fact, experiments proved that blunt-shaped re-entry bodies make the most efficient heat shields when returning to earth from space. Blunt-shaped re-entry vehicles have been used for all manned-spaceflights, including the Mercury, Gemini, Apollo and Space Shuttle missions.*[4] The Michelson-Morley experiment sought to measure the velocity of the Earth through the aether as it revolved around the Sun. The result was that it has no aether velocity at all. Relativity theory later explained the results, replacing the conventional notions of aether and separate space, time, mass, and energy with a counterintuitive four-dimensional non-Euclidean universe.*[5]

23.2 Examples

Some further counterintuitive examples are: In science:

• Gödel's incompleteness theorems - for thousands of years, it was confidently assumed that arithmetic, and therefore similar systems of logic, were completely solid in terms of being reliable for deductions. Gödel

87 88 CHAPTER 23. COUNTERINTUITIVE

proved that such systems could not be both complete and consistent. • Wave–particle duality / photoelectric effect - As demonstrated by the double slit experiment light and quantum particles behave as both waves and particles. • A significant number of people find it difficult to accept the mathematical fact that 0.999... equals 1 • The poses a simple yes-or-no question from probability that even professionals can find difficult to reconcile with their intuition. • orbits in orbital mechanics • That light may pass through two perpendicularly oriented polarizing filters if a third filter, not oriented per- pendicular to either of the other two, is placed between them.*[6] • The Mpemba effect, in which, under certain circumstances, a warmer body of water will freeze faster than a cooler body in the same environment. • That water vapor is lighter than air and is the reason clouds float and barometers work.*[7]

In politics and economics:

• The violation of the monotonicity criterion in voting systems • David Ricardo's theory of that suggests that comparative advantage is in general more important than advantage

Many examples of cognitive bias, such as:

• The clustering illusion that suggests that significant patterns exist in a set of random points when no other cause than chance is present • That alignments of random points on a plane are vastly easier to find than intuition would suggest

23.3 See also

• Common sense • Folk physics • Folk psychology • Paradox • •

23.4 References

[1] http://dictionary.reference.com/browse/counter+intuitive "Counterintuitive: –adjective. Counter to what intuition would lead one to expect: The direction we had to follow was counterintuitive—we had to go north first before we went south.” Retrieved: 09 NOV 2010.

[2] http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3979650/ - Memory for Expectation-Violating Concepts: The Effects of Agents and Cultural Familiarity

[3] New Scientist, July 2005

[4] http://www.freshbrainz.com/2009/12/counterintuitive-science-fast-speed-fat.html Fresh Brainz. Counterintuitive Sci- ence: Fast Speed, Fat Shape. “It turns out that pointy-nosed spaceships perform well on their way out of the atmosphere, but not when they have to come BACK.”Retrieved 09 NOV 2010. 23.5. FURTHER READING 89

[5] http://galileoandeinstein.physics.virginia.edu/lectures/michelson.html The Michelson-Morley Experiment. “As a result of Michelson’s efforts in 1879, the was known to be 186,350 miles per second with a likely error of around 30 miles per second. This measurement, made by timing a flash of light travelling between mirrors in Annapolis, agreed well with less direct measurements based on astronomical . Still, this did not really clarify the nature of light. Two hundred years earlier, Newton had suggested that light consists of tiny particles generated in a hot object, which spray out at very high speed, bounce off other objects, and are detected by our eyes. Newton’s arch-enemy Robert Hooke, on the other hand, thought that light must be a kind of wave motion, like sound. To appreciate his point of view, let us briefly review the nature of sound.”Retrieved: 09 NOV 2010.

[6] http://alienryderflex.com/polarizer/

[7] http://www.atmos.umd.edu/~{}stevenb/vapor/

23.5 Further reading

• Alvermann, D. E.; Hague, S. A. (1989). “Comprehension of counterintuitive science text: Effects of prior knowledge and text structure”. The Journal of Educational Research 82 (4): 197–202. JSTOR 27540341.

• Forrester, J. W. (1971). "Counterintuitive Behavior of Social Systems". Technology Review (Alumni Associ- ation of the Massachusetts Institute of Technology). CiteSeerX: 10 .1 .1 .35 .4776. Based on testimony given to the Subcommittee on Urban Growth of the Committee on Banking and Currency, U.S. House of Representa- tives. Updated March, 1995. • Havil, J. (2008). Impossible?: surprising solutions to counterintuitive conundrums. Princeton University Press. ISBN 978-0-691-13131-3. • Norenzayan, A.; Atran, S.; Faulkner, J.; Schaller, M. (2006). “The cultural selection of minimally coun- terintuitive narratives”. 30: 531–553. doi:10.1207/s15516709cog0000_68. CiteSeerX: 10 .1 .1 .88 .187. Chapter 24

Cratylism

Cratylism is a philosophical theory based on the teachings of , also known as Kratylos. Vaguely exegetical, it holds that the fluid nature of ideas, words, and leaves them fundamentally baseless, and possibly unable to support logic and reason. Cratylism reaches similar conclusions about the nature of reality and communication that and Buddhism also confronted. How can a mind in flux, in a flowing world, hold on to any solid 'truth' and convey it to another mind? A fellow Greek , Gorgias, came to an equally ironic cul de sac conclusion about the nature of human epis- temological understanding: "Nothing exists. Even if something did exist, nothing can be known about it; and Even if something can be known about it, knowledge about it cannot be communicated to others. And, finally, Even if it can be communicated, it cannot be understood. "

24.1 See also

• (disambiguation)

24.2 External links

• What Was Cratylus Trying To Say?

90 Chapter 25

Critical thinking

Critical thinking, also called critical analysis, is clear, rational thinking involving critique. Its details vary amongst those who define it. According to Barry K. Beyer (1995), critical thinking means making clear, reasoned judgments. During the process of critical thinking, ideas should be reasoned, well thought out, and judged.*[1] The National Council for Excellence in Critical Thinking*[2] defines critical thinking as the intellectually disciplined process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and/or evaluating information gathered from, or generated by, , experience, reflection, reasoning, or communication, as a guide to belief and action.'*[3]

25.1 Etymology

In the term critical thinking, the word critical, (Grk. κριτικός = kritikos = “critic”) derives from the word critic and implies a critique; it identifies the intellectual capacity and the means “of judging”, “of judgement”, “for judging”, and of being “able to discern”.*[4]

25.2 Definitions

Critical thinking is variously defined as:

•“the process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and evaluating infor- mation to reach an answer or conclusion”*[5] •“disciplined thinking that is clear, rational, open-minded, and informed by evidence”*[5] •“reasonable, reflective thinking focused on deciding what to believe or do”*[6] •“purposeful, self-regulatory judgment which results in interpretation, analysis, evaluation, and inference, as well as explanation of the evidential, conceptual, methodological, criteriological, or contextual considerations upon which that judgment is based”*[7] •“includes a commitment to using reason in the formulation of our beliefs”*[8] • in critical social theory, it is the commitment to the social and political practice of participatory democracy; willingness to imagine or to remain open to considering alternative perspectives; willingness to integrate new or revised perspectives into our ways of thinking and acting; and willingness to foster criticality in others.*[9] • the skill and propensity to engage in an activity with reflective scepticism (McPeck, 1981) • disciplined, self-directed thinking which exemplifies the perfection of thinking appropriate to a particular mode or domain of thinking (Paul, 1989, p. 214) • thinking about one's thinking in a manner designed to organize and clarify, raise the efficiency of, and recognize errors and biases in one's own thinking. Critical thinking is not 'hard' thinking nor is it directed at solving problems (other than 'improving' one's own thinking). Critical thinking is inward-directed with the intent of

91 92 CHAPTER 25. CRITICAL THINKING

maximizing the rationality of the thinker. One does not use critical thinking to solve problems —one uses critical thinking to improve one's process of thinking.*[10]

25.3 Skills

The list of core critical thinking skills includes observation, interpretation, analysis, inference, evaluation, explanation, and metacognition. According to Reynolds (2011), an individual or group engaged in a strong way of critical thinking gives due consideration to establish for instance:*[11]

• Evidence through reality

• Context skills to isolate the problem from context

• Relevant criteria for making the judgment well

• Applicable methods or techniques for forming the judgment

• Applicable theoretical constructs for understanding the problem and the question at hand

In addition to possessing strong critical-thinking skills, one must be disposed to engage problems and decisions using those skills. Critical thinking employs not only logic but broad intellectual criteria such as clarity, credibility, accuracy, precision, relevance, depth, breadth, significance, and fairness.*[12]

25.4 Procedure

Critical thinking calls for the ability to:

• Recognize problems, to find workable means for meeting those problems

• Understand the importance of prioritization and order of precedence in problem solving

• Gather and marshal pertinent (relevant) information

• Recognize unstated assumptions and values

• Comprehend and use language with accuracy, clarity, and discernment

• Interpret data, to appraise evidence and evaluate arguments

• Recognize the existence (or non-existence) of logical relationships between propositions

• Draw warranted conclusions and generalizations

• Put to test the conclusions and generalizations at which one arrives

• Reconstruct one's patterns of beliefs on the basis of wider experience

• Render accurate judgments about specific things and qualities in everyday life

In sum: “A persistent effort to examine any belief or supposed form of knowledge in the light of the evidence that supports or refutes it and the further conclusions to which it tends.”*[13] 25.5. HABITS OR TRAITS OF MIND 93

25.5 Habits or traits of mind

The habits of mind that characterize a person strongly disposed toward critical thinking include a desire to fol- low reason and evidence wherever they may lead, a systematic approach to problem solving, inquisitiveness, even- handedness, and confidence in reasoning.*[14] According to a definition analysis by Kompf & Bond (2001), critical thinking involves problem solving, decision making, metacognition, rationality, rational thinking, reasoning, knowledge, intelligence and also a moral component such as reflective thinking. Critical thinkers therefore need to have reached a level of maturity in their development, possess a certain attitude as well as a set of taught skills.

25.6 Research

Edward M. Glaser proposed that the ability to think critically involves three elements:*[13]

1. An attitude of being disposed to consider in a thoughtful way the problems and subjects that come within the range of one's experiences 2. Knowledge of the methods of logical inquiry and reasoning 3. Some skill in applying those methods.

Educational programs aimed at developing critical thinking in children and adult learners, individually or in group problem solving and decision making contexts, continue to address these same three central elements. Contemporary regards human reasoning as a complex process that is both reactive and reflec- tive.*[15] The relationship between critical thinking skills and critical thinking dispositions is an empirical question. Some people have both in abundance, some have skills but not the disposition to use them, some are disposed but lack strong skills, and some have neither. A measure of critical thinking dispositions is the California Measure of Mental Motivation.*[16]

25.7 Education

John Dewey is one of many educational leaders who recognized that a curriculum aimed at building thinking skills would benefit the individual learner, the community, and the entire democracy.*[17] Critical thinking is significant in academics due to being significant in learning. Critical thinking is significant in the learning process of internalization, in the construction of basic ideas, principles, and theories inherent in content. And critical thinking is significant in the learning process of application, whereby those ideas, principles, and theories are implemented effectively as they become relevant in learners' lives. Good teachers cultivate critical thinking (intellec- tually engaged thinking) at every stage of learning, including initial learning. This process of intellectual engagement is at the heart of the Oxford, Durham, Cambridge and London School of Economics tutorials. The tutor questions the students, often in a Socratic manner (see Socratic questioning). The key is that the teacher who fosters critical thinking fosters reflectiveness in students by asking questions that stimulate thinking essential to the construction of knowledge. Each discipline adapts its use of critical thinking concepts and principles (principles like in school). The core con- cepts are always there, but they are embedded in subject-specific content. For students to learn content, intellectual engagement is crucial. All students must do their own thinking, their own construction of knowledge. Good teachers recognize this and therefore focus on the questions, readings, activities that stimulate the mind to take ownership of key concepts and principles underlying the subject. In the UK school system, Critical Thinking is offered as a subject that 16- to 18-year-olds can take as an A-Level. Under the OCR exam board, students can sit two exam papers for the AS:“Credibility of Evidence”and“Assessing and Developing Argument”. The full Advanced GCE is now available: in addition to the two AS units, candidates sit the two papers“Resolution of Dilemmas”and“Critical Reasoning”. The A-level tests candidates on their ability to think critically about, and analyze, arguments on their deductive or inductive validity, as well as producing their own 94 CHAPTER 25. CRITICAL THINKING

arguments. It also tests their ability to analyze certain related topics such as credibility and ethical decision-making. However, due to its comparative lack of subject content, many universities do not accept it as a main A-level for admissions.*[18] Nevertheless, the AS is often useful in developing reasoning skills, and the full Advanced GCE is useful for degree courses in politics, philosophy, history or theology, providing the skills required for critical analysis that are useful, for example, in biblical study. There used to also be an Advanced Extension Award offered in Critical Thinking in the UK, open to any A-level student regardless of whether they have the Critical Thinking A-level. Cambridge International Examinations have an A-level in Thinking Skills.*[19] From 2008, Assessment and Qualifications Alliance has also been offering an A-level Critical Thinking specifica- tion;*[20] OCR exam board have also modified theirs for 2008. Many examinations for university entrance set by universities, on top of A-level examinations, also include a critical thinking component, such as the LNAT, the UKCAT, the BioMedical Admissions Test and the Thinking Skills Assessment. In the , critical thinking has been identified as one of the Four Cs of 21st century learning. In its 2012 platform, the Republican Party of Texas rejected the teaching of “Higher Order Thinking Skills... critical thinking skills and similar programs,”giving as a reason that this sort of teaching has“the purpose of challenging the student's fixed beliefs and undermining parental authority.”Media ridicule led to a response from RPT Communications Director Chris Elam that the inclusion of the term “critical thinking skills”was an oversight which cannot be corrected until 2014, when the next state convention will occur.*[21]*[22] In Qatar, Critical thinking was offered by AL-Bairaq which is an out-reach, non-traditional educational program that targets high school students and focuses on a curriculum based on STEM fields. The idea behind AL-Bairaq is to offer high school students the opportunity to connect with the research environment in the Center for Advanced Materials (CAM) at Qatar University. Faculty members train and mentor the students and help develop and enhance their critical thinking, problem-solving, and teamwork skills.*[23]

25.7.1 Efficacy

In 1995, a meta-analysis of the literature on teaching effectiveness in higher education was undertaken.*[24] The study noted concerns from higher education, politicians and business that higher education was failing to meet so- ciety's requirements for well-educated citizens. It concluded that although faculty may aspire to develop students' thinking skills, in practice they have tended to aim at facts and concepts utilizing lowest levels of cognition, rather than developing intellect or values.

25.8 Importance in academia

Critical thinking is an important element of all professional fields and academic disciplines (by referencing their re- spective sets of permissible questions, evidence sources, criteria, etc.). Within the framework of scientific , the process of critical thinking involves the careful acquisition and interpretation of information and use of it to reach a well-justified conclusion. The concepts and principles of critical thinking can be applied to any context or case but only by reflecting upon the nature of that application. Critical thinking forms, therefore, a system of related, and overlapping, modes of thought such as anthropological thinking, sociological thinking, historical thinking, political thinking, psychological thinking, philosophical thinking, mathematical thinking, chemical thinking, biological think- ing, ecological thinking, legal thinking, ethical thinking, musical thinking, thinking like a painter, sculptor, engineer, business person, etc. In other words, though critical thinking principles are universal, their application to disciplines requires a process of reflective contextualization. Critical thinking is considered important in the academic fields because it enables one to analyze, evaluate, explain, and restructure their thinking, thereby decreasing the risk of adopting, acting on, or thinking with, a false belief. How- ever, even with knowledge of the methods of logical inquiry and reasoning, mistakes can happen due to a thinker's inability to apply the methods or because of character traits such as egocentrism. Critical thinking includes identifica- tion of prejudice, bias, propaganda, self-deception, distortion, misinformation, etc.*[25] Given research in cognitive psychology, some educators believe that schools should focus on teaching their students critical thinking skills and cultivation of intellectual traits.*[26] Critical thinking skills can be used to help nurses during the assessment process. Through the use of critical thinking, 25.9. SEE ALSO 95

nurses can question, evaluate, and reconstruct the nursing care process by challenging the established theory and practice. Critical thinking skills can help nurses problem solve, reflect, and make a conclusive decision about the current situation they face. Critical thinking creates“new possibilities for the development of the nursing knowledge.” *[27] Due to the sociocultural, environmental, and political issues that are affecting healthcare delivery, it would be helpful to embody new techniques in nursing. Nurses can also engage their critical thinking skills through the Socratic method of dialogue and reflection. This practice standard is even part of some regulatory organizations such as the College of Nurses of Ontario - Professional Standards for Continuing Competencies (2006).*[28] It requires nurses to engage in Reflective Practice and keep records of this continued professional development for possible review by the College. Critical thinking also is considered important for education for . The Declaration of Principles on Tolerance adopted by UNESCO in 1995 affirms that“education for tolerance could aim at countering factors that lead to fear and exclusion of others, and could help young people to develop capacities for independent judgement, critical thinking and ethical reasoning.”*[29] Critical thinking is used as a way of deciding whether a claim is true, partially true, or false. It is a tool by which one can come about reasoned conclusions based on a reasoned process.

25.9 See also

• Cognitive bias mitigation

•

• Freedom of thought

•

• Outline of thought - topic tree that identifies many types of thoughts, types of thinking, aspects of thought, related fields, and more.

• Outline of human intelligence - topic tree presenting the traits, capacities, models, and research fields of human intelligence, and more.

• Sapere Aude

25.10 Notes

[1] “The Critical Thinking Movement: Alternating Currents in One Teacher's Thinking”. myweb.wvnet.edu. 22 March 2014.

[2] “Critical Thinking Index Page”.

[3] “Defining Critical Thinking”.

[4] Brown, Lesley. (ed.) The New Shorter Oxford English Dictionary (1993) p. 551.

[5] “Critical - Define Critical at Dictionary.com”. Dictionary.com. Retrieved 2016-02-24.

[6] “SSConceptionCT.html”.

[7] Facione, Peter A. (2011). “Critical Thinking: What It is and Why It Counts” (PDF). insightassessment.com. p. 26.

[8] Mulnix, J. W. (2010).“Thinking critically about critical thinking”. Educational Philosophy and Theory: 471. doi:10.1111/j.1469- 5812.2010.00673.x.

[9] Raiskums, B.W. (2008). An Analysis of the Concept Criticality in Adult Education. Capella University. ISBN 0549778349

[10] Carmichael, Kirby; letter to Olivetti, Laguna Salada Union School District, May 1997.

[11] Reynolds, Martin (2011). Critical thinking and systems thinking: towards a critical literacy for systems thinking in practice. In: Horvath , Christopher P. and Forte, James M. eds. Critical Thinking. New York, USA: Nova Science Publishers, pp. 37–68. 96 CHAPTER 25. CRITICAL THINKING

[12] Jones, Elizabeth A., & And Others (1995). National Assessment of College Student Learning: Identifying College Graduates' Essential Skills in Writing, Speech and Listening, and Critical Thinking. Final Project Report (ISBN 0-16-048051-5; NCES- 95-001) (PDF). from National Center on Postsecondary Teaching, Learning, and Assessment, University Park, PA.; Office of Educational Research and Improvement (ED), Washington, DC.; U.S. Government Printing Office, Superintendent of Documents, Mail Stop: SSOP, Washington, DC 20402-9328. PUB TYPE - Reports Research/Technical (143) pg. 14-15. Retrieved 2016-02-24.

[13] Edward M. Glaser (1941). An Experiment in the Development of Critical Thinking. New York, Bureau of Publications, Teachers College, Columbia University. ISBN 0-404-55843-7.

[14] The National Assessment of College Student Learning: Identification of the Skills to be Taught, Learned, and Assessed, NCES 94–286, US Dept of Education, Addison Greenwood (Ed), Sal Carrallo (PI). See also, Critical thinking: A statement of expert consensus for purposes of educational assessment and instruction. ERIC Document No. ED 315–423

[15] Solomon, S.A. (2002) “Two Systems of Reasoning,”in Heuristics and Biases: The Psychology of Intuitive Judgment, Govitch, Griffin, Kahneman (Eds), Cambridge University Press. ISBN 978-0-521-79679-8; Thinking and Reasoning in Human Decision Making: The Method of Argument and Heuristic Analysis, Facione and Facione, 2007, California Academic Press. ISBN 978-1-891557-58-3

[16] Research on Sociocultural Influences on Motivation and Learning, p. 46

[17] Dewey, John. (1910). How we think. Lexington, MA: D.C. Heath & Co.

[18] Critical Thinking FAQs from Oxford Cambridge and RSA Examinations Archived 11 April 2008 at the Wayback Machine.

[19] “Cambridge International AS and A Level subjects”.

[20] “New GCEs for 2008”, Assessment and Qualifications Alliance Archived 17 February 2008 at the Wayback Machine.

[21] Strauss, Valerie (9 July 2012). “Texas GOP rejects 'critical thinking' skills. Really”. Washington Post.

[22] Lach, Eric (29 June 2012). “Texas GOP's 2012 Platform Accidentally Opposes Teaching Of 'Critical Thinking Skills'". TPM Muckraker.

[23] https://web.archive.org/20140419220638/http://www.qu.edu.qa/offices/research/CAM/dmsprogram/index.php. Archived from the original on 19 April 2014. Retrieved 5 July 2014. Missing or empty |title= (help)

[24] Lion Gardiner, Redesigning Higher Education: Producing Dramatic Gains in Student Learning, in conjunction with: ERIC Clearinghouse on Higher Education, 1995

[25] Lau, Joe; Chan, Jonathan. "[F08] Cognitive biases”. Critical thinking web. Retrieved 2016-02-01.

[26] “Critical Thinking, Moral Integrity and Citizenship”. Criticalthinking.org. Retrieved 2016-02-01.

[27] Catching the wave: understanding the concept of critical thinking (1999) doi:10.1046/j.1365-2648.1999.00925.x

[28] College of Nurses of Ontario - Professional Standards for Continuing Competencies (2006)

[29] “International Day for Tolerance . Declaration of Principles on Tolerance, Article 4, 3”. UNESCO. Retrieved 2016-02-24.

25.11 References

• Le Cornu, Alison. (2009). “Meaning, Internalization and Externalization: Towards a fuller understanding of the process of reflection and its role in the construction of the self”. Adult Education Quarterly 59 (4): 279–297. doi:10.1177/0741713609331478

25.12 Further reading

• Cederblom, J & Paulsen, D.W. (2006) Critical Reasoning: Understanding and criticizing arguments and the- ories, 6th edn. (Belmont, CA, ThomsonWadsworth).

• College of Nurses of Ontario Professional Standards (2006) - Continuing Competencies

• Damer, T. Edward. (2005) Attacking Faulty Reasoning, 6th Edition, Wadsworth. ISBN 0-534-60516-8 25.13. EXTERNAL LINKS 97

• Dauer, Francis Watanabe. Critical Thinking: An Introduction to Reasoning, 1989, ISBN 978-0-19-504884-1 • Facione, P. 2007. Critical Thinking: What It Is and Why It Counts – 2007 Update • Fisher, Alec and Scriven, Michael. (1997) Critical Thinking: Its Definition and Assessment, Center for Research in Critical Thinking (UK) / Edgepress (US). ISBN 0-9531796-0-5 • Hamby, B.W. (2007) The Philosophy of Anything: Critical Thinking in Context. Kendall Hunt Publishing Company, Dubuque Iowa. ISBN 978-0-7575-4724-9 • Vincent F. Hendricks. (2005) Thought 2 Talk: A Crash Course in Reflection and Expression, New York: Automatic Press / VIP. ISBN 87-991013-7-8 • Kompf, M., & Bond, R. (2001). Critical reflection in adult education. In T. Barer-Stein & M. Kompf(Eds.), The craft of teaching adults (pp. 21–38). Toronto, ON: Irwin. • McPeck, J. (1992). Thoughts on subject specificity. In S. Norris (Ed.), The generalizability of critical thinking (pp. 198–205). New York: Teachers College Press. • Moore, Brooke Noel and Parker, Richard. (2012) Critical Thinking. 10th ed. Published by McGraw-Hill. ISBN 0-07-803828-6. • Mulnix, J. W. (2010). “Thinking critically about critical thinking”. Educational Philosophy and Theory. doi:10.1111/j.1469-5812.2010.00673.x. • Paul, R (1982). “Teaching critical thinking in the strong sense: A focus on self-deception, world views and a dialectical mode of analysis”. Informal Logic Newsletter 4 (2): 2–7. • Paul, Richard. (1995) Critical Thinking: How to Prepare Students for a Rapidly Changing World. 4th ed. Foundation for Critical Thinking. ISBN 0-944583-09-1. • Paul, Richard and Elder, Linda. (2006) Critical Thinking Tools for Taking Charge of Your Learning and Your Life, New Jersey: Prentice Hall Publishing. ISBN 0-13-114962-8. • Paul, Richard; Elder, Linda. (2002) Critical Thinking: Tools for Taking Charge of Your Professional and Personal Life. Published by Financial Times Prentice Hall. ISBN 0-13-064760-8. • Pavlidis, Periklis. (2010) Critical Thinking as Dialectics: a Hegelian-Marxist Approach. Journal for Critical Education Policy Studies.Vol.8(2) • Sagan, Carl. (1995) The Demon-Haunted World: Science As a Candle in the Dark. Ballantine Books. ISBN 0-345-40946-9 • Theodore Schick & Lewis Vaughn “How to Think About Weird Things: Critical Thinking for a ” (2010) ISBN 0-7674-2048-9 • Twardy, Charles R. (2003) Argument Maps Improve Critical Thinking. Teaching Philosophy 27:2 June 2004. • van den Brink-Budgen, R (2010) 'Critical Thinking for Students', How To Books. ISBN 978-1-84528-386-5 • Whyte, J. (2003) Bad Thoughts – A Guide to Clear Thinking, Corvo. ISBN 0-9543255-3-2. • Zeigarnik, B.V. (1927). On finished and unfinished tasks. In English translation Edited by Willis D. Ellis ; with an introduction by Kurt Koffka. (1997). A source book of gestalt psychology xiv, 403 p. : ill. ; 22 cmHighland, N.Y: Gestalt Journal Press. “This Gestalt Journal Press edition is a verbatim reprint of the book as originally published in 1938”—T.p. verso. ISBN 9780939266302. OCLC 38755142

25.13 External links

Media related to Critical thinking at Wikimedia Commons Quotations related to Critical thinking at Wikiquote

• Critical thinking at PhilPapers • Critical thinking at the Indiana Philosophy Ontology Project 98 CHAPTER 25. CRITICAL THINKING

• Informal logic entry in the Stanford Encyclopedia of Philosophy

• Critical thinking at DMOZ • Critical Thinking Web – Online tutorials and teaching material on critical thinking.

• Critical Thinking: What Is It Good for? (In Fact, What Is It?) by Howard Gabennesch, Skeptical Inquirer magazine.

• Glossary of Critical Thinking Terms Chapter 26

Deductive reasoning

Deductive reasoning, also deductive logic, logical deduction or, informally, "top-down" logic,*[1] is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.*[2] It differs from or . Deductive reasoning links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true. Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of a closed , narrowing the range under consideration until only the conclusion(s) is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from, i.e., there is epistemic uncertainty. How- ever, the inductive reasoning mentioned here is not the same as induction used in mathematical proofs – is actually a form of deductive reasoning.

26.1 Simple example

An example of a deductive argument:

1. All men are mortal. 2. Socrates is a man. 3. Therefore, Socrates is mortal.

The first premise states that all objects classified as“men”have the attribute“mortal”. The second premise states that“Socrates”is classified as a“man”– a member of the set“men”. The conclusion then states that“Socrates” must be “mortal”because he inherits this attribute from his classification as a “man”.

26.2 Law of detachment

Main article: Modus ponens

The law of detachment (also known as affirming the antecedent and Modus ponens) is the first form of deductive reasoning. A single conditional statement is made, and a hypothesis (P) is stated. The conclusion (Q) is then deduced from the statement and the hypothesis. The most basic form is listed below:

1. P → Q (conditional statement) 2. P (hypothesis stated) 3. Q (conclusion deduced)

99 100 CHAPTER 26. DEDUCTIVE REASONING

In deductive reasoning, we can conclude Q from P by using the law of detachment.*[3] However, if the conclusion (Q) is given instead of the hypothesis (P) then there is no definitive conclusion. The following is an example of an argument using the law of detachment in the form of an if-then statement:

1. If an angle satisfies 90° < A < 180°, then A is an obtuse angle. 2. A = 120°. 3. A is an obtuse angle.

Since the measurement of angle A is greater than 90° and less than 180°, we can deduce that A is an obtuse angle. If however, we are given the conclusion that A is an obtuse angle we cannot deduce the premise that A = 120°.

26.3 Law of syllogism

The law of syllogism takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. Here is the general form:

1. P → Q 2. Q → R 3. Therefore, P → R.

The following is an example:

1. If Larry is sick, then he will be absent. 2. If Larry is absent, then he will miss his classwork. 3. Therefore, if Larry is sick, then he will miss his classwork.

We deduced the final statement by combining the hypothesis of the first statement with the conclusion of the sec- ond statement. We also allow that this could be a false statement. This is an example of the transitive property in mathematics. The transitive property is sometimes phrased in this form:

1. A = B. 2. B = C. 3. Therefore, A = C.

26.4 Law of contrapositive

Main article:

The law of contrapositive states that, in a conditional, if the conclusion is false, then the hypothesis must be false also. The general form is the following:

1. P → Q. 2. ~Q. 3. Therefore, we can conclude ~P.

The following are examples:

1. If it is raining, then there are clouds in the sky. 2. There are no clouds in the sky. 3. Thus, it is not raining. 26.5. VALIDITY AND SOUNDNESS 101

26.5 Validity and soundness

Argument terminology

Deductive arguments are evaluated in terms of their validity and soundness. An argument is valid if it is impossible for its premises to be true while its conclusion is false. In other words, the conclusion must be true if the premises are true. An argument can be valid even though the premises are false. An argument is sound if it is valid and the premises are true. It is possible to have a deductive argument that is logically valid but is not sound. Fallacious arguments often take that form. The following is an example of an argument that is valid, but not sound:

1. Everyone who eats carrots is a quarterback. 2. John eats carrots. 3. Therefore, John is a quarterback.

The example's first premise is false – there are people who eat carrots and are not quarterbacks – but the conclusion must be true, so long as the premises are true (i.e. it is impossible for the premises to be true and the conclusion false). Therefore, the argument is valid, but not sound. Generalizations are often used to make invalid arguments, such as “everyone who eats carrots is a quarterback.”Not everyone who eats carrots is a quarterback, thus proving the flaw of such arguments. In this example, the first statement uses categorical reasoning, saying that all carrot-eaters are definitely quarterbacks. This theory of deductive reasoning – also known as – was developed by Aristotle, but was superseded by propositional (sentential) logic and predicate logic. Deductive reasoning can be contrasted with inductive reasoning, in regards to validity and soundness. In cases of inductive reasoning, even though the premises are true and the argument is“valid”, it is possible for the conclusion to be false (determined to be false with a counterexample or other means).

26.6 History

Aristotle started documenting deductive reasoning in the 4th century BC.*[4] 102 CHAPTER 26. DEDUCTIVE REASONING

26.7 Education

Deductive reasoning is generally considered to be a skill that develops without any formal teaching or training. As a result of this belief, deductive reasoning skills are not taught in secondary schools, where students are expected to use reasoning more often and at a higher level.*[5] It is in high school, for example, that students have an abrupt introduction to mathematical proofs – which rely heavily on deductive reasoning.*[5]

26.8 See also

• Abductive reasoning

• Analogical reasoning

• Argument (logic)

• Correspondence theory of truth

• Decision making

• Decision theory

• Defeasible reasoning

• Fallacy

• Fault Tree Analysis

• Geometry

• Hypothetico-deductive method

• Inductive reasoning

• Inference

• Inquiry

• Logic

• Logical consequence

• Mathematical induction

• Mathematical logic

• Natural deduction

• Propositional calculus

• Retroductive reasoning

• Scientific method

• Soundness

• Syllogism

• Theory of justification 26.9. REFERENCES 103

26.9 References

[1] Deduction & Induction, Research Methods Knowledge Base

[2] Sternberg, R. J. (2009). Cognitive Psychology. Belmont, CA: Wadsworth. p. 578. ISBN 978-0-495-50629-4.

[3] Guide to Logic

[4] Evans, Jonathan St. B. T.; Newstead, Stephen E.; Byrne, Ruth M. J., eds. (1993). Human Reasoning: The Psychology of Deduction (Reprint ed.). Psychology Press. p. 4. ISBN 9780863773136. Retrieved 2015-01-26. In one sense [...] one can see the psychology of deductive reasoning as being as old as the study of logic, which originated in the writings of Aristotle.

[5] Stylianides, G. J.; Stylianides (2008).“A. J.”. Mathematical Thinking and Learning 10 (2): 103–133. doi:10.1080/10986060701854425.

26.10 Further reading

• Vincent F. Hendricks, Thought 2 Talk: A Crash Course in Reflection and Expression, New York: Automatic Press / VIP, 2005, ISBN 87-991013-7-8

• Philip Johnson-Laird, Ruth M. J. Byrne, Deduction, Psychology Press 1991, ISBN 978-0-86377-149-1 • Zarefsky, David, Argumentation: The Study of Effective Reasoning Parts I and II, The Teaching Company 2002

• Bullemore, Thomas, * The Pragmatic Problem of Induction.

26.11 External links

• Deductive reasoning at PhilPapers • Deductive reasoning at the Indiana Philosophy Ontology Project

• Deductive reasoning entry in the Internet Encyclopedia of Philosophy Chapter 27

Defeasible reasoning

Defeasible reasoning is a kind of reasoning that is based on reasons that are defeasible, as opposed to the indefeasible reasons of deductive logic. Defeasible reasoning is a particular kind of non-demonstrative reasoning, where the reasoning does not produce a full, complete, or final demonstration of a claim, i.e., where fallibility and corrigibility of a conclusion are acknowledged. In other words defeasible reasoning produces a contingent statement or claim. Other kinds of non-demonstrative reasoning are probabilistic reasoning, inductive reasoning, statistical reasoning, abductive reasoning, and paraconsistent reasoning. Defeasible reasoning is also a kind of ampliative reasoning because its conclusions reach beyond the pure meanings of the premises. The differences between these kinds of reasoning correspond to differences about the conditional that each kind of reasoning uses, and on what premise (or on what authority) the conditional is adopted:

• Deductive (from meaning postulate, axiom, or contingent assertion): if p then q (i.e., q or not-p)

• Defeasible (from authority): if p then (defeasibly) q

• Probabilistic (from combinatorics and indifference): if p then (probably) q

• Statistical (from data and presumption): the frequency of qs among ps is high (or inference from a model fit to data); hence, (in the right context) if p then (probably) q

• Inductive (theory formation; from data, coherence, simplicity, and confirmation): (inducibly) “if p then q"; hence, if p then (deducibly-but-revisably) q

• Abductive (from data and theory): p and q are correlated, and q is sufficient for p; hence, if p then (abducibly) q as cause

Defeasible reasoning finds its fullest expression in , ethics and moral philosophy, epistemology, pragmatics and conversational conventions in linguistics, constructivist decision theories, and in knowledge representation and planning in artificial intelligence. It is also closely identified with prima facie (presumptive) reasoning (i.e., reasoning on the “face”of evidence), and ceteris paribus (default) reasoning (i.e., reasoning, all things “being equal”).

27.1 History

Though Aristotle differentiated the forms of reasoning that are valid for logic and philosophy from the more general ones that are used in everyday life (see dialectics and rhetoric), 20th century philosophers mainly concentrated on deductive reasoning. At the end of the 19th century, logic texts would typically survey both demonstrative and non- demonstrative reasoning, often giving more space to the latter. However, after the blossoming of mathematical logic at the hands of Bertrand Russell, and , latter-20th century logic texts paid little attention to the non-deductive modes of inference. There are several notable exceptions. wrote his dissertation on non-demonstrative reason- ing, and influenced the thinking of Ludwig Wittgenstein on this subject. Wittgenstein, in turn, had many admirers, including the positivist legal scholar H.L.A. Hart and the linguist John L. Austin, Stephen Toulmin in

104 27.2. POLITICAL AND JUDICIAL USE 105

rhetoric (Chaim Perelman too), the moral theorists W.D. Ross and C.L. Stevenson, and the epistemolo- gist/ontologist . The etymology of defeasible usually refers to Middle English law of contracts, where a condition of defeasance is a clause that can invalidate or annul a contract or deed. Though defeat, dominate, defer, defy, deprecate and derogate are often used in the same contexts as defeasible, the verbs annul and invalidate (and nullify, overturn, rescind, vacate, repeal, debar, void, cancel, countermand, preempt, etc.) are more properly correlated with the concept of defeasibility than those words beginning with the letter d. Many dictionaries do contain the verb, to defease with past participle, defeased. Philosophers in moral theory and rhetoric had taken defeasibility largely for granted when American epistemologists rediscovered Wittgenstein's thinking on the subject: John Ladd, , Roderick Firth, Ernest Sosa, , and John L. Pollock all began writing with new conviction about how appearance as red was only a defeasible reason for believing something to be red. More importantly Wittgenstein's orientation toward language- games (and away from semantics) emboldened these epistemologists to manage rather than to expurgate prima facie logical inconsistency. At the same time (in the mid-1960s), two more students of Hart and Austin at Oxford, Brian Barry and David Gauthier, were applying defeasible reasoning to political argument and practical reasoning (of action), respectively. Joel Feinberg and Joseph Raz were beginning to produce equally mature works in ethics and jurisprudence informed by defeasibility. By far the most significant works on defeasibility by the mid-1970s were in epistemology, where John Pollock's 1974 Knowledge and Justification popularized his terminology of undercutting and rebutting (which mirrored the analysis of Toulmin). Pollock's work was significant precisely because it brought defeasibility so close to philosophical logicians. The failure of logicians to dismiss defeasibility in epistemology (as Cambridge's logicians had done to Hart decades earlier) landed defeasible reasoning in the philosophical mainstream. Defeasibility had always been closely related to argument, rhetoric, and law, except in epistemology, where the chains of reasons, and the origin of reasons, were not often discussed. 's Dialectics is an example of how difficult it was for philosophers to contemplate more complex systems of defeasible reasoning. This was in part because proponents of informal logic became the keepers of argument and rhetoric while insisting that formalism was anathema to argument. About this time, researchers in artificial intelligence became interested in non-monotonic reasoning and its semantics. With philosophers such as Pollock and Donald Nute (e.g., defeasible logic), dozens of computer scientists and lo- gicians produced complex systems of defeasible reasoning between 1980 and 2000. No single system of defeasible reasoning would emerge in the same way that Quine's system of logic became a de facto standard. Nevertheless, the 100-year headstart on non-demonstrative logical calculi, due to George Boole, Charles Sanders Peirce, and Gottlob Frege was being closed: both demonstrative and non-demonstrative reasoning now have formal calculi. There are related (and slightly competing) systems of reasoning that are newer than systems of defeasible reason- ing, e.g., belief revision and dynamic logic. The dialogue logics of Charles Hamblin and Jim Mackenzie, and their colleagues, can also be tied closely to defeasible reasoning. Belief revision is a non-constructive specification of the desiderata with which, or constraints according to which, epistemic change takes place. Dynamic logic is related mainly because, like paraconsistent logic, the reordering of premises can change the set of justified conclusions. Di- alogue logics introduce an adversary, but are like belief revision theories in their adherence to deductively consistent states of belief.

27.2 Political and judicial use

Many political philosophers have been fond of the word indefeasible when referring to rights, e.g., that were in- alienable, divine, or indubitable. For example, in the 1776 Virginia Declaration of Rights, “community hath an indubitable, inalienable, and indefeasible right to reform, alter or abolish government...”(also attributed to James Madison); and John Adams, “The people have a right, an indisputable, unalienable, indefeasible, divine right to that most dreaded and envied kind of knowledge – I mean of the character and conduct of their rulers.”Also, Lord Aberdeen: “indefeasible right inherent in the British Crown”and Gouverneur Morris: “the Basis of our own Con- stitution is the indefeasible Right of the People.”Scholarship about Abraham Lincoln often cites these passages in the justification of secession. Philosophers who use the word defeasible have historically had different world views from those who use the word indefeasible (and this distinction has often been mirrored by Oxford and Cambridge zeitgeist); hence it is rare to find authors who use both words. 106 CHAPTER 27. DEFEASIBLE REASONING

In judicial opinions, the use of defeasible is commonplace. There is however disagreement among legal logicians whether defeasible reasoning is central, e.g., in the consideration of open texture, precedent, exceptions, and ratio- nales, or whether it applies only to explicit defeasance clauses. H.L.A. Hart in The Concept of Law gives two famous examples of defeasibility: “No vehicles in the park”(except during parades); and “Offer, acceptance, and memo- randum produce a contract”(except when the contract is illegal, the parties are minors, inebriated, or incapacitated, etc.).

27.3 Specificity

One of the main disputes among those who produce systems of defeasible reasoning is the status of a rule of specificity. In its simplest form, it is the same rule as subclass inheritance preempting class inheritance: (R1) if r then (defeasibly) q e.g., if bird, then can fly (R2) if p then (defeasibly) not-q e.g., if penguin, then cannot fly (O1) if p then (deductively) r e.g., if penguin, then bird (M1) arguably, p e.g., arguably, penguin (M2) R2 is a more specific reason than R1 e.g., R2 is better than R1 (M3) therefore, arguably, not-q e.g., therefore, arguably, not-flies Approximately half of the systems of defeasible reasoning discussed today adopt a rule of specificity, while half expect that such preference rules be written explicitly by whoever provides the defeasible reasons. For example, Rescher's dialectical system uses specificity, as do early systems of multiple inheritance (e.g., David Touretzky) and the early argument systems of Donald Nute and of Guillermo Simari and Ronald Loui. Defeasible reasoning accounts of precedent (stare decisis and case-based reasoning) also make use of specificity (e.g., Joseph Raz and the work of Kevin D. Ashley and Edwina Rissland). Meanwhile, the argument systems of Henry Prakken and Giovanni Sartor, of Bart Verheij and Jaap Hage, and the system of Phan Minh Dung do not adopt such a rule.

27.4 Nature of defeasibility

There is a distinct difference between those who theorize about defeasible reasoning as if it were a system of con- firmational revision (with affinities to belief revision), and those who theorize about defeasibility as if it were the result of further (non-empirical) investigation. There are at least three kinds of further non-empirical investigation: in a lexical/syntactic process, progress in a computational process, and progress in an adversary or legal proceeding. Defeasibility as corrigibility: Here, a person learns something new that annuls a prior inference. In this case, defea- sible reasoning provides a constructive mechanism for belief revision, like a truth maintenance system as envisioned by Jon Doyle. Defeasibility as shorthand for preconditions: Here, the author of a set of rules or legislative code is writing rules with exceptions. Sometimes a set of defeasible rules can be rewritten, with more cogency, with explicit (local) pre- conditions instead of (non-local) competing rules. Many non-monotonic systems with fixed-point or preferential semantics fit this view. However, sometimes the rules govern a process of argument (the last view on this list), so that they cannot be re-compiled into a set of deductive rules lest they lose their force in situations with incomplete knowledge or incomplete derivation of preconditions. Defeasibility as an anytime algorithm: Here, it is assumed that calculating arguments takes time, and at any given time, based on a subset of the potentially constructible arguments, a conclusion is defeasibly justified. Isaac Levi has protested against this kind of defeasibility, but it is well-suited to the heuristic projects of, for example, Herbert A. Simon. On this view, the best move so far in a chess-playing program's analysis at a particular depth is a defeasibly justified conclusion. This interpretation works with either the prior or the next semantical view. Defeasibility as a means of controlling an investigative or social process: Here, justification is the result of the right kind of procedure (e.g., a fair and efficient hearing), and defeasible reasoning provides impetus for pro and con responses to each other. Defeasibility has to do with the alternation of verdict as locutions are made and cases presented, not the changing of a mind with respect to new (empirical) discovery. Under this view, defeasible reasoning and defeasible argumentation refer to the same phenomenon. 27.5. SEE ALSO 107

27.5 See also

• Defeasible estate

• Indefeasible rights of use • Argument (logic)

• Prima facie

• Practical reasoning • Pragmatics

• Non-monotonic reasoning

27.6 References

• Defeasible logic, Donald Nute, Lecture Notes in Computer Science, Springer, 2003.

• Logical models of argument, Carlos Chesnevar, et al., ACM Computing Surveys 32:4, 2000. • Logics for defeasible argumentation, Henry Prakken and Gerard Vreeswijk, in Handbook of Philosophical Logic, Dov M. Gabbay, Franz Guenthner, eds., Kluwer, 2002. • Dialectics, Nicholas Rescher, SUNY Press, 1977.

• Defeasible reasoning, John Pollock, Cognitive Science, 1987. • Knowledge and Justification, John Pollock, Princeton University Press, 1974.

• Abstract argumentation systems, Gerard Vreeswijk, Artificial Intelligence, 1997. • Hart's critics on defeasible concepts and ascriptivism, Ronald Loui, Proc. 5th Intl. Conf. on AI and Law, 1995.

• Political argument, Brian Barry, Routledge & Kegan Paul, 1970. • The uses of argument, Stephen Toulmin, Cambridge University Press, 1958.

• Discourse relations and defeasible knowledge, Alex Lascarides and Nicholas Asher, Proc. of the 29th Meeting of the Assn. for Comp. Ling., 1991.

• Defeasible logic programming: an argumentative approach, Alejandro Garcia and Guillermo Simari, Theory and Practice of Logic Programming 4:95–138, 2004.

• Philosophical foundations of and the logic of defeasible conditionals, Carlos Alchourron, in De- ontic logic in computer science: normative system specification, J. Meyer, R. Wieringa, eds., Wiley, 1994.

• A Mathematical Treatment of Defeasible Reasoning and its Implementation. Guillermo Simari, Ronald Loui, Artificial Intelligence Journal, 53(2–3): 125–157 (1992).

27.7 External links

• Article on Defeasible Reasoning in the Stanford Encyclopedia of Philosophy • An example of defeasible reasoning in action Chapter 28

Definable set

In mathematical logic, a definable set is an n-ary relation on the domain of a structure whose elements are precisely those elements satisfying some formula in the language of that structure. A set can be defined with or without parameters, which are elements of the domain that can be referenced in the formula defining the relation.

28.1 Definition

Let L be a first-order language, M an L -structure with domain M , X a fixed subset of M , and m a natural number. Then:

m • A set A ⊆ M is definable in M with parameters from X if and only if there exists a formula φ[x1, . . . , xm, y1, . . . , yn] and elements b1, . . . , bn ∈ X such that for all a1, . . . , am ∈ M ,

(a1, . . . , am) ∈ A if and only if M |= φ[a1, . . . , am, b1, . . . , bn] The bracket notation here indicates the semantic evaluation of the free variables in the formula.

• A set A is definable in M without parameters if it is definable in M with parameters from the empty set (that is, with no parameters in the defining formula).

• A function is definable in M (with parameters) if its graph is definable (with those parameters) in M .

• An element a is definable in M (with parameters) if the singleton set {a} is definable in M (with those parameters).

28.2 Examples

28.2.1 The natural numbers with only the order relation

Let N = (N, <) be the structure consisting of the natural numbers with the usual ordering. Then every natural number is definable in N without parameters. The number 0 is defined by the formula φ(x) stating that there exist no elements less than x: φ = ¬∃y(y < x), and a natural n > 0 is defined by the formula φ(x) stating there exist exactly n elements less than x:

φ = ∃x0 · · · ∃xn−1(x0 < x1 ∧ · · · ∧ xn−1 < x ∧ ∀y(y < x → (y ≡ x0 ∨ · · · ∨ y ≡ xn−1))) In contrast, one cannot define any specific integer without parameters in the structure Z = (Z, <) consisting of the integers with the usual ordering (see the section on automorphisms below).

108 28.3. INVARIANCE UNDER AUTOMORPHISMS 109

28.2.2 The natural numbers with their arithmetical operations

Let N = (N, +, ·, <) be the first-order structure consisting of the natural numbers and their usual arithmetic oper- ations and order relation. The sets definable in this structure are known as the arithmetical sets, and are classified in the arithmetical hierarchy. If the structure is considered in second-order logic instead of first-order logic, the de- finable sets of natural numbers in the resulting structure are classified in the analytical hierarchy. These hierarchies reveal many relationships between definability in this structure and computability theory, and are also of interest in descriptive .

28.2.3 The field of real numbers

Let R = (R, 0, 1, +, ·) be the structure consisting of the field of real numbers. Although the usual ordering relation is not directly included in the structure, there is a formula that defines the set of nonnegative reals, since these are the only reals that possess square roots: φ = ∃y(y · y ≡ x). Thus any a ∈ R is nonnegative if and only if R |= φ[a] . In conjunction with a formula that defines the additive inverse of a real number in R , one can use φ to define the usual ordering in R : for a, b ∈ R , set a ≤ b if and only if b − a is nonnegative. The enlarged structure R≤ = (R, 0, 1, +, ·, ≤) s is called a definitional extension of the original structure. It has the same expressive power as the original structure, in the sense that a set is definable over the enlarged structure from a set of parameters if and only if it is definable over the original structure from that same set of parameters. The theory of R≤ has quantifier elimination. Thus the definable sets are Boolean combinations of solutions to polynomial equalities and inequalities; these are called semi-algebraic sets. Generalizing this property of the real line leads to the study of o-minimality.

28.3 Invariance under automorphisms

An important result about definable sets is that they are preserved under automorphisms.

Let M be an L -structure with domain M , X ⊆ M , and A ⊆ M m definable in M with parameters from X . Let π : M → M be an automorphism of M which is the identity on X . Then for all a1, . . . , am ∈ M ,

(a1, . . . , am) ∈ A if and only if (π(a1), . . . , π(am)) ∈ A

This result can sometimes be used to classify the definable subsets of a given structure. For example, in the case of Z = (Z, <) above, any translation of Z is an automorphism preserving the empty set of parameters, and thus it is impossible to define any particular integer in this structure without parameters in Z . In fact, since any two integers are carried to each other by a translation and its inverse, the only sets of integers definable in Z without parameters are the empty set and Z itself. In contrast, there are infinitely many definable sets of pairs (or indeed n-tuples for any fixed n>1) of elements of Z , since any automorphism (translation) preserves the“distance”between two elements.

28.4 Additional results

The Tarski–Vaught test is used to characterize the elementary substructures of a given structure.

28.5 References

• Hinman, Peter. Fundamentals of Mathematical Logic, A. K. Peters, 2005.

• Marker, David. : An Introduction, Springer, 2002. 110 CHAPTER 28. DEFINABLE SET

• Rudin, Walter. Principles of , 3rd. ed. McGraw-Hill, 1976.

• Slaman, Theodore A. and W. Hugh Woodin. Mathematical Logic: The Berkeley Undergraduate Course. Spring 2006. Chapter 29

Definitions of logic

This article divides the definitions into two classes: first are the simple definitions, that consist of a pithy sentence characterising the topic; second are theoretical definitions, where the definition of logic turns on an analysis the definer provides.

29.1 Formal definitions

W. V. Quine (1940, pp. 2–3) defined logic in terms of a logical vocabulary, which in turn is identified by an argument that the many particular vocabularies —Quine mentions geological vocabulary—are used in their particular discourses together with a common, topic-independent kernel of terms.*[1] These terms, then, constitute the logical vocabulary, and the logical truths are those truths common to all particular topics. Hofweber (2004) lists several definitions of logic, and goes on to claim that all definitions of logic are of one of four sorts. These are that logic is the study of: (i) artificial formal structures, (ii) sound inference (e.g., Poinsot), (iii) tautologies (e.g., Watts), or (iv) general features of thought (e.g., Frege). He argues then that these definitions are related to each other, but do not exhaust each other, and that an examination of formal ontology shows that these mismatches between rival definitions are due to tricky issues in ontology.

29.2 Informal and colloquial definitions

Arranged in approximate chronological order.

• The tool for distinguishing between the true and the false ().*[2]

• The science of reasoning, teaching the way of investigating unknown truth in connection with a thesis ().

• The art whose function is to direct the reason lest it err in the manner of inferring or knowing (John Poinsot).

• The art of conducting reason well in knowing things (Antoine Arnauld).

• The right use of reason in the inquiry after truth (Isaac Watts).

• The Science, as well as the Art, of reasoning (Richard Whately).

• The science of the operations of the understanding which are subservient to the estimation of evidence (John Stuart Mill).

• The science of the laws of discursive thought (James McCosh).

• The science of the most general laws of truth (Gottlob Frege).

111 112 CHAPTER 29. OF LOGIC

29.3 See also

• Universal logic

29.4 Notes

[1] Cf. Ferreiros, 2001

[2] Zekai Sen Philosophical, Logical and Scientific Perspectives in Engineering page 114

29.5 References

• Ferreiros, J. (2001). The road to modern logic: an interpretation. In Bulletin of Symbolic Logic 7(4):441-483.

• Frege, G. (1897). Logic. transl. Long, P. & White, R., Posthumous Writings. • Hofweber, T. (2004). Logic and ontology. Stanford Encyclopedia of Philosophy.

• Joyce, G.H. (1908). Principles of Logic. London. • Kilwardby, R. The Nature of Logic, from De Ortu Scientarum, transl. Kretzmann, in Kretzmann N. & Stump E., The Cambridge Translation of Medieval Philosophical Texts, Vol I. Cambridge 1988, pp. 262 ff.) • McCosh, J. (1870). The Laws of Discursive Thought. London.

• Mill, J.S. (1904). A System of Logic. 8th edition. London. • Poinsot, J. (1637/1955). 'Outlines of Formal Logic'. In his Ars Logica, 1637, ed. and transl. F.C. Wade, 1955. • Quine, W.V.O. (1940/1981). Mathematical Logic. Third edition. Harvard University Press.

• Watts, I. (1725). Logick. • Whateley, R.(1826). Elements of Logic. Chapter 30

Degree of truth

In standard mathematics, propositions can typically be considered unambiguously true or false. For instance, the proposition zero belongs to the set { 1 } is regarded as simply false; while the proposition one belongs to the set { 1 } is regarded as simply true. However, some mathematicians, computer scientists, and philosophers have been attracted to the idea that a proposition might be more or less true, rather than simply true or simply false. Consider My coffee is hot. In mathematics, this idea can be developed in terms of fuzzy logic. In computer science, it has found application in artificial intelligence. In philosophy, the idea has proved particularly appealing in the case of vagueness. Degrees of truth is an important concept in law.

30.1 See also

• Artificial intelligence • Bivalence

• Fuzzy logic

• • Half-truth

• Multi-valued logic • Paradox of the heap

• Truth • Truth value

• Vagueness

30.2 Bibliography

• Zadeh, L.A. (1965).“Fuzzy sets”. Information and Control 8 (3): 338–353. doi:10.1016/S0019-9958(65)90241- X. ISSN 0019-9958.

113 Chapter 31

Denying the antecedent

Denying the antecedent, sometimes also called inverse error or fallacy of the inverse, is a of inferring the inverse from the original statement. It is committed by reasoning in the form:

If P, then Q. Not P. Therefore, not Q. which may also be phrased as

P → Q

∴ ¬P → ¬Q Arguments of this form are invalid. Informally, this means that arguments of this form do not give good reason to establish their conclusions, even if their premises are true. The name denying the antecedent derives from the premise“not P", which denies the“if”clause of the conditional premise. One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example:

If it is raining, then the grass is wet. It is not raining. Therefore, the grass is not wet.

The argument is invalid because there are other reasons for which the grass could be wet (being sprayed with water by a hose, for example). Another example:

If Queen Elizabeth is an American citizen, then she is a human being. Queen Elizabeth is not an American citizen. Therefore, Queen Elizabeth is not a human being.

That argument is intentionally bad, but arguments of the same form can sometimes seem superficially convincing, as in the following example offered, with apologies for its lack of logical rigour, by in the article "Computing Machinery and Intelligence":

If each man had a definite set of rules of conduct by which he regulated his life he would be no better than a machine. But there are no such rules, so men cannot be machines.*[1]

114 31.1. SEE ALSO 115

However, men could still be machines that do not follow a definite set of rules. Thus this argument (as Turing intends) is invalid. It is possible that an argument that denies the antecedent could be valid, if the argument instantiates some other valid form. For example, if the claims P and Q express the same proposition, then the argument would be trivially valid, as it would beg the question. In everyday discourse, however, such cases are rare, typically only occurring when the “if-then”premise is actually an "if and only if" claim (i.e., a biconditional/equality). For example:

If I am President of the United States, then I can veto Congress. I am not President. Therefore, I cannot veto Congress.

The above argument is not valid, but would be if the first premise ended thus: "...and if I can veto Congress, then I am the U.S. President”(as is in fact true). More to the point, the validity of the new argument stems not from denying the antecedent, but modus tollens (denying the consequent).

31.1 See also

• Affirming the consequent • Modus ponens

• Modus tollens • Necessary and sufficient conditions

31.2 References

[1] Turing, Alan (October 1950),“Computing Machinery and Intelligence”, Mind LIX (236): 433–460, doi:10.1093/mind/LIX.236.433, ISSN 0026-4423, retrieved 2008-08-18

31.3 External links

• FallacyFiles.org: Denying the Antecedent • safalra.com: Denying The Antecedent Chapter 32

Diagrammatic reasoning

Diagrammatic reasoning is reasoning by means of visual representations. The study of diagrammatic reasoning is about the understanding of concepts and ideas, visualized with the use of diagrams and imagery instead of by linguistic or algebraic means.

32.1 Related topics

32.1.1 Characteristica universalis

Characteristica universalis, commonly interpreted as universal characteristic, or universal character in English, is a universal and imagined by the German philosopher Gottfried Leibniz able to express mathematical, scientific, and metaphysical concepts. Leibniz thus hoped to create a language usable within the framework of a universal logical calculation or calculus ratiocinator. Since the characteristica universalis is diagrammatic and employs pictograms (below left), the diagrams in Leibniz's work warrant close study. On at least two occasions, Leibniz illustrated his philosophical reasoning with diagrams. One diagram, the frontispiece to his 1666 De Arte Combinatoria (On the Art of Combinations), represents the Aris- totelian theory of how all material things are formed from combinations of the elements earth, water, air, and fire. These four elements make up the four corners of a diamond (see picture to right). Opposing pairs of these are joined by a bar labeled “contraries”(earth-air, fire-water). At the four corners of the superimposed square are the four qualities defining the elements. Each adjacent pair of these is joined by a bar labeled “possible combination"; the diagonals joining them are labeled “impossible combination.”Starting from the top, fire is formed from the combination of dryness and heat; air from wetness and heat; water from coldness and wetness; earth from coldness and dryness.*[1]

32.1.2 Diagram

A diagram is a 2D geometric symbolic representation of information according to some visualization technique. Sometimes, the technique uses a 3D visualization which is then projected onto the 2D surface. The term diagram in common sense can have two meanings.

• visual information device: Like the term "illustration" the diagram is used as a collective term standing for the whole class of technical genres, including graphs, technical drawings and tables.*[2]

• specific kind of visual display: This is only the genre, that shows qualitative data with shapes that are connected by lines, arrows, or other visual links.

In science you will find the term used in both ways. For example Anderson (1997) stated more general “diagrams are pictorial, yet abstract, representations of information, and maps, line graphs, bar charts, engineering blueprints, and architects' sketches are all examples of diagrams, whereas photographs and video are not”.*[3] On the other

116 32.1. RELATED TOPICS 117

Leibniz's diagrammatic reasoning.

Basic elements of Leibniz's pictograms. hand Lowe (1993) defined diagrams as specifically“abstract graphic portrayals of the subject matter they represent” .*[4] In the specific sense diagrams and charts contrast computer graphics, technical illustrations, infographics, maps, and technical drawings, by showing “abstract rather than literal representations of information”.*[2] The of a diagram can be seen as:*[2]

• a form of visual formatting devices • a display that does not show quantitative data, but rather relationships and abstract information • with building blocks such as geometrical shapes that are connected by lines, arrows, or other visual links. 118 CHAPTER 32. DIAGRAMMATIC REASONING

Adding an article to Wikipedia

Search Wikipedia

Is it Is there a found? No related term? No Yes Yes

Think of Create Create another a new a term article redirect

Sample flowchart representing the decision process to add a new article to Wikipedia.

Or in Hall's (1996) words“diagrams are simplified figures, caricatures in a way, intended to convey essential meaning” .*[5] According to Jan V. White (1984) “the characteristics of a good diagram are elegance, clarity, ease, pattern, simplicity, and validity”.*[2] Elegance for White means that what you are seeing in the diagram is “the simplest and most fitting solution to a problem”.*[6]

32.1.3

A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. In his papers on qualitative logic, entitative graphs, and existential graphs, Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic. In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures.

Conceptual graph

A conceptual graph (CG) is a notation for logic based on the existential graphs of Charles Sanders Peirce and the semantic networks of artificial intelligence. In the first published paper on conceptual graphs, John F. Sowa used them to represent the conceptual schemas used in database systems. His first book*[7] applied them to a wide range 32.1. RELATED TOPICS 119

of topics in artificial intelligence, computer science, and cognitive science. A linear notation, called the Conceptual Graph Interchange Format (CGIF), has been standardized in the ISO standard for Common Logic.

CAT: Elsie Sitting MAT

agent agent

Elsie the cat is sitting on a mat

The diagram on the right is an example of the display form for a conceptual graph. Each box is called a concept node, and each oval is called a relation node. In CGIF, this CG would be represented by the following statement:

[Cat Elsie] [Sitting *x] [Mat *y] (agent ?x Elsie) (location ?x ?y)

In CGIF, brackets enclose the information inside the concept nodes, and parentheses enclose the information inside the relation nodes. The letters x and y, which are called coreference labels, show how the concept and relation nodes are connected. In the Common Logic Interchange Format (CLIF), those letters are mapped to variables, as in the following statement:

(exists ((x Sitting) (y Mat)) (and (Cat Elsie) (agent x Elsie) (location x y)))

As this example shows, the asterisks on the coreference labels *x and *y in CGIF map to existentially quantified variables in CLIF, and the question marks on ?x and ?y map to bound variables in CLIF. A universal quantifier, represented @every*z in CGIF, would be represented forall (z) in CLIF.

Entitative graph

An entitative graph is an element of the graphical syntax for logic that Charles Sanders Peirce developed under the name of qualitative logic beginning in the 1880s, taking the coverage of the formalism only as far as the propositional or sentential aspects of logic are concerned.*[8] The syntax is:

• The blank page; • Single letters, phrases; • Objects (subgraphs) enclosed by a simple closed curve called a cut. A cut can be empty.

The semantics are:

• The blank page denotes False; • Letters, phrases, subgraphs, and entire graphs can be True' or False; • To surround objects with a cut is equivalent to Boolean complementation. Hence an empty cut denotes Truth;

• All objects within a given cut are tacitly joined by disjunction.

A “proof”manipulates a graph, using a short list of rules, until the graph is reduced to an empty cut or the blank page. A graph that can be so reduced is what is now called a (or the complement thereof). Graphs that cannot be simplified beyond a certain point are analogues of the satisfiable formulas of first-order logic. 120 CHAPTER 32. DIAGRAMMATIC REASONING

Existential graph

An is a type of diagrammatic or visual notation for logical expressions, proposed by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914. Peirce proposed three systems of existential graphs:

• alpha – isomorphic to sentential logic and the two-element Boolean algebra;

• beta – isomorphic to first-order logic with identity, with all formulas closed;

• gamma – (nearly) isomorphic to normal modal logic.

Alpha nests in beta and gamma. Beta does not nest in gamma, quantified modal logic being more than even Peirce could envisage.

PQ P

P Q PQ

PQ

Alpha Graphs

In alpha the syntax is:

• The blank page;

• Single letters or phrases written anywhere on the page;

• Any graph may be enclosed by a simple closed curve called a cut or sep. A cut can be empty. Cuts can nest and concatenate at will, but must never intersect. 32.2. SEE ALSO 121

Any well-formed part of a graph is a subgraph. The semantics are:

• The blank page denotes Truth;

• Letters, phrases, subgraphs, and entire graphs may be True or False;

• To enclose a subgraph with a cut is equivalent to logical negation or Boolean complementation. Hence an empty cut denotes False;

• All subgraphs within a given cut are tacitly conjoined.

Hence the alpha graphs are a minimalist notation for sentential logic, grounded in the expressive adequacy of And and Not. The alpha graphs constitute a radical simplification of the two-element Boolean algebra and the truth functors.

32.1.4 The Venn-II reasoning system

In the early 1990s Sun-Joo Shin presented an extension of Existential Graphs called Venn-II.*[9] Syntax and semantics are given formally, together with a set of Rules of Transformation which are shown to be sound and complete. Proofs proceed by applying the rules (which remove or add syntactic elements to or from diagrams) sequentially. Venn-II is equivalent in expressive power to a first-order monadic language.

32.2 See also

32.3 References

[1] This diagram is reproduced in several texts including Saemtliche Schriften und Briefe, Reihe VI, Band 1: 166, Loemker 1969: 83, 366, Karl Popp and Erwin Stein 2000: 33.

[2] Brasseur, Lee E. (2003). Visualizing technical information: a cultural critique. Amityville, N.Y.: Baywood Pub. ISBN 0-89503-240-6.

[3] Michael Anderson (1997). “Introduction to Diagrammatic Reasoning”. Retrieved 21 July 2008.

[4] Lowe, Richard K. (1993).“Diagrammatic information: techniques for exploring its and processing” . Information Design Journal 7 (1): 3–18. doi:10.1075/idj.7.1.01low.

[5] Bert S. Hall (1996). “The Didactic and the Elegant: Some Thoughts on Scientific and Technological Illustrations in the Middle Ages and Renaissance”. in: B. Braigie (ed.) Picturing knowledge: historical and philosophical problems concerning the use of art in science. Toronto: Press. p.9

[6] White, Jan V. (1984). Using charts and graphs: 1000 ideas for visual persuasion. New York: Bowker. ISBN 0-8352- 1894-5.

[7] John F. Sowa (1984). Conceptual Structures: Information Processing in Mind and Machine. Addison-Wesley, Reading, MA, 1984.

[8] See 3.468, 4.434, and 4.564 in Peirce's Collected Papers.

[9] Shin, Sun-Joo. 1994. The Logical Status of Diagrams. Cambridge: Cambridge University Press.

32.4 Further reading

• Gerard Allwein and Jon Barwise (ed.) (1996). Logical Reasoning with Diagrams. Oxford University Press.

• Michael Anderson, Peter Cheng, Volker Haarslev (Eds.) (2000). Theory and Application of Diagrams: First International Conference, Diagrams 2000. Edinburgh, , UK, September 1–3, 2000. Proceedings. 122 CHAPTER 32. DIAGRAMMATIC REASONING

• Micheal Anderson and R. McCartney (2003). Diagram Processing: Computing with Diagrams. In: Artificial Intelligence, Volume 145 , Issue 1-2 , April, 2003. • James Robert Brown (1999). Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures. Routledge. • Janice Glasgow, N. Hari Narayanan, and B. Chandrasekaran (ed) (1995). Diagrammatic Reasoning: Cognitive and Computational Perspectives. AAAI Press. • Kulpa, Zenon."Diagrammatic representation and reasoning.”Machine GRAPHICS & VISION 3 (1/2. 1994.

• Gem Stapleton A Survey of Reasoning Systems Based on Euler Diagrams. Electronic Notes in Theoretical Computer Science. 2005.

32.5 External links

• Diagrammatic Reasoning Site from the University of Hartford,Connecticut, USA

• Lecture about Universal Algebra and Diagrammatic Reasoning by John Baez, 3 Feb 2006. • Homepage of Sun-Joo Shin.

• Visual Modelling Group at the University of Brighton, UK. Chapter 33

Dialectica space

Dialectica spaces are a categorical way of constructing models of linear logic. They were introduced by , Martin Hyland's student, in her doctoral thesis, as a way of modeling both linear logic and Gödel's dialectica interpretation—hence the name. Given a category C and a specific object K of C with certain (logical) properties, one can construct the category of Dialectica spaces over C, whose objects are pairs of objects of C, related by a C-morphism into the given object. Morphisms of Dialectica spaces are similar to Chu space morphisms, but instead of an equality condition, they have an condition, which is read as a logical implication, the first object implies the second.

33.1 References

• K. Gödel. “Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes - Dialectica”, 1958. (Translation and analysis in Collected Works, Vol II, Publications, 1937-1974—eds S. Feferman et al., 1990).

• V. de Paiva. “The Dialectica Categories”. In Proc. of Categories in Computer Science and Logic, Boulder, CO, 1987. Contemporary Mathematics, vol 92, American Mathematical Society, 1989 (eds. J. Gray and A. Scedrov)

• V. de Paiva. “A dialectica-like model of linear logic”. In Proc. Conf. on and Computer Science, Springer-Verlag Lecture Notes in Computer Science 389, pp. 341–356, Manchester, September 1989.

123 Chapter 34

Diamond 25

Diamond 25 is a mathematical puzzle in which one must use the digits '1' through '9' to get 25, within each diamond composed of 6 pieces without repeating the same digit or summing more than or less than 25. There are 16 diamonds within the larger diamond.

34.1 See also

• 36 cube

• Algorithmics of Sudoku • Blendoku

• Constraint satisfaction problem • Futoshiki

• Hidato • Kakuro

• KenKen • List of Nikoli puzzle types

• List of Sudoku terms and jargon

• Logic puzzle • Mathematics of Sudoku

• Nonogram (aka Paint by numbers, O'ekaki) • Str8ts sequential numbers

34.2 Further reading

• Papp International Inc, , Brain Twisters 2015.

124 Chapter 35

Dichotomy

This article is about in logic and related topics. For usage of “dichotomous”in botany, see Glossary of botanical terms. A dichotomy is a partition of a whole (or a set) into two parts (subsets) that are: U A

AC

In this image, the U (the entire rectangle) is dichotomized into the two sets A (in pink) and its complement A*c (in gray).

• jointly exhaustive: everything must belong to one part or the other, and • mutually exclusive: nothing can belong simultaneously to both parts.

Such a partition is also frequently called a bipartition. The two parts thus formed are complements. In logic, the partitions are opposites if there exists a proposition such that it holds over one and not the other. Treating continuous variables or multicategorical variables as binary variables is called dichotomization. The discretization error inherent in dichotomization is temporarily ignored for modeling purposes.

125 126 CHAPTER 35. DICHOTOMY

35.1 Etymology

The term dichotomy is from the Greek language διχοτομία dichotomía“dividing in two”from δίχα dícha“in two, asunder”and τομή tomḗ “a cutting, incision”.

35.2 Usage and examples

• The above applies directly when the term is used in mathematics, philosophy, literature, or linguistics. For example, if there is a concept A, and it is split into parts B and not-B, then the parts form a dichotomy: they are mutually exclusive, since no part of B is contained in not-B and vice versa, and they are jointly exhaustive, since they cover all of A, and together again give A.

• In set theory, a dichotomous relation R is such that either aRb, bRa, but not both.*[1]

• In statistics, dichotomous data may only exist at first two levels of measurement, namely at the nominal level of measurement (such as “British”vs “American”when measuring nationality) and at the ordinal level of measurement (such as “tall”vs “short”, when measuring height). A variable measured dichotomously is called a dummy variable. If a dependent variable in a regression is dichotomous, then logistic regression or probit regression is employed.

• In the classification of mental disorders in psychiatry or clinical psychology, dichotomous classification or refers to the use of cut-offs intended to separate disorder from non-disorder at some level of abnormality, severity or disability

• A false dichotomy is an consisting of a supposed dichotomy which fails one or both of the conditions: it is not jointly exhaustive and/or not mutually exclusive. In its most common form, two entities are presented as if they are exhaustive, when in fact other alternatives are possible. In some cases, they may be presented as if they are mutually exclusive although there is a broad middle ground (see also undistributed middle).

• The divine dichotomy is mentioned in the Conversations With God series of books by religious author Neale Donald Walsch

• In economics, the classical dichotomy is the division between the real side of the economy and the monetary side. According to the classical dichotomy, changes in monetary variables do not affect real values such as output, employment, and the real interest rate. Money is therefore neutral in the sense that its quantity cannot affect these real variables.

• In biology, a dichotomy is a division of organisms into two groups, typically based on a characteristic present in one group and absent in the other. Such dichotomies are used as part of the process of identifying species, as part of a dichotomous key, which asks a series of questions, each of which narrows down the set of organisms. A well known dichotomy is the question “does it have a backbone?" used to divide species into vertebrates and invertebrates.

• In botany, a dichotomy is a mode of branching by repeated bifurcation - thus a focus on branching rather than on division

• In computer science, more specifically in programming-language engineering, the term dichotomy can denote fundamental dualities in a language's design. For instance, C++ has a dichotomy in its memory model (heap versus stack), whereas Java has a dichotomy in its type system (references versus primitive data types).

• In the anthropological field of theology and in philosophy, dichotomy is the belief that humans consist of a soul and a body. (See Mind-body dichotomy.) This stands in contrast to trichotomy.

• Perceived dichotomies are common in Western thought. C. P. Snow believes that Western society has become an argument culture (The Two Cultures). In The Argument Culture (1998), Deborah Tannen suggests that the dialogue of Western culture is characterized by a warlike atmosphere in which the winning side has truth (like a trophy). Such a dialogue virtually ignores the middle alternatives. 35.3. SEE ALSO 127

• In sociology and semiotics, dichotomies (also sometimes called 'binaries' and/or 'binarisms') are the subject of attention because they may form the basis to divisions and inequality. For example, the domestic–public dichotomy divides men's and women's roles in a society; the East-West dichotomy contrasts the Orient and the Occident. Some social scientists attempt to deconstruct dichotomies in order to address the divisions and in- equalities they create: for instance Judith Butler's of the gender-dichotomy and Val Plumwood's deconstruction of the human-environment dichotomy. • The I Ching and taijitu represent the yin yang theories of traditional Chinese culture. However, these do not represent a true dichotomy as the symbol incorporates a portion of each in the other, representing a dialectic. • In dialectical behavioral therapy, a treatment shown to have some success in treating some clients with Borderline Personality Disorder, an essential tool used is based on the idea of dichotomy. Dichotomy, in this case, is a self-defeating behavior using“all-or-nothing”or“black-and-white”thinking. The therapy teaches the patient how to change the dichotomy to a more "dialectical" (or “seeing the middle ground”) way of thinking. • One type of dichotomy is dichotomous classification - classifying objects by recursively splitting them into two groups until all are separated and in their own unique category.

• Astronomy defines a dichotomy as “the phase of the moon or an inferior planet in which half its disk appears illuminated”

35.3 See also

• Binary opposition • Bipartite (disambiguation)

• Borderline personality disorder • Class (set theory)

• Dialectical behavior therapy • Dichotomy paradox

• Dilemma • Dualism

• Law of excluded middle, which in logic asserts the existence of a dichotomy • Polychotomy

• Trichotomy (disambiguation)

• Yin and yang

35.4 Notes and references

[1] Komjath, Peter; Totik, Vilmos (2006). Problems and Theorems in Classical Set Theory. Google Books (Springer Science & Business Media). p. 497. Retrieved 17 September 2014.

35.5 External links

• The dictionary definition of dichotomy at Wiktionary Chapter 36

Difference (philosophy)

Difference is a key concept of philosophy, denoting the process or set of properties by which one entity is distin- guished from another within a relational field or a given . In the Western philosophical system, difference is traditionally viewed as being opposed to identity, following the Principles of Leibniz, and in particular, his Law of the identity of indiscernibles. In structuralist and poststructuralist accounts, however, difference is under- stood to be constitutive of both meaning and identity. In other words, because identity (particularly, but not limited to, personal identity) is viewed in non-essentialist terms as a construct, and because constructs only produce mean- ing through the interplay of differences (see below), it is the case that for both and poststructuralism, identity cannot be said to exist without difference.

36.1 Difference in Leibniz's Law

Gottfried Leibniz's Principle of the identity of indiscernibles states that two things are identical if and only if they share the same and only the same properties. This is a principle which defines identity rather than difference, although it established the tradition in logic and analytical philosophy of conceiving of identity and difference as oppositional.

36.2 Kant's critique

In his , argues that it is necessary to distinguish between the thing in itself and its appearance. Even if two objects have completely the same properties, if they are at two different places at the same time, they are numerically different:

Identity and Difference.—[...] Thus, in the case of two drops of water, we may make complete ab- straction of all internal difference (quality and quantity), and, the fact that they are intuited at the same time in different places, is sufficient to justify us in holding them to be numerically different. Leibnitz [sic] regarded phaenomena as things in themselves, consequently as intelligibilia, that is, objects of pure understanding [...], and in this case his principle of the indiscernible (principium identatis indiscernibil- ium) is not to be impugned. But, as phaenomena are objects of sensibility, and, as the understanding, in respect of them, must be employed empirically and not purely or transcendentally, plurality and numer- ical difference are given by space itself as the condition of external phaenomena. For one part of space, although it may be perfectly similar and equal to another part, is still without it, and for this reason alone is different from the latter [...]. It follows that this must hold good of all things that are in the different parts of space at the same time, however similar and equal one may be to another.*[1]

36.3 Difference in structuralism

Structural linguistics, and subsequently structuralism proper, are founded on the idea that meaning can only be pro- duced differentially in signifying systems (such as language). This concept first came to prominence in the structuralist

128 36.4. AND DIFFÉRANCE IN POSTSTRUCTURALISM 129

writings of Swiss linguist and was developed for the analysis of social and mental structures by French anthropologist Claude Lévi-Strauss. The former was concerned to question the prevailing view of meaning“inhering”in words, or the idea that language is a bearing a one-to-one correspondence to the real. Instead, Saussure argues that meaning arises through differentiation of one sign from another, or even of one phoneme from another:

In language there are only differences. Even more important: a difference generally implies positive terms between which the difference is set up; but in language there are only differences without positive terms. Whether we take the signified or the signifier, language has neither ideas nor sounds that existed before the linguistic system, but only conceptual and phonic differences that have issued from the system. The idea or phonic substance that a sign contains is of less importance than the other signs that surround it. [...] A linguistic system is a series of differences of sound combined with a series of differences of ideas; but the pairing of a certain number of acoustical signs with as many cuts made from the mass thought engenders a system of values.*[2]

In his Structural Anthropology, Claude Lévi-Strauss applied this concept to the anthropological study of mental struc- tures, kinship and belief systems, examining the way in which social meaning emerges through a series of structural oppositions between paired/opposed kinship groups, for example, or between basic oppositional categories (such as friend and enemy, life and death, or in a later volume, the raw and the cooked).*[3]*[4]

36.4 Difference and différance in poststructuralism

The French philosopher both extended and profoundly critiqued structuralist thought on the processes by which meaning is produced through the interplay of difference in language, and in particular, writing. Whereas structuralist linguistics had recognized that meaning is differential, much structuralist thought, such as narratology, had become too focused on identifying and producing a typology of the fixed differential structures and binary oppositions at work in any given system. In his work, Derrida sought to show how the differences on which any signifying system depends are not fixed, but get caught up and entangled with each other. Writing itself becomes the prototype of this process of entanglement, and in (1967) and “Différance”(in Margins of Philosophy, 1972) Derrida shows how the concept of writing (as the paradoxical absence or de-presencing of the living voice) has been subordinated to the desired “full presence”of speech within the Western philosophical tradition.*[5]*[6] His early thought on the relationship between writing and difference is collected in his book of essays entitled Writing and Difference (1967).*[7] Elsewhere, Derrida coined the term différance (a deliberate misspelling of différence) in order to provide a conceptual hook for his thinking on the meaning processes at work within writing/language.*[6] This neologism is a clever play on the two meanings of the French word différer: to differ and to defer. Derrida thereby argues that meaning does not arise out of fixed differences between static elements in a structure, but that the meanings produced in language and other signifying systems are always partial, provisional and infinitely deferred along a chain of differing/deferring signifiers. At the same time, the word différance itself performs this entanglement and confusion of differential meanings, for it depends on a minimal difference (the substitution of the letter“a”for the letter“e”) which cannot be apprehended in oral speech, since the suffixes "-ance”and "-ence”have the same pronunciation in French. The “phonemic”(non-)difference between différence and différance can only be observed in writing, hence producing differential meaning only in a partial, deferred and entangled manner. Différance has been defined as“the non-originary, constituting-disruption of presence": spatially, it differs, creating spaces, ruptures, and differences and temporally, it defers, delaying presence from ever being fully attained.*[8] Derrida's criticism of essentialist ontology draws on the differential ontology of (who introduced the concept of Verschiedenheit“, difference”, in his unpublished manuscripts (KSA 11:35[58], p. 537)) and Emmanuel Levinas (who proposed an ethics of the Other).*[6]*[9] In a similar vein, 's Difference and Repetition (1968) was an attempt to think difference as having an ontological privilege over identity, inverting the traditional relationship between those two concepts and implying that identities are only produced through processes of differentiation. 130 CHAPTER 36. DIFFERENCE (PHILOSOPHY)

36.5 See also

• Deconstruction

36.6 References

[1] Kant, Immanuel (1781 [trans. 1855]). Critique of Pure Reason. Trans. by J. M. D. Meiklejohn. London: Henry G. Bohn. p. 191. Check date values in: |date= (help)

[2] Saussure, Ferdinand de (1916 [trans. 1959]). Course in General Linguistics. New York: New York Philosophical Library. pp. 121–22. Check date values in: |date= (help)

[3] Lévi-Strauss, Claude (1958 [trans. 1963]). Structural Anthropology. London: Allen Lane. Check date values in: |date= (help)

[4] Lévi-Strauss, Claude (1964 [trans. 1970]). The Raw and the Cooked. London: Cape. Check date values in: |date= (help)

[5] Derrida, Jacques (1967 [trans. 1976]). Of Grammatology. Johns Hopkins University Press. Check date values in: |date= (help)

[6] Derrida, Jacques (1972 [trans. 1982]). Margins of Philosophy. University of Chicago Press. pp. 3–27. Check date values in: |date= (help)

[7] Derrida, Jacques (1967 [trans. 1978]). Writing and Difference. London: Routledge and Kegan Paul. Check date values in: |date= (help)

[8] “Differential Ontology” at the Internet Encyclopedia of Philosophy

[9] Douglas L. Donkel, The Theory of Difference: Readings in Contemporary Continental Thought, SUNY Press, 2001, p. 295. Chapter 37

Digital timing diagram

A digital timing diagram is a representation of a set of signals in the time domain. A timing diagram can contain many rows, usually one of them being the clock. It is a tool that is ubiquitous in , hardware debug- ging, and digital communications. Besides providing an overall description of the timing relationships, the digital timing diagram can help find and diagnose digital logic hazards.

37.1 Diagram convention

Most timing diagrams use the following conventions:

• Higher value is a logic one

• Lower value is a logic zero

• A slot showing a high and low is an either or (such as on a data line)

• A Z indicates high impedance

• A greyed out slot is a don't-care or indeterminate.

37.2 Example: SPI bus timing

The timing diagram example on the right describes the Serial Peripheral Interface (SPI) Bus. Most SPI master nodes have the ability to set the clock polarity (CPOL) and clock phase (CPHA) with respect to the data. This timing diagram shows the clock for both values of CPOL and the values for the two data lines (MISO & MOSI) for each value of CPHA. Note that when CPHA=1 then the data is delayed by one-half clock cycle. SPI operates in the following way:

• The master determines an appropriate CPOL & CPHA value

• The master pulls down the slave select (SS) line for a specific slave chip

• The master clocks SCK at a specific frequency

• During each of the 8 clock cycles the transfer is full duplex:

• The master writes on the MOSI line and reads the MISO line • The slave writes on the MISO line and reads the MOSI line

• When finished the master can continue with another byte transfer or pull SS high to end the transfer

131 132 CHAPTER 37. DIGITAL TIMING DIAGRAM

CPOL=0 SCK CPOL=1 SS

Cycle # 1 2 3 4 5 6 7 8 CPHA=0 MISO z 1 2 3 4 5 6 7 8 z MOSI z 1 2 3 4 5 6 7 8 z

Cycle # 1 2 3 4 5 6 7 8 CPHA=1 MISO z 1 2 3 4 5 6 7 8 z MOSI z 1 2 3 4 5 6 7 8 z

A timing diagram for the Serial Peripheral Interface Bus

When a slave's SS line is high then both of its MISO and MOSI line should be high impedance so to avoid disrupting a transfer to a different slave. Prior to SS being pulled low, the MISO & MOSI lines are indicated with a “z”for high impedance. Also, prior to the SS being pulled low the “cycle #" row is meaningless and is shown greyed-out. Note that for CPHA=1 the MISO & MOSI lines are undefined until after the first clock edge and are also shown greyed-out before that. A more typical timing diagram has just a single clock and numerous data lines

37.3 External Links

• Wavedrom is an online timing diagram editor.

• Waves has Windows binary. Chapter 38

Don't-care term

In digital logic, a don't-care term for a function is an input-sequence (a series of bits) for which the function output does not matter. An input that is known never to occur is a can't-happen term. Both these types of conditions are treated the same way in logic design and may be referred to collectively as don't-care conditions for brevity.*[1] The designer of a logic circuit to implement the function need not care about such inputs, but can choose the circuit's output arbitrarily, usually such that the simplest circuit results (minimization). Examples of don't-care terms are the binary values 1010 through 1111 (10 through 15 in decimal) for a function that takes a binary-coded decimal (BCD) value, because a BCD value never takes on such values (so called pseudo-tetrades); in the pictures, the circuit computing the lower left bar of a 7-segment display can be minimized to a b + a c by an appropriate choice of circuit outputs for dcba=1010...1111. Don't-care terms are important to consider in minimizing logic circuit design, using Karnaugh maps and the Quine– McCluskey algorithm. Don't care optimization can also be used in the development of highly size-optimized assembly or machine code taking advantage of side effects.

38.1 X value

“Don't care”may also refer to an unknown value in a multi-valued logic system, in which case it may also be called an X value. In the Verilog hardware description language such values are denoted by the letter“X”. In the VHDL hardware description language such values are denoted (in the standard logic package) by the letter “X”(forced unknown) or the letter “W”(weak unknown).*[2] An X value does not exist in hardware. In simulation, an X value can result from two or more sources driving a signal simultaneously, or the stable output of a flip-flop (electronics) not having been reached. In synthesized hardware, however, the actual value of such a signal will be either 0 or 1, but will not be determinable from the circuit's inputs.*[2]

38.2 See also

• Decision table

38.3 References

[1] J.A. Strong, Basic Digital Electronics, pp. 28-29, Springer, 2013 ISBN 940113118X.

[2] David Naylor and Simon Jones (1997). Vhdl: A Approach. Springer. pp. 14–15,219,221. ISBN 0-412- 61650-5.

133 Chapter 39

Embedded dependency

In relational database theory, an embedded dependency (ED) is a certain kind of constraint on a relational database. It is the most general type of constraint used in practice, including both tuple-generating dependencies (TGDs) and equality-generating dependencies (EGDs). EDs can express functional dependencies, join dependencies, multi-valued dependencies, inclusion dependencies, foreign key dependencies, and many more besides. An ED is a sentence in first-order logic of the form: ∀x1 ... xn, P(x1, ..., xn) → ∃z1, ..., zk, Q(y1, ..., ym) where {z1, ..., zk} = {y1, ..., ym} \ {x1, ..., xn}, and P is a possibly empty and Q is a non-empty conjunction of relational and equality atoms. A relational atom has the form R(w1, ..., wh) and an equality atom has the form wi = wj where each of the w, ..., wh, wi, wj, are variables or constants. When all atoms in Q are equalities, the ED is an EGD, and when all atoms in Q are relational, the ED is a TGD. Every ED is equivalent to an EGD and a TGD. An algorithm known as the chase takes as input an instance that may or may not satisfy a set of EDs, and, if it terminates (which is a priori undecidable), output an instance that does satisfy the EDs.

39.1 References

• Serge Abiteboul, Richard B. Hull, Victor Vianu: Foundations of Databases. Addison-Wesley, 1995.

• Alin Deutsch, FOL Modeling of Integrity Constraints, http://db.ucsd.edu/pubsFileFolder/305.pdf

134 Chapter 40

Empty name

In the philosophy of language, an is a that has no . The problem of empty names is that empty names have a meaning that it seems they shouldn't have. The name "Pegasus" is empty; there is nothing to which it refers. Yet, though there is no Pegasus, we know what the sentence “Pegasus has two wings”means. We can even understand the sentence “There is no such thing as Pegasus.”But, what can the meaning of a proper name be, except the object to which it refers? There are three broad ways which philosophers have tried to approach this problem.

1. The meaning of a proper name is not the same as the object (if there is any) it refers to. Hence, though “Pegasus”refers to nothing, it still has a meaning. The German philosopher Gottlob Frege seems to have held a theory of this sort. He says that the sentence “Odysseus was set ashore at Ithaca while sound asleep” obviously has a sense. ... the thought [expressed by that sentence] remains the same whether “Odysseus” has reference or not.”Bertrand Russell may also have held a similar theory, that a proper name is a disguised definite description that signifies some unique characteristic. If any object has this characteristic feature, the name has a referent. Otherwise it is empty. Perhaps "Aristotle" means“the teacher of Alexander". Since there was such a person,“Aristotle”refers to that person. By contrast,“Pegasus”may mean“the winged horse of Bellerophon". Since there was no such horse, the name has no referent. This is the so-called description theory of names. The difficulty with this account is that we may always use a proper name to deny that the individual bearing the name actually has some characteristic feature. So, we can meaningfully say that Aristotle was not the teacher of Alexander. But if“Aristotle”means“teacher of Alexander”, it would follow that this assertion is self-contradictory, which it is not. proposed this argument in a series of influential papers in the 1970s. Another difficulty is that different people may have different ideas about the defining characteristics of any individual. Yet we all understand what the name means. The sole information carried by the name seems to be the identity of the individual that it belongs to. This information therefore cannot be descriptive, it cannot describe the individual. As John Stuart Mill argued, a proper name tells us the identity of its bearer, without telling us anything else about it. Naming is rather like pointing. 2. A theory that became influential following Kripke's attack is that empty proper names, have, strictly speaking, no meaning. This is the so-called direct-reference theory. Versions of this theory have been defended by , David Kaplan, , and others. The problem with the direct-reference theory is that names appear to be meaningful independently of whether they are empty. Furthermore, negative existential statements using empty names are both true and apparently meaningful. How can “Pegasus does not exist”be true if the name “Pegasus”, as used in that sentence, has no meaning? 3. There are no empty names. All names have a referent. The difficulty with this theory is how to distinguish names like“Pegasus”from names like“Aristotle”. Any coherent account of this distinction seems to require that there are, i.e. there exist, objects that do not exist. Given that “Pegasus does not exist”is true, it follows that the referent of “Pegasus”does not exist. Hence there is something —the referent of “Pegasus”—that does not exist.

Some philosophers, such as Alexius Meinong have argued that there are two senses of the verb“exists”, exemplified by the sentence“there are things that do not exist”. The first, signified by“there are”, is the so-called“wide sense” , including Pegasus, the golden mountain, the round square &c. The second, signified by “exist”is the so-called

135 136 CHAPTER 40. EMPTY NAME

“narrow sense”, encompassing only things that are real or existent. The difficulty with this “two sense”theory is that there is no strong evidence that there really are two such distinct senses of the verb “to be”.

40.1 See also

• Meinong's jungle

• Ontological commitment

• Meta-ontology Chapter 41

Enumerative definition

An enumerative definition of a concept or term is a special type of extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question. Enumerative definitions are only possible for finite sets and only practical for relatively small sets. An example of an enumerative definition would be, [types of Car brands] Ford, Chevrolet, Volkswagen, Toyota, etc...

41.1 See also

• Definition • Extension

• Extensional definition • Set notation

• Enumeration

137 Chapter 42

Epicureanism

“Epicurean”redirects here. For other uses, see Epicurean (disambiguation). is a system of philosophy based upon the teachings of the philosopher , founded around 307 BC. Epicurus was an atomic materialist, following in the steps of . His led him to a general attack on superstition and divine intervention. Following Aristippus—about whom very little is known—Epicurus believed that what he called “”is the greatest good, but the way to attain such pleasure is to live modestly and to gain knowledge of the workings of the world and the limits of one's desires. This led one to attain a state of tranquility () and freedom from fear, as well as absence of bodily pain (). The combination of these two states is supposed to constitute happiness in its highest form. Although Epicureanism is a form of , insofar as it declares pleasure to be the sole intrinsic good, its conception of absence of pain as the greatest pleasure and its advocacy of a simple life makes it different from“hedonism”as it is commonly understood. Epicureanism was originally a challenge to , though later it became the main opponent of . Epi- curus and his followers shunned politics. After the death of Epicurus, his school was headed by ; later many Epicurean societies flourished in the Late Hellenistic era and during the Roman era (such as those in Antiochia, Alexandria, Rhodes, and Ercolano). Its best-known Roman proponent was the poet . By the end of the Ro- man Empire, being opposed by (mainly Neo-Platonism) that were now in the ascendance, Epicureanism had all but died out, and would be resurrected in the 17th century by the atomist , who adapted it to the Christian doctrine. Some writings by Epicurus have survived. Some scholars consider the epic poem On the Nature of Things by Lucretius to present in one unified work the core arguments and theories of Epicureanism. Many of the papyrus scrolls un- earthed at the Villa of the Papyri at Herculaneum are Epicurean texts. At least some are thought to have belonged to the Epicurean .

42.1 History

The school of Epicurus, called “The Garden,”was based in Epicurus' home and garden. It had a small but devoted following in his lifetime. Its members included Hermarchus, Idomeneus, , Polyaenus, and Metrodorus. Epi- curus emphasized friendship as an important ingredient of happiness, and the school seems to have been a moderately ascetic community which rejected the political limelight of Athenian philosophy. They were fairly cosmopolitan by Athenian standards, including women and slaves. Some members were also vegetarians as Epicurus did not eat meat, although no prohibition against eating meat was made.*[1]*[2] The school's popularity grew and it became, along with Stoicism and Skepticism, one of the three dominant schools of , lasting strongly through the later Roman Empire.*[3] Another major source of information is the Roman politician and philosopher Cicero, although he was highly critical, denouncing the Epicureans as unbridled hedonists, devoid of a sense of and duty, and guilty of withdrawing from public life. Another ancient source is Diogenes of , who composed a large inscription at Oenoanda in Lycia. A library in the Villa of the Papyri, in Herculaneum, was perhaps owned by Julius Caesar's father-in-law, Lucius Calpurnius Piso Caesoninus. The scrolls which the library consisted of were preserved albeit in carbonized form by the eruption of Vesuvius in 79 AD. Several of these Herculaneum papyri which are unrolled and deciphered were found to contain a large number of works by Philodemus, a late Hellenistic Epicurean, and Epicurus himself,

138 42.2. RELIGION 139

attesting to the school's enduring popularity. The task of unrolling and deciphering the over 1800 charred papyrus scrolls continues today. With the dominance of the Neo-Platonism and philosophy (and later ), Epicureanism declined. By the late third century AD, there was very little trace of its existence.*[4] The early Christian writer Lactantius criticizes Epicurus at several points throughout his Divine Institutes. In Dante's Divine Comedy, the Epicureans are depicted as heretics suffering in the sixth circle of hell. In fact, Epicurus appears .(אפיקורוס)”to represent the ultimate heresy. The word for a heretic in the Talmudic literature is “Apiqoros By the 16th century, the works of Diogenes Laertius were being printed in . In the 17th century the French Franciscan priest, scientist and philosopher Pierre Gassendi wrote two books forcefully reviving Epicureanism. Shortly thereafter, and clearly influenced by Gassendi, Walter Charleton published several works on Epicureanism in English. Attacks by Christians continued, most forcefully by the Cambridge Platonists. In the Modern Age, scientists adopted atomist theories, while materialist philosophers embraced Epicurus' hedonist ethics and restated his objections to natural teleology.

42.2 Religion

Epicureanism emphasizes the neutrality of the , that they do not interfere with human lives. It states that gods, matter, and are all made up of atoms. Souls are made from atoms, and gods possess souls, but their souls adhere to their bodies without escaping. Humans have the same kind of souls, but the forces binding human atoms together do not hold the soul forever. The Epicureans also used the atomist theories of Democritus and to assert that man has . They held that all thoughts are merely atoms swerving randomly. The Riddle of Epicurus, or , is a famous argument against the existence of an all-powerful and provi- dential God or gods. As recorded by Lactantius:

God either wants to eliminate bad things and cannot, or can but does not want to, or neither wishes to nor can, or both wants to and can. If he wants to and cannot, then he is weak – and this does not apply to god. If he can but does not want to, then he is spiteful – which is equally foreign to god's nature. If he neither wants to nor can, he is both weak and spiteful, and so not a god. If he wants to and can, which is the only thing fitting for a god, where then do bad things come from? Or why does he not eliminate them? —Lactantius, De Ira Deorum*[5]

This type of trilemma argument (God is omnipotent, God is good, but Evil exists) was one favoured by the an- cient Greek skeptics, and this argument may have been wrongly attributed to Epicurus by Lactantius, who, from his Christian perspective, regarded Epicurus as an atheist.*[6] According to Reinhold F. Glei, it is settled that the argument of is from an academical source which is not only not Epicurean, but even anti-Epicurean.*[7] The earliest extant version of this trilemma appears in the writings of the skeptic .*[8] Epicurus' view was that there were gods, but that they were neither willing nor able to prevent evil. This was not because they were malevolent, but because they lived in a perfect state of ataraxia, a state everyone should strive to emulate; it is not the gods who are upset by evils, but people.*[6] Epicurus conceived the gods as blissful and immortal yet material beings made of atoms inhabiting the metakosmia: empty spaces between worlds in the vastness of infinite space. In spite of his recognition of the gods, the practical effect of this materialistic explanation of the gods' existence and their complete non-intervention in human affairs renders his philosophy akin in divine effects to the attitude of . In Dante's Divine Comedy, the flaming tombs of the Epicureans are located within the sixth circle of hell (Inferno, Canto X). They are the first heretics seen and appear to represent the ultimate, if not quintessential, heresy.*[9] Similarly, according to Jewish Mishnah, Epicureans (apiqorsim, people who share the beliefs of the movement) are among the people who do not have a share of the “World-to-Come”( or the world of the Messianic era). Parallels may be drawn to Buddhism, which similarly emphasizes a lack of divine interference and aspects of its . Buddhism also resembles Epicureanism in its temperateness, including the belief that great excess leads to great dissatisfaction. 140 CHAPTER 42. EPICUREANISM

42.3 Philosophy

The philosophy originated by Epicurus flourished for seven centuries. It propounded an ethic of individual pleasure as the sole or chief good in life. Hence, Epicurus advocated living in such a way as to derive the greatest amount of plea- sure possible during one's lifetime, yet doing so moderately in order to avoid the suffering incurred by overindulgence in such pleasure. The emphasis was placed on of the mind rather than on physical pleasures. Therefore, according to Epicurus, with whom a person eats is of greater importance than what is eaten. Unnecessary and, es- pecially, artificially produced desires were to be suppressed. Since learning, culture, and civilization as well as social and political involvements could give rise to desires that are difficult to satisfy and thus result in disturbing one's peace of mind, they were discouraged. Knowledge was sought only to rid oneself of religious fears and superstitions, the two primary fears to be eliminated being fear of the gods and of death. Viewing marriage and what attends it as a threat to one's peace of mind, Epicurus lived a celibate life but did not impose this restriction on his followers. The philosophy was characterized by an absence of divine principle. Lawbreaking was counseled against because of both the shame associated with detection and the punishment it might bring. Living in fear of being found out or punished would take away from pleasure, and this made even secret wrongdoing inadvisable. To the Epicureans, virtue in itself had no value and was beneficial only when it served as a means to gain happiness. Reciprocity was recommended, not because it was divinely ordered or innately noble, but because it was personally beneficial. Friend- ships rested on the same mutual basis, that is, the pleasure resulting to the possessors. Epicurus laid great emphasis on developing friendships as the basis of a satisfying life.

of all the things which wisdom has contrived which contribute to a blessed life, none is more impor- tant, more fruitful, than friendship —quoted by Cicero*[10]

While the pursuit of pleasure formed the focal point of the philosophy, this was largely directed to the “static pleasures”of minimizing pain, anxiety and suffering. In fact, Epicurus referred to life as a “bitter gift”.

When we say . . . that pleasure is the end and aim, we do not mean the pleasures of the prodigal or the pleasures of sensuality, as we are understood to do by some through ignorance, prejudice or wilful misrepresentation. By pleasure we mean the absence of pain in the body and of trouble in the soul. It is not by an unbroken succession of drinking bouts and of revelry, not by sexual lust, nor the enjoyment of fish and other delicacies of a luxurious table, which produce a pleasant life; it is sober reasoning, searching out the grounds of every choice and avoidance, and banishing those beliefs through which the greatest tumults take possession of the soul. —Epicurus, “Letter to Menoeceus”*[11]

The Epicureans believed in the existence of the gods, but believed that the gods were made of atoms just like every- thing else. It was thought that the gods were too far away from the earth to have any interest in what man was doing; so it did not do any good to pray or to sacrifice to them. The gods, they believed, did not create the universe, nor did they inflict punishment or bestow blessings on anyone, but they were supremely happy; this was the goal to strive for during one's own human life. "Live unknown was one of [key] maxims. This was completely at odds with all previous ideas of seeking fame and glory, or even wanting something so apparently decent as honor.”*[12] Epicureanism rejects immortality and ; it believes in the soul, but suggests that the soul is as mortal as the body. Epicurus rejected any possibility of an afterlife, while still contending that one need not fear death: “Death is nothing to us; for that which is dissolved, is without sensation, and that which lacks sensation is nothing to us.” *[13] From this doctrine arose the Epicurean Epitaph: Non fui, fui, non sum, non curo (“I was not; I was; I am not; I do not care”), which is inscribed on the gravestones of his followers and seen on many ancient gravestones of the Roman Empire. This quotation is often used today at humanist funerals.*[14]

42.4 Ethics

Epicurus was an early thinker to develop the notion of justice as a . He defined justice as an agreement “neither to harm nor be harmed”. The point of living in a society with laws and punishments is to be protected from 42.5. EPICUREAN PHYSICS 141

harm so that one is free to pursue happiness. Because of this, laws that do not contribute to promoting human happi- ness are not just. He gave his own unique version of the Ethic of Reciprocity, which differs from other formulations by emphasizing minimizing harm and maximizing happiness for oneself and others:

It is impossible to live a pleasant life without living wisely and well and justly (agreeing “neither to harm nor be harmed”*[15]), and it is impossible to live wisely and well and justly without living a pleasant life.*[16]

Epicureanism incorporated a relatively full account of the social contract theory, following after a vague description of such a society in Plato's Republic. The social contract theory established by Epicureanism is based on mutual agreement, not divine decree. The human soul is mortal because, like everything, it is composed of atoms, but made up the most perfect, rounded and smooth. It disappears with the destruction of the body. We don't have to fear death because, firstly, nothing follows after the disappearance of the body, and, secondly, the experience of death is not so: “the most terrible evil, death, is nothing for us, since when we exist, death does not exist, and when death exists, we do not exist "(Epicurus,” Letter to Menoeceus "). Nature has set a target of every actions of living beings (including men) seeking pleasure, as shown by the fact that children instinctively and animals tend to shy away from pleasure and pain. Pleasure and pain are the main reasons for each actions of living beings. Pure pleasure is the highest good, pain the supreme evil. The pleasures and pains are the result of the realization or impairment of appetites. Epicurus distinguishes three kinds of appetites:

• Natural and necessary: eating, drinking, sleeping; They are easy to please.

• Natural but not necessary: as the erotic; they are not difficult to master and are not needed for happiness.

• Those who are not natural nor necessary: we must reject them completely.

Types of pleasures: since man is composed of body and soul there are two general types of pleasures:

• Pleasures of the body: Although considered to be the most important, in the background the proposal is to give up these pleasures and seek the lack of body pain. There are soul aches and pains of the body, but the body is bad because the pain of the soul is directly or indirectly related to body aches occurring in the present or to anticipations of future pains. Epicurus believed there was no need to fear bodily pain because when it is intense and unbearable, it is also usually shorter. When it lasts longer and is less intense, it is more bearable. He also believed one should relieve physical pain with the memory of past joys, and in extreme cases, to suicide.

• Pleasures of the soul: the pleasure of the soul is greater than the pleasure of the body: pleasures of the body are effective in the present, but those of the soul are more durable; the pleasures of the soul, Epicurus believed, can eliminate or reduce bodily pains or displeasures.

42.5 Epicurean physics

Epicurus' philosophy of the physical world is found in his Letter to : Diogenes Laertius 10.34–83. If the sum of all matter (“the totality”) was limited and existed within an unlimited void, it would be scattered and constantly becoming more diffuse, because the finite collection of bodies would travel forever, having no obstacles. Conversely, if the totality was unlimited it could not exist within a limited void, for the unlimited bodies would not all have a place to be in. Therefore, either both the void and the totality must be limited or both must be unlimited and – as is mentioned later – the totality is unlimited (and therefore so is the void). Forms can change, but not their inherent qualities, for change can only affect their shape. Some things can be changed and some things cannot be changed because forms that are unchangeable cannot be destroyed if certain attributes can be removed; for attributes not only have the intention of altering an unchangeable form, but also the inevitable possibility of becoming—in relation to the form's disposition to its present environment—both an armor and a vul- nerability to its stability. 142 CHAPTER 42. EPICUREANISM

Further proof that there are unchangeable forms and their inability to be destroyed, is the concept of the “non- evident.”A form cannot come into being from the void—which is nothing; it would be as if all forms come into being spontaneously, needless of reproduction. The implied meaning of “destroying”something is to undo its existence, to make it not there anymore, and this cannot be so: if the void is that which does not exist, and if this void is the implied destination of the destroyed, then the thing in reality cannot be destroyed, for the thing (and all things) could not have existed in the first place (as said, ex nihilo nihil fit: nothing comes from nothing). This totality of forms is eternal and unchangeable. Atoms move, in the appropriate way, constantly and for all time. Forms first come to us in images or“projections”— outlines of their true selves. For an image to be perceived by the human eye, the “atoms”of the image must cross a great distance at enormous speed and must not encounter any conflicting atoms along the way. The presence of atomic resistance equal atomic slowness; whereas, if the path is deficient of atomic resistance, the traversal rate is much faster (and clearer). Because of resistance, forms must be unlimited (unchangeable and able to grasp any point within the void) because, if they weren't, a form's image would not come from a single place, but fragmented and from several places. This confirms that a single form cannot be at multiple places at the same time. Epicurus for the most part follows Democritean atomism but differs in proclaiming the (swerve or decli- nation). Imagining atoms to be moving under an external force, Epicurus conceives an occasional atom “swerving” for reasons peculiar to itself, i.e. not by external compulsion but by“free will”. In this, his view absolutely opposes Democritean as well as developed Stoicism. Otherwise he conceives of atoms as does Democritus – in that they have position, number, and shape. To Democritus' differentiating criteria, Epicurus adds “weight”, but maintains Democritus' view that atoms are necessarily indivisible and hence possess no demonstrable internal space. And the senses warrant us other means of perception: hearing and smelling. As in the same way an image traverses through the air, the atoms of sound and smell traverse the same way. This perceptive experience is itself the flow of the moving atoms; and like the changeable and unchangeable forms, the form from which the flow traverses is shed and shattered into even smaller atoms, atoms of which still represent the original form, but they are slightly disconnected and of diverse magnitudes. This flow, like that of an echo, reverberates (off one's senses) and goes back to its start; meaning, one's sensory perception happens in the coming, going, or arch, of the flow; and when the flow retreats back to its starting position, the atomic image is back together again: thus when one smells something one has the ability to see it too [because atoms reach the one who smells or sees from the object.] And this leads to the question of how atomic speed and motion works. Epicurus says that there are two kinds of motion: the straight motion and the curved motion, and its motion traverse as fast as the speed of thought. Epicurus proposed the idea of 'the space between worlds' (metakosmia) the relatively empty spaces in the infinite void where worlds had not been formed by the joining together of the atoms through their endless motion.

42.6 Epistemology

Epicurean epistemology has three criteria of truth: sensations (aisthêsis), preconceptions (prolepsis), and feelings (pathê). Prolepsis is sometimes translated as“basic grasp”but could also be described as“universal ideas": concepts that are understood by all. An example of prolepsis is the word “man”because every person has a preconceived notion of what a man is. Sensations or sense perception is knowledge that is received from the senses alone. Much like modern science, Epicurean philosophy posits that can be used to sort truth from falsehood. Feelings are more related to ethics than Epicurean physical theory. Feelings merely tell the individual what brings about pleasure and what brings about pain. This is important for the Epicurean because these are the basis for the entire Epicurean ethical doctrine. According to Epicurus, the basic means for our understanding of things are the“sensations”(aestheses),“concepts” (prolepsis),“emotions”(pathe), and the“focusing of thought into an impression”(phantastikes epiboles tes dianoias). Epicureans reject dialectic as confusing (parelkousa) because for the physical philosophers it is sufficient to use the correct words which refer to the concepts of the world. Epicurus then, in his work On the Canon, says that the criteria of truth are the senses, the preconceptions and the feelings. Epicureans add to these the focusing of thought into an impression. He himself is referring to those in his Epitome to Herodotus and in Principal Doctrines.*[17] The senses are the first criterion of truth, since they create the first impressions and testify the existence of the external world. Sensory input is neither subjective nor deceitful, but the misunderstanding comes when the mind adds to or subtracts something from these impressions through our preconceived notions. Therefore, our sensory input alone cannot lead us to inaccuracy, only the concepts and opinions that come from our interpretations of our sensory input 42.6. EPISTEMOLOGY 143

can. Therefore, our sensory data is the only truly accurate thing which we have to rely for our understanding of the world around us.

And whatever image we receive by direct understanding by our mind or through our sensory organs of the shape or the essential properties that are the true form of the solid object, since it is created by the constant repetition of the image or the impression it has left behind. There is always inaccuracy and error involved in bringing into a judgment an element that is additional to sensory impressions, either to confirm [what we sensed] or deny it. —Letter to Herodotus, 50

Epicurus said that all the tangible things are real and each impression comes from existing objects and is determined by the object that causes the sensations. —Sextus Empiricus, To Rationals, 8.63

Therefore all the impressions are real, while the preconceived notions are not real and can be modi- fied. —Sextus Empiricus, To Rationals, 7.206–45

If you battle with all your sensations, you will be unable to form a standard for judging which of them are incorrect. —Principal Doctrines, 23

The concepts are the categories which have formed mentally according to our sensory input, for example the concepts “man”, “warm”, and “sweet”, etc. These concepts are directly related to memory and can be recalled at any time, only by the use of the respective word. (Compare the anthropological Sapir–Whorf hypothesis). Epicurus also calls them“the meanings that underlie the words”(hypotetagmena tois phthongois: semantic substance of the words) in his letter to Herodotus. The feelings or emotions (pathe) are related to the senses and the concepts. They are the inner impulses that make us feel like or dislike about certain external objects, which we perceive through the senses, and are associated with the preconceptions that are recalled.

In this moment that the word“man”is spoken, immediately due to the concept [or category of the idea] an image is projected in the mind which is related to the sensory input data. —Diogenes Laertius, Lives of Eminent Philosophers, X, 33

First of all Herodotus, we must understand the meanings that underlie the words, so that by referring to them, we may be able to reach judgments about our opinions, matters of inquiry, or problems and leave everything undecided as we can argue endlessly or use words that have no clearly defined meaning. —Letter to Herodotus, 37

Apart from these there is the assumption (hypolepsis), which is either the hypothesis or the opinion about something (matter or action), and which can be correct or incorrect. The assumptions are created by our sensations, concepts and emotions. Since they are produced automatically without any rational analysis and verification (see the modern idea of the subconscious) of whether they are correct or not, they need to be confirmed (epimarteresis: confirmation), a process which must follow each assumption.

For beliefs they [the Epicureans] use the word hypolepsis which they claim can be correct or incor- rect. —Diogenes Laertius, Lives of Eminent Philosophers, X, 34 144 CHAPTER 42. EPICUREANISM

Referring to the “focusing of thought into an impression”or else “intuitive understandings of the mind”, they are the impressions made on the mind that come from our sensations, concepts and emotions and form the basis of our assumptions and beliefs. All this unity (sensation – concept or category – emotion – focusing of thought into an impression) leads to the formation of a certain assumption or belief (hypolepsis). (Compare the modern anthropological concept of a "world view".) Following the lead of Aristotle, Epicurus also refers to impressions in the form of mental images which are projected on the mind. The “correct use of impressions”was something adopted later by the Stoics. Our assumptions and beliefs have to be 'confirmed', which actually proves if our opinions are either accurate or inaccurate. This verification and confirmation (epimarteresis) can only be done by means of the “evident reason” (henargeia), which means what is self-evident and obvious through our sensory input. An example is when we see somebody approaching us, first through the sense of eyesight, we perceive that an object is coming closer to us, then through our preconceptions we understand that it is a human being, afterwards through that assumption we can recognize that he is someone we know, for example . This assumption is associated with pleasant or unpleasant emotions accompanied by the respective mental images and impressions (the focusing of our thoughts into an impression), which are related to our feelings toward each other. When he gets close to us, we can confirm (verify) that he is Socrates and not Theaetetus through the proof of our eyesight. Therefore, we have to use the same method to understand everything, even things which are not observable and obvious (adela, imperceptible), that is to say the confirmation through the evident reason (henargeia). In the same way we have to reduce () each assumption and belief to something that can be proved through the self-evident reason (empirically verified). Verification theory and reductionism have been adopted, as we know, by the of science. In this way, one can get rid of the incorrect assumptions and beliefs (biases) and finally settle on the real (confirmed) facts.

Consequently the confirmation and lack of disagreement is the criterion of accuracy of something, while non-confirmation and disagreement is the criterion of its inaccuracy. The basis and foundation of [understanding] everything are the obvious and self-evident [facts]. —Sextus Empiricus, To Rationals, 7.211–6

All the above-mentioned criteria of knowledge form the basic principles of the [scientific] method, that Epicurus followed in order to find the truth. He described this method in his work On the Canon or On the Criteria.

If you reject any sensation and you do not distinguish between the opinion based on what awaits confirmation and evidence already available based on the senses, the feelings and every intuitive faculty of the mind, you will send the remaining sensations into a turmoil with your foolish opinions, thus getting rid of every standard for judging. And if among the perceptions based on beliefs are things that are verified and things that are not, you are guaranteed to be in error since you have kept everything that leads to uncertainty concerning the correct and incorrect.*[18]

(Based on excerpt from Epicurus' Gnoseology Handbook of Greek Philosophy: From Thales to the Stoics Analysis and Fragments, Nikolaos Bakalis, Trafford Publishing 2005, ISBN 1-4120-4843-5)

42.7

Main article: Tetrapharmakos Tetrapharmakos, or “The four-part cure”, is Epicurus' basic guideline as to how to live the happiest possible life. This poetic doctrine was handed down by an anonymous Epicurean who summed up Epicurus' philosophy on happiness in four simple lines:

Don't fear god, Don't worry about death; What is good is easy to get, and What is terrible is easy to endure. —Philodemus, Herculaneum Papyrus, 1005, 4.9–14 42.8. NOTABLE EPICUREANS 145

42.8 Notable Epicureans

One of the earliest Roman writers espousing Epicureanism was Amafinius. Other adherents to the teachings of Epicurus included the poet Horace, whose famous statement Carpe Diem“( Seize the Day”) illustrates the philosophy, as well as Lucretius, as he showed in his . The poet Virgil was another prominent Epicurean (see Lucretius for further details). The Epicurean philosopher Philodemus of Gadara, until the 18th century only known as a poet of minor importance, rose to prominence as most of his work along with other Epicurean material was discovered in the Villa of the Papyri. Julius Caesar leaned considerably toward Epicureanism, which e.g. led to his plea against the death sentence during the trial against Catiline, during the Catiline conspiracy where he spoke out against the Stoic Cato.*[19] In modern times Thomas Jefferson referred to himself as an Epicurean.*[20] Other modern-day Epicureans were Gassendi, Walter Charleton, François Bernier, Saint-Evremond, Ninon de l'Enclos, , Frances Wright and . Christopher Hitchens referred to himself as an Epicurean.*[21] In France, where perfumer/restaurateur Gérald Ghislain refers to himself as an Epicurean,*[22] is developing a post-modern approach to Epi- cureanism.*[23] In his recent book titled The Swerve, Stephen Greenblatt identified himself as strongly sympathetic to Epicureanism and Lucretius.

42.9 Modern usage and misconceptions

In modern popular usage, an epicurean is a connoisseur of the arts of life and the refinements of sensual pleasures; epicureanism implies a love or knowledgeable enjoyment especially of good and drink—see the definition of gourmet at Wiktionary. Because Epicureanism posits that pleasure is the ultimate good (telos), it has been commonly misunderstood since ancient times as a doctrine that advocates the partaking in fleeting pleasures such as constant partying, sexual excess and decadent food. This is not the case. Epicurus regarded ataraxia (tranquility, freedom from fear) and aponia (absence of pain) as the height of happiness. He also considered prudence an important virtue and perceived excess and overindulgence to be contrary to the attainment of ataraxia and aponia.*[11]

42.10 See also

• Epicurea • Epicurean paradox • Epikoros (Judaism) • • Lucretius • List of English translations of De rerum natura • • Cārvāka, a hedonic Indian school • Separation of church and state • • Dehellenization

42.11 Notes

[1] The Hidden History of Greco-Roman Vegetarianism

[2] The Philosophy of Vegetarianism – Daniel A. Dombrowski 146 CHAPTER 42. EPICUREANISM

[3] Erlend D. MacGillivray “The Popularity of Epicureanism in Late-Republic Roman Society”The Ancient World, XLIII (2012) 151–72.

[4] Michael Frede, Epilogue, The Cambridge History of Hellenistic Philosophy pp. 795–96;

[5] Lactantius, De Ira Deorum, 13.19 (Epicurus, Frag. 374, Usener). paraphrased this passage in his Dialogues Concerning Natural Religion: “EPICURUS's old questions are yet unanswered. Is he willing to prevent evil, but not able? then is he impotent. Is he able, but not willing? then is he malevolent. Is he both able and willing? whence then is evil?"

[6] Mark Joseph Larrimore, (2001), The Problem of Evil, pp. xix-xxi. Wiley-Blackwell

[7] Reinhold F. Glei, Et invidus et inbecillus. Das angebliche Epikurfragment bei Laktanz, De ira dei 13,20–21, in: Vigiliae Christianae 42 (1988), pp. 47–58

[8] Sextus Empiricus, Outlines of , 175: “those who firmly maintain that god exists will be forced into impiety; for if they say that he [god] takes care of everything, they will be saying that god is the cause of evils, while if they say that he takes care of some things only or even nothing, they will be forced to say that he is either malevolent or weak”

[9] Trans. Robert Pinsky, The Inferno of Dante, p. 320 n. 11.

[10] On Goals, 1.65

[11] Epicurus, “Letter to Menoeceus”, contained in Diogenes Laertius, Lives of Eminent Philosophers, Book X

[12] The Story of Philosophy: The Essential Guide to the History of Western Philosophy. Bryan Magee. DK Publishing, Inc. 1998.

[13] Russell, Bertrand. A History of Western Philosophy, pp. 239–40

[14] Epicurus (c 341–270 BC) British Humanist Association

[15] Tim O'Keefe, Epicurus on Freedom, Cambridge University Press, 2005, p. 134

[16] Epicurus Principal Doctrines tranls. by Robert Drew Hicks (1925)

[17] Diogenes Laertius, Lives of Eminent Philosophers, X, 31.

[18] Principal Doctrines, 24.

[19] Cf. Sallust, The War With Catiline, Caesar's speech: 51.29 & Cato's reply: 52.13).

[20] Letter to William Short, 11 Oct. 1819 in The Writings of Thomas Jefferson : 1816–1826 by Thomas Jefferson, Paul Leicester Ford, G.P. Putnam's Sons, 1899

[21] Townhall.com::Talk Radio Online::Radio Show

[22] Anon., Gérald Ghislain—Creator of The Scent of Departure. IdeaMensch, July 14, 2011.

[23] Michel Onfray, La puissance d'exister: Manifeste hédoniste, Grasset, 2006

42.12 Further reading

• Dane R. Gordon and David B. Suits, Epicurus. His Continuing Influence and Contemporary Relevance, Rochester, N.Y.: RIT Cary Graphic Arts Press, 2003. • Holmes, Brooke & Shearin, W. H. Dynamic Reading: Studies in the Reception of Epicureanism, New York: Oxford University Press, 2012. • Jones, Howard. The Epicurean Tradition, New York: Routledge, 1989. • Neven Leddy and Avi S. Lifschitz, Epicurus in the Enlightenment, Oxford: Voltaire Foundation, 2009. • Long, A.A.& Sedley, D.N. The Hellenistic Philosophers Volume 1, Cambridge: Cambridge University Press, 1987. (ISBN 0-521-27556-3) • Long, Roderick (2008). “Epicureanism”. In Hamowy, Ronald. The Encyclopedia of . Thou- sand Oaks, CA: SAGE; Cato Institute. p. 153. ISBN 978-1-4129-6580-4. LCCN 2008009151. OCLC 750831024. 42.13. EXTERNAL LINKS 147

• Martin Ferguson Smith (ed.), Diogenes of Oinoanda. The Epicurean inscription, edited with introduction, translation, and notes, : Bibliopolis, 1993. • Martin Ferguson Smith, Supplement to Diogenes of Oinoanda. The Epicurean Inscription, Naples: Bibliopolis, 2003. • Warren, James (ed.) The Cambridge Companion to Epicureanism, Cambridge: Cambridge University Press, 2009. • Wilson, Catherine. Epicureanism at the Origins of Modernity, New York: Oxford University Press, 2008.

• Zeller, Eduard; Reichel, Oswald J., The Stoics, Epicureans and Sceptics, Longmans, Green, and Co., 1892

42.13 External links

• Society of Friends of Epicurus • Epicureans on PhilPapers

• Epicurus.info – Epicurean Philosophy Online • Epicurus.net – Epicurus and Epicurean Philosophy

• 's Notebooks on Epicurean Philosophy • Marx's Doctoral Dissertation On the Difference between the Democritean and Epicurean Philosophy of Nature

• NewEpicurean.com • Commentary on the 40 Principal Doctrines by Nikos

• Jules Evans' Epicureans piece for his Philosophy for Life series 148 CHAPTER 42. EPICUREANISM

Epicurus 42.13. EXTERNAL LINKS 149

Part of Herculaneum Papyrus 1005 (P.Herc.1005), col. 5. Contains Epicurean tetrapharmakos from Philodemus' Adversus Sophis- tas. 150 CHAPTER 42. EPICUREANISM

De rerum natura manuscript, copied by an Augustinian friar for Pope Sixtus IV, c. 1483, after the discovery of an early manuscript in 1417 by the humanist and papal secretary Poggio Bracciolini. Chapter 43

Equality-generating dependency

In relational database theory, an equality-generating dependency (EGD) is a certain kind of constraint on data. It is a subclass of the class of embedded dependencies (ED). An ED is a sentence in first-order logic of the form: ∀x1 ... xn, P(x1, ..., xn) → ∃z1, ..., zk, Q(y1, ..., ym) where {z1, ..., zk} = {y1, ..., ym} \ {x1, ..., xn}, and P is a possibly empty and Q is a non-empty conjunction of equality atoms. A n equality atom has the form wi = wj where each of the w, ..., wh, wi, wj, are variables or constants. An algorithm known as the chase takes as input an instance that may or may not satisfy a set of EGDs (or more generally a set of EDs), and, if it terminates (which is a priori undecidable), output an instance that does satisfy the EGDs.

43.1 References

• Serge Abiteboul, Richard B. Hull, Victor Vianu: Foundations of Databases. Addison-Wesley, 1995.

• Alin Deutsch, FOL Modeling of Integrity Constraints, http://db.ucsd.edu/pubsFileFolder/305.pdf

151 Chapter 44

Erotetics

Erotetics is a part of logic, devoted to logical analysis of questions. The idea was originally developed by Richard Whately, whose work was popularized by Eugeniu Sperantia, and then such philosophers like or Nuel D. Belnap, Jr. were interested by this subject.

44.1 Bibliography

• Nuel D. Belnap, Jr., Questions, Answers, and ,"The Journal of Philosophy" Vol. 63, No. 20, American Philosophical Association Eastern Division Sixty-Third Annual Meeting (Oct., 1966), pp. 609–611. • Mary Prior, Arthur Prior, Erotetic Logic,"The Philosophical Review" Vol. 64, No. 1 (Jan., 1955), pp. 43–59.

• Eugeniu Sperantia, Remarques sur les propositions interrogatives. Projet d'une“logique du problème”, Actes du Congrès International de Philosophie Scientifique, VII Logique, Paris 1936, pp. 18–28.

• Whately, Richard, Elements of Logic, Longman, Greens and Co. (9th Edition, London, 1875)

• Richard Whately, Elements of Rhetoric

152 Chapter 45

Existential graph

An existential graph is a type of diagrammatic or visual notation for logical expressions, proposed by Charles Sanders Peirce, who wrote on graphical logic as early as 1882,*[1] and continued to develop the method until his death in 1914.

45.1 The graphs

Peirce proposed three systems of existential graphs:

• alpha, isomorphic to sentential logic and the two-element Boolean algebra;

• beta, isomorphic to first-order logic with identity, with all formulas closed;

• gamma, (nearly) isomorphic to normal modal logic.

Alpha nests in beta and gamma. Beta does not nest in gamma, quantified modal logic being more than even Peirce could envisage.

45.1.1 Alpha

The syntax is:

• The blank page;

• Single letters or phrases written anywhere on the page;

• Any graph may be enclosed by a simple closed curve called a cut or sep. A cut can be empty. Cuts can nest and concatenate at will, but must never intersect.

Any well-formed part of a graph is a subgraph. The semantics are:

• The blank page denotes Truth;

• Letters, phrases, subgraphs, and entire graphs may be True or False;

• To enclose a subgraph with a cut is equivalent to logical negation or Boolean complementation. Hence an empty cut denotes False;

• All subgraphs within a given cut are tacitly conjoined.

153 154 CHAPTER 45. EXISTENTIAL GRAPH

PQ P

P Q PQ

PQ

Alpha Graphs

Hence the alpha graphs are a minimalist notation for sentential logic, grounded in the expressive adequacy of And and Not. The alpha graphs constitute a radical simplification of the two-element Boolean algebra and the truth functors. The depth of an object is the number of cuts that enclose it. Rules of inference:

• Insertion - Any subgraph may be inserted into an odd numbered depth. • Erasure - Any subgraph in an even numbered depth may be erased.

Rules of equivalence:

• Double cut - A pair of cuts with nothing between them may be drawn around any subgraph. Likewise two nested cuts with nothing between them may be erased. This rule is equivalent to Boolean involution. • Iteration/Deiteration – To understand this rule, it is best to view a graph as a tree structure having nodes and ancestors. Any subgraph P in node n may be copied into any node depending on n. Likewise, any subgraph P in node n may be erased if there exists a copy of P in some node ancestral to n (i.e., some node on which n depends). For an equivalent rule in an algebraic context, see C2 in .

A proof manipulates a graph by a series of steps, with each step justified by one of the above rules. If a graph can be reduced by steps to the blank page or an empty cut, it is what is now called a tautology (or the complement thereof). Graphs that cannot be simplified beyond a certain point are analogues of the satisfiable formulas of first-order logic. 45.2. PEIRCE'S ROLE 155

45.1.2 Beta

Peirce notated predicates using intuitive English phrases; the standard notation of contemporary logic, Latin letters, may also be employed. A dot asserts the existence of some individual in the domain of discourse. Multiple instances of the same object are linked by a line, called the “line of identity”. There are no literal variables or quantifiers in the sense of first-order logic. A line of identity connecting two or more predicates can be read as asserting that the predicates share a common variable. The presence of lines of identity requires modifying the alpha rules of Equivalence. The beta graphs can be read as a system in which all formula are to be taken as closed, because all variables are implicitly quantified. If the “shallowest”part of a line of identity has even (odd) depth, the associated variable is tacitly existentially (universally) quantified. Zeman (1964) was the first to note that the beta graphs are isomorphic to first-order logic with equality (also see Zeman 1967). However, the secondary literature, especially Roberts (1973) and Shin (2002), does not agree on just how this is so. Peirce's writings do not address this question, because first-order logic was first clearly articulated only some years after his death, in the 1928 first edition of and Wilhelm Ackermann's Principles of Mathematical Logic.

45.1.3 Gamma

Add to the syntax of alpha a second kind of simple closed curve, written using a dashed rather than a solid line. Peirce proposed rules for this second style of cut, which can be read as the primitive unary operator of modal logic. Zeman (1964) was the first to note that straightforward emendations of the gamma graph rules yield the well-known modal logics S4 and S5. Hence the gamma graphs can be read as a peculiar form of normal modal logic. This finding of Zeman's has gone unremarked to this day, but is nonetheless included here as a point of interest.

45.2 Peirce's role

The existential graphs are a curious offspring of Peirce the logician/mathematician with Peirce the founder of a major strand of semiotics. Peirce's graphical logic is but one of his many accomplishments in logic and mathematics. In a series of papers beginning in 1867, and culminating with his classic paper in the 1885 American Journal of Mathematics, Peirce developed much of the two-element Boolean algebra, propositional calculus, quantification and the predicate calculus, and some rudimentary set theory. Model theorists consider Peirce the first of their kind. He also extended De Morgan's . He stopped short of (which eluded even Principia Mathematica). But Peirce's evolving semiotic theory led him to doubt the value of logic formulated using conventional linear notation, and to prefer that logic and mathematics be notated in two (or even three) dimensions. His work went beyond Euler's diagrams and Venn's revision thereof. Frege's 1879 Begriffsschrift also employed a two-dimensional notation for logic, but one very different from Peirce's. Peirce's first published paper on graphical logic (reprinted in Vol. 3 of his Collected Papers) proposed a system dual (in effect) to the alpha existential graphs, called the entitative graphs. He very soon abandoned this formalism in favor of the existential graphs. The graphical logic went unremarked during his lifetime, and was invariably denigrated or ignored after his death, until the Ph.D. theses by Roberts (1964) and Zeman (1964).

45.3 See also

• Ampheck

• Conceptual graph

• Entitative graph

• Logical graph 156 CHAPTER 45. EXISTENTIAL GRAPH

45.4 Notes

[1] Peirce, C. S., "[On Junctures and Fractures in Logic]" (editors' title for MS 427 (the new numbering system), Fall–Winter 1882), and“Letter, Peirce to O. H. Mitchell”(L 294, 21 December 1882), Writings of Charles S. Peirce, v. 4,“Junctures” on pp. 391–3 (Google preview) and the letter on pp. 394–9 (Google preview). See Sowa, John F. (1997), “Matching Logical Structure to Linguistic Structure”, Studies in the Logic of Charles Sanders Peirce, Nathan Houser, Don D. Roberts, and James Van Evra, editors, Bloomington and Indianopolis: Indiana University Press, pp. 418–44, see 420, 425, 426, 428.

45.5 References

45.5.1 Primary literature

• 1931-35 & 1958. The Collected Papers of Charles Sanders Peirce. Volume 4, Book II: “Existential Graphs” , consists of paragraphs 347–584. A discussion also begins in paragraph 617.

• Paragraphs 347–349 (II.1.1. “Logical Diagram”)—Peirce's definition“Logical Diagram (or Graph)" in Baldwin's Dictionary of Philosophy and Psychology (1902), v. 2, p. 28. Classics in the History of Psychology Eprint. • Paragraphs 350–371 (II.1.2. “Of Euler's Diagrams”)—from “Graphs”(manuscript 479) c. 1903. • Paragraphs 372–584 Eprint. • Paragraphs 372–393 (II.2. “Symbolic Logic”)—Peirce's part of “Symbolic Logic”in Baldwin's Dictionary of Philosophy and Psychology (1902) v. 2, pp. 645–650, beginning (near second column's top) with “If symbolic logic be defined...”. Paragraph 393 (Baldwin's DPP2 p. 650) is by Peirce and Christine Ladd-Franklin (“C.S.P., C.L.F.”). • Paragraphs 394–417 (II.3.“Existential Graphs”)—from Peirce's pamphlet A Syllabus of Certain Topics of Logic, pp. 15–23, Alfred Mudge & Son, Boston (1903). • Paragraphs 418–509 (II.4. “On Existential Graphs, Euler's Diagrams, and Logical Algebra”)—from “Logical Tracts, No. 2”(manuscript 492), c. 1903. • Paragraphs 510–529 (II.5. “The Gamma Part of Existential Graphs”)—from “Lowell Lectures of 1903,”Lecture IV (manuscript 467). • Paragraphs 530–572 (II.6.)—"Prolegomena To an Apology For Pragmaticism”(1906), The Monist, v. XVI, n. 4, pp. 492−546. Corrections (1907) in The Monist v. XVII, p. 160. • Paragraphs 573–584 (II.7. “An Improvement on the Gamma Graphs”)—from “For the National Academy of Science, 1906 April Meeting in Washington”(manuscript 490). • Paragraphs 617–623 (at least) (in Book III, Ch. 2, §2, paragraphs 594–642)—from “Some Amazing Mazes: Explanation of Curiosity the First”, The Monist, v. XVIII, 1908, n. 3, pp. 416−464, see starting p. 440.

• 1992. “Lecture Three: The Logic of Relatives”, Reasoning and the Logic of Things, pp. 146–64. Ket- ner, Kenneth Laine (editing and introduction), and (commentary). Harvard University Press. Peirce's 1898 lectures in Cambridge, Massachusetts.

• 1977, 2001. Semiotic and Significs: The Correspondence between C.S. Peirce and Victoria Lady Welby. Hard- wick, C.S., ed. Lubbock TX: Texas Tech University Press. 2nd edition 2001.

• A transcription of Peirce's MS 514 (1909), edited with commentary by John Sowa.

Currently, the chronological critical edition of Peirce's works, the Writings, extends only to 1892. Much of Peirce's work on logical graphs consists of manuscripts written after that date and still unpublished. Hence our understanding of Peirce's graphical logic is likely to change as the remaining 23 volumes of the chronological edition appear. 45.6. EXTERNAL LINKS 157

45.5.2 Secondary literature

• Hammer, Eric M. (1998), “Semantics for Existential Graphs,”Journal of Philosophical Logic 27: 489-503. • Ketner, Kenneth Laine

• (1981), “The Best Example of Semiosis and Its Use in Teaching Semiotics”, American Journal of Semiotics v. I, n. 1-2, pp. 47–83. Article is an introduction to existential graphs. • (1990), Elements of Logic: An Introduction to Peirce's Existential Graphs, Texas Tech University Press, Lubbock, TX, 99 pages, spiral-bound. • Queiroz, João & Stjernfelt, Frederik

• (2011), “Diagrammatical Reasoning and Peircean Logic Representation”, Semiotica vol. 186 (1/4). (Special issue on Peirce's diagrammatic logic.)

• Roberts, Don D.

• (1964), “Existential Graphs and Natural Deduction”in Moore, E. C., and Robin, R. S., eds., Studies in the Philosophy of C. S. Peirce, 2nd series. Amherst MA: University of Massachusetts Press. The first publication to show any sympathy and understanding for Peirce's graphical logic. • (1973). The Existential Graphs of C.S. Peirce. John Benjamins. An outgrowth of his 1963 thesis. • Shin, Sun-Joo (2002), The Iconic Logic of Peirce's Graphs. MIT Press.

• Zalamea, Fernando. Peirce's Logic of Continuity. Docent Press, Boston MA. 2012. ISBN 9 780983 700494. • Part II: Peirce's Existential Graphs, pp. 76-162.

• Zeman, J. J. • (1964), The Graphical Logic of C.S. Peirce. Unpublished Ph.D. thesis submitted to the University of Chicago. • (1967), “A System of Implicit Quantification,”Journal of Symbolic Logic 32: 480-504.

45.6 External links

• Stanford Encyclopedia of Philosophy: Peirce's Logic by Sun-Joo Shin and Eric Hammer.

• Dau, Frithjof, Peirce's Existential Graphs --- Readings and Links. An annotated bibliography on the existential graphs.

• Gottschall, Christian, Proof Builder —Java applet for deriving Alpha graphs.

• Liu, Xin-Wen, "The literature of C.S. Peirce’s Existential Graphs" (via Wayback Machine), Institute of Philosophy, Chinese Academy of Social Sciences, Beijing, PRC.

• Sowa, John F.,“Laws, Facts, and Contexts: Foundations for Multimodal Reasoning”accessdate=2009-10-23 Existential graphs and conceptual graphs.

• Van Heuveln, Bram, "Existential Graphs." Dept. of Cognitive Science, Rensselaer Polytechnic Institute. Alpha only.

• Zeman, Jay J., "Existential Graphs". With four online papers by Peirce. Chapter 46

Extensional context

In philosophy of language, a context in which a sub-sentential expression e appears is called extensional if and only if e can be replaced by an expression with the same extension and necessarily preserve truth-value. The extension of a term is the set of objects that that term denotes. Take the case of Clark Kent, who is secretly Superman. Suppose that Lois Lane fell out of a window and Superman caught her. Thus the statement, “Clark Kent caught Lois Lane,”is true because it has an extensional context. The names “Superman”and “Clark Kent”have the same extension, which is to say that they both refer to the same person, i.e., that superhero who is vulnerable to kryptonite. Anybody that Superman caught, Clark Kent caught. In opposition to extensional contexts are intensional contexts, where synonymous terms cannot be substituted in without potentially compromising the truth-value. Suppose that Lois Lane believes that Clark Kent will investigate a news story with her. The statement, “Lois Lane believes that Superman will investigate a news story with her,”is false, even though Superman is Clark Kent. This is because 'believes' is typically an intensional context.

46.1 See also

• Extension (semantics) • Extension (predicate logic)

• Extensional definition

158 Chapter 47

Extensional definition

An extensional definition of a concept or term formulates its meaning by specifying its extension, that is, every object that falls under the definition of the concept or term in question. For example, an extensional definition of the term “nation of the world”might be given by listing all of the nations of the world, or by giving some other means of recognizing the members of the corresponding class. An explicit listing of the extension, which is only possible for finite sets and only practical for relatively small sets, is a type of enumerative definition. Extensional definitions are used when listing examples would give more applicable information than other types of definition, and where listing the members of a set tells the questioner enough about the nature of that set. This is similar to an ostensive definition, in which one or more members of a set (but not necessarily all) are pointed out as examples. The opposite approach is the intensional definition, which defines by listing properties that a thing must have in order to be part of the set captured by the definition.

47.1 See also

• Extensional context • Extension (predicate logic)

• Intensional definition

159 Chapter 48

Extensionalism

Extensionalism, in the philosophy of language, in logic and semantics, is the view that all languages or at least all scientific languages should be extensional. Rudolf Carnap (in his earlier work) and Willard Van Orman Quine were prominent proponents of this view.

48.1 See also

• Extension (semantics)

• Extension (predicate logic)

160 Chapter 49

Fa (concept)

Fa (Chinese: 法;Mandarin pronunciation: [fà]) is a concept in that covers ethics, logic, and law. It can be translated as“law”in some contexts, but more often as“model”or“standard.”First gaining importance in the Mohist , the concept was principally elaborated in method-focused Chinese referred to as Legalism. In the philosophy Han Fei the king is the sole source of fa (law), taught to the common people so that there would be harmonious society free of chance occurrences, disorder, and “appeal to privilege” . High officials were not to be held above fa (law or protocol), nor were they to be allowed to independently create their own fa, uniting both executive fiat and rule of law.*[1] Xunzi, a philosopher that would end up being foundational in Han dynasty , also took up fa, suggesting that it could only be properly assessed by the Confucian sage (ruler), and that the most important fa were the very rituals that had ridiculed for their ostentatious waste and lack of benefit for the people at large.*[2]

49.1 and the

The concept of fa first gained importance in the Mohist school of thought. To Mozi, a standard must stand “three tests”in order to determine its efficacy and .*[3] The first of these tests was its origin; if the standard had precedence in the actions or thought of the semi-mythological sage kings of the Xia dynasty whose examples are frequently cited in classical Chinese philosophy. The second test was one of validity; does the model stand up to evidence in the estimation of the people? The third and final test was one of applicability; this final one is a utilitarian estimation of the net good that, if implemented, the standard would have on both the people and the state.*[4] The third test speaks to the fact that to the Mohists, a fa was not simply an abstract model, but an active tool. The real- world use and practical application of fa were vital. Yet fa as models were also used in later Mohist logic as principles used in deductive reasoning. As classical Chinese philosophical logic was based on analogy rather than syllogism, fa were used as benchmarks to determine the validity of logical claims through comparison. There were three fa in particular that were used by these later Mohists (or "Logicians") to assess such claims, which were mentioned earlier. The first was considered a“root”standard, a concern for precedence and origin. The second, a“source”, a concern for empiricism. The third, a“use”, a concern for the consequence and pragmatic of a standard. These three fa were used by the Mohists to both promote social welfare and denounce ostentation or wasteful spending.*[5]

49.2 See also

• Logic in China

49.3 References

[1] Han Fei. (2003). Basic Writings. Columbia University Press: New York, p. 7, 21- 28, 40, 91

[2] Robins, Dan, “Xunzi”, The Stanford Encyclopedia of Philosophy (Fall 2008 Edition), Edward N. Zalta (ed.), URL =

161 162 CHAPTER 49. FA (CONCEPT)

[3]

[4] Mozi. (2003). Basic Writings. Burton Watson, Ed. Columbia University Press: New York, p. 122

[5] Fraser, Chris, “Mohism”, The Stanford Encyclopedia of Philosophy (Summer 2010 Edition), Edward N. Zalta (ed.), URL = Chapter 50

Finite model property

In logic, we say a logic L has the finite model property (fmp for short) if there is a class of models M of L (i.e. each model in M is a model of L) such that any non-theorem of L is falsified by some finite model in M. Another way of putting this is to say that L has the fmp if for every formula A of L, A is an L-theorem iff A is a theorem of the theory of finite models of L. If L is finitely axiomatizable (and has a recursive set of recursive rules) and has the fmp, then it is decidable. However, the strengthened claim that if L is recursively axiomatizable and the fmp then it is decidable, is false. Even if there are only finitely many finite models to choose from (up to isomorphism) there is still the problem of checking whether the underlying frames of such models validate the logic, and this may not be decidable when the logic is not finitely axiomatizable, even when it is recursively axiomatizable. (Note that a logic is recursively enumerable iff it is recursively axiomatizable, a result known as Craig's theorem.)

50.1 Example

A first-order formula with one universal quantification has the fmp. A first-order formula without functional symbols, where all existential quantifications appear first in the formula, also has the fmp.*[1]

50.2 See also

• Kripke semantics

50.3 References

• Blackburn P., de Rijke M., Venema Y. Modal Logic. Cambridge University Press, 2001.

• A Urquhart. Decidability and the Finite Model Property. Journal of Philosophical Logic, 10 (1981), 367-370.

[1] Leonid Libkin, Elements of finite model theory, chapter 14

163 Chapter 51

Fluidics

Fluidics, or fluidic logic, is the use of a fluid to perform analog or digital operations similar to those performed with electronics. The physical basis of fluidics is pneumatics and hydraulics, based on the theoretical foundation of fluid dynamics. The term fluidics is normally used when devices have no moving parts, so ordinary hydraulic components such as hydraulic cylinders and spool valves are not considered or referred to as fluidic devices. The 1960s saw the application of fluidics to sophisticated control systems, with the introduction of the fluidic amplifier. A jet of fluid can be deflected by a weaker jet striking it at the side. This provides nonlinear amplification, similar to the transistor used in electronic digital logic. It is used mostly in environments where electronic digital logic would be unreliable, as in systems exposed to high levels of electromagnetic interference or ionizing radiation. Nanotechnology considers fluidics as one of its instruments. In this domain, effects such as fluid-solid and fluid-fluid interface forces are often highly significant. Fluidics have also been used for military applications.

51.1 Amplifier

Fluidic amplifier, showing flow in both states. From U.S. Patent #4,000,757.

The basic concept of the fluidic amplifier is shown here. A fluid supply, which may be air, water, or hydraulic fluid, enters at the bottom. Pressure applied to the control ports C1 or C2 deflects the stream, so that it exits via either port O1 or O2. The stream entering the control ports may be much weaker than the stream being deflected, so the device has gain. Given this basic device, flip flops and other fluidic logic elements can be constructed. Simple systems of digital logic

164 51.2. TRIODE 165

can thus be built. Fluidic amplifiers typically have bandwidths in the low kilohertz range, so systems built from them are quite slow compared to electronic devices.

51.2 Triode

The fluidic triode is an amplification device that uses a fluid to convey the signal. Although much studied in the laboratory they have few practical applications. Many expect them to be key elements of nanotechnology. Fluidic triodes were used as the final stage in the main Public Address system at the 1964 New York World's Fair.

The Fluidic Triode was invented in 1962 by Murray O. Meetze, Jr., a high school student in Heath Springs, S.C. He also built a fluid diode, a fluid oscillator and a variety of hydraulic“circuits,”including one that has no electronic counterpart. As a result he was invited to the National Science Fair, held this year at the Seattle Century 21 Exposition. There his project won an award.

(Scientific American, Aug. 1962)

51.3 Logic elements

Logic gates can be built that use water instead of electricity to power the gating function. These are reliant on being positioned in one orientation to perform correctly. An OR gate is simply two pipes being merged, a NOT gate consists of “A”deflecting a supply stream to produce Ā. An could also be implemented with the XOR gate, as A XOR 1 = Ā. *[1] Bubble logic is another kind of fluidic logic. Bubble logic gates conserve the number of bits entering and exiting the device, since bubbles are neither produced nor destroyed in the logic operation, analogous to billiard-ball computer gates. *[2]

51.4 Uses

The MONIAC Computer built in 1949 was a fluid-based analogue computer used for teaching economic principles as it could recreate complex simulations that digital computers could not. Twelve to fourteen were built and acquired by businesses and teaching establishments. Fluidic components appear in some hydraulic and pneumatic systems, including some automotive automatic trans- missions. As digital logic has become more accepted in industrial control, the role of fluidics in industrial control has declined. In the consumer , fluidically controlled products are increasing in both popularity and presence, installed in items ranging from toy spray guns through shower heads and hot tub jets; all provide oscillating streams of air and/or water. Fluid logic can be used to create a valve with no moving parts such as in some anaesthetic machines.

51.4.1 Research

Fluidic injection is being researched for use in aircraft to control direction, in two ways: circulation control and thrust vectoring. In both, larger more complex mechanical parts are replaced by fluidic systems, in which larger forces in fluids are diverted by smaller jets or flows of fluid intermittently, to change the direction of vehicles. In circulation control, near the trailing edges of wings, aircraft flight control systems such as ailerons, elevators, elevons, flaps and flaperons are replaced by slots which emit fluid flows.*[3]*[4]*[5] 166 CHAPTER 51. FLUIDICS

AND

XOR

The two gates AND and XOR in one module. The bucket in the center collects the AND output, and the output at the bottom is A XOR B.

In thrust vectoring, in jet engine nozzles, swiveling parts are replaced by slots which inject fluid flows into jets.*[6] Such systems divert thrust via fluid effects. Tests show that air forced into a jet engine exhaust stream can deflect thrust up to 15 degrees. 51.5. SEE ALSO 167

In such uses, fluidics is desirable for lower: mass, cost (up to 50% less), drag (up to 15% less during use), inertia (for faster, stronger control response), complexity (mechanically simpler, fewer or no moving parts or surfaces, less maintenance), and radar cross section for stealth. This will likely be used in many unmanned aerial vehicles (UAVs), 6th generation fighter aircraft, and ships.

51.5 See also

• Microfluidics

51.6 References

[1] A four-bit Adder made using fluidic logic [2] Manu Prakash. “Bubble Logic”. MIT 2007. [3] P John (2010).“The flapless air vehicle integrated industrial research (FLAVIIR) programme in aeronautical engineering” . Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering (London: Mechanical Engineering Publications) 224 (4): 355–363. doi:10.1243/09544100JAERO580. ISSN 0954-4100. [4] “Showcase UAV Demonstrates Flapless Flight”. BAE Systems. 2010. Retrieved 2010-12-22. [5] “Demon UAV jets into history by flying without flaps”. Metro.co.uk (London: Associated Newspapers Limited). 28 September 2010. [6] P. J. Yagle, D. N. Miller, K. B. Ginn, J. W. Hamstra (2001). “Demonstration of Fluidic Throat Skewing for Thrust Vectoring in Structurally Fixed Nozzles”. Journal of Engineering for Gas Turbines and Power 123 (3): 502–508. doi:10.1115/1.1361109.

51.7 Further reading

• Jagdish Lal (1963). “Hydraulics”. Metropolitan Book. ASIN B0007IWSX2. • O.Lew wood (June 1964). “Pure Fluid Device”. Machine Design: 154–180. • E.F.Tarumoto; D.H.Humphrey (1965). “Fluidics”. Fluid Amplifier Associates, Inc. ASIN B000HBV114. • RE.Bowles; EM.Dexter (October 1965). “A second Generation of Fluid System Application”. Fluid Am- plification Symposium: 213. • Forbes T., Brown (1967). Advances in Fluidics. The American Society of Mechanical Engineers. ASIN B000I3ZC7K. • Fluidics: Components and Circuits Foster, Kenneth John Wiley & Sons （1970）ISBN 9780471267706 • Jagdish Lal (1975). “Hydraulic Machines”. Metropolitan Book. ASIN B000HBV114. • A Guide to Fluidics A. Conway ISBN 9780444196019 • U.S. Patent 3,417,770 • U.S. Patent 3,495,253 • U.S. Patent 3,503,410 • U.S. Patent 3,612,085 • U.S. Patent 4,854,176

51.8 External links

• Scanned article available online from Google Books: Popular Science June 1967, Fluidics: How They've Taught A Stream of Air to Think pp. 118–121,196.197, illustrating several switch designs and discussing applications Chapter 52

Formal ontology

This article concerns formal ontology in philosophy. In , a formal ontology is an ontology (in- formation science) that is defined by axioms in a formal language, typically an ontology language. Those formal ontologies may or may not be based on the kind of formal upper level ontology described here.

In philosophy, the term formal ontology is used to refer to an ontology defined by axioms in a formal language with the goal to provide an unbiased (domain- and application-independent) view on reality, which can help the modeler of domain- or application-specific ontologies (information science) to avoid possibly erroneous ontological assumptions encountered in modeling large-scale ontologies. By maintaining an independent view on reality a formal (upper level) ontology gains the following properties:

• indefinite expandability: the ontology remains consistent with increasing content. • content and context independence: any kind of 'concept' can find its place. • accommodate different levels of granularity.

Theories on how to conceptualize reality date back as far as Plato and Aristotle.

52.1 Existing formal upper level ontologies (foundational ontologies)

Main article: (information science)

• BFO - Basic Formal Ontology • CIDOC Conceptual Reference Model • Cyc (Cyc is not just an upper ontology, it also contains many mid-level and specialized ontologies as well) • UMBEL - Upper Mapping and Binding Exchange Layer, a subset of OpenCyc • DOLCE - Descriptive Ontology for Linguistic and Cognitive Engineering • GFO - General Formal Ontology • UFO - Unified Foundational Ontology • OCHRE - Object-Centered High-level REference ontology • SUMO - Suggested Upper Merged Ontology

168 52.2. COMMON TERMS IN FORMAL (UPPER-LEVEL) ONTOLOGIES 169

• Business Objects Reference Ontology

• YAMATO - Top ontology with objectives similar to those of DOLCE, BFO, or GFO

52.2 Common terms in formal (upper-level) ontologies

The differences in terminology used between separate formal upper-level ontologies can be quite substantial, but most formal upper-level ontologies apply one foremost dichotomy: that between endurants and perdurants.

52.2.1 Endurant

Also known as continuants, or in some cases as “substance”, endurants are those entities that can be observed- perceived as a complete concept, at no matter which given snapshot of time. Were we to freeze time we would still be able to perceive/conceive the entire endurant. Examples include material objects (such as an apple or a human), and abstract“fiat”objects (such as an organization, or the border of a country).

52.2.2 Perdurant

Also known as occurrents, accidents or happenings, perdurants are those entities for which only a part exists if we look at them at any given snapshot in time. When we freeze time we can only see a part of the perdurant. Perdurants are often what we know as processes, for example:“running”. If we freeze time then we only see a part of the running, without any previous knowledge one might not even be able to determine the actual process as being a process of running. Other examples include an activation, a kiss, or a procedure.

52.2.3 Qualities

In a broad sense, qualities can also be known as properties or tropes. Qualities do not exist on their own, but they need another entity (in many formal ontologies this entity is restricted to be an endurant) which they occupy. Examples of qualities and the values they assume include colors (red color), or temperatures (warm). Most formal upper-level ontologies recognize qualities, attributes, tropes, or something related, although the exact classification may differ. Some see qualities and the values they can assume (sometimes called quale) as a separate hierarchy besides endurants and perdurants (example: DOLCE). Others classify qualities as a subsection of endurants, e.g. the dependent endurants (example: BFO). Others consider property-instances or tropes that are single charac- teristics of individuals as the atoms of the ontology, the simpler entities of which all other entities are composed, so that all the entities are sums or bundles of tropes (example: OCHRE).

52.3 Formal versus nonformal

In information science an ontology is formal if it is specified in a formal language, otherwise it is informal. In philosophy, a separate distinction between formal and nonformal ontologies exists, which does not relate to the use of a formal language.

52.3.1 Example

An ontology might contain a concept representing 'mobility of the arm'. In a nonformal ontology a concept like this can often be classified as for example a 'finding of the arm', right next to other concepts such as 'bruising of the arm'. This method of modeling might create problems with increasing amounts information, as there is no foolproof way to keep hierarchies like this, or their descendant hierarchies (one is a process, the other is a quality) from entangling or knotting. 170 CHAPTER 52. FORMAL ONTOLOGY

In a formal ontology, there is an optimal way to properly classify this concept, it is a kind of 'mobility', which is a kind of quality/property (see above). As a quality, it is said to inhere in independent endurant entities (see above), as such, it cannot exist without a bearer (in the case the arm).

52.4 Applications for formal (upper-level) ontologies

52.4.1 Formal ontology as a template to create novel specific domain ontologies

Having a formal ontology at your disposal, especially when it consists of a Formal upper layer enriched with concrete domain-independent 'middle layer' concepts, can really aid the creation of a domain specific ontology. It allows the modeller to focus on the content of the domain specific ontology without having to worry on the exact higher structure or abstract philosophical framework that gives his ontology a rigid backbone. Disjoint axioms at the higher level will prevent many of the commonly made ontological mistakes made when creating the detailed layer of the ontology.

52.4.2 Formal ontology as a crossmapping hub: crossmapping taxonomies, databases and nonformal ontologies

Aligning terminologies and ontologies is not an easy task. The divergence of the underlying meaning of word descrip- tions and terms within different information sources is a well known obstacle for direct approaches to data integration and mapping. One single description may have a completely different meaning in one data source when compared with another. This is because different databases/terminologies often have a different viewpoint on similar items. They are usually built with a specific application-perspective in mind and their hierarchical structure represents this. A formal ontology, on the other hand, represents entities without a particular application scope. Its hierarchy reflects ontological principles and a basic class-subclass relation between its concepts. A consistent framework like this is ideal for crossmapping data sources. However, one cannot just integrate these external data sources in the formal ontology. A direct incorporation would lead to corruption of the framework and principles of the formal ontology. A formal ontology is a great crossmapping hub only if a complete distinction between the content and structure of the external information sources and the formal ontology itself is maintained. This is possible by specifying a mapping relation between concepts from a chaotic external information source and a concept in the formal ontology that corresponds with the meaning of the former concept. Where two or more external information sources map to one and the same formal ontology concept a crossmap- ping/translation is achieved, as you know that those concepts - no matter what their phrasing is - mean the same thing.

52.4.3 Formal ontology to empower natural language processing

In ontologies designed to serve natural language processing (NLP) and natural language understanding (NLU) sys- tems, ontology concepts are usually connected and symbolized by terms. This kind of connection represents a lin- guistic realization. Terms are words or a combination of words (multi-word units), in different languages, used to describe in natural language an element from reality, and hence connected to that formal ontology concept that frames this element in reality. The lexicon, the collection of terms and their inflections assigned to the concepts and relationships in an ontology, forms the ‘ontology interface to natural language’, the channel through which the ontology can be accessed from a natural language input.

52.4.4 Formal ontology to normalize database/instance data

The great thing about a formal ontology, in contrast to rigid taxonomies or classifications, is that it allows for indefinite expansion. Given proper modeling, just about any kind of conceptual information, no matter the content, can find its place. To disambiguate a concept's place in the ontology, often a context model is useful to improve the classification power. The model typically applies rules to surrounding elements of the context to select the most valid classification. 52.5. SEE ALSO 171

52.5 See also

• Ontology (information science)

• Upper ontology (information science)

52.6 References

52.7 External links

• Laboratory of Applied Ontology (LOA) homepage

• Research Group Ontologies in Medicine (Onto-Med) homepage • National Center for Biomedical Ontology (NCBO)

• National Center for Ontological Research (NCOR) • International Association for Ontology and its Applications Chapter 53

Forward chaining

Forward chaining is one of the two main methods of reasoning when using an inference engine and can be described logically as repeated application of modus ponens. Forward chaining is a popular implementation strategy for expert systems, business and production rule systems. The opposite of forward chaining is backward chaining. Forward chaining starts with the available data and uses inference rules to extract more data (from an end user, for example) until a goal is reached. An inference engine using forward chaining searches the inference rules until it finds one where the antecedent (If clause) is known to be true. When such a rule is found, the engine can conclude, or infer, the consequent (Then clause), resulting in the addition of new information to its data.*[1] Inference engines will iterate through this process until a goal is reached. For example, suppose that the goal is to conclude the color of a pet named Fritz, given that he croaks and eats flies, and that the rule base contains the following four rules:

1. If X croaks and X eats flies - Then X is a frog

2. If X chirps and X sings - Then X is a canary

3. If X is a frog - Then X is green

4. If X is a canary - Then X is yellow

Let us illustrate forward chaining by following the pattern of a computer as it evaluates the rules. Assume the following facts:

• Fritz croaks

• Fritz eats flies

With forward reasoning, the inference engine can derive that Fritz is green in a series of steps: 1. Since the base facts indicate that “Fritz croaks”and “Fritz eats flies”, the antecedent of rule #1 is satisfied by substituting Fritz for X, and the inference engine concludes: Fritz is a frog 2. The antecedent of rule #3 is then satisfied by substituting Fritz for X, and the inference engine concludes: Fritz is green The name“forward chaining”comes from the fact that the inference engine starts with the data and reasons its way to the answer, as opposed to backward chaining, which works the other way around. In the derivation, the rules are used in the opposite order as compared to backward chaining. In this example, rules #2 and #4 were not used in determining that Fritz is green. Because the data determines which rules are selected and used, this method is called data-driven, in contrast to goal- driven backward chaining inference. The forward chaining approach is often employed by expert systems, such as CLIPS.

172 53.1. SEE ALSO 173

One of the advantages of forward-chaining over backward-chaining is that the reception of new data can trig- ger new , which makes the engine better suited to dynamic situations in which conditions are likely to change.*[2]*[3]

53.1 See also

• Backward chaining

• Inference engine • Opportunistic reasoning

• Rete algorithm

53.2 References

[1] Feigenbaum, Edward (1988). The Rise of the Expert Company. Times Books. p. 318. ISBN 0-8129-1731-6.

[2] Hayes-Roth, Frederick; Donald Waterman; Douglas Lenat (1983). Building Expert Systems. Addison-Wesley. ISBN 0- 201-10686-8.

[3] Kaczor, Krzystof; Szymon Bobek; Grzegorz J. Nalepa (12.05.2010). “Overview of Expert System Shells” (PDF). http://geist.agh.edu.pl/. Krakow, Poland: Institute of Automatics: AGH University of Science and Technology, Poland. Retrieved 5 December 2013. Check date values in: |date= (help)

53.3 External links

• Forward vs. Backward Chaining Explained at SemanticWeb.com Chapter 54

Freethought

Not to be confused with freedom of thought or free will. For the Ukrainian language newspaper published in , see The Free Thought. For the Dutch freethinkers association “The Free Thought”, see De Vrije Gedachte.

Freethought —or free thought*[1] —is a philosophical viewpoint which holds that positions regarding truth should be formed on the basis of logic, reason, and empiricism, rather than authority, tradition, revelation, or other .*[1]*[2]*[3] The cognitive application of freethought is known as “freethinking”, and practitioners of freethought are known as“freethinkers”.*[1]*[4] The term first came into use in the 17th century to indicate people who inquired into the basis of traditional religious beliefs. A free thinker is defined as a person who forms his or her own opinions about important subjects (such as religion and politics) instead of accepting what others say.*[1] Freethinkers are heavily committed to the use of scientific inquiry, and logic. The skeptical application of science implies freedom from the intellectually limiting effects of confirmation bias, cognitive bias, , popular culture, prejudice, or sectarianism. Atheist author Adam Lee defines freethought as thinking independent of revelation, tradition, established belief, and authority,*[5] also defining it as a “broader umbrella”than “that embraces a rainbow of unorthodoxy, religious , skepticism, and unconventional thinking.”*[6] The basic summarizing statement of the essay The Ethics of Belief by the 19th-century British mathematician and philosopher William Kingdon Clifford is: “It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence.”*[7] The essay became a rallying cry for freethinkers when published in the 1870s, and has been described as a point when freethinkers grabbed the moral high ground.*[8] Clifford was himself an organizer of freethought gatherings, the driving force behind the Congress of Liberal Thinkers held in 1878. Regarding religion, freethinkers often hold that there is insufficient evidence to support the existence of supernatural phenomena.*[9] According to the Freedom from Religion Foundation, “No one can be a freethinker who demands conformity to a bible, creed, or messiah. To the freethinker, revelation and are invalid, and orthodoxy is no guarantee of truth.”and “Freethinkers are convinced that religious claims have not withstood the tests of reason. Not only is there nothing to be gained by believing an untruth, but there is everything to lose when we sacrifice the indispensable tool of reason on the altar of superstition. Most freethinkers consider religion to be not only untrue, but harmful.”*[10] However, philosopher Bertrand Russell in his 1957 essay “The Value of Free Thought”wrote

What makes a freethinker is not his beliefs but the way in which he holds them. If he holds them because his elders told him they were true when he was young, or if he holds them because if he did not he would be unhappy, his thought is not free; but if he holds them because, after careful thought he finds a balance of evidence in their favor, then his thought is free, however odd his conclusions may seem.

Fred Edwords, former executive of the American Humanist Association, suggests that by Russell's definition, even liberal religionists who have challenged established orthodoxies might be considered freethinkers.*[11] In the 18th and 19th century, many thinkers regarded as freethinkers were deists, arguing that the nature of God can only be known from a study of nature rather than from religious revelation. In the 18th century,“deism”was as much

174 54.1. SYMBOL 175

of a 'dirty word' as“atheism”, and deists were often stigmatized as either atheists or at least as freethinkers by their Christian opponents.*[12]*[13] Deists today regard themselves as freethinkers, but are now arguably less prominent in the freethought movement than atheists.

54.1 Symbol

The pansy, symbol of freethought

The pansy serves as the long-established and enduring symbol of freethought; literature of the American Secular Union inaugurated its usage in the late 1800s. The reasoning behind the pansy as the symbol of freethought lies both in the flower's name and in its appearance. The pansy derives its name from the French word pensée, which means “thought”. It allegedly received this name because the flower is perceived by some to bear resemblance to a human face, and in mid-to-late summer it nods forward as if deep in thought.*[14]

54.2 History

54.2.1 Pre-modern movement

Critical thought has flourished in the Hellenistic Mediterranean, in the repositories of knowledge and wisdom in Ireland and in the Iranian civilizations (for example in the era of Khayyam (1048–1131) and his unorthodox Sufi Rubaiyat poems), and in other civilizations, such as the Chinese (note for example the seafaring renaissance of the Southern Song dynasty of 420–479),*[15] and on through heretical thinkers on esoteric or astrology, to the Renaissance and the Protestant Reformation. French physician and writer Rabelais celebrated “rabelaisian”freedom as well as good feasting and drinking (an 176 CHAPTER 54. FREETHOUGHT

expression and a symbol of freedom of the mind) in defiance of the hypocrisies of conformist orthodoxy in his utopian Thelema Abbey (from θέλημα: free “will”), the device of which was Do What Thou Wilt:

“So had Gargantua established it. In all their rule and strictest tie of their order there was but this one clause to be observed, Do What Thou Wilt; because free people ... act virtuously and avoid vice. They call this honor.”

When Rabelais's hero Pantagruel journeys to the “Oracle of The Div(in)e Bottle”, he learns the lesson of life in one simple word: “Trinch!", Drink! Enjoy the simple life, learn wisdom and knowledge, as a free human. Beyond puns, irony, and satire, Gargantua's prologue-metaphor instructs the reader to “break the bone and suck out the substance-full marrow”("la substantifique moëlle"), the core of wisdom.

54.2.2 Modern movements

The year 1600 is considered a landmark of the era of modern freethought. It was the year of the execution in Italy of , a former Dominican Monk, by the Inquisition.*[16]*[17]*[18]

England

The term free-thinker emerged toward the end of the 17th century in England to describe those who stood in opposi- tion to the institution of the Church, and of literal belief in the Bible. The beliefs of these individuals were centered on the concept that people could understand the world through consideration of nature. Such positions were formally documented for the first time in 1697 by William Molyneux in a widely publicized letter to , and more extensively in 1713, when Anthony Collins wrote his Discourse of Free-thinking, which gained substantial popularity. This essay attacks clergy of all churches and is a plea for deism. The Freethinker magazine was first published in Britain in 1881.

France

In France, the concept first appeared in publication in 1765 when Denis Diderot, Jean le Rond d'Alembert, and Voltaire included an article on Liberté de penser in their Encyclopédie.*[19] The European freethought concepts spread so widely that even places as remote as the Jotunheimen, in Norway, had well-known freethinkers such as Jo Gjende by the 19th century. François-Jean Lefebvre de la Barre (September 12, 1745 – July 1, 1766) was a young French nobleman, famous for having been tortured and beheaded before his body was burnt on a pyre along with Voltaire's Philosophical Dictionary. La Barre is often said to have been executed for not saluting a Roman Catholic religious procession, but the elements of the case were far more complex. In France, Lefebvre de la Barre is widely regarded a symbol of the victims of Christian religious intolerance; La Barre along with Jean Calas and Pierre-Paul Sirven, was championed by Voltaire. A second replacement statue to de la Barre stands nearby the Basilica of the Sacred Heart of Jesus of Paris at the summit of the butte Montmartre (itself named from the Temple of Mars), the highest point in Paris and an 18th arrondissement street nearby the Sacré-Cœur is also named after Lefebvre de la Barre.

Germany

In Germany, during the period 1815–1848 and before the March Revolution, the resistance of citizens against the dogma of the church increased. In 1844, under the influence of Johannes Ronge and Robert Blum, belief in the rights of man, tolerance among men, and grew, and by 1859 they had established the Bund Freireligiöser Gemeinden Deutschlands (literally Union of Free Religious Communities of Germany), an association of persons who consider themselves to be religious without adhering to any established and institutionalized church or sacerdotal cult. This union still exists today, and is included as a member in the umbrella organization of free humanists. In 1881 in Frankfurt am Main, Ludwig Büchner established the Deutscher Freidenkerbund (German Freethinkers League) as the first German organization for atheists and agnostics. In 1892 the Freidenker-Gesellschaft and in 1906 the Deutscher Monistenbund were formed.*[20] 54.2. HISTORY 177

Freethought organizations developed the “Jugendweihe”(literally Youth consecration), a secular "confirmation" ceremony, and atheist funeral rites.*[20]*[21] The Union of Freethinkers for was founded in 1905, and the Central Union of German Proletariat Freethinker in 1908. The two groups merged in 1927, becoming the German Freethinking Association in 1930.*[22] More “bourgeois”organizations declined after World War I, and “proletarian”Freethought groups proliferated, becoming an organization of socialist parties.*[20]*[23] European socialist freethought groups formed the Interna- tional of Proletarian Freethinkers (IPF) in 1925.*[24] Activists agitated for Germans to disaffiliate from their re- spective Church and for seculari-zation of elementary schools; between 1919–21 and 1930–32 more than 2.5 million Germans, for the most part supporters of the Social Democratic and Communist parties, gave up church member- ship.*[25] Conflict developed between radical forces including the Soviet League of the Militant Godless and Social Democratic forces in Western Europe led by Theodor Hartwig and Max Sievers.*[24] In 1930 the Soviet and allied delegations, following a walk-out, took over the IPF and excluded the former leaders.*[24] Following Hitler's rise to power in 1933, most freethought organizations were banned, though some right-wing groups that worked with so- called Völkische Bünde (literally “ethnic”associations with nationalist, xenophobic and very often racist ) were tolerated by the Nazis until the mid-1930s.*[20]*[23]

Belgium

Main article: Organized

The Université Libre de Bruxelles and the Vrije Universiteit Brussel, along with the two Circles of Free Inquiry (Dutch and French speaking), defend the freedom of critical thought, lay philosophy and ethics, while rejecting the argument of authority.

Netherlands

In the Netherlands, freethought has existed in organized form since of De Dageraad (now known as De Vrije Gedachte) in 1856. Among its most notable subscribing 19th century individuals were Johannes van Vloten, Multatuli, Adriaan Gerhard and Domela Nieuwenhuis. In 2009, Frans van Dongen established the Atheist-Secular Party, which takes a considerably restrictive view of religion and public religious expressions. Since the 19th century, Freethought in the Netherlands has become more well known as a political phenomenon through at least three currents: liberal freethinking, conservative freethinking, and classical freethinking. In other words, parties which identify as freethinking tend to favor non-doctrinal, rational approaches to their preferred ide- ologies, and arose as secular alternatives to both clerically aligned parties as well as labor-aligned parties. Common themes among freethinking political parties are “freedom”, “”, and "".

United States

The Free Thought movement first organized itself in the United States of America as the “Free Press Association” in 1827 in defense of George , publisher of The Correspondent, an early journal of Biblical criticism in an era when blasphemy convictions were still possible. Houston had helped found an Owenite community at Haverstraw, New York in 1826–27. The short-lived Correspondent was superseded by the Free Enquirer, the official organ of 's New Harmony community in Indiana, edited by Robert Dale Owen and by Fanny Wright between 1828 and 1832 in New York. During this time Robert Dale Owen sought to introduce the philosophic skepticism of the Free Thought movement into the Workingmen's Party in . The Free Enquirer's annual civic celebrations of Paine's birthday after 1825 finally coalesced in 1836 in the first national Free Thinkers organization, the“United States Moral and Philosophical Society for the General Diffusion of Useful Knowledge”. It was founded on August 1, 1836, at a national convention at the Lyceum in Saratoga Springs, with Isaac S. Smith of Buffalo, New York, as president. He was also the 1836 Equal Rights Party's candidate for Governor of New York. Smith had also been the Workingmen's Party candidate for Lt. Governor of New York in 1830. The Moral and Philosophical Society published The Beacon, edited by Gilbert Vale.*[26] Driven by the in the German states, the 19th century saw an immigration of German freethinkers and anti-clericalists to the United States (see Forty-Eighters). In the United States, they hoped to be able to live by 178 CHAPTER 54. FREETHOUGHT

Robert G. Ingersoll*[27]

their principles, without interference from government and church authorities.*[28] Many Freethinkers settled in German immigrant strongholds, including St. Louis, Indianapolis, Wisconsin, and Texas, where they founded the town of Comfort, Texas, as well as others.*[28] These groups of German Freethinkers referred to their organizations as Freie Gemeinden, or “free congregations” .*[28] The first Freie Gemeinde was established in St. Louis in 1850.*[29] Others followed in Pennsylvania, Califor- nia, Washington, D.C., New York, Illinois, Wisconsin, Texas, and other states.*[28]*[29] 54.2. HISTORY 179

Freethinkers tended to be liberal, espousing ideals such as racial, social, and sexual equality, and the abolition of .*[28] The “Golden Age of Freethought”in the US came in the late 1800s. The dominant organization was the National Liberal League which formed in 1876 in Philadelphia. This group re-formed itself in 1885 as the American Secular Union under the leadership of the eminent agnostic orator Robert G. Ingersoll. Following Ingersoll's death in 1899 the organization declined, in part due to lack of effective leadership.*[30] Freethought in the United States declined in the early twentieth century. Its anti-religious views alienated would-be sympathizers. The movement also lacked cohesive goals or beliefs. By the early twentieth century, most Freethought congregations had disbanded or joined other mainstream churches. The longest continuously operating Freethought congregation in America is the Free Congregation of Sauk County, Wisconsin, which was founded in 1852 and is still active as of 2016. It affiliated with the American Unitarian Association (now the Unitarian Universalist Association) in 1955.*[31] D. M. Bennett was the founder and publisher of The Truth Seeker in 1873, a radical freethought and reform American periodical. German Freethinker settlements were located in:

• Burlington, Racine County, Wisconsin*[28]

• Belleville, St. Clair County, Illinois

• Castell, Llano County, Texas

• Comfort, Kendall County, Texas

• Fond du Lac, Fond du Lac County, Wisconsin*[28]

• Frelsburg, Colorado County, Texas

• Hermann, Gasconade County, Missouri

• Jefferson, Jefferson County, Wisconsin*[28]

• Indianapolis, Indiana*[32]

• Latium, Washington County, Texas

• Manitowoc, Manitowoc County, Wisconsin*[28]

• Meyersville, DeWitt County, Texas

• Milwaukee, Wisconsin*[28]

• Millheim, Austin County, Texas

• Oshkosh, Winnebago County, Wisconsin*[28]

• Ratcliffe, DeWitt County, Texas

• Sauk City, Sauk County, Wisconsin*[28]*[31]

• Shelby, Austin County, Texas

• Sisterdale, Kendall County, Texas

• St. Louis, Missouri

• Tusculum, Kendall County, Texas

• Two Rivers, Manitowoc County, Wisconsin*[28]

• Watertown, Dodge County, Wisconsin*[28] 180 CHAPTER 54. FREETHOUGHT

Canada

The earliest known secular organization in English Canada is the Toronto Freethought Association, founded in 1873 by a handful of secularists. Reorganized in 1877 and again in 1881, when it was renamed the Toronto Secular Society, the group formed the nucleus of the Canadian Secular Union, established in 1884 to bring together freethinkers from across the country. A significant number of the early members appear to have been drawn from the educated labour “aristocracy,” including Alfred F. Jury, J. Ick Evans and J. I. Livingstone, all of whom were leading labour activists and secularists. The second president of the Toronto association was T. Phillips Thompson, a central figure in the city's labour and social reform movements during the 1880s and 1890s and arguably Canada's foremost late nineteenth-century labour intellectual. By the early 1880s, freethought organizations were scattered throughout southern Ontario and parts of Quebec, and elicited both urban and rural support. The principal organ of the freethought movement in Canada was Secular Thought (Toronto, 1887–1911). Founded and edited by English freethinker Charles Watts (1835–1906) during its first several years, the editorship was assumed by Toronto printer and publisher James Spencer Ellis in 1891 when Watts returned to England. In 1968 the Humanist Association of Canada was formed to serve as an umbrella group for Humanists, atheists, freethinkers, and to champion social justice issues and oppose religious influence on —most notably in the fight to make access to abortion free and legal in Canada. HAC, also known as Humanist Canada, is an active voice for Humanism in Canada and supports the activities of groups who wish to raise awareness about secular issues. The Canadian Secular Alliance is an active community.

Anarchism

In the United States,“freethought was a basically anti-christian, anti-clerical movement, whose purpose was to make the individual politically and spiritually free to decide for himself on religious matters. A number of contributors to Liberty were prominent figures in both freethought and . The individualist anarchist George MacDonald was a co-editor of Freethought and, for a time, The Truth Seeker. E.C. Walker was co-editor of the freethought/free love journal Lucifer, the Light-Bearer.”*[33]“Many of the anarchists were ardent freethinkers; reprints from freethought papers such as Lucifer, the Light-Bearer, Freethought and The Truth Seeker appeared in Liberty...The church was viewed as a common ally of the state and as a repressive force in and of itself.”*[33] In Europe, a similar development occurred in French and Spanish individualist anarchist circles.“Anticlericalism, just as in the rest of the libertarian movement, in another of the frequent elements which will gain relevance related to the measure in which the (French) Republic begins to have conflicts with the church...Anti-clerical discourse, frequently called for by the French individualist André Lorulot, will have its impacts in Estudios (a Spanish individualist anarchist publication). There will be an attack on institutionalized religion for the responsibility that it had in the past on negative developments, for its irrationality which makes it a counterpoint of philosophical and scientific progress. There will be a criticism of proselitism and ideological manipulation which happens on both believers and agnostics” .*[34] These tendencies will continue in French in the work and activism of Charles-Auguste Bontemps and others. In the Spanish individualist anarchist magazines Ética and Iniciales “there is a strong interest in publishing scientific news, usually linked to a certain atheist and anti-theist obsession, philosophy which will also work for pointing out the incompatibility between science and religion, faith, and reason. In this way there will be a lot of talk on Darwin´s theories or on the negation of the existence of the soul”.*[35] In 1901, Catalan anarchist and freethinker Francesc Ferrer i Guàrdia established“modern”or progressive schools in Barcelona in defiance of an educational system controlled by the Catholic Church.*[36] The schools' stated goal was to "educate the working class in a rational, secular and non-coercive setting”. Fiercely anti-clerical, Ferrer believed in “freedom in education”, education free from the authority of church and state.*[37] Ferrer's ideas generally, formed the inspiration for a series of Modern Schools in the United States,*[36] Cuba, South America and London. The first of these was started in New York City in 1911. It also inspired the Italian newspaper Università popolare, founded in 1901.*[36]

54.3 See also

• 54.3. SEE ALSO 181

• • Anti- • Apologetics • Atheism • Brights movement • Camp Quest • Conflict thesis • Critical • Critical thinking • Deism • Ethical movement • Evidentialism • Freedom from Religion Foundation • Freedom of thought • Freethought Association of Canada • Freethought Day • GAMPAC (Godless Americans PAC) • Golden Age of Freethought • Individualism • Infidel • Internet Infidels • • Libertine • (philosophy) • Naturalistic • • Occam's razor • • Pantheism • Philosophical • Positivism • Rationalism • • • • Secular Review 182 CHAPTER 54. FREETHOUGHT

• Secular Thought • Secularism • Skepticism • Spiritual but not religious • The Freethinker (journal) • Unitarian Universalism

54.4 Notes and references

[1] “Freethinker - Definition of freethinker by Merriam-Webster”. Retrieved 12 June 2015.

[2] “Free thought - Define Free thought at Dictionary.com”. Dictionary.com. Retrieved 12 June 2015.

[3] http://www.iheu.org/glossary/12#letterf

[4] “Nontracts”. Retrieved 12 June 2015.

[5] “What Is Freethought?". Daylight Atheism. Retrieved 12 June 2015.

[6] Adam Lee. “9 Great Freethinkers and Religious Dissenters in History”. Big Think. Retrieved 12 June 2015.

[7] William Kingdon Clifford, The Ethics of Belief (1879 [1877]).

[8] Becker, Lawrence and Charlotte (2013). Encyclopedia of Ethics (article on “agnosticism”). Routledge. p. 44. ISBN 9781135350963.

[9] Hastings, James. Encyclopedia of Religion.

[10] “What is a Freethinker? - Freedom From Religion Foundation”. Retrieved 12 June 2015.

[11] “Saga Of Freethought And Its Pioneers”. American Humanist Association. Retrieved 12 June 2015.

[12] James E. Force, Introduction (1990) to An Account of the Growth of Deism in England (1696) by William Stephens

[13] Aveling, Francis, ed. (1908). “Deism”. The Catholic Encyclopedia. Retrieved 2012-10-10. The deists were what nowadays would be called freethinkers, a name, indeed, by which they were not infrequently known; and they can only be classed together wholly in the main attitude that they adopted, viz. in agreeing to cast off the trammels of authoritative religious teaching in favour of a free and purely rationalistic speculation.... Deism, in its every manifestation was opposed to the current and traditional teaching of revealed religion.

[14] A Pansy For Your Thoughts, by Annie Laurie Gaylor, Freethought Today, June/July 1997

[15] Chinese History – Song Dynasty 宋 (www.chinaknowledge.de)

[16] Gatti, Hilary (2002). Giordano Bruno and Renaissance Science: Broken Lives and Organizational Power. Ithaca, New York: Cornell University Press. pp. 18–19. Retrieved 21 March 2014. For Bruno was claiming for the philosopher a principle of free thought and inquiry which implied an entirely new concept of authority: that of the individual intellect in its serious and continuing pursuit of an autonomous inquiry…It is impossible to understand the issue involved and to evaluate justly the stand made by Bruno with his life without appreciating the question of free thought and liberty of expression. His insistence on placing this issue at the center of both his work and of his defense is why Bruno remains so much a figure of the modern world. If there is, as many have argued, an intrinsic link between science and liberty of inquiry, then Bruno was among those who guaranteed the future of the newly emerging sciences, as well as claiming in wider terms a general principle of free thought and expression.

[17] Montano, Aniello (24 November 2007). Antonio Gargano, ed. Le deposizioni davanti al tribunale dell'Inquisizione. Napoli: La Città del Sole. p. 71. In Rome, Bruno was imprisoned for seven years and subjected to a difficult trial that analyzed, minutely, all his philosophical ideas. Bruno, who in had been willing to recant some theses, become increasingly resolute and declared on 21 December 1599 that he 'did not wish to repent of having too little to repent, and in fact did not know what to repent.' Declared an unrepentant heretic and excommunicated, he was burned alive in the Campo dei Fiori in Rome on 17 February 1600. On the stake, along with Bruno, burned the hopes of many, including philosophers and scientists of good faith like Galileo, who thought they could reconcile religious faith and scientific research, while belonging to an ecclesiastical organization declaring itself to be the custodian of absolute truth and maintaining a cultural militancy requiring continual commitment and suspicion. 54.5. FURTHER READING 183

[18] Birx, James (11 November 1997). “Giordano Bruno”. Mobile Alabama Harbinger. Retrieved 28 April 2014. To me, Bruno is the supreme martyr for both free thought and critical inquiry…Bruno's critical writings, which pointed out the hypocrisy and bigotry within the Church, along with his tempestuous personality and undisciplined behavior, easily made him a victim of the religious and philosophical intolerance of the 16th century. Bruno was excommunicated by the Catholic, Lutheran and Calvinist Churches for his heretical beliefs. The Catholic hierarchy found him guilty of infidelity and many errors, as well as serious crimes of heresy…Bruno was burned to death at the stake for his pantheistic stance and cosmic perspective. [19] “ARTFL Encyclopédie Search Results”. Retrieved 12 June 2015. [20] Bock, Heike (2006). “Secularization of the modern conduct of life? Reflections on the religiousness of early modern Europe”. In Hanne May. Religiosität in der säkularisierten Welt. VS Verlag fnr Sozialw. p. 157. ISBN 3-8100-4039-8. [21] Reese, Dagmar (2006). Growing up female in . Ann Arbor, Mich: University of Press. p. 160. ISBN 0-472-06938-1. [22] Reinhalter, Helmut (1999). “Freethinkers”. In Bromiley, Geoffrey William; Fahlbusch, Erwin. The encyclopedia of Christianity. Grand Rapids, MI: Wm. B. Eerdmans. ISBN 90-04-11695-8. [23] Kaiser, Jochen-Christoph (2003). Christel Gärtner, ed. Atheismus und religiöse Indifferenz. Organisierter Atheismus. VS Verlag. ISBN 978-3-8100-3639-1. [24] Peris, Daniel (1998). Storming the heavens: the Soviet League of the Militant Godless. Ithaca, N.Y: Cornell University Press. pp. 110–11. ISBN 0-8014-3485-8. [25] Lamberti, Marjorie (2004). Politics Of Education: Teachers and School Reform in Weimar Germany (Monographs in German History). Providence: Berghahn Books. p. 185. ISBN 1-57181-299-7. [26] Hugins, Walter (1960). Jacksonian Democracy and the Working Class: A Study of the New York Workingmen's Movement 1829-1837. Stanford: Stanford University Press. pp. 36–48. [27] Brandt, Eric T., and Timothy Larsen (2011). “The Old Atheism Revisited: Robert G. Ingersoll and the Bible”. Journal of the Historical Society 11 (2): 211–238. doi:10.1111/j.1540-5923.2011.00330.x. [28] “Freethinkers in Wisconsin”. Dictionary of Wisconsin History. 2008. Retrieved 2008-07-27. [29] Demerath, N. J. III and Victor Thiessen, “On Spitting Against the Wind: Organizational Precariousness and American Irreligion,”The American Journal of Sociology, 71: 6 (May, 1966), 674–687. [30] “National Liberal League”. The Freethought Trail. freethought-trail.org. Retrieved 9 March 2014. [31] “History of the Free Congregation of Sauk County: The “Freethinkers”Story”. Free Congregation of Sauk County. April 2009. Retrieved 2012-02-05. [32] “The Turners, Forty-eighters and Freethinkers”. Freedom from Religion Foundation. July 2002. Retrieved 2008-07-27. [33] “The Journal of Libertarian Studies” (PDF). Mises Institute. Retrieved 12 June 2015. [34] Xavier Diez. El anarquismo individualista en España (1923-1939) Virus Editorial. 2007. pg. 143 [35] Xavier Diez. El anarquismo individualista en España (1923-1939) Virus Editorial. 2007. pg. 152 [36] Geoffrey C. Fidler (Spring–Summer 1985). “The Escuela Moderna Movement of Francisco Ferrer: “Por la Verdad y la Justicia"". History of Education Quarterly (History of Education Society) 25 (1/2): 103–132. doi:10.2307/368893. JSTOR 368893. [37] “Francisco Ferrer's Modern School”. Flag.blackened.net. Retrieved 2010-09-20.

54.5 Further reading

• Jacoby, Susan (2004). Freethinkers: a history of American secularism. New York: Metropolitan Books. ISBN 0-8050-7442-2 • Royle, Edward (1974). Victorian Infidels: the origins of the British Secularist Movement, 1791–1866. Manch- ester: Manchester University Press. ISBN 0-7190-0557-4 • Royle, Edward (1980). Radicals, Secularists and Republicans: popular freethought in Britain, 1866–1915. Manchester: Manchester University Press. ISBN 0-7190-0783-6 • Tribe, David (1967). 100 Years of Freethought. London: Elek Books. 184 CHAPTER 54. FREETHOUGHT

54.6 External links

• Freethinker

• A History of Freethought • Young Freethought

• "Freethinker". New International Encyclopedia. 1905. Chapter 55

Graphoid

A graphoid is a set of statements of the form, "X is irrelevant to Y given that we know Z" where X, Y and Z are sets of variables. The notion of “irrelevance”and “given that we know”may obtain different interpretations, including probabilistic, relational and correlational, depending on the application. These interpretations share com- mon properties that can be captured by paths in graphs (hence the name “graphoid”). The theory of graphoids characterizes these properties in a finite set of axioms that are common to informational irrelevance and its graphical representations.

55.1 History

Judea Pearl and Azaria Paz*[1] coined the term “graphoids”after discovering that a set of axioms that govern conditional independence in probability theory is shared by undirected graphs. Variables are represented as nodes in a graph in such a way that variable sets X and Y are independent conditioned on Z in the distribution whenever node set Z separates X from Y in the graph. Axioms for conditional independence in probability were derived earlier by A. Philip Dawid*[2] and Wolfgang Spohn.*[3] The correspondence between dependence and graphs was later extended to directed acyclic graphs (DAGs)*[4]*[5]*[6] and to other models of dependency.*[1]*[7]

55.2 Definition

A dependency model M is a subset of triplets (X,Z,Y) for which the predicate I(X,Z,Y): X is independent of Y given Z, is true. A graphoid is defined as a dependency model that is closed under the following five axioms:

1. Symmetry: I(X,Z,Y ) ⇔ I(Y,Z,X)

2. Decomposition: I(X,Z,Y ∪ W ) ⇒ I(X,Z,Y ) & I(X,Z,W )

3. Weak Union: I(X,Z,Y ∪ W ) ⇒ I(X,Z ∪ W, Y )

4. Contraction: I(X,Z,Y ) & I(X,Z ∪ Y,W ) ⇒ I(X,Z,Y ∪ W )

5. Intersection: I(X,Z ∪ W, Y ) & I(X,Z ∪ Y,W ) ⇒ I(X,Z,Y ∪ W )

A semi-graphoid is a dependency model closed under (i)–(iv). These five axioms together are known as the graphoid axioms.*[8] Intuitively, the weak union and contraction properties mean that irrelevant information should not alter the relevance status of other propositions in the system; what was relevant remains relevant and what was irrelevant remains irrelevant.*[8]

55.3 Types of graphoids

185 186 CHAPTER 55. GRAPHOID

55.3.1 Probabilistic graphoids*[1]*[7]

Conditional independence, defined as

I(X,Z,Y ) iff P (X | Y,Z) = P (X | Z)

is a semi-graphoid which becomes a full graphoid when P is strictly positive.

55.3.2 Correlational graphoids*[1]*[7]

A dependency model is a correlational graphoid if in some probability function we have,

Ic(X,Y,Z) ⇔ ρxy.z = 0 every for x ∈ X and y ∈ Y

where ρxy.z is the partial correlation between x and y given set Z. In other words, the linear estimation error of the variables in X using measurements on Z would not be reduced by adding measurements of the variables in Y, thus making Y irrelevant to the estimation of X. Correlational and probabilistic dependency models coincide for normal distributions.

55.3.3 Relational graphoids*[1]*[7]

A dependency model is a relational graphoid if it satisfies

P (X,Z) > 0 & P (Y,Z) > 0 =⇒ P (X,Y,Z) > 0.

In words, the range of values permitted for X is not restricted by the choice of Y, once Z is fixed. Independence statements belonging to this model are similar to embedded multi-valued dependencies (EMVD s) in databases.

55.3.4 Graph-induced graphoids

If there exists an undirected graph G such that,

I(X,Z,Y ) ⇔ ⟨X,Z,Y ⟩G, then the graphoid is called as graph-induced. In other words, there exists an undirected graph G such that every independence statement in M is reflected as a vertex separation in G and vice versa. A necessary and sufficient condition for a dependency model to be a graph induced graphoid is that it satisfies the following axioms: symmetry, decomposition, intersection, strong union and transitivity. Strong union states that,

I(X,Z,Y ) =⇒ I(X,Z ∪ W, Y )

Transitivity states that

I(X,Z,Y ) =⇒ I(X, Z, γ) or I(γ, Z, Y ) ∀ γ ∈/ X ∪ Y ∪ Z.

Graphoid axioms constitute a complete characterization of undirected graphs.*[9] 55.4. INCLUSION AND CONSTRUCTION 187

55.3.5 DAG-induced graphoids

A graphoid is termed DAG-induced if there exists a directed acyclic graph D such that I(X,Z,Y ) ⇔ ⟨X,Z,Y ⟩D where ⟨X,Z,Y ⟩D stands for d-separation in D. d-separation (d-connotes“directional”) extends the notion of vertex separation from undirected graphs to directed acyclic graphs. It permits the reading of conditional independencies from the structure of Bayesian networks. However, conditional independencies in a DAG cannot be completely characterized by a finite set of axioms. *[10]

55.4 Inclusion and construction

Graph-induced and DAG-induced graphoids are both contained in probabilistic graphoids.*[11] This means that for every graph G there exists a probability distribution P such that every conditional independence in P is represented in G, and vice versa. The same is true for DAGs. However, there are probabilistic distributions that are not semi- graphoids and, moreover, there is no finite axiomatization for probabilistic dependencies. *[12] Thomas Verma showed that every semi-graphoid has a recursive way of constructing a DAG in which every d- separation is valid.*[13] The construction is similar to that used in Bayes networks and goes as follows:

1. Arrange the variables in some arbitrary order 1, 2,...,i,...,N and, starting with i = 1,

2. choose for each node i a set of nodes PAi such that i is independent on all its predecessors, 1, 2,...,i − 1, conditioned on PAi.

3. Draw arrows from PAi to i and continue.

The DAG created by this construction will represent all the conditional independencies that follow from those used in the construction. Furthermore, every d-separation shown in the DAG will be a valid conditional independence in the graphoid used in the construction.

55.5 References

[1] Pearl, Judea; Paz, Azaria (1985). “Graphoids: A Graph-Based Logic for Reasoning About Relevance Relations”.

[2] Dawid, A. Philip (1979). “Conditional independence in statistical theory”. Journal of the Royal Statistical Society, Series B (Methodological): 1–31.

[3] Spohn, Wolfgang (1980). “Stochastic independence, causal independence, and shieldability”. Journal of Philosophical Logic 9: 73–99. doi:10.1007/bf00258078.

[4] Pearl, Judea (1986). “Fusion, propagation and structuring in belief networks”. Artificial Intelligence 29 (3): 241–288. doi:10.1016/0004-3702(86)90072-x.

[5] Verma, Thomas; Pearl, Judea (1988).“Causal networks: Semantics and expressiveness”. Proceedings of the 4th Workshop on Uncertainty in Artificial Intelligence: 352–359.

[6] Lauritzen, S.L. (1996). Graphical Models. Oxford: Clarendon Press.

[7] Geiger, Dan (1990). “Graphoids: A Qualitative Framework for Probabilistic Inference”(PhD Dissertation, Technical Report R-142, Computer Science Department, University of California, Los Angeles). line feed character in |format= at position 59 (help)

[8] Pearl, Judea (1988). Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann.

[9] A. Paz, J. Pearl, and S. Ur,“A New Characterization of Graphs Based on Interception Relations”Journal of Graph Theory, Vol. 22, No. 2, 125-136, 1996.

[10] Geiger, D. (1987).“The non-axiomatizability of dependencies in directed acyclic graphs”(PDF). UCLA Computer Science Tech Report R-83.

[11] Geiger, D.; Pearl, J. (1993). “Logical and algorithmic properties of conditional independence and graphical models”. The Annals of Statistics 21 (4): 2001–2021. doi:10.1214/aos/1176349407. 188 CHAPTER 55. GRAPHOID

[12] Studeny, M. (1992). Kubik, S.; Visek, J.A., eds. “Conditional independence relations have no finite complete charac- terization”. Information Theory, Statistical Decision Functions and Random Processes. Transactions of the 11th Prague Conference (Dordrecht: Kluwer) B: 377–396.

[13] Verma, T.; Pearl, J. (1990). Shachter, R.; Levitt, T.S.; Kanal, L.N., eds.“Causal Networks: Semantics and Expressiveness” . Uncertainty in AI 4 (Elsevier Science Publishers): 69–76. Chapter 56

Herbrandization

The Herbrandization of a logical formula (named after Jacques Herbrand) is a construction that is dual to the Skolemization of a formula. Thoralf Skolem had considered the Skolemizations of formulas in prenex form as part of his proof of the Löwenheim-Skolem theorem (Skolem 1920). Herbrand worked with this dual notion of Her- brandization, generalized to apply to non-prenex formulas as well, in order to prove Herbrand's theorem (Herbrand 1930). The resulting formula is not necessarily equivalent to the original one. As with Skolemization which only preserves satisfiability, Herbrandization being Skolemization's dual preserves validity: the resulting formula is valid if and only if the original one is. Let F be a formula in the language of first-order logic. We may assume that F contains no variable that is bound by two different quantifier occurrences, and that no variable occurs both bound and free. (That is, F could be relettered to ensure these conditions, in such a way that the result is an equivalent formula). The Herbrandization of F is then obtained as follows:

• First, replace any free variables in F by constant symbols.

• Second, delete all quantifiers on variables that are either (1) universally quantified and within an even number of negation signs, or (2) existentially quantified and within an odd number of negation signs.

• Finally, replace each such variable v with a function symbol fv(x1, . . . , xk) , where x1, . . . , xk are the variables that are still quantified, and whose quantifiers govern v .

For instance, consider the formula F := ∀y∃x[R(y, x) ∧ ¬∃zS(x, z)] . There are no free variables to replace. The variables y, z are the kind we consider for the second step, so we delete the quantifiers ∀y and ∃z . Finally, we then replace y with a constant cy (since there were no other quantifiers governing y ), and we replace z with a function symbol fz(x) :

H F = ∃x[R(cy, x) ∧ ¬S(x, fz(x))]. The Skolemization of a formula is obtained similarly, except that in the second step above, we would delete quantifiers on variables that are either (1) existentially quantified and within an even number of , or (2) universally quantified and within an odd number of negations. Thus, considering the same F from above, its Skolemization would be:

S F = ∀y[R(y, fx(y)) ∧ ¬∃zS(fx(y), z)]. To understand the significance of these constructions, see Herbrand's theorem or the Löwenheim-Skolem theorem.

56.1 See also

• Predicate functor logic

189 190 CHAPTER 56. HERBRANDIZATION

56.2 References

• Skolem, T. “Logico-combinatorial investigations in the satisfiability or provability of mathematical propo- sitions: A simplified proof of a theorem by L. Löwenheim and generalizations of the theorem”. (In van Heijenoort 1967, 252-63.)

• Herbrand, J. “Investigations in proof theory: The properties of true propositions”. (In van Heijenoort 1967, 525-81.)

• van Heijenoort, J. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press, 1967. Chapter 57

History of logic

The history of logic is the study of the development of the science of valid inference (logic). Formal logics were developed in ancient times in China, , and Greece. Greek methods, particularly Aristotelian logic (or term logic) as found in the Organon, found wide application and acceptance in science and mathematics for millenia.*[1] The Stoics, especially , were the first to develop predicate logic. Aristotle's logic was further developed by Christian and Islamic philosophers in the Middle Ages, such as or , reaching a high point in the mid-fourteenth century. The period between the fourteenth century and the beginning of the nineteenth century was largely one of decline and neglect, and is regarded as barren by at least one historian of logic.*[2] Empirical methods seemed to rule the day, as evidenced by Bacon's Novum Organon. Logic was revived in the mid-nineteenth century, at the beginning of a revolutionary period when the subject devel- oped into a rigorous and formal discipline whose exemplar was the exact method of proof used in mathematics, a hearkening back to the Greek tradition.*[3] The development of the modern “symbolic”or “mathematical”logic during this period by the likes of Boole, Frege, Russell, and Peano is the most significant in the two-thousand-year history of logic, and is arguably one of the most important and remarkable events in human intellectual history.*[4] Progress in mathematical logic in the first few decades of the twentieth century, particularly arising from the work of Gödel and Tarski, had a significant impact on analytic philosophy and philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic, , deontic logic, and relevance logic.

57.1 Logic in the West

57.1.1 Prehistory of logic

Valid reasoning has been employed in all periods of . However, logic studies the principles of valid reasoning, inference and demonstration. It is probable that the idea of demonstrating a conclusion first arose in connection with geometry, which originally meant the same as “land measurement”.*[5] The ancient Egyptians discovered geometry, including the formula for the volume of a truncated pyramid.*[6] Ancient Babylon was also skilled in mathematics. Esagil-kin-apli's medical Diagnostic Handbook in the 11th century BC was based on a logical set of axioms and assumptions,*[7] while Babylonian astronomers in the 8th and 7th centuries BC employed an internal logic within their predictive planetary systems, an important contribution to the philosophy of science.*[8]

57.1.2

While the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative proof. Both Thales and seem aware of geom- etry's methods. Further evidence that early Greek thinkers were concerned with the principles of reasoning is found in the fragment called dissoi logoi, probably written at the beginning of the fourth century BC. This is part of a protracted debate about truth and falsity.*[9] In the case of the classical Greek city-states, interest in argumentation was also stimulated by the activities of the Rhetoricians or Orators and the , who used arguments to defend or attack a thesis,

191 192 CHAPTER 57. HISTORY OF LOGIC

A

B C

Proof of the in Euclid's Elements

both in legal and political contexts.*[10]

Thales

It is said Thales, most widely regarded as the first philosopher in the Greek tradition,*[11]*[12] measured the height of the pyramids by their shadows at the moment when his own shadow was equal to his height. Thales was said to have had a sacrifice in celebration of discovering Thales' Theorem just as Pythagoras had the Pythagorean Theorem.*[13] Thales is the first known individual to use deductive reasoning applied to geometry, by deriving four corollaries to his theorem, and the first known individual to whom a mathematical discovery has been attributed.*[14] Indian and Babylonian mathematicians knew his theorem for special cases before he proved it.*[15] It is believed that Thales learned that an angle inscribed in a semicircle is a right angle during his travels to Babylon.*[16] 57.1. LOGIC IN THE WEST 193

Thales Theorem

Pythagoras

Before 520 BC, on one of his visits to or Greece, Pythagoras might have met the c. 54 years older Thales.*[17] The systematic study of proof seems to have begun with the school of Pythagoras (i. e. the Pythagoreans) in the late sixth century BC.*[6] Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize form rather than matter.*[18] Fragments of early proofs are preserved in the works of Plato and Aristotle,*[19] and the idea of a deductive system was probably known in the Pythagorean school and the Platonic Academy.*[6] The proofs of Euclid of Alexandria are a of Greek geometry. The three basic principles of geometry are as follows:

• Certain propositions must be accepted as true without demonstration; such a proposition is known as an axiom of geometry.

• Every proposition that is not an axiom of geometry must be demonstrated as following from the axioms of geometry; such a demonstration is known as a proof or a “derivation”of the proposition.

• The proof must be formal; that is, the derivation of the proposition must be independent of the particular subject matter in question.*[6]

The

Separately from geometry, the idea of a standard argument pattern is found in the method of proof known as , which was used by , a pre-Socratic philosopher of the fifth century BC. This is the 194 CHAPTER 57. HISTORY OF LOGIC

Parmenides technique of drawing an obviously false (that is,“absurd”) conclusion from an assumption, thus demonstrating that the assumption is false.*[20] His teacher, the elder Parmenides has been called the inventor of logic,*[21] with the student Zeno as the first to apply this art.*[22] Parmenides pondered that nonbeing must in some sense be, otherwise what is it that there is not? Plato's Parmenides portrays Zeno as claiming to have written a book defending the of Parmenides by demonstrating the absurd consequence of assuming that there is plurality. Zeno famously used this method to develop his paradoxes in his arguments against motion. Other philosophers who practised such dialectic reasoning were the “minor Socratics”, including , who were probably followers of Parmenides and Zeno. The members of this school were called “”(from a Greek word meaning “to discuss”). 57.1. LOGIC IN THE WEST 195

Heraclitus

The writing of (c. 535 – c. 475 BC) was the first place where the word logos was given special attention in ,*[23] Parmenides held that all is one and nothing changes; Heraclitus held everything changes. He is known for his obscure sayings.

This logos holds always but humans always prove unable to understand it, both before hearing it and when they have first heard it. For though all things come to be in accordance with this logos, humans are like the inexperienced when they experience such words and deeds as I set out, distinguishing each in accordance with its nature and saying how it is. But other people fail to notice what they do when awake, just as they forget what they do while asleep. —Diels-Kranz, 22B1

Plato Let no one ignorant of geometry enter —Inscribed over the entrance to Plato's Academy.

None of the surviving works of the great fourth-century philosopher Plato (428–347 BC) include any formal logic,*[24] but they include important contributions to the field of philosophical logic. Plato raises three questions:

• What is it that can properly be called true or false? • What is the nature of the connection between the assumptions of a valid argument and its conclusion? • What is the nature of definition?

The first question arises in the dialogue Theaetetus, where Plato identifies thought or opinion with talk or discourse (logos).*[25] The second question is a result of Plato's theory of Forms. Forms are not things in the ordinary sense, nor strictly ideas in the mind, but they correspond to what philosophers later called universals, namely an abstract entity common to each set of things that have the same name. In both the Republic and the Sophist, Plato suggests that the necessary connection between the assumptions of a valid argument and its conclusion corresponds to a necessary connection between “forms”.*[26] The third question is about definition. Many of Plato's dialogues concern the search for a definition of some important concept (justice, truth, the Good), and it is likely that Plato was impressed by the importance of definition in mathematics.*[27] What underlies every definition is a Platonic Form, the common nature present in different particular things. Thus, a definition reflects the ultimate object of understanding, and is the foundation of all valid inference. This had a great influence on Plato's student Aristotle, in particular Aristotle's notion of the essence of a thing.*[28]

Aristotle

The logic of Aristotle, and particularly his theory of the syllogism, has had an enormous influence in Western thought.*[29] His logical works, called the Organon, are the earliest formal study of logic that have come down to modern times. Though it is difficult to determine the dates, the probable order of writing of Aristotle's logical works is:

• The Categories, a study of the ten kinds of primitive term. • The Topics (with an appendix called On Sophistical Refutations), a discussion of dialectics. • On Interpretation, an analysis of simple categorical propositions into simple terms, negation, and signs of quan- tity. It also contains a comprehensive treatment of the notions of opposition and conversion; chapter 7 is at the origin of the square of opposition (or logical square); chapter 9 contains the beginning of modal logic. • The , a formal analysis of what makes a syllogism (a valid argument, according to Aristotle). • The Posterior Analytics, a study of scientific demonstration, containing Aristotle's mature views on logic. 196 CHAPTER 57. HISTORY OF LOGIC

Plato's academy

These works are of outstanding importance in the history of logic. Aristotle was the first logician to attempt a systematic analysis of logical syntax, of noun (or term), and of verb. He was the first formal logician, in that he demonstrated the principles of reasoning by employing variables to show the underlying of an argument. He was looking for relations of dependence which characterize necessary inference, and distinguished the validity of these relations, from the truth of the premises (the soundness of the argument). He was the first to deal with the principles of contradiction and excluded middle in a systematic way.*[30] In the Categories, he attempts to discern all the possible things to which a term can refer; this idea underpins his philosophical work Metaphysics, which itself had a profound influence on Western thought. The Prior Analytics contains his exposition of the“syllogism”, where three important principles are applied for the first time in history: the use of variables, a purely formal treatment, and the use of an axiomatic system. He also developed a theory of non-formal logic (i.e., the theory of fallacies), which is presented in Topics and Sophistical Refutations.*[30]

Stoics

The other great school of Greek logic is that of the Stoics.*[31] traces its roots back to the late 5th century BC philosopher Euclid of Megara, a pupil of Socrates and slightly older contemporary of Plato. His pupils and successors were called "Megarians", or “Eristics”, and later the “Dialecticians”. The two most important 57.1. LOGIC IN THE WEST 197

Aristotle

dialecticians of the were Diodorus Cronus and Philo, who were active in the late 4th century BC. The Stoics adopted the Megarian logic and systemized it. The most important member of the school was Chrysippus (c. 278–c. 206 BC), who was its third head, and who formalized much of Stoic doctrine. He is supposed to have written over 700 works, including at least 300 on logic, almost none of which survive.*[32]*[33] Unlike with Aristotle, we have no complete works by the Megarians or the early Stoics, and have to rely mostly on accounts (sometimes hostile) by later sources, including prominently Diogenes Laertius, Sextus Empiricus, , Aulus Gellius, Alexander of Aphrodisias, and Cicero.*[34] 198 CHAPTER 57. HISTORY OF LOGIC

Aristotle's logic was still influential in the Renaissance

Three significant contributions of the Stoic school were (i) their account of modality, (ii) their theory of the , and (iii) their account of meaning and truth.*[35]

• Modality. According to Aristotle, the Megarians of his day claimed there was no distinction between potentiality and actuality.*[36] Diodorus Cronus defined the possible as that which either is or will be, the impossible as what will not be true, and the contingent as that which either is already, or will be false.*[37] Diodorus is also famous for what is known as his Master argument, which states that each pair of the following 3 propositions contradicts the third proposition:

• Everything that is past is true and necessary. • The impossible does not follow from the possible. • What neither is nor will be is possible. Diodorus used the plausibility of the first two to prove that nothing is possible if it neither is nor will be true.*[38] Chrysippus, by contrast, denied the second premise and said that the impossible could follow from the possible.*[39]

• Conditional statements. The first logicians to debate conditional statements were Diodorus and his pupil Philo of Megara. Sextus Empiricus refers three times to a debate between Diodorus and Philo. Philo regarded a conditional as true unless it has both a true antecedent and a false consequent. Precisely, let T0 and T1 be true statements, and let F0 and F1 be false statements; then, according to Philo, each of the following conditionals is a true statement, because it is not the case that the consequent is false while the antecedent is true (it is not the case that a false statement is asserted to follow from a true statement):

• If T0, then T1

• If F0, then T0

• If F0, then F1 57.1. LOGIC IN THE WEST 199

Chrysippus of Soli

The following conditional does not meet this requirement, and is therefore a false statement according to Philo:

• If T0, then F0 Indeed, Sextus says “According to [Philo], there are three ways in which a conditional may be true, and one in which it may be false.”*[40] Philo's criterion of truth is what would now be called a truth- functional definition of “if ... then"; it is the definition used in modern logic.

In contrast, Diodorus allowed the validity of conditionals only when the antecedent clause could never lead to an untrue conclusion.*[40]*[41]*[42] A century later, the Stoic philosopher Chrysippus attacked 200 CHAPTER 57. HISTORY OF LOGIC

the assumptions of both Philo and Diodorus.

• Meaning and truth. The most important and striking difference between Megarian-Stoic logic and Aristotelian logic is that Megarian-Stoic logic concerns propositions, not terms, and is thus closer to modern propositional logic.*[43] The Stoics distinguished between utterance (phone), which may be noise, speech (lexis), which is articulate but which may be meaningless, and discourse (logos), which is meaningful utterance. The most original part of their theory is the idea that what is expressed by a sentence, called a lekton, is something real; this corresponds to what is now called a proposition. Sextus says that according to the Stoics, three things are linked together: that which signifies, that which is signified, and the object; for example, that which signifies is the word Dion, and that which is signified is what Greeks understand but barbarians do not, and the object is Dion himself.*[44]

57.2 Medieval logic

57.2.1 Logic in the Middle East

Main article: Logic in See also: Avicennian logic The works of Al-Kindi, Al-Farabi, , Al-Ghazali, Averroes and other Muslim logicians were based on Aristotelian logic and were important in communicating the ideas of the ancient world to the medieval West.*[45] Al- Farabi (Alfarabi) (873–950) was an Aristotelian logician who discussed the topics of future contingents, the number and relation of the categories, the relation between logic and grammar, and non-Aristotelian forms of inference.*[46] Al-Farabi also considered the theories of conditional syllogisms and analogical inference, which were part of the Stoic tradition of logic rather than the Aristotelian.*[47] Ibn Sina (Avicenna) (980–1037) was the founder of Avicennian logic, which replaced Aristotelian logic as the dom- inant system of logic in the Islamic world,*[48] and also had an important influence on Western medieval writers such as Albertus Magnus.*[49] Avicenna wrote on the hypothetical syllogism*[50] and on the propositional calculus, which were both part of the Stoic logical tradition.*[51] He developed an original“temporally modalized”syllogistic theory, involving temporal logic and modal logic.*[46] He also made use of inductive logic, such as the methods of agreement, difference, and concomitant variation which are critical to the scientific method.*[50] One of Avicenna's ideas had a particularly important influence on Western logicians such as William of Ockham: Avicenna's word for a meaning or notion (ma'na), was translated by the scholastic logicians as the Latin intentio; in medieval logic and epistemology, this is a sign in the mind that naturally represents a thing.*[52] This was crucial to the development of Ockham's : A universal term (e.g., “man”) does not signify a thing existing in reality, but rather a sign in the mind (intentio in intellectu) which represents many things in reality; Ockham cites Avicenna's commentary on Metaphysics V in support of this view.*[53] Fakhr al-Din al-Razi (b. 1149) criticised Aristotle's "first figure" and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill (1806–1873).*[54] Al-Razi's work was seen by later Islamic scholars as marking a new direction for Islamic logic, towards a Post-Avicennian logic. This was further elaborated by his student Afdaladdîn al-Khûnajî (d. 1249), who developed a form of logic revolving around the subject matter of conceptions and assents. In response to this tradition, Nasir al-Din al-Tusi (1201–1274) began a tradition of Neo-Avicennian logic which remained faithful to Avicenna's work and existed as an alternative to the more dominant Post-Avicennian school over the following centuries.*[55] The Illuminationist school was founded by Shahab al-Din Suhrawardi (1155–1191), who developed the idea of“deci- sive necessity”, which refers to the reduction of all modalities (necessity, possibility, contingency and impossibility) to the single mode of necessity.*[56] Ibn al-Nafis (1213–1288) wrote a book on Avicennian logic, which was a commen- tary of Avicenna's Al-Isharat (The Signs) and Al-Hidayah (The Guidance).*[57] Ibn Taymiyyah (1263–1328), wrote the Ar-Radd 'ala al-Mantiqiyyin, where he argued against the usefulness, though not the validity, of the syllogism*[58] and in favour of inductive reasoning.*[54] Ibn Taymiyyah also argued against the certainty of syllogistic arguments and in favour of analogy; his argument is that concepts founded on induction are themselves not certain but only prob- able, and thus a syllogism based on such concepts is no more certain than an argument based on analogy. He further claimed that induction itself is founded on a process of analogy. His model of analogical reasoning was based on that of juridical arguments.*[59]*[60] This model of analogy has been used in the recent work of John F. Sowa.*[60] The Sharh al-takmil fi'l-mantiq written by Muhammad ibn Fayd Allah ibn Muhammad Amin al-Sharwani in the 15th century is the last major Arabic work on logic that has been studied.*[61] However, “thousands upon thousands of 57.2. MEDIEVAL LOGIC 201

A text by Avicenna, founder of Avicennian logic

pages”on logic were written between the 14th and 19th centuries, though only a fraction of the texts written during this period have been studied by historians, hence little is known about the original work on Islamic logic produced during this later period.*[55]

57.2.2 Logic in medieval Europe

“Medieval logic”(also known as “Scholastic logic”) generally means the form of Aristotelian logic developed in medieval Europe throughout roughly the period 1200–1600.*[1] For centuries after Stoic logic had been formulated, it was the dominant system of logic in the classical world. When the study of logic resumed after the Dark Ages, the main source was the work of the Christian philosopher Boethius, who was familiar with some of Aristotle's logic, but almost none of the work of the Stoics.*[62] Until the twelfth century, the only works of Aristotle available in the West were the Categories, On Interpretation, and Boethius's translation of the Isagoge of Porphyry (a commentary on the Categories). These works were known as the “Old Logic”(Logica Vetus or Ars Vetus). An important work in this tradition was the Logica Ingredientibus of Peter Abelard (1079–1142). His direct influence was small,*[63] but his influence through pupils such as John of Salisbury was great, and his method of applying rigorous logical analysis to theology shaped the way that theological criticism developed in the period that followed.*[64] By the early thirteenth century, the remaining works of Aristotle's Organon (including the Prior Analytics, Posterior Analytics, and the Sophistical Refutations) had been recovered in the West.*[65] Logical work until then was mostly 202 CHAPTER 57. HISTORY OF LOGIC

Brito's questions on the Old Logic paraphrasis or commentary on the work of Aristotle.*[66] The period from the middle of the thirteenth to the middle of the fourteenth century was one of significant developments in logic, particularly in three areas which were original, with little foundation in the Aristotelian tradition that came before. These were:*[67]

• The theory of supposition. deals with the way that predicates (e.g., 'man') range over a domain of individuals (e.g., all men).*[68] In the proposition 'every man is an animal', does the term 'man' range over or 'supposit for' men existing just in the present, or does the range include past and future men? Can a term supposit for a non-existing individual? Some medievalists have argued that this idea is a precursor of modern first-order logic.*[69] “The theory of supposition with the associated theories of copulatio (sign- capacity of adjectival terms), ampliatio (widening of referential domain), and distributio constitute one of the most original achievements of Western medieval logic”.*[70]

• The theory of syncategoremata. Syncategoremata are terms which are necessary for logic, but which, un- like categorematic terms, do not signify on their own behalf, but 'co-signify' with other words. Examples of syncategoremata are 'and', 'not', 'every', 'if', and so on.

• The theory of consequences. A consequence is a hypothetical, conditional proposition: two propositions joined by the terms 'if ... then'. For example, 'if a man runs, then God exists' (Si homo currit, Deus est).*[71] A fully developed theory of consequences is given in Book III of William of Ockham's work Summa Logicae. There, Ockham distinguishes between 'material' and 'formal' consequences, which are roughly equivalent to the modern material implication and logical implication respectively. Similar accounts are given by Jean Buridan and Albert of Saxony.

The last great works in this tradition are the Logic of John Poinsot (1589–1644, known as John of St Thomas), the Metaphysical Disputations of Francisco Suarez (1548–1617), and the Logica Demonstrativa of Giovanni Girolamo Saccheri (1667–1733). 57.3. TRADITIONAL LOGIC 203

57.3 Traditional logic

57.3.1 The textbook tradition

Dudley Fenner's Art of Logic (1584) 204 CHAPTER 57. HISTORY OF LOGIC

Traditional logic generally means the textbook tradition that begins with Antoine Arnauld's and Pierre Nicole's Logic, or the Art of Thinking, better known as the Port-Royal Logic.*[72] Published in 1662, it was the most influential work on logic in England until the nineteenth century.*[73] The book presents a loosely Cartesian doctrine (that the proposition is a combining of ideas rather than terms, for example) within a framework that is broadly derived from Aristotelian and medieval term logic. Between 1664 and 1700, there were eight editions, and the book had considerable influence after that.*[73] The account of propositions that Locke gives in the Essay is essentially that of Port-Royal:“Verbal propositions, which are words, [are] the signs of our ideas, put together or separated in affirmative or negative sentences. So that proposition consists in the putting together or separating these signs, according as the things which they stand for agree or disagree.”(Locke, An Essay Concerning Human Understanding, IV. 5. 6) Another influential work was the Novum Organum by , published in 1620. The title translates as“new instrument”. This is a reference to Aristotle's work known as the Organon. In this work, Bacon rejects the syllogistic method of Aristotle in favor of an alternative procedure “which by slow and faithful toil gathers information from things and brings it into understanding”.*[74] This method is known as inductive reasoning, a method which starts from empirical observation and proceeds to lower axioms or propositions; from these lower axioms, more general ones can be induced. For example, in finding the cause of a phenomenal nature such as heat, 3 lists should be constructed:

• The presence list: a list of every situation where heat is found. • The absence list: a list of every situation that is similar to at least one of those of the presence list, except for the lack of heat. • The variability list: a list of every situation where heat can vary.

Then, the form nature (or cause) of heat may be defined as that which is common to every situation of the presence list, and which is lacking from every situation of the absence list, and which varies by degree in every situation of the variability list. Other works in the textbook tradition include Isaac Watts's Logick: Or, the Right Use of Reason (1725), Richard Whately's Logic (1826), and John Stuart Mill's A System of Logic (1843). Although the latter was one of the last great works in the tradition, Mill's view that the foundations of logic lie in *[75] influenced the view that logic is best understood as a branch of psychology, a view which dominated the next fifty years of its development, especially in Germany.*[76]

57.3.2 Logic in Hegel's philosophy

G.W.F. Hegel indicated the importance of logic to his philosophical system when he condensed his extensive into a shorter work published in 1817 as the first volume of his Encyclopaedia of the Philosophical Sciences. The“Shorter”or“Encyclopaedia”Logic, as it is often known, lays out a series of transitions which leads from the most empty and abstract of categories—Hegel begins with “Pure Being”and “Pure Nothing”—to the "Absolute, the category which contains and resolves all the categories which preceded it. Despite the title, Hegel's Logic is not really a contribution to the science of valid inference. Rather than deriving conclusions about concepts through valid inference from premises, Hegel seeks to show that thinking about one concept compels thinking about another concept (one cannot, he argues, possess the concept of “Quality”without the concept of “Quantity”); this compulsion is, supposedly, not a matter of individual psychology, because it arises almost organically from the content of the concepts themselves. His purpose is to show the rational structure of the “Absolute”—indeed of rationality itself. The method by which thought is driven from one concept to its contrary, and then to further concepts, is known as the Hegelian dialectic. Although Hegel's Logic has had little impact on mainstream logical studies, its influence can be seen elsewhere:

• Carl von Prantl's Geschichte der Logik in Abendland (1855–1867).*[77] • The work of the British Idealists, such as F.H. Bradley's Principles of Logic (1883). • The economic, political, and philosophical studies of Karl Marx, and in the various schools of .

57.3.3 Logic and psychology

Between the work of Mill and Frege stretched half a century during which logic was widely treated as a descriptive science, an empirical study of the structure of reasoning, and thus essentially as a branch of psychology.*[78] The 57.3. TRADITIONAL LOGIC 205

Georg Wilhelm Friedrich Hegel

German psychologist , for example, discussed deriving “the logical from the psychological laws of thought”, emphasizing that “psychological thinking is always the more comprehensive form of thinking.”*[79] This view was widespread among German philosophers of the period:

• Theodor Lipps described logic as “a specific discipline of psychology”.*[80]

• Christoph von Sigwart understood logical necessity as grounded in the individual's compulsion to think in a certain way.*[81]

• Benno Erdmann argued that “logical laws only hold within the limits of our thinking”.*[82]

Such was the dominant view of logic in the years following Mill's work.*[83] This psychological approach to logic was rejected by Gottlob Frege. It was also subjected to an extended and destructive critique by in the first volume of his Logical Investigations (1900), an assault which has been described as“overwhelming”.*[84] 206 CHAPTER 57. HISTORY OF LOGIC

Husserl argued forcefully that grounding logic in psychological observations implied that all logical truths remained unproven, and that skepticism and relativism were unavoidable consequences. Such criticisms did not immediately extirpate what is called "psychologism". For example, the American philosopher Josiah Royce, while acknowledging the force of Husserl's critique, remained “unable to doubt”that progress in psychology would be accompanied by progress in logic, and vice versa.*[85]

57.4 Rise of modern logic

The period between the fourteenth century and the beginning of the nineteenth century had been largely one of decline and neglect, and is generally regarded as barren by historians of logic.*[2] The revival of logic occurred in the mid-nineteenth century, at the beginning of a revolutionary period where the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in mathematics. The development of the modern “symbolic”or “mathematical”logic during this period is the most significant in the 2000-year history of logic, and is arguably one of the most important and remarkable events in human intellectual history.*[4] A number of features distinguish modern logic from the old Aristotelian or traditional logic, the most important of which are as follows:*[86] Modern logic is fundamentally a calculus whose rules of operation are determined only by the shape and not by the meaning of the symbols it employs, as in mathematics. Many logicians were impressed by the “success”of mathematics, in that there had been no prolonged dispute about any truly mathematical result. C.S. Peirce noted*[87] that even though a mistake in the evaluation of a definite by Laplace led to an error concerning the moon's orbit that persisted for nearly 50 years, the mistake, once spotted, was corrected without any serious dispute. Peirce contrasted this with the disputation and uncertainty surrounding traditional logic, and especially reasoning in metaphysics. He argued that a truly “exact”logic would depend upon mathematical, i.e., “diagrammatic”or“iconic”thought. “Those who follow such methods will ... escape all error except such as will be speedily corrected after it is once suspected”. Modern logic is also“constructive”rather than“abstractive"; i.e., rather than abstracting and formalising theorems derived from ordinary language (or from psychological intuitions about validity), it constructs theorems by formal methods, then looks for an interpretation in ordinary language. It is entirely symbolic, meaning that even the logical constants (which the medieval logicians called "syncategoremata") and the categoric terms are expressed in symbols.

57.5 Modern logic

The development of modern logic falls into roughly five periods:*[88]

• The embryonic period from Leibniz to 1847, when the notion of a logical calculus was discussed and devel- oped, particularly by Leibniz, but no schools were formed, and isolated periodic attempts were abandoned or went unnoticed. • The algebraic period from Boole's Analysis to Schröder's Vorlesungen. In this period, there were more prac- titioners, and a greater continuity of development. • The logicist period from the Begriffsschrift of Frege to the Principia Mathematica of Russell and Whitehead. The aim of the“logicist school”was to incorporate the logic of all mathematical and scientific discourse in a single unified system which, taking as a fundamental principle that all mathematical truths are logical, did not accept any non-logical terminology. The major logicists were Frege, Russell, and the early Wittgenstein.*[89] It culminates with the Principia, an important work which includes a thorough examination and attempted solution of the which had been an obstacle to earlier progress. • The metamathematical period from 1910 to the 1930s, which saw the development of metalogic, in the finitist system of Hilbert, and the non-finitist system of Löwenheim and Skolem, the combination of logic and metalogic in the work of Gödel and Tarski. Gödel's incompleteness theorem of 1931 was one of the greatest achievements in the history of logic. Later in the 1930s, Gödel developed the notion of set-theoretic constructibility. • The period after World War II, when mathematical logic branched into four inter-related but separate areas of research: model theory, proof theory, computability theory, and set theory, and its ideas and methods began to influence philosophy. 57.5. MODERN LOGIC 207

57.5.1 Embryonic period

Leibniz

The idea that inference could be represented by a purely mechanical process is found as early as Raymond Llull, who proposed a (somewhat eccentric) method of drawing conclusions by a system of concentric rings. The work of logicians such as the Oxford Calculators*[90] led to a method of using letters instead of writing out logical calculations (calculationes) in words, a method used, for instance, in the Logica magna by Paul of Venice. Three hundred years after Llull, the English philosopher and logician suggested that all logic and reasoning could be reduced to the mathematical operations of addition and subtraction.*[91] The same idea is found in the work of Leibniz, who had read both Llull and Hobbes, and who argued that logic can be represented through a combinatorial process or calculus. But, like Llull and Hobbes, he failed to develop a detailed or comprehensive system, and his 208 CHAPTER 57. HISTORY OF LOGIC

work on this topic was not published until long after his death. Leibniz says that ordinary languages are subject to “countless ”and are unsuited for a calculus, whose task is to expose mistakes in inference arising from the forms and structures of words;*[92] hence, he proposed to identify an alphabet of human thought comprising fundamental concepts which could be composed to express complex ideas,*[93] and create a calculus ratiocinator that would make all arguments“as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate.”*[94] Gergonne (1816) said that reasoning does not have to be about objects about which one has perfectly clear ideas, because algebraic operations can be carried out without having any idea of the meaning of the symbols involved.*[95] Bolzano anticipated a fundamental idea of modern proof theory when he defined logical consequence or“deducibility” in terms of variables:*[96]

Hence I say that propositions M , N , O ,…are deducible from propositions A , B , C , D ,…with respect to variable parts i , j ,…, if every class of ideas whose substitution for i , j ,…makes all of A , B , C , D ,…true, also makes all of M , N , O ,…true. Occasionally, since it is customary, I shall say that propositions M , N , O ,…follow, or can be inferred or derived, from A , B , C , D ,…. Propositions A , B , C , D ,…I shall call the premises, M , N , O ,…the conclusions.

This is now known as semantic validity.

57.5.2 Algebraic period

Modern logic begins with what is known as the “algebraic school”, originating with Boole and including Peirce, Jevons, Schröder, and Venn.*[97] Their objective was to develop a calculus to formalise reasoning in the area of classes, propositions, and . The school begins with Boole's seminal work Mathematical Analysis of Logic which appeared in 1847, although De Morgan (1847) is its immediate precursor.*[98] The fundamental idea of Boole's system is that algebraic formulae can be used to express logical relations. This idea occurred to Boole in his teenage years, working as an usher in a private school in Lincoln, Lincolnshire.*[99] For example, let x and y stand for classes let the symbol = signify that the classes have the same members, xy stand for the class containing all and only the members of x and y and so on. Boole calls these elective symbols, i.e. symbols which select certain objects for consideration.*[100] An expression in which elective symbols are used is called an elective function, and an equation of which the members are elective functions, is an elective equation.*[101] The theory of elective functions and their“development”is essentially the modern idea of truth-functions and their expression in disjunctive normal form.*[100] Boole's system admits of two interpretations, in class logic, and propositional logic. Boole distinguished between “primary propositions”which are the subject of syllogistic theory, and “secondary propositions”, which are the subject of propositional logic, and showed how under different “interpretations”the same algebraic system could represent both. An example of a primary proposition is “All inhabitants are either Europeans or Asiatics.”An example of a secondary proposition is “Either all inhabitants are Europeans or they are all Asiatics.”*[102] These are easily distinguished in modern propositional calculus, where it is also possible to show that the first follows from the second, but it is a significant disadvantage that there is no way of representing this in the Boolean system.*[103] In his Symbolic Logic (1881), John Venn used diagrams of overlapping areas to express Boolean relations between classes or truth-conditions of propositions. In 1869 Jevons realised that Boole's methods could be mechanised, and constructed a “logical machine”which he showed to the Royal Society the following year.*[100] In 1885 Allan Marquand proposed an electrical version of the machine that is still extant (picture at the Firestone Library). The defects in Boole's system (such as the use of the letter v for existential propositions) were all remedied by his followers. Jevons published Pure Logic, or the Logic of Quality apart from Quantity in 1864, where he suggested a symbol to signify exclusive or, which allowed Boole's system to be greatly simplified.*[104] This was usefully exploited by Schröder when he set out theorems in parallel columns in his Vorlesungen (1890–1905). Peirce (1880) showed how all the Boolean elective functions could be expressed by the use of a single primitive , "neither ... nor ..." and equally well "not both ... and ...",*[105] however, like many of Peirce's innovations, this remained unknown or unnoticed until Sheffer rediscovered it in 1913.*[106] Boole's early work also lacks the idea of the logical sum which originates in Peirce (1867), Schröder (1877) and Jevons (1890),*[107] and the concept of inclusion, first suggested by Gergonne (1816) and clearly articulated by Peirce (1870). The success of Boole's algebraic system suggested that all logic must be capable of algebraic representation, and there were attempts to express a logic of relations in such form, of which the most ambitious was Schröder's monumental 57.5. MODERN LOGIC 209

George Boole

Vorlesungen über die Algebra der Logik (“Lectures on the Algebra of Logic”, vol iii 1895), although the original idea was again anticipated by Peirce.*[108] Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic John Corcoran in an acces- sible introduction to Laws of Thought*[109] Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought.*[110] According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were “to go under, over, and beyond”Aristotle's logic by 1) providing it with mathematical foundations involving 210 CHAPTER 57. HISTORY OF LOGIC

Charles Sanders Peirce

equations, 2) extending the class of problems it could treat —from assessing validity to solving equations —and 3) expanding the range of applications it could handle —e.g. from propositions having only two terms to those having arbitrarily many. More specifically, Boole agreed with what Aristotle said; Boole's 'disagreements', if they might be called that, con- cern what Aristotle did not say. First, in the realm of foundations, Boole reduced the four propositional forms of Aristotelian logic to formulas in the form of equations —by itself a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving to logic —another revolutionary idea —involved Boole's doctrine that Aristotle's rules of inference (the“perfect syllogisms”) must be supplemented by rules for equation solving. Third, in the realm of applications, Boole's system could handle multi-term propositions and arguments whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce “No quadrangle that is a square is a rectangle that is a rhombus”from “No square that is a quadrangle 57.5. MODERN LOGIC 211

UNIVERSE (Integers from 0 to 30) 1 7 11 13 17 19 23 29 SET A (Even numbers/Multiples of 2)

2 4 8 14 16 22 26 28

SET AB (Multiples of 6) 6 12 18 24 SET AC SET ABC SET B (Multiples of 10) (Multiples of 30) (Multiples of 3) 10 20 0 30 SET BC (Multiples of 15) 5 25 15 3 9 21 27

SET C (Multiples of 5)

Boolean multiples

is a rhombus that is a rectangle”or from “No rhombus that is a rectangle is a square that is a quadrangle”.

57.5.3 Logicist period

After Boole, the next great advances were made by the German mathematician Gottlob Frege. Frege's objective was the program of Logicism, i.e. demonstrating that arithmetic is identical with logic.*[111] Frege went much further than any of his predecessors in his rigorous and formal approach to logic, and his calculus or Begriffsschrift is important.*[111] Frege also tried to show that the concept of number can be defined by purely logical means, so that (if he was right) logic includes arithmetic and all branches of mathematics that are reducible to arithmetic. He was not the first writer to suggest this. In his pioneering work Die Grundlagen der Arithmetik (The Foundations of Arithmetic), sections 15–17, he acknowledges the efforts of Leibniz, J.S. Mill as well as Jevons, citing the latter's claim that “algebra is a highly developed logic, and number but logical discrimination.”*[112] Frege's first work, the Begriffsschrift (“concept script”) is a rigorously axiomatised system of propositional logic, relying on just two connectives (negational and conditional), two rules of inference (modus ponens and substitution), and six axioms. Frege referred to the “completeness”of this system, but was unable to prove this.*[113] The most significant innovation, however, was his explanation of the quantifier in terms of mathematical functions. Traditional logic regards the sentence“Caesar is a man”as of fundamentally the same form as“all men are mortal.”Sentences with a proper name subject were regarded as universal in character, interpretable as“every Caesar is a man”.*[114] At the outset Frege abandons the traditional “concepts subject and predicate", replacing them with argument and function respectively, which he believes “will stand the test of time. It is easy to see how regarding a content as a function of an argument leads to the formation of concepts. Furthermore, the demonstration of the connection between the meanings of the words if, and, not, or, there is, some, all, and so forth, deserves attention”.*[115] Frege argued that the quantifier expression“all men”does not have the same logical or semantic form as“all men”, and that the universal proposition “every A is B”is a complex proposition involving two functions, namely ' – is A' and ' – is B' such that whatever satisfies the first, also satisfies the second. In modern notation, this would be expressed as

( ) ∀ x A(x) → B(x)

In English, “for all x, if Ax then Bx”. Thus only singular propositions are of subject-predicate form, and they are irreducibly singular, i.e. not reducible to a general proposition. Universal and particular propositions, by contrast, are 212 CHAPTER 57. HISTORY OF LOGIC

Gottlob Frege.

not of simple subject-predicate form at all. If“all mammals”were the logical subject of the sentence“all mammals are land-dwellers”, then to negate the whole sentence we would have to negate the predicate to give “all mammals are not land-dwellers”. But this is not the case.*[116] This functional analysis of ordinary-language sentences later had a great impact on philosophy and linguistics. This means that in Frege's calculus, Boole's “primary”propositions can be represented in a different way from “secondary”propositions. “All inhabitants are either men or women”is

( ( )) ∀ x I(x) → M(x) ∨ W (x)

whereas “All the inhabitants are men or all the inhabitants are women”is 57.5. MODERN LOGIC 213 x F(x)

Frege's “Concept Script”

( ) ( ) ∀ x I(x) → M(x) ∨ ∀ x I(x) → W (x)

As Frege remarked in a critique of Boole's calculus:

“The real difference is that I avoid [the Boolean] division into two parts ... and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is like the mirror image of the other, but for that very reason stands in no organic relation to it'*[117]

As well as providing a unified and comprehensive system of logic, Frege's calculus also resolved the ancient problem of multiple generality. The ambiguity of “every girl kissed a boy”is difficult to express in traditional logic, but Frege's logic resolves this through the different scope of the quantifiers. Thus

( ( )) ∀ x G(x) → ∃ y B(y) ∧ K(x, y)

means that to every girl there corresponds some boy (any one will do) who the girl kissed. But

( ( )) ∃ x B(x) ∧ ∀ y G(y) → K(y, x)

means that there is some particular boy whom every girl kissed. Without this device, the project of logicism would have been doubtful or impossible. Using it, Frege provided a definition of the ancestral relation, of the many-to-one relation, and of mathematical induction.*[118] This period overlaps with the work of what is known as the“mathematical school”, which included Dedekind, Pasch, Peano, Hilbert, Zermelo, Huntington, Veblen and Heyting. Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory. Most notable was Hilbert's Program, which sought to ground all of mathematics to a finite set of axioms, proving its consistency by “finitistic”means and providing a procedure which would decide the truth or falsity of any mathematical statement. The standard axiomatization of the natural numbers is named the Peano axioms in his honor. Peano maintained a clear distinction between mathematical and logical symbols. While unaware of Frege's work, he independently recreated his logical apparatus based on the work of Boole and Schröder.*[119] The logicist project received a near-fatal setback with the discovery of a paradox in 1901 by Bertrand Russell. This proved Frege's naive set theory led to a contradiction. Frege's theory contained the axiom that for any formal criterion, there is a set of all objects that meet the criterion. Russell showed that a set containing exactly the sets that are not members of themselves would contradict its own definition (if it is not a member of itself, it is a member of itself, and if it is a member of itself, it is not).*[120] This contradiction is now known as Russell's paradox. One important method of resolving this paradox was proposed by Ernst Zermelo.*[121] Zermelo set theory was the first axiomatic set theory. It was developed into the now-canonical Zermelo–Fraenkel set theory (ZF). Russell's paradox symbolically is as follows:

LetR = {x | x ̸∈ x} then ,R ∈ R ⇐⇒ R ̸∈ R 214 CHAPTER 57. HISTORY OF LOGIC

Peano

The monumental Principia Mathematica, a three-volume work on the foundations of mathematics, written by Russell and Alfred North Whitehead and published 1910–13 also included an attempt to resolve the paradox, by means of an elaborate system of types: a set of elements is of a different type than is each of its elements (set is not the element; one element is not the set) and one cannot speak of the "set of all sets". The Principia was an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic. 57.5. MODERN LOGIC 215

Ernst Zermelo

57.5.4 Metamathematical period

The names of Gödel and Tarski dominate the 1930s,*[122] a crucial period in the development of metamathematics – the study of mathematics using mathematical methods to produce metatheories, or mathematical theories about other mathematical theories. Early investigations into metamathematics had been driven by Hilbert's program. Work on metamathematics culminated in the work of Gödel, who in 1929 showed that a given first-order sentence is deducible if and only if it is logically valid – i.e. it is true in every structure for its language. This is known as Gödel's 216 CHAPTER 57. HISTORY OF LOGIC

Kurt Goedel completeness theorem. A year later, he proved two important theorems, which showed Hibert's program to be unattainable in its original form. The first is that no consistent system of axioms whose theorems can be listed by an effective procedure such as an algorithm or computer program is capable of proving all facts about the natural numbers. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second is that if such a system is also capable of proving certain basic facts about the natural numbers, then the system cannot prove the consistency of the system itself. These two results are known as Gödel's incompleteness theorems, or simply Gödel's Theorem. Later in the decade, Gödel developed the concept of set-theoretic constructibility, as part of his proof that the and the continuum hypothesis are consistent with Zermelo–Fraenkel set theory. 57.5. MODERN LOGIC 217

Alonzo Church

In proof theory, Gerhard Gentzen developed natural deduction and the sequent calculus. The former attempts to model logical reasoning as it 'naturally' occurs in practice and is most easily applied to intuitionistic logic, while the latter was devised to clarify the derivation of logical proofs in any . Since Gentzen's work, natural deduction and sequent calculi have been widely applied in the fields of proof theory, mathematical logic and computer science. Gentzen also proved normalization and cut-elimination theorems for intuitionistic and classical logic which could be used to reduce logical proofs to a normal form.*[123]*[124] 218 CHAPTER 57. HISTORY OF LOGIC

Alfred Tarski

Alfred Tarski, a pupil of Łukasiewicz, is best known for his definition of truth and logical consequence, and the semantic concept of logical satisfaction. In 1933, he published (in Polish) The concept of truth in formalized languages, in which he proposed his semantic theory of truth: a sentence such as “snow is white”is true if and only if snow is white. Tarski's theory separated the metalanguage, which makes the statement about truth, from the object language, which contains the sentence whose truth is being asserted, and gave a correspondence (the T-schema) between phrases in the object language and elements of an interpretation. Tarski's approach to the difficult idea of explaining truth has been enduringly influential in logic and philosophy, especially in the development of model theory.*[125] Tarski also produced important work on the of deductive systems, and on fundamental principles such as completeness, decidability, consistency and definability. According to Anita Feferman, Tarski“changed the face of logic in the twentieth century”.*[126] Alonzo Church and Alan Turing proposed formal models of computability, giving independent negative solutions to Hilbert's Entscheidungsproblem in 1936 and 1937, respectively. The Entscheidungsproblem asked for a procedure that, given any formal mathematical statement, would algorithmically determine whether the statement is true. Church and Turing proved there is no such procedure; Turing's paper introduced the halting problem as a key example of a mathematical problem without an algorithmic solution. Church's system for computation developed into the modern λ-calculus, while the Turing machine became a standard model for a general-purpose computing device. It was soon shown that many other proposed models of computation were equivalent in power to those proposed by Church and Turing. These results led to the Church–Turing thesis that any deterministic algorithm that can be carried out by a human can be carried out by a Turing machine. Church proved additional undecidability results, showing that both Peano arithmetic and first-order logic are undecidable. Later work by Emil Post and Stephen Cole Kleene in the 1940s extended the scope of computability theory and introduced the concept of degrees of unsolvability. The results of the first few decades of the twentieth century also had an impact upon analytic philosophy and philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic, temporal logic, deontic logic, and relevance logic. 57.5. MODERN LOGIC 219

Saul Kripke

57.5.5 Logic after WWII

After World War II, mathematical logic branched into four inter-related but separate areas of research: model theory, proof theory, computability theory, and set theory.*[127] In set theory, the method of forcing revolutionized the field by providing a robust method for constructing models and obtaining independence results. Paul Cohen introduced this method in 1962 to prove the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory.*[128] His technique, which was simplified and extended soon after its introduction, has since been applied to many other problems in all areas of mathematical logic. Computability theory had its roots in the work of Turing, Church, Kleene, and Post in the 1930s and 40s. It devel- oped into a study of abstract computability, which became known as recursion theory.*[129] The priority method, discovered independently by Albert Muchnik and Richard Friedberg in the 1950s, led to major advances in the un- derstanding of the degrees of unsolvability and related structures. Research into higher-order computability theory demonstrated its connections to set theory. The fields of and computable analysis were devel- oped to study the effective content of classical mathematical theorems; these in turn inspired the program of reverse mathematics. A separate branch of computability theory, computational complexity theory, was also characterized in logical terms as a result of investigations into descriptive complexity. Model theory applies the methods of mathematical logic to study models of particular mathematical theories. Alfred Tarski published much pioneering work in the field, which is named after a series of papers he published under the 220 CHAPTER 57. HISTORY OF LOGIC

title Contributions to the theory of models. In the 1960s, Abraham Robinson used model-theoretic techniques to develop calculus and analysis based on infinitesimals, a problem that first had been proposed by Leibniz. In proof theory, the relationship between classical mathematics and intuitionistic mathematics was clarified via tools such as the realizability method invented by Georg Kreisel and Gödel's Dialectica interpretation. This work inspired the contemporary area of proof mining. The Curry-Howard correspondence emerged as a deep analogy between logic and computation, including a correspondence between systems of natural deduction and typed lambda calculi used in computer science. As a result, research into this class of formal systems began to address both logical and computational aspects; this area of research came to be known as modern type theory. Advances were also made in ordinal analysis and the study of independence results in arithmetic such as the Paris–Harrington theorem. This was also a period, particularly in the 1950s and afterwards, when the ideas of mathematical logic begin to influence philosophical thinking. For example, tense logic is a formalised system for representing, and reasoning about, propositions qualified in terms of time. The philosopher Arthur Prior played a significant role in its de- velopment in the 1960s. Modal logics extend the scope of formal logic to include the elements of modality (for example, possibility and necessity). The ideas of Saul Kripke, particularly about possible worlds, and the formal system now called Kripke semantics have had a profound impact on analytic philosophy.*[130] His best known and most influential work is (1980).*[131] Deontic logics are closely related to modal logics: they attempt to capture the logical features of obligation, permission and related concepts. Although some basic novelties syncretizing mathematical and philosophical logic were shown by Bolzano in the early 1800s, it was , a pupil of Alexius Meinong, who was to propose the first formal deontic system in his Grundgesetze des Sollens, based on the syntax of Whitehead's and Russell's propositional calculus. Another logical system founded after World War II was fuzzy logic by Azerbaijani mathematician Lotfi Asker Zadeh in 1965.

57.6 Logic in the East

57.6.1 Logic in India

Main article:

Logic began independently in ancient India and continued to develop to early modern times without any known influ- ence from Greek logic.*[132] Medhatithi Gautama (c. 6th century BC) founded the school of logic.*[133] The Mahabharata (12.173.45), around the 5th century BC, refers to the anviksiki and tarka schools of logic. Pāṇini (c. 5th century BC) developed a form of logic (to which Boolean logic has some similarities) for his formulation of grammar. Logic is described by (c. 350-283 BC) in his as an independent field of inquiry.*[134] Two of the six Indian schools of thought deal with logic: and . The Nyaya of Aksapada Gautama (c. 2nd century AD) constitute the core texts of the Nyaya school, one of the six orthodox schools of . This realist school developed a rigid five-member schema of inference involving an initial premise, a reason, an example, an application, and a conclusion.*[135] The idealist became the chief opponent to the Naiyayikas. (c. 150-250 AD), the founder of the Madhyamika“( ”) developed an analysis known as the catuṣkoṭi (Sanskrit), a“four-cornered”system of argumentation that involves the systematic examination and rejection of each of the 4 possibilities of a proposition, P:

1. P; that is, being.

2. not P; that is, not being.

3. P and not P; that is, being and not being.

4. not (P or not P); that is, neither being nor not being. It is interesting to note that under propositional logic, De Morgan's laws imply that this is equivalent to the third case (P and not P), and is therefore superfluous; there are actually only 3 cases to consider.

However, Dignaga (c 480-540 AD) is sometimes said to have developed a formal syllogism,*[136] and it was through him and his successor, , that Buddhist logic reached its height; it is contested whether their analysis 57.7. SEE ALSO 221 actually constitutes a formal syllogistic system. In particular, their analysis centered on the definition of an inference- warranting relation, "vyapti", also known as invariable concomitance or pervasion.*[137] To this end, a doctrine known as“apoha”or differentiation was developed.*[138] This involved what might be called inclusion and exclusion of defining properties. The difficulties involved in this enterprise, in part, stimulated the neo-scholastic school of Navya-Nyāya, which de- veloped a formal analysis of inference in the sixteenth century. This later school began around eastern India and Bengal, and developed theories resembling modern logic, such as Gottlob Frege's “distinction between of proper names”and his“definition of number,”as well as the Navya-Nyaya theory of“restrictive con- ditions for universals”anticipating some of the developments in modern set theory.*[139] Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such as Charles Babbage, Augustus De Morgan, and particularly George Boole, as confirmed by his wife Mary Everest Boole, who wrote in 1901 an “open letter to Dr Bose”, which was titled “Indian Thought and Western Science in the Nineteenth Century”and stated:*[140]*[141] “Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan and George Boole on the mathematical atmosphere of 1830-1865”. Dignāga's famous “wheel of reason”() is a method of indicating when one thing (such as smoke) can be taken as an invariable sign of another thing (like fire), but the inference is often inductive and based on past observation. Matilal remarks that Dignāga's analysis is much like John Stuart Mill's Joint Method of Agreement and Difference, which is inductive.*[142] In addition, the traditional five-member Indian syllogism, though deductively valid, has repetitions that are unneces- sary to its logical validity. As a result, some commentators see the traditional Indian syllogism as a rhetorical form that is entirely natural in many cultures of the world, and yet not as a logical form—not in the sense that all logically unnecessary elements have been omitted for the sake of analysis.

57.6.2 Logic in China

Main article: Logic in China

In China, a contemporary of , Mozi, “Master ”, is credited with founding the Mohist school, whose canons dealt with issues relating to valid inference and the conditions of correct conclusions. In particular, one of the schools that grew out of Mohism, the Logicians, are credited by some scholars for their early investigation of formal logic. Due to the harsh rule of Legalism in the subsequent Qin Dynasty, this line of investigation disappeared in China until the introduction of by Buddhists.

57.7 See also

• History of the function concept

• History of Mathematics

• History of Philosophy

• Plato's beard

• Timeline of mathematical logic

57.8 Notes

[1] Boehner p. xiv

[2] Oxford Companion p. 498; Bochenski, Part I Introduction, passim

[3] Gottlob Frege. The Foundations of Arithmetic (PDF). p. 1.

[4] Oxford Companion p. 500

[5] Kneale, p. 2 222 CHAPTER 57. HISTORY OF LOGIC

[6] Kneale p. 3

[7] H. F. J. Horstmanshoff, Marten Stol, Cornelis Tilburg (2004), Magic and Rationality in Ancient Near Eastern and Graeco- Roman Medicine, p. 99, Brill Publishers, ISBN 90-04-13666-5.

[8] D. Brown (2000), Mesopotamian Planetary Astronomy-Astrology , Styx Publications, ISBN 90-5693-036-2.

[9] Kneale, p. 16

[10] Encyclopedia Britannica

[11] Aristotle, Metaphysics Alpha, 983b18.

[12] Smith, Sir William (1870). Dictionary of Greek and Roman biography and mythology. p. 1016.

[13] Prof.T.Patronis & D.Patsopoulos The Theorem of Thales: A Study of the naming of theorems in school Geometry textbooks. Patras University. Retrieved 2012-02-12.

[14] (Boyer 1991, “Ionia and the Pythagoreans”p. 43)

[15] de Laet, Siegfried J. (1996). History of Humanity: Scientific and Cultural Development. UNESCO, Volume 3, p. 14. ISBN 92-3-102812-X

[16] Boyer, Carl B. and Merzbach, Uta c. (2010). A History of Mathematics. John Wiley and Sons, Chapter IV. ISBN 0-470- 63056-6

[17] C. B. Boyer (1968)

[18] Samuel Enoch Stumpf. Socrates to Sartre. p. 11.

[19] Heath, Mathematics in Aristotle, cited in Kneale, p. 5

[20] Kneale p. 15

[21] R. J. Hollingdale (1974). Western Philosophy: an introduction. p. 73.

[22] https://books.google.com/books?id=DPoqAAAAMAAJ&pg=PA170&lpg=PA170

[23] F.E. Peters, Greek Philosophical Terms, New York University Press, 1967.

[24] Kneale p. 17

[25]“forming an opinion is talking, and opinion is speech that is held not with someone else or aloud but in silence with oneself” Theaetetus 189E–190A

[26] Kneale p. 20. For example, the proof given in the that the square on the diagonal is double the area of the original square presumably involves the forms of the square and the triangle, and the necessary relation between them

[27] Kneale p. 21

[28] Zalta, Edward N. "Aristotle's Logic". Stanford University, 18 March 2000. Retrieved 13 March 2010.

[29] See e.g. Aristotle's logic, Stanford Encyclopedia of Philosophy

[30] Bochenski p. 63

[31]“Throughout later antiquity two great schools of logic were distinguished, the Peripatetic which was derived from Aristotle, and the Stoic which was developed by Chrysippus from the teachings of the Megarians”– Kneale p. 113

[32] Oxford Companion, article “Chrysippus”, p. 134

[33] Stanford Encyclopedia of Philosophy: Susanne Bobzien, Ancient Logic

[34] K. Huelser, Die Fragmente zur Dialektik der Stoiker, 4 vols, Stuttgart 1986-7

[35] Kneale 117–158

[36] Metaphysics Eta 3, 1046b 29

[37] Boethius, Commentary on the Perihermenias, Meiser p. 234

[38] , Dissertationes ed. Schenkel ii. 19. I. 57.8. NOTES 223

[39] Alexander p. 177

[40] Sextus Empiricus, Adv. Math. viii, Section 113

[41] Sextus Empiricus, Hypotyp. ii. 110, comp.

[42] Cicero, Academica, ii. 47, de Fato, 6.

[43] See e.g. Lukasiewicz p. 21

[44] Sextus Bk viii., Sections 11, 12

[45] See e.g. Routledge Encyclopedia of Philosophy Online Version 2.0, article 'Islamic philosophy'

[46] History of logic: Arabic logic, Encyclopædia Britannica.

[47] Feldman, Seymour (1964-11-26). “Rescher on Arabic Logic”. The Journal of Philosophy (Journal of Philosophy, Inc.) 61 (22): 724–734. doi:10.2307/2023632. ISSN 0022-362X. JSTOR 2023632. [726]. Long, A. A.; D. N. Sedley (1987). The Hellenistic Philosophers. Vol 1: Translations of the principal sources with philosophical commentary. Cambridge: Cambridge University Press. ISBN 0-521-27556-3.

[48] Dag Nikolaus Hasse (September 19, 2008). “Influence of Arabic and Islamic Philosophy on the Latin West”. Stanford Encyclopedia of Philosophy. Retrieved 2009-10-13.

[49] Richard F. Washell (1973),“Logic, Language, and Albert the Great”, Journal of the History of Ideas 34 (3), pp. 445–450 [445].

[50] Goodman, Lenn Evan (2003), Islamic Humanism, p. 155, Oxford University Press, ISBN 0-19-513580-6.

[51] Goodman, Lenn Evan (1992); Avicenna, p. 188, Routledge, ISBN 0-415-01929-X.

[52] Kneale p. 229

[53] Kneale: p. 266; Ockham: Summa Logicae i. 14; Avicenna: Avicennae Opera Venice 1508 f87rb

[54] Muhammad Iqbal, The Reconstruction of Religious Thought in , “The Spirit of Muslim Culture”(cf. and )

[55] Tony Street (July 23, 2008). “Arabic and Islamic Philosophy of Language and Logic”. Stanford Encyclopedia of Philosophy. Retrieved 2008-12-05.

[56] Dr. Lotfollah Nabavi, Sohrevardi's Theory of Decisive Necessity and kripke's QSS System, Journal of Faculty of Literature and Human Sciences.

[57] Dr. Abu Shadi Al-Roubi (1982), “Ibn Al-Nafis as a philosopher”, Symposium on Ibn al-Nafis, Second International Conference on Islamic Medicine: Islamic Medical Organization, Kuwait (cf. Ibn al-Nafis As a Philosopher, Encyclopedia of Islamic World).

[58] See pp. 253–254 of Street, Tony (2005). “Logic”. In Peter Adamson and Richard C. Taylor (edd.). The Cambridge Companion to Arabic Philosophy. Cambridge University Press. pp. 247–265. ISBN 978-0-521-52069-0.

[59] Ruth Mas (1998). “Qiyas: A Study in Islamic Logic” (PDF). Folia Orientalia 34: 113–128. ISSN 0015-5675.

[60] John F. Sowa; Arun K. Majumdar (2003). “Analogical reasoning”. Conceptual Structures for Knowledge Creation and Communication, Proceedings of ICCS 2003. Berlin: Springer-Verlag., pp. 16-36

[61] Nicholas Rescher and Arnold vander Nat,“The Arabic Theory of Temporal Modal Syllogistic”, in George Fadlo Hourani (1975), Essays on Islamic Philosophy and Science, pp. 189–221, State University of New York Press, ISBN 0-87395-224-3.

[62] Kneale p. 198

[63] Stephen Dumont, article “Peter Abelard”in Gracia and Noone p. 492

[64] Kneale, pp. 202–3

[65] See e.g. Kneale p. 225

[66] Boehner p. 1

[67] Boehner pp. 19–76

[68] Boehner p. 29 224 CHAPTER 57. HISTORY OF LOGIC

[69] Boehner p. 30

[70] Ebbesen 1981

[71] Boehner pp. 54–5

[72] Oxford Companion p. 504, article “Traditional logic”

[73] Buroker xxiii

[74] Farrington, 1964, 89

[75] N. Abbagnano, “Psychologism”in P. Edwards (ed) The Encyclopaedia of Philosophy, MacMillan, 1967

[76] Of the German literature in this period, Robert Adamson wrote "Logics swarm as bees in springtime..."; Robert Adamson, A Short History of Logic, Wm. Blackwood & Sons, 1911, page 242

[77] Carl von Prantl (1855-1867), Geschichte von Logik in Abendland, Leipsig: S. Hirzl, anastatically reprinted in 1997, Hildesheim: Georg Olds.

[78] See e.g. Psychologism, Stanford Encyclopedia of Philosophy

[79] Wilhelm Wundt, Logik (1880–1883); quoted in Edmund Husserl, Logical Investigations, translated J.N. Findlay, Routledge, 2008, Volume 1, pp. 115–116.

[80] Theodor Lipps, Grundzüge der Logik (1893); quoted in Edmund Husserl, Logical Investigations, translated J.N. Findlay, Routledge, 2008, Volume 1, p. 40

[81] Christoph von Sigwart, Logik (1873–78); quoted in Edmund Husserl, Logical Investigations, translated J.N. Findlay, Rout- ledge, 2008, Volume 1, p. 51

[82] Benno Erdmann, Logik (1892); quoted in Edmund Husserl, Logical Investigations, translated J.N. Findlay, Routledge, 2008, Volume 1, p. 96

[83] Dermot Moran,“Introduction"; Edmund Husserl, Logical Investigations, translated J.N. Findlay, Routledge, 2008, Volume 1, p. xxi

[84] , “Preface"; Edmund Husserl, Logical Investigations, translated J.N. Findlay, Routledge, 2008, Volume 1, p. xvii

[85] Josiah Royce, “Recent Logical Enquiries and their Psychological Bearings”(1902) in John J. McDermott (ed) The Basic Writings of Josiah Royce Volume 2, Fordham University Press, 2005, p. 661

[86] Bochenski, p. 266

[87] Peirce 1896

[88] See Bochenski p. 269

[89] Oxford Companion p. 499

[90] Edith Sylla (1999), “Oxford Calculators”, in The Cambridge Dictionary of Philosophy, Cambridge, Cambridgeshire: Cambridge.

[91] El. philos. sect. I de corp 1.1.2.

[92] Bochenski p. 274

[93] Rutherford, Donald, 1995,“Philosophy and language”in Jolley, N., ed., The Cambridge Companion to Leibniz. Cambridge Univ. Press.

[94] Wiener, Philip, 1951. Leibniz: Selections. Scribner.

[95] Essai de dialectique rationelle, 211n, quoted in Bochenski p. 277.

[96] Bolzano, Bernard (1972). George, Rolf, ed. The Theory of Science: Die Wissenschaftslehre oder Versuch einer Neuen Darstellung der Logik. Translated by George Rolf. University of California Press. p. 209. ISBN 9780520017870.

[97] See e.g. Bochenski p. 296 and passim 57.8. NOTES 225

[98] Before publishing, he wrote to De Morgan, who was just finishing his work Formal Logic. De Morgan suggested they should publish first, and thus the two books appeared at the same time, possibly even reaching the bookshops on the same day. cf. Kneale p. 404

[99] Kneale p. 404

[100] Kneale p. 407

[101] Boole (1847) p. 16

[102] Boole 1847 pp. 58–9

[103] Beaney p. 11

[104] Kneale p. 422

[105] Peirce,“A Boolean Algebra with One Constant”, 1880 MS, Collected Papers v. 4, paragraphs 12–20, reprinted Writings v. 4, pp. 218-21. Google Preview.

[106] Trans. Amer. Math. Soc., xiv (1913), pp. 481–8. This is now known as the Sheffer stroke

[107] Bochenski 296

[108] See CP III

[109] George Boole. 1854/2003. , facsimile of 1854 edition, with an introduction by J. Corcoran. Buffalo: Prometheus Books (2003). Reviewed by James van Evra in Philosophy in Review.24 (2004) 167–169.

[110] JOHN CORCORAN, Aristotle's Prior Analytics and Boole's Laws of Thought, History and , vol. 24 (2003), pp. 261–288.

[111] Kneale p. 435

[112] Jevons, The Principles of Science, London 1879, p. 156, quoted in Grundlagen 15

[113] Beaney p. 10 – the completeness of Frege's system was eventually proved by Jan Łukasiewicz in 1934

[114] See for example the argument by the medieval logician William of Ockham that singular propositions are universal, in Summa Logicae III. 8 (??)

[115] Frege 1879 in van Heijenoort 1967, p. 7

[116]“On concept and object”p. 198; Geach p. 48

[117] BLC p. 14, quoted in Beaney p. 12

[118] See e.g. The Internet Encyclopedia of Philosophy, article “Frege”

[119] Van Heijenoort 1967, p. 83

[120] See e.g. Potter 2004

[121] Zermelo 1908

[122] Feferman 1999 p. 1

[123] Girard, Jean-Yves; Paul Taylor; Yves Lafont (1990) [1989]. Proofs and Types. Cambridge University Press (Cambridge Tracts in Theoretical Computer Science, 7). ISBN 0-521-37181-3.

[124] Alex Sakharov, “Cut Elimination Theorem”, MathWorld.

[125] Feferman and Feferman 2004, p. 122, discussing “The Impact of Tarski's Theory of Truth”.

[126] Feferman 1999, p. 1

[127] See e.g. Barwise, Handbook of Mathematical Logic

[128] The Independence of the Continuum Hypothesis, II Paul J. Cohen Proceedings of the National Academy of Sciences of the United States of America, Vol. 51, No. 1. (Jan. 15, 1964), pp. 105-110.

[129] Many of the foundational papers are collected in The Undecidable (1965) edited by Martin Davis

[130] , "Water's water everywhere", London Review of Books, 21 October 2004 226 CHAPTER 57. HISTORY OF LOGIC

[131] See in the Twentieth Century: Volume 2: The Age of Meaning, Scott Soames: "Naming and Necessity is among the most important works ever, ranking with the classical work of Frege in the late nineteenth century, and of Russell, Tarski and Wittgenstein in the first half of the twentieth century”. Cited in Byrne, Alex and Hall, Ned. 2004. 'Necessary Truths'. Boston Review October/November 2004

[132] Bochenski p. 446

[133] S. C. Vidyabhusana (1971). A History of Indian Logic: Ancient, Mediaeval, and Modern Schools.

[134] R. P. Kangle (1986). The Kautiliya Arthashastra (1.2.11). Motilal Banarsidass.

[135] Bochenski p. 417 and passim

[136] Bochenski pp. 431–7

[137] Matilal, Bimal Krishna (1998). The Character of Logic in India. Albany, NY: State University of New York Press. pp. 12, 18. ISBN 9780791437407.

[138] Bochenksi p. 441

[139] Kisor Kumar Chakrabarti (June 1976). “Some Comparisons Between Frege's Logic and Navya-Nyaya Logic”. Philos- ophy and Phenomenological Research (International Phenomenological Society) 36 (4): 554–563. doi:10.2307/2106873. JSTOR 2106873. This paper consists of three parts. The first part deals with Frege's distinction between sense and ref- erence of proper names and a similar distinction in Navya-Nyaya logic. In the second part we have compared Frege's definition of number to the Navya-Nyaya definition of number. In the third part we have shown how the study of the so-called 'restrictive conditions for universals' in Navya-Nyaya logic anticipated some of the developments of modern set theory.

[140] Boole, Mary Everest“Collected Works”eds E M Cobham and E S Dummer London, Daniel 1931. Letter also published in the Ceylon National Review in 1909, and published as a separate pamphlet “The Psychologic Aspect of Imperialism” in 1911.

[141] Jonardon Ganeri (2001). Indian logic: a reader. Routledge. p. vii. ISBN 0-7007-1306-9

[142] Matilal, 17

57.9 References

Primary Sources

• Alexander of Aphrodisias, In Aristotelis An. Pr. Lib. I Commentarium, ed. Wallies, Berlin, C.I.A.G. vol. II/1, 1882.

• Avicenna, Avicennae Opera Venice 1508.

• Boethius Commentary on the Perihermenias, Secunda Editio, ed. Meiser, Leipzig, Teubner, 1880.

• Bolzano, Bernard Wissenschaftslehre, (1837) 4 Bde, Neudr., hrsg. W. Schultz, Leipzig I-II 1929, III 1930, IV 1931 (Theory of Science, four volumes, translated by Rolf George and Paul Rusnock, New York: Oxford University Press, 2014).

• Bolzano, Bernard Theory of Science (Edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell – D. Reidel Publishing Company, Dordrecht and Boston 1973).

• Boole, George (1847) The Mathematical Analysis of Logic (Cambridge and London); repr. in Studies in Logic and Probability, ed. R. Rhees (London 1952).

• Boole, George (1854) The Laws of Thought (London and Cambridge); repr. as Collected Logical Works. Vol. 2, (Chicago and London: Open Court, 1940).

• Epictetus, Epicteti Dissertationes ab Arriano digestae, edited by Heinrich Schenkl, Leipzig, Teubner. 1894.

• Frege, G., Boole's Logical Calculus and the Concept Script, 1882, in Posthumous Writings transl. P. Long and R. White 1969, pp. 9–46. 57.9. REFERENCES 227

• Gergonne, Joseph Diaz, (1816) Essai de dialectique rationelle, in Annales de mathématiques, pures et appliées 7, 1816/7, 189–228.

• Jevons, W.S. The Principles of Science, London 1879.

• Ockham's Theory of Terms: Part I of the Summa Logicae, translated and introduced by Michael J. Loux (Notre Dame, IN: University of Notre Dame Press 1974). Reprinted: South Bend, IN: St. Augustine's Press, 1998.

• Ockham's Theory of Propositions: Part II of the Summa Logicae, translated by Alfred J. Freddoso and Henry Schuurman and introduced by Alfred J. Freddoso (Notre Dame, IN: University of Notre Dame Press, 1980). Reprinted: South Bend, IN: St. Augustine's Press, 1998.

• Peirce, C.S., (1896),“The Regenerated Logic”, The Monist, vol. VII, No. 1, p pp. 19−40, The Open Court Publishing Co., Chicago, IL, 1896, for the Hegeler Institute. Reprinted (CP 3.425–455). The Monist 7.

• Sextus Empiricus, Against the Logicians. (Adversus Mathematicos VII and VIII). Richard Bett (trans.) Cam- bridge: Cambridge University Press, 2005. ISBN 0-521-53195-0.

• Zermelo, Ernst (1908). “Untersuchungen über die Grundlagen der Mengenlehre I”. Mathematische Annalen 65 (2): 261–281. doi:10.1007/BF01449999. English translation in Heijenoort, Jean van (1967). “Investiga- tions in the foundations of set theory”. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Source Books in the History of the Sciences. Harvard Univ. Press. pp. 199–215. ISBN 978-0-674-32449-7..

Secondary Sources

• Barwise, Jon, (ed.), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, Amsterdam, North Holland, 1982 ISBN 978-0-444-86388-1 .

• Beaney, Michael, The Frege Reader, London: Blackwell 1997.

• Bochenski, I.M., A History of Formal Logic, Indiana, Notre Dame University Press, 1961.

• Boehner, Philotheus, Medieval Logic, Manchester 1950.

• Buroker, Jill Vance (transl. and introduction), A. Arnauld, P. Nicole Logic or the Art of Thinking, Cambridge University Press, 1996, ISBN 0-521-48249-6.

• Church, Alonzo, 1936-8. “A bibliography of symbolic logic”. Journal of Symbolic Logic 1: 121–218; 3:178–212.

• Ebbesen, Sten “Early supposition theory (12th–13th Century)" Histoire, Épistémologie, Langage 3/1: 35–48 (1981).

• Farrington, B., The Philosophy of Francis Bacon, Liverpool 1964.

• Feferman, Anita B. (1999). “Alfred Tarski”. American National Biography. 21. Oxford University Press. pp. 330–332. ISBN 978-0-19-512800-0.

• Feferman, Anita B.; Feferman, Solomon (2004). Alfred Tarski: Life and Logic. Cambridge University Press. ISBN 978-0-521-80240-6. OCLC 54691904.

• Gabbay, Dov and John Woods, eds, Handbook of the History of Logic 2004. 1. Greek, Indian and Arabic logic; 2. Mediaeval and Renaissance logic; 3. The rise of modern logic: from Leibniz to Frege; 4. British logic in the Nineteenth century; 5. Logic from Russell to Church; 6. Sets and extensions in the Twentieth century; 7. Logic and the modalities in the Twentieth century; 8. The many-valued and nonmonotonic turn in logic; 9. Computational Logic; 10. Inductive logic; 11. Logic: A history of its central concepts; Elsevier, ISBN 0-444-51611-5.

• Geach, P.T. Logic Matters, Blackwell 1972.

• Goodman, Lenn Evan (2003). Islamic Humanism. Oxford University Press, ISBN 0-19-513580-6.

• Goodman, Lenn Evan (1992). Avicenna. Routledge, ISBN 0-415-01929-X. 228 CHAPTER 57. HISTORY OF LOGIC

• Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870–1940. Princeton University Press.

• Gracia, J.G. and Noone, T.B., A Companion to Philosophy in the Middle Ages, London 2003. • Haaparanta, Leila (ed.) 2009. The Development of Modern Logic Oxford University Press.

• Heath, T.L., 1949. Mathematics in Aristotle Oxford University Press. • Heath, T.L., 1931, A Manual of Greek Mathematics, Oxford (Clarendon Press).

• Honderich, Ted (ed.). The Oxford Companion to Philosophy (New York: Oxford University Press, 1995) ISBN 0-19-866132-0.

• Kneale, William and Martha, 1962. The development of logic. Oxford University Press, ISBN 0-19-824773-7. • Lukasiewicz, Aristotle's Syllogistic, Oxford University Press 1951.

• Potter, Michael (2004), Set Theory and its Philosophy, Oxford University Press.

57.10 External links

• The History of Logic from Aristotle to Gödel and Its Relationship with Ontology with annotated bibliographies on the history of logic

• Ancient Logic entry by Susanne Bobzien in the Stanford Encyclopedia of Philosophy • Peter of entry by Joke Spruyt in the Stanford Encyclopedia of Philosophy

• Paul Spade's“Thoughts Words and Things”An Introduction to Late Mediaeval Logic and Semantic Theory • Insights, Images, and Bios of 145 logicians by David Marans Chapter 58

HPO formalism

The History Projection Operator (HPO) formalism is an approach to temporal quantum logic developed by Chris Isham. It deals with the logical structure of quantum mechanical propositions asserted at different points in time.

58.1 Introduction

In standard quantum mechanics a physical system is associated with a Hilbert space H . States of the system at a fixed time are represented by normalised vectors in the space and physical observables are represented by Hermitian operators on H . A physical proposition P about the system at a fixed time can be represented by a projection operator Pˆ on H (See quantum logic). This representation links together the lattice operations in the lattice of logical propositions and the lattice of projection operators on a Hilbert space (See quantum logic). The HPO formalism is a natural extension of these ideas to propositions about the system that are concerned with more than one time.

58.2 History Propositions

58.2.1 Homogeneous Histories

A homogeneous history proposition α is a sequence of single-time propositions αti specified at different times t1 < t2 < . . . < tn . These times are called the temporal support of the history. We shall denote the proposition α as (α1, α2, . . . , αn) and read it as

" αt1 at time t1 is true and then αt2 at time t2 is true and then ... and then αtn at time tn is true”

58.2.2 Inhomogeneous Histories

Not all history propositions can be represented by a sequence of single-time propositions are different times. These are called inhomogeneous history propositions. An example is the proposition α OR β for two homogeneous histories α, β .

58.3 History Projection Operators

The key observation of the HPO formalism is to represent history propositions by projection operators on a history Hilbert space. This is where the name “History Projection Operator”(HPO) comes from.

For a homogeneous history α = (α1, α2, . . . , αn) we can use the tensor product to define a projector ⊗ ⊗ ⊗ αˆ :=α ˆt1 αˆt2 ... αˆtn

229 230 CHAPTER 58. HPO FORMALISM

H where αˆti is the projection operator on that represents the proposition αti at time ti . This αˆ is a projection operator on the tensor product “history Hilbert space”H = H ⊗ H ⊗ ... ⊗ H Not all projection operators on H can be written as the sum of tensor products of the form αˆ . These other projection operators are used to represent inhomogeneous histories by applying lattice operations to homogeneous histories.

58.4 Temporal Quantum Logic

Representing history propositions by projectors on the history Hilbert space naturally encodes the logical structure of history propositions. The lattice operations on the set of projection operations on the history Hilbert space H can be applied to model the lattice of logical operations on history propositions. If two homogeneous histories α and β don't share the same temporal support they can be modified so that they do. If ti is in the temporal support of α but not β (for example) then a new homogeneous history proposition which differs from β by including the “always true”proposition at each time ti can be formed. In this way the temporal supports of α, β can always be joined together. What shall therefore assume that all homogeneous histories share the same temporal support. We now present the logical operations for homogeneous history propositions α and β such that αˆβˆ = βˆαˆ

58.4.1 Conjunction (AND)

If α and β are two homogeneous histories then the history proposition " α and β " is also a homogeneous history. It is represented by the projection operator α\∧ β :=α ˆβˆ (= βˆαˆ)

58.4.2 Disjunction (OR)

If α and β are two homogeneous histories then the history proposition " α or β " is in general not a homogeneous history. It is represented by the projection operator α\∨ β :=α ˆ + βˆ − αˆβˆ

58.4.3 Negation (NOT)

The negation operation in the lattice of projection operators takes Pˆ to ¬Pˆ := I − Pˆ where I is the identity operator on the Hilbert space. Thus the projector used to represent the proposition ¬α (i.e. “not α ") is ¬cα := I − αˆ where I is the identity operator on the history Hilbert space.

58.4.4 Example: Two-time history

As an example, consider the negation of the two-time homogeneous history proposition α = (α1, α2) . The projector to represent the proposition ¬α is

¬cα = I ⊗ I − αˆ1 ⊗ αˆ2 = (I − αˆ1) ⊗ αˆ2 +α ˆ1 ⊗ (I − αˆ2) + (I − αˆ1) ⊗ (I − αˆ2) The terms which appear in this expression:

• (I − αˆ1) ⊗ αˆ2

• αˆ1 ⊗ (I − αˆ2) 58.5. REFERENCES 231

• (I − αˆ1) ⊗ (I − αˆ2) . can each be interpreted as follows:

• α1 is false and α2 is true

• α1 is true and α2 is false

• both α1 is false and α2 is false

These three homogeneous histories, joined together with the OR operation, include all the possibilities for how the proposition " α1 and then α2 " can be false. We therefore see that the definition of ¬cα agrees with what the proposition ¬α should mean.

58.5 References

• C.J. Isham, Quantum Logic and the Histories Approach to Quantum Theory, J.Math.Phys. 35 (1994) 2157- 2185, arXiv:gr-qc/9308006v1 Chapter 59

Imperative logic

Imperative logic is the field of logic concerned with arguments containing sentences in the imperative mood. In contrast to sentences in the declarative mood, imperatives are neither true nor false. This leads to a number of logical dilemmas, puzzles, and paradoxes. Unlike classical logic, there is almost no consensus on any aspect of imperative logic.

59.1 Jørgensen's Dilemma

One of a logic's principal concerns is logical validity. It seems that arguments with imperatives can be valid. Consider:

P1. Take all the books off the table! P2. Foundations of Arithmetic is on the table. C1. Therefore, take Foundations of Arithmetic off the table!

However, an argument is valid if the conclusion follows from the premises. This means the premises give us reason to believe the conclusion, or, alternatively, the truth of the premises determines truth of the conclusion. Since imperatives are neither true nor false and since they are not proper objects of belief, none of the standard accounts of logical validity apply to arguments containing imperatives. Here is the dilemma. Either arguments containing imperatives can be valid or not. On the one hand, if such arguments can be valid, we need a new or expanded account of logical validity and the concomitant details. Providing such an account has proved challenging. On the other hand, if such arguments cannot be valid (either because such arguments are all invalid or because validity is not a notion that applies to imperatives), then our logical intuitions regarding the above argument (and others similar to it) are mistaken. Since either answer seems problematic, this has come to be known as Jorgensen's dilemma, named after Jørgen Jørgensen (da). While this problem was first noted in a footnote by Frege, it received a more developed formulation by Jørgensen.*[1]*[2]

59.2 Ross' Paradox

Alf Ross observed that there is a potential problem for any account of imperative inference.*[3]*[4] Classical logic validates the following inference:

P1. The room is clean. C1. Therefore, the room is clean or grass is green.

This inference is called disjunction introduction. However, a similar inference does not seem to be valid for impera- tives. Consider:

232 59.3. MIXED INFERENCES 233

P1. Clean your room! C1. Therefore, clean your room or burn the house down!

Ross' paradox highlights the challenge faced by anyone who wants to modify or add to the standard account of validity. The challenge is what we mean by a valid imperative inference. For valid declarative inference, the premises give you a reason to believe the conclusion. One might think that for imperative inference, the premises give you a reason to do as the conclusion says. While Ross's paradox seems to suggest otherwise, its severity has been subject of much debate.

59.3 Mixed Inferences

The following is an example of a pure imperative inference:

P1. Do both of the following: wash the dishes and clean your room! C1. Therefore, clean your room!

In this case, all the sentences making up the argument are imperatives. Not all imperative inferences are of this kind. Consider again:

P1. Take all the books off the table! P2. Foundations of Arithmetic is on the table. C1. Therefore, take Foundations of Arithmetic off the table!

Notice that this argument is composed of both imperatives and declaratives and has an imperative conclusion. Mixed inferences are of special interest to logicians. For instance, Henri Poincaré held that no imperative conclusion can be validly drawn from a set of premises which does not contain at least one imperative.*[5] While R.M. Hare held that no declarative conclusion can be validly drawn from a set of premises which cannot validly be drawn from the declaratives among them alone.*[6] There is no consensus among logicians about the truth or falsity of these (or similar) claims and mixed imperative and declarative inference remains vexed.

59.4 Applications

Aside from intrinsic interest, imperative logic has other applications. The use of imperatives in moral theory should make imperative inference an important subject for ethics and metaethics. Also, many major computer programming languages are imperative programming languages.

59.5 See also

• Deontic logic

• List of paradoxes#Logic

• Pragmatics

• Speech acts

• Temporal logic 234 CHAPTER 59. IMPERATIVE LOGIC

59.6 References

[1] Frege, G. (1892) 'On sense and reference', in Geach and Black (eds.) Translations from the Philosophical Writings of Gottlob Frege Oxford: Blackwell.

[2] Jørgensen, J. (1938) 'Imperatives and logic', Erkenntnis 7: 288-98.

[3] Ross, A. (1941) ‘Imperatives and Logic’, Theoria 7: 53–71. doi:10.1111/j.1755-2567.1941.tb00034.x

[4] Ross, A. (1944) ‘Imperatives and Logic’, Philosophy of Science 11: 30–46.

[5] Poincaré, Henri (1913). Dernières Pensées. Paris: Ernest Flammarion.

[6] Hare, Richard M. (1967). Some alleged differences between imperatives and indicatives. Mind, 76, 309-326.

59.7 Further reading

• Charles Leonard Hamblin (1987). Imperatives. Basil Blackwell. ISBN 978-0-631-15193-7. • Peter B. M. Vranas (2010), IMPERATIVES, LOGIC OF*, Entry for The International Encyclopedia of Ethics

• Harry J. Gensler (2010). Introduction to Logic (2nd ed.). Taylor & Francis. Chapter 12: Deontic and Impera- tive Logic. ISBN 978-0-415-99650-1. Covers mostly the approach of Héctor-Neri Castañeda.

59.8 External links

• Mitchell S. Green, Imperative Logic, University of Virginia Chapter 60

Inclusion (logic)

In logic and mathematics, inclusion is the concept that all the contents of one object are also contained within a second object.*[1] The modern symbol for inclusion first appears in Gergonne (1816), who defines it as one idea 'containing' or being 'contained' by another, using the backward letter 'C' to express this. Peirce articulated this clearly in 1870, arguing also that inclusion was a wider concept than equality, and hence a logically simpler one.*[2] Schröder (also Frege) calls the same concept 'subordination'.*[3]

60.1 References

[1] Quine, W. V. (December 1937). “Logic based on inclusion and abstraction”. The Journal of Symbolic Logic 2 (4). pp. 145–152. doi:10.2307/2268279.

[2]“Descr. of a notation”, CP III 28.

[3] Vorlesungen I., 127.

235 Chapter 61

Index of logic articles

61.1 Contents

• A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

See also: List of logicians

61.2 A

A System of Logic -- A priori and a posteriori -- Abacus logic -- Abduction (logic) -- Abductive validation -- Academia Analitica -- Accuracy and precision -- Ad captandum -- Ad hoc hypothesis -- Ad hominem -- Affine logic -- Affirming the antecedent -- Affirming the consequent -- Algebraic logic -- Ambiguity -- Analysis -- Analysis (journal) -- Analytic reasoning -- Analytic–synthetic distinction -- Anangeon -- -- Antecedent (logic) -- Antepredicament -- Anti-psychologism -- -- Apophasis -- -- -- Archive for Mathematical Logic -- Arché -- Argument -- Argument by example -- Argument form -- -- Argument map -- Argumentation ethics -- Argumentation theory -- -- Argumentum e contrario -- Ariadne's thread (logic) -- Aristotelian logic -- Aristotle -- Association for Informal Logic and Critical Thinking -- Association for Logic, Language and Information -- Association for Sym- bolic Logic -- Attacking Faulty Reasoning -- Australasian Association for Logic -- Axiom -- Axiom independence -- Axiom of reducibility -- Axiomatic system -- Axiomatization --

61.3 B

Backward chaining -- Barcan formula -- -- Begriffsschrift -- Belief -- Belief bias -- Belief revision -- Benson Mates -- Bertrand Russell Society -- Biconditional elimination -- Biconditional introduction -- Bivalence and related laws -- -- Boole's syllogistic -- Boolean algebra (logic) -- Boolean algebra (structure) -- Boolean network --

61.4 C

Canon (basic principle) -- Canonical form -- Canonical form (Boolean algebra) -- Cartesian circle -- Case-based reasoning -- -- Categories (Aristotle) -- Categories (Peirce) -- Category mistake -- Catuṣkoṭi -- Center for Critical Thinking -- Circular definition -- -- Circular reference -- Circular report- ing -- Circumscription (logic) -- Circumscription (taxonomy) -- Classical logic -- Clocked logic -- Cognitive bias -- Cointerpretability -- College logic -- Colorless green ideas sleep furiously -- -- Combinatory logic -- -- Common knowledge (logic) -- Commutativity of conjunction -- Completeness (logic) -- Composition of Causes -- Compossibility -- Comprehension (logic) -- Computability logic -- Concept -- Conceptualism

236 61.5. D 237

-- Condensed detachment -- Conditional disjunction -- Conditional probability -- Conditional proof -- Conditional quantifier -- Confirmation bias -- Conflation -- Confusion of the inverse -- Conjunction elimination -- -- Conjunction introduction -- Conjunctive normal form -- -- Connotation -- Consequent -- Consistency -- -- Contra principia negantem non est disputandum -- Contradiction -- Contrapositive -- Control logic -- -- Converse (logic) -- Converse Barcan formula -- Correlative-based fallacies -- Cotolerance -- Cotolerant sequence -- Counterexample -- Counterfactual conditional -- Counterintuitive -- Cratylism -- Credibility -- Criteria of truth -- Critical-Creative Thinking and Behavioral Research Laboratory -- Critical peda- gogy -- Critical reading -- Critical thinking -- Critique of Pure Reason -- Curry's paradox -- Cyclic negation --

61.5 D

Dagfinn Føllesdal -- De Interpretatione -- De Morgan's laws -- Decidability (logic) -- Decidophobia -- Decision making -- Decisional balance sheet -- Deductive closure -- Deduction theorem -- Deductive fallacy -- Deductive reasoning -- Default logic -- Defeasible logic -- Defeasible reasoning -- Definable set -- Definist fallacy -- Definition -- Definitions of logic -- -- Denying the antecedent -- Denying the correlative -- Deontic logic -- Description -- -- Descriptive fallacy -- Deviant logic -- Dharmakirti -- Diagrammatic reasoning -- Dialectica - - Dialectica space -- Dialetheism -- Dichotomy -- Difference (philosophy) -- Digital timing diagram -- Dignāga -- Dilemma -- Disjunction elimination -- Disjunction introduction -- Disjunctive normal form -- -- Dispositional and occurrent belief -- Disquotational principle -- Dissoi logoi -- Division of Logic, Methodology, and Philosophy of Science -- Don't-care term -- Donald Davidson (philosopher) -- Double counting (fallacy) -- -- Double negative -- Double negative elimination -- Doxa -- Drinking the Kool-Aid --

61.6 E

EL++ -- -- Effective method -- Elimination rule -- Emotional reasoning -- Emotions in decision- making -- Empty name -- Encyclopedia of the Philosophical Sciences -- End term -- Engineered language -- Entailment -- Entitative graph -- Enumerative definition -- Epicureanism -- Epilogism -- Epistemic closure -- Equisatisfiability -- Erotetics -- Eternal statement -- -- European Summer School in Logic, Language and Infor- mation -- Evidence -- Evolutionary logic -- Exclusive nor -- Exclusive or -- -- Existential graph -- Existential quantification -- Expert -- Explanandum -- Explanation -- Explanatory power -- Extension (semantics) -- Extensional context -- Extensional definition --

61.7 F

Fa (concept) -- Fact -- Fallacies of definition -- Fallacy -- Fallacy of distribution -- Fallacy of four terms -- Fallacy of -- Fallacy of the four terms -- -- -- -- False premise -- Fictionalism -- Finitary relation -- Finite model property -- First-order logic -- First-order predicate -- First-order predicate calculus -- First-order resolution -- Fitch-style calculus -- Fluidic logic -- Fluidics -- Formal fallacy -- Formal ontology -- Formal system -- Formalism (philosophy) -- Forward chaining -- Free logic -- Free variables and bound variables -- Function and Concept -- Fuzzy logic --

61.8 G

Game semantics -- Ganto's Ax -- Geometry of interaction -- Gilles-Gaston Granger -- Gongsun Long -- Grammaticality -- Greedy reductionism -- Grundlagen der Mathematik --

61.9 H

HPO formalism -- Halo effect -- Handbook of Automated Reasoning -- Hanlon's razor -- Hasty generalization - - Herbrandization -- Hetucakra -- -- Higher-order predicate -- Higher-order thinking -- Historian's 238 CHAPTER 61. INDEX OF LOGIC ARTICLES fallacy -- Historical fallacy -- History of logic -- History of the function concept -- Hold come what may -- Homunculus argument -- Horn clause -- Hume's fork -- Hume's principle -- Hypothetical syllogism --

61.10 I

Identity (philosophy) -- Identity of indiscernibles -- Idola fori -- Idola specus -- Idola theatri -- Idola tribus -- If-by- whiskey -- Iff -- -- -- -- Immutable truth -- Imperative logic -- Implicant -- Inclusion (logic) -- Incomplete comparison -- Inconsistent comparison -- Inconsistent triad -- Independence-friendly logic -- Indian logic -- Inductive logic -- Inductive logic programming -- Inference -- Inference procedure -- Inference rule -- Inferential role semantics -- Infinitary logic -- Infinite regress -- Infinity -- Informal fallacy -- Informal logic -- Inquiry -- Inquiry (philosophy journal) -- Insolubilia -- Institute for Logic, Language and Computation -- Intellectual responsibility -- Intended interpretation -- -- Intensional fallacy -- Intensional logic -- Intensional state- ment -- Intentional Logic -- -- Interpretability -- Interpretability logic -- Interpretive discussion -- Introduction rule -- Introduction to Mathematical Philosophy -- Intuitionistic linear logic -- Intuitionistic logic -- Invalid proof -- Inventor's paradox -- Inverse (logic) -- Inverse consequences -- Irreducibility -- Is logic empirical? -- Isagoge -- Ivor Grattan-Guinness --

61.11 J

Jacobus Naveros -- -- Jingle-jangle fallacies -- John Corcoran (logician) -- John W. Dawson, Jr -- Journal of Applied Non-Classical Logics -- Journal of Automated Reasoning -- Journal of Logic, Language and Information -- Journal of Logic and Computation -- Journal of Mathematical Logic -- Journal of Philosophical Logic -- Journal of Symbolic Logic -- Judgment (mathematical logic) -- Judgmental language -- Just-so story --

61.12 K

Karnaugh map -- Kinetic logic -- Knowing and the Known -- Kripke semantics -- Kurt Gödel Society --

61.13 L

Language -- Language, Proof and Logic -- Lateral thinking -- Law of excluded middle -- -- Law of non-contradiction -- Law of noncontradiction -- Law of thought -- Laws of Form -- Laws of logic -- Leap of faith -- Lemma (logic) -- Lexical definition -- Linear logic -- Linguistic and Philosophical Investigations -- Linguistics and Philosophy -- -- List of incomplete proofs -- List of logic journals -- List of paradoxes -- Logic -- Logic Lane -- Logic Spectacles -- -- Logic in China -- Logic in Islamic philosophy -- Logic of class -- Logic of information -- Logic programming -- Logica Universalis -- Logica nova -- Logical Analysis and History of Philosophy -- Logical Investigations (Husserl) -- Logical Methods in Computer Science -- Logical abacus -- Logical argument -- Logical assertion -- -- -- Logical conditional -- -- -- -- -- Logical equivalence -- Logical extreme -- Logical form -- Logical harmony -- Logical -- Logical nand -- Logical nor -- Logical operator -- -- -- Logicism -- Logico-linguistic modeling -- Logos -- Loosely associated statements -- Łoś–Tarski preservation theorem -- Ludic fallacy -- Lwów–Warsaw school of logic --

61.14 M

Main contention -- Major term -- Markov's principle -- Martin Gardner bibliography -- Masked man fallacy -- Material conditional -- Mathematical fallacy -- Mathematical logic -- Meaning (linguistics) -- Meaning (non-linguistic) -- Meaning (philosophy of language) -- Meaningless statement -- Megarian school -- Mental model theory of reasoning -- Mereology -- Meta-communication -- Metalanguage -- Metalogic -- Metamathematics -- Metasyntactic variable -- Metatheorem -- Metavariable -- Middle term -- -- Minor premise -- Miscellanea Logica -- Missing dol- lar riddle -- Modal fictionalism -- Modal logic -- Model theory -- Modus ponens -- Modus tollens -- Moral reasoning 61.15. N 239

-- Motivated reasoning -- -- Multigrade predicate -- Multi-valued logic -- Multiple-conclusion logic -- Mutatis mutandis -- Mutual knowledge (logic) -- Mutually exclusive events -- Münchhausen trilemma --

61.15 N

Naive set theory -- Name -- Narrative logic -- Natural deduction -- Natural kind -- Natural language -- Necessary and sufficient -- Necessity and sufficiency -- Negation -- Neutrality (philosophy) -- fallacy -- Nixon diamond -- -- Nominal identity -- Non-Aristotelian logic -- Non-classical logic -- Non-monotonic logic -- Non- -- Non sequitur (logic) -- Noneism -- Nonfirstorderizability -- Nordic Journal of Philosophical Logic -- Normal form (natural deduction) -- Novum Organum -- Nyaya -- Nyāya Sūtras --

61.16 O

Object language -- -- Object theory -- Occam's razor -- On Formally Undecidable Propositions of Principia Mathematica and Related Systems -- One-sided argument -- Ontological commitment -- Open sentence -- Opinion -- Opposing Viewpoints series -- Ordered logic -- Organon -- Original proof of Gödel's completeness theorem -- Osmund Lewry -- Ostensive definition -- Outline of logic -- Overbelief -- Oxymoron --

61.17 P

Package-deal fallacy -- Panlogism -- Paraconsistent logic -- Paraconsistent logics -- -- Paradox -- Pars destruens/pars construens -- Pathetic fallacy -- Per fas et nefas -- Persuasive definition -- Peter Simons (academic) -- Philosophia Mathematica -- Philosophical logic -- Philosophy of logic -- Pierce's law -- Plural quantification -- -- Polarity item -- Polish Logic -- -- Politician's syllogism -- Polychotomous key -- Polylogism -- Polysyllogism -- Port-Royal Logic -- Possible world -- Post's lattice -- Post disputation argument -- Post hoc ergo propter hoc -- Posterior Analytics -- Practical syllogism -- Pragmatic mapping -- -- Pragmatic theory of truth -- Pramāṇa -- Pramāṇa-samuccaya -- Precising definition -- Precision questioning -- Predicable -- Predicate (logic) -- Predicate abstraction -- Predicate logic -- Preferential entailment -- Preintuitionism -- Prescriptivity -- Presentism (literary and historical analysis) -- -- Principia Mathematica -- Principle of bivalence -- Principle of explosion -- Principle of nonvacuous contrast -- Principle of sufficient reason -- Principles of Mathematical Logic -- Prior Analytics -- Private Eye Project -- Pro hominem -- Probabilistic logic -- Probabilistic logic network -- Problem of future contingents -- Problem of induction -- Process of elimination -- Project Reason -- Proof- theoretic semantics -- Proof (truth) -- -- Proof theory -- -- Proposition -- Propositional calculus -- Propositional function -- Propositional representation -- Propositional variable -- Prosecutor's fallacy -- Provability logic -- Proving too much -- Prudence -- Pseudophilosophy -- Psychologism -- Psychologist's fallacy --

61.18 Q

Q.E.D. -- Quantification -- Quantization (linguistics) -- Quantum logic --

61.19 R

Ramism -- Rationality -- Razor (philosophy) -- Reason -- Reductio ad absurdum -- Reference -- Reflective equilibrium -- -- Regular modal logic -- Reification (fallacy) -- -- Relevance -- Relevance logic -- Relevant logic -- Remarks on the Foundations of Mathematics -- Retroduction -- Retrospective determinism -- Revolutions in Mathematics -- Rhetoric -- Rigour -- Rolandas Pavilionis -- Round square copula -- Rudolf Carnap -- Rule bank -- Rule of inference -- Rvachev function -- 240 CHAPTER 61. INDEX OF LOGIC ARTICLES

61.20 S

SEE-I -- Salva congruitate -- Salva veritate -- Satisfiability -- Scholastic logic -- School of Names -- Science of Logic -- Scientific temper -- Second-order predicate -- Segment addition postulate -- Self-reference -- Self-refuting idea -- Self-verifying theories -- Semantic theory of truth -- Semantics -- Sense and reference -- Sequent -- Sequent calculus -- -- Set (mathematics) -- Seven Types of Ambiguity (Empson) -- Sheffer stroke -- -- Simple non-inferential passage -- Singular term -- Situation -- Situational analysis -- Skeptic's Toolbox -- Slingshot argument -- Social software (social procedure) -- Socratic questioning -- Soku hi -- Some Remarks on Logical Form -- Sophism -- Sophistical Refutations -- Soundness -- Source credibility -- Source criticism -- Special case -- Specialization (logic) -- Speculative reason -- Spurious relationship -- Square of opposition -- State of affairs (philosophy) -- Statement (logic) -- Straight and Crooked Thinking -- Straight face test -- -- Strength (mathematical logic) -- -- Strict implication -- Strict logic -- -- Studia Logica -- Studies in Logic, Grammar and Rhetoric -- Subjective logic -- Substitution (logic) -- -- Sufficient condition -- Sum of Logic -- Sunk costs -- Supertask -- Supervaluationism -- Supposition theory -- Survivorship bias -- Syllogism -- -- Symbol (formal) -- Syntactic Structures -- Syntax (logic) -- Synthese -- Systems of Logic Based on Ordinals --

61.21 T

T-schema -- -- Tarski's undefinability theorem -- Tautology (logic) -- Temporal logic -- Temporal parts -- Teorema (journal) -- Term (argumentation) -- Term logic -- Ternary logic -- -- -- Textual case based reasoning -- The False Subtlety of the Four Syllogistic Figures -- The Foundations of Arithmetic -- The of Thought -- The Laws of Thought -- The Paradoxes of the Infinite -- Theorem -- Theoretical definition -- Theory and Decision -- Theory of justification -- Theory of obligationes -- Third-cause fallacy -- Three men make a tiger -- Tolerance (in logic) -- Topical logic -- Topics (Aristotle) -- Tractatus Logico-Philosophicus -- Train of thought -- Trairūpya -- Transferable belief model -- Transparent Intensional Logic -- TregoED -- Trikonic -- Trilemma -- Trivial objections -- -- Truth -- Truth-bearer -- Truth claim -- Truth condition -- -- Truth value -- Truthiness -- Truthmaker -- Type (model theory) -- Type theory -- Type–token distinction --

61.22 U

Ultrafinitism -- Unification (computer science) -- Unifying theories in mathematics -- Uniqueness quantification -- Universal logic -- Universal quantification -- Univocity -- Unspoken rule -- Use–mention distinction --

61.23 V

Vacuous truth -- Vagrant predicate -- Vagueness -- Validity -- Valuation-based system -- Van Gogh fallacy -- Venn diagram -- Vicious circle principle --

61.24 W

Warnier/Orr diagram -- Weak mindedness -- Well-formed formula -- What the Tortoise Said to Achilles -- Willard Van Orman Quine -- William Kneale -- Window operator -- -- Witness (mathematics) -- Wolfram axiom -- Word sense --

61.25 Z

Zhegalkin polynomial -- 61.26. SEE ALSO 241

61.26 See also

• List of logicians

• List of rules of inference • List of mathematical logic topics

• There is a list of paradoxes on the paradox page.

• There is a list of fallacies on the logical fallacy page. • Modern mathematical logic is at the list of mathematical logic topics page.

• For introductory set theory and other supporting material see the list of basic discrete mathematics topics. Chapter 62

Informal logic

Informal logic, intuitively, refers to the principles of logic and logical thought outside of a formal setting. However, perhaps because of the “informal”in the title, the precise definition of “informal logic”is a matter of some dispute.*[1] Ralph H. Johnson and J. Anthony Blair define informal logic as“a branch of logic whose task is to develop non-formal standards, criteria, procedures for the analysis, interpretation, evaluation, criticism and construction of argumentation.”*[2] This definition reflects what had been implicit in their practice and what others*[3]*[4]*[5] were doing in their informal logic texts. Informal logic is associated with (informal) fallacies, critical thinking, the Thinking Skills Movement*[6] and the interdisciplinary inquiry known as argumentation theory. Frans H. van Eemeren writes that the label “informal logic”covers a “collection of normative approaches to the study of reasoning in ordinary language that remain closer to the practice of argumentation than formal logic.”*[7]

62.1 History

Informal logic as a distinguished enterprise under this name emerged roughly in the late 1970s as a sub-field of philosophy. The naming of the field was preceded by the appearance of a number of textbooks that rejected the symbolic approach to logic on pedagogical grounds as inappropriate and unhelpful for introductory textbooks on logic for a general audience, for example Howard Kahane's Logic and Contemporary Rhetoric, subtitled “The Use of Reason in Everyday Life”, first published in 1971. Kahane's textbook was described on the notice of his death in the Proceedings And Addresses of the American Philosophical Association (2002) as “a text in informal logic, [that] was intended to enable students to cope with the misleading rhetoric one frequently finds in the media and in political discourse. It was organized around a discussion of fallacies, and was meant to be a practical instrument for dealing with the problems of everyday life. [It has] ... gone through many editions; [it is] ... still in print; and the thousands upon thousands of students who have taken courses in which his text [was] ... used can thank Howard for contributing to their ability to dissect arguments and avoid the deceptions of deceitful rhetoric. He tried to put into practice the ideal of discourse that aims at truth rather than merely at persuasion. (Hausman et al. 2002)"*[8]*[9] Other textbooks from the era taking this approach were Michael Scriven's Reasoning (Edgepress, 1976) and Logical Self-Defense by Ralph Johnson and J. Anthony Blair, first published in 1977.*[8] Earlier precursors in this tradition can be considered Monroe Beardsley's Practical Logic (1950) and Stephen Toulmin's The Uses of Argument (1958).*[10] The field perhaps became recognized under its current name with the First International Symposium on Informal Logic held in 1978. Although initially motivated by a new pedagogical approach to undergraduate logic textbooks, the scope of the field was basically defined by a list of 13 problems and issues which Blair and Johnson included as an appendix to their keynote address at this symposium:*[8]*[11]

• the theory of logical criticism

• the theory of argument

• the theory of fallacy

• the fallacy approach vs. the critical thinking approach

242 62.2. PROPOSED DEFINITIONS 243

• the viability of the inductive/deductive dichotomy • the ethics of argumentation and logical criticism • the problem of assumptions and missing premises • the problem of context • methods of extracting arguments from context • methods of displaying arguments • the problem of pedagogy • the nature, division and scope of informal logic • the relationship of informal logic to other inquiries

David Hitchcock argues that the naming of the field was unfortunate, and that philosophy of argument would have been more appropriate. He argues that more undergraduate students in North America study informal logic than any other branch of philosophy, but that as of 2003 informal logic (or philosophy of argument) was not recognized as separate sub-field by the World Congress of Philosophy.*[8] Frans H. van Eemeren wrote that “informal logic”is mainly an approach to argumentation advanced by a group of US and Canadian philosophers and largely based on the previous works of Stephen Toulmin and to a lesser extent those of Chaïm Perelman.*[7] Alongside the symposia, since 1983 the journal Informal Logic has been the publication of record of the field, with Blair and Johnson as initial editors, with the editorial board now including two other colleagues from the University of Windsor—Christopher Tindale and Hans V. Hansen.*[12] Other journals that regularly publish articles on informal logic include Argumentation (founded in 1986), Philosophy and Rhetoric, Argumentation and Advocacy (the journal of the American Forensic Association), and Inquiry: Critical Thinking Across the Disciplines (founded in 1988).*[13]

62.2 Proposed definitions

Johnson and Blair (2000) proposed the following definition: “Informal logic designates that branch of logic whose task is to develop non-formal2 standards, criteria, procedures for the analysis, interpretation, evaluation, critique and construction of argumentation in everyday discourse.”Their meaning of non-formal2 is taken from Barth and Krabbe (1982), which is explained below. To understand the definition above, one must understand “informal”which takes its meaning in contrast to its counterpart “formal.”(This point was not made for a very long time, hence the nature of informal logic remained opaque, even to those involved in it, for a period of time.) Here it is helpful to have recourse*[14] to Barth and Krabbe (1982:14f) where they distinguish three senses of the term“form.”By“form1,”Barth and Krabbe mean the sense of the term which derives from the Platonic idea of form—the ultimate metaphysical unit. Barth and Krabbe claim that most traditional logic is formal in this sense. That is, syllogistic logic is a logic of terms where the terms could naturally be understood as place-holders for Platonic (or Aristotelian) forms. In this first sense of “form,”almost all logic is informal (not-formal). Understanding informal logic this way would be much too broad to be useful. By “form2,”Barth and Krabbe mean the form of sentences and statements as these are understood in modern systems of logic. Here validity is the focus: if the premises are true, the conclusion must then also be true. Now validity has to do with the logical form of the statement that makes up the argument. In this sense of “formal,” most modern and contemporary logic is “formal.”That is, such logics canonize the notion of logical form, and the notion of validity plays the central normative role. In this second sense of form, informal logic is not-formal, because it abandons the notion of logical form as the key to understanding the structure of arguments, and likewise retires validity as normative for the purposes of the evaluation of argument. It seems to many that validity is too stringent a requirement, that there are good arguments in which the conclusion is supported by the premises even though it does not follow necessarily from them (as validity requires). An argument in which the conclusion is thought to be “beyond reasonable doubt, given the premises”is sufficient in law to cause a person to be sentenced to death, even though it does not meet the standard of logical validity. This type of argument, based on accumulation of evidence rather than pure deduction, is called a conductive argument. By “form3,”Barth and Krabbe mean to refer to “procedures which are somehow regulated or regimented, which take place according to some set of rules.”Barth and Krabbe say that“we do not defend formality3 of all kinds and 244 CHAPTER 62. INFORMAL LOGIC

under all circumstances.”Rather “we defend the thesis that verbal dialectics must have a certain form (i.e., must proceed according to certain rules) in order that one can speak of the discussion as being won or lost”(19). In this third sense of“form”, informal logic can be formal, for there is nothing in the informal logic enterprise that stands opposed to the idea that argumentative discourse should be subject to norms, i.e., subject to rules, criteria, standards or procedures. Informal logic does present standards for the evaluation of argument, procedures for detecting missing premises etc. Johnson and Blair (2000) noticed a limitation of their own definition, particularly with respect to“everyday discourse” , which could indicate that it does not seek to understand specialized, domain-specific arguments made in natural languages. Consequently, they have argued that the crucial divide is between arguments made in formal languages and those made in natural languages. Fisher and Scriven (1997) proposed a more encompassing definition, seeing informal logic as “the discipline which studies the practice of critical thinking and provides its intellectual spine”. By “critical thinking”they understand “skilled and active interpretation and evaluation of observations and communications, information and argumenta- tion.”*[15]

62.3 Criticisms

Some hold the view that informal logic is not a branch or subdiscipline of logic, or even the view that there cannot be such a thing as informal logic.*[16]*[17]*[18] Massey criticizes informal logic on the grounds that it has no theory underpinning it. Informal logic, he says, requires detailed classification schemes to organize it, which in other disciplines is provided by the underlying theory. He maintains that there is no method of establishing the invalidity of an argument aside from the formal method, and that the study of fallacies may be of more interest to other disciplines, like psychology, than to philosophy and logic.*[16]

62.4 Relation to critical thinking

See also: Critical thinking

Since the 1980s, informal logic has been partnered and even equated,*[19] in the minds of many, with critical think- ing. The precise definition of "critical thinking" is a subject of much dispute.*[20] Critical thinking, as defined by Johnson, is the evaluation of an intellectual product (an argument, an explanation, a theory) in terms of its strengths and weaknesses.*[20] While critical thinking will include evaluation of arguments and hence require skills of argu- mentation including informal logic, critical thinking requires additional abilities not supplied by informal logic, such as the ability to obtain and assess information and to clarify meaning. Also, many believe that critical thinking re- quires certain dispositions.*[21] Understood in this way,“critical thinking”is a broad term for the attitudes and skills that are involved in analyzing and evaluating arguments. The critical thinking movement promotes critical thinking as an educational ideal. The movement emerged with great force in the '80s in North America as part of an ongoing critique of education as regards the thinking skills not being taught.

62.5 Relation to argumentation theory

See also: Argumentation theory

The social, communicative practice of argumentation can and should be distinguished from implication (or entailment) —a relationship between propositions; and from inference—a mental activity typically thought of as the drawing of a conclusion from premises. Informal logic may thus be said to be a logic of argumentation, as distinguished from implication and inference.*[22] Argumentation theory (or the theory of argumentation) has come to be the term that designates the theoretical study of argumentation. This study is interdisciplinary in the sense that no one discipline will be able to provide a complete account. A full appreciation of argumentation requires insights from logic (both formal and informal), rhetoric, communication theory, linguistics, psychology, and, increasingly, computer science. Since the 1970s, there has been significant agreement that there are three basic approaches to argumentation theory: the logical, the rhetorical and the 62.6. SEE ALSO 245 dialectical. According to Wenzel,*[23] the logical approach deals with the product, the dialectical with the process, and the rhetorical with the procedure. Thus, informal logic is one contributor to this inquiry, being most especially concerned with the norms of argument.

62.6 See also

• Argument map • Informal fallacy • Inference objection • Lemma • Philosophy of language • Semantics

62.7 Footnotes

[1] See Johnson 1999 for a survey of definitions. [2] Johnson, Ralph H., and Blair, J. Anthony (1987), “The Current State of Informal Logic”, Informal Logic, 9(2–3), 147– 151. Johnson & Blair added "... in everyday discourse”but in (2000), modified their definition, and broadened the focus now to include the sorts of argument that occurs not just in everyday discourse but also disciplined inquiry—what Weinstein (1990) calls “stylized discourse.” [3] Scriven, 1976 [4] Munson, 1976 [5] Fogelin, 1978 [6] Resnick, 1989 [7] Frans H. van Eemeren (2009). “The Study of Argumentation”. In Andrea A. Lunsford, Kirt H. Wilson, Rosa A. Eberly. The SAGE handbook of rhetorical studies. SAGE. p. 117. ISBN 978-1-4129-0950-1. [8] David Hitchcock, Informal logic 25 years later in Informal Logic at 25: Proceedings of the Windsor Conference (OSSA 2003) [9] JSTOR 3218569 [10] Fisher (2004) p. vii [11] J. Anthony Blair and Ralph H. Johnson (eds.), Informal Logic: The First International Symposium, 3-28. Pt. Reyes, CA: Edgepress [12] http://ojs.uwindsor.ca/ojs/leddy/index.php/informal_logic/about/editorialTeam [13] Johnson and Blair (2000), p. 100 [14] As Johnson (1999) does. [15] Johnson and Blair (2000), p. 95 [16] Massey, 1981 [17] Woods, 1980 [18] Woods, 2000 [19] Johnson (2000) takes the conflation to be part of the Network Problem and holds that settling the issue will require a theory of reasoning. [20] Johnson, 1992 [21] Ennis, 1987 [22] Johnson, 1999 [23] Wenzel (1990) 246 CHAPTER 62. INFORMAL LOGIC

62.8 References

• Barth, E. M., & Krabbe, E. C. W. (Eds.). (1982). From axiom to dialogue: A philosophical study of logics and argumentation. Berlin: Walter De Gruyter. • Blair, J. A & Johnson, R.H. (1980). The recent development of informal logic. In J. Anthony Blair and Ralph H. Johnson (Eds.). Informal logic: The first international symposium, (pp. 3–28). Inverness, CA: Edgepress. • Ennis, R.H. (1987). A taxonomy of critical thinking dispositions and abilities. In J.B. Baron and R.J. Sternberg (Eds.), Teaching critical thinking skills: Theory and practice, (pp. 9–26). New York: Freeman. • Eemeren, F. H. van, & Grootendorst, R. (1992). Argumentation, communication and fallacies. Hillsdale, NJ: Lawrence Erlbaum Associates. • Fisher, A. and Scriven, M. (1997). Critical thinking: Its definition and assessment. Point Reyes, CA: Edgepress • Fisher, Alec (2004). The logic of real arguments (2nd ed.). Cambridge University Press. ISBN 978-0-521- 65481-4. • Govier, T. (1987). Problems in argument analysis and evaluation. Dordrecht: Foris. • Govier, T. (1999). The Philosophy of Argument. Newport News, VA: Vale Press. • Groarke, L. (2006). Informal Logic. Stanford Encyclopedia of Philosophy, from http://plato.stanford.edu/ entries/logic-informal/ • Hitchcock, David (2007). “Informal logic and the concept of argument”. In Jacquette, Dale. Philosophy of logic. Elsevier. ISBN 978-0-444-51541-4. preprint • Johnson, R. H. (1992). The problem of defining critical thinking. In S. P. Norris (Ed.), The generalizability of critical thinking (pp. 38–53). New York: Teachers College Press. (Reprinted in Johnson (1996).) • Johnson, R. H. (1996). The rise of informal logic. Newport News, VA: Vale Press • Johnson, R. H. (1999). The relation between formal and informal logic. Argumentation, 13(3) 265-74. • Johnson, R. H. (2000). Manifest rationality: A pragmatic theory of argument. Mahwah, NJ: Lawrence Erl- baum Associates. • Johnson, R. H. & Blair, J. A. (1987). The current state of informal logic. Informal Logic 9, 147-51. • Johnson, R. H. & Blair, J. A. (1996). Informal logic and critical thinking. In F. van Eemeren, R. Grootendorst, & F. Snoeck Henkemans (Eds.), Fundamentals of argumentation theory (pp. 383–86). Mahwah, NJ: Lawrence Erlbaum Associates • Johnson, R. H. & Blair, J. A. (2002). Informal logic and the reconfiguration of logic. In D. Gabbay, R. H. Johnson, H.-J. Ohlbach and J. Woods (Eds.). Handbook of the logic of argument and inference: The turn towards the practical (pp. 339–396). Elsivier: North Holland. • MacFarlane, J. (2005). Logical Constants. Stanford Encyclopedia of Philosophy. • Massey, G. (1981). The fallacy behind fallacies. Midwest Studies of Philosophy, 6, 489-500. • Munson, R. (1976). The way of words: an informal logic. Boston: Houghton Mifflin. • Resnick, L. (1987). Education and learning to think. Washington, DC: National Academy Press.. • Walton, D. N. (1990). What is reasoning? What is an argument? The Journal of Philosophy, 87, 399-419. • Weinstein, M. (1990) Towards a research agenda for informal logic and critical thinking. Informal Logic, 12, 121-143. • Wenzel, J. 1990 Three perspectives on argumentation. In R Trapp and J Scheutz, (Eds.), Perspectives on argumentation: Essays in honour of Wayne Brockreide, 9-26 Waveland Press: Prospect Heights, IL • Woods, J. (1980). What is informal logic? In J.A. Blair & R. H. Johnson (Eds.), Informal Logic: The First International Symposium (pp. 57–68). Point Reyes, CA: Edgepress. 62.9. EXTERNAL LINKS 247

62.8.1 Special journal issue

The open access issue 20(2) of Informal Logic from year 2000 groups a number of papers addressing foundational issues, based on the Panel on Informal Logic that was held at the 1998 World Congress of Philosophy, including:

• Hitchcock, D. (2000) The significance of informal logic for philosophy. Informal Logic 20(2), 129-138. • Johnson, R. H. & Blair, J. A. (2000). Informal logic: An overview. Informal Logic 20(2): 93-99.

• Woods, J. (2000). How Philosophical is Informal Logic? Informal Logic 20(2): 139-167. 2000

62.8.2 Textbooks

• Kahane, H. (1971). Logic and contemporary rhetoric:The use of reasoning in everyday life. Belmont: Wadsworth. Still in print as Nancy Cavender; Howard Kahane (2009). Logic and Contemporary Rhetoric: The Use of Reason in Everyday Life (11th ed.). Cengage Learning. ISBN 978-0-495-80411-6.

• Scriven, M. (1976). Reasoning. New York. McGraw Hill. • Johnson, R. H. & Blair, J. A. (1977). Logical self-defense. Toronto: McGraw-Hill Ryerson. US Edition. (2006). New York: Idebate Press. • Fogelin, R.J. (1978). Understanding arguments: An introduction to informal logic. New York: Harcourt, Brace, Jovanovich. Still in print as Sinnott-Armstrong, Walter; Fogelin, Robert (2010), Understanding Argu- ments: An Introduction to Informal Logic (8th ed.), Belmont, California: Wadsworth Cengage Learning, ISBN 978-0-495-60395-5

• Stephen N. Thomas (1997). Practical reasoning in natural language (4th ed.). Prentice Hall. ISBN 978-0-13- 678269-8.

• Irving M. Copi; Keith Burgess-Jackson (1996). Informal logic (3rd ed.). Prentice Hall. ISBN 978-0-13- 229048-7.

• Woods, John, Andrew Irvine and Douglas Walton, 2004. Argument: Critical Thinking, Logic and the Fallacies. Toronto: Prentice Hall

• Groarke, Leo and Christopher Tindale, 2004. Good Reasoning Matters! (3rd edition). Toronto: Oxford University Press

• Douglas N. Walton (2008). Informal logic: a pragmatic approach (2nd ed.). Cambridge University Press. ISBN 978-0-521-71380-1.

• Trudy Govier (2009). A Practical Study of Argument (7th ed.). Cengage Learning. ISBN 978-0-495-60340-5.

62.9 External links

• Informal Logic entry by Leo Groarke in the Stanford Encyclopedia of Philosophy Chapter 63

Inquiry

“Enquiry”redirects here. For the 1990 Malayalam film, see Enquiry (film). For other uses, see Inquiry (disambiguation). An inquiry is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem.A theory of inquiry is an account of the various types of inquiry and a treatment of the ways that each type of inquiry achieves its aim.

63.1 Classical sources

63.1.1 Deduction When three terms are so related to one another that the last is wholly contained in the middle and the middle is wholly contained in or excluded from the first, the extremes must admit of perfect syllogism. By 'middle term' I mean that which both is contained in another and contains another in itself, and which is the middle by its position also; and by 'extremes' (a) that which is contained in another, and (b) that in which another is contained. For if A is predicated of all B, and B of all C, A must necessarily be predicated of all C. ... I call this kind of figure the First. (Aristotle, Prior Analytics, 1.4)

63.1.2 Induction Inductive reasoning consists in establishing a relation between one extreme term and the middle term by means of the other extreme; for example, if B is the middle term of A and C, in proving by means of C that A applies to B; for this is how we effect inductions. (Aristotle, Prior Analytics, 2.23)

63.1.3 Abduction

The locus classicus for the study of abductive reasoning is found in Aristotle's Prior Analytics, Book 2, Chapt. 25. It begins this way:

We have Reduction (απαγωγη, abduction): 1. When it is obvious that the first term applies to the middle, but that the middle applies to the last term is not obvious, yet is nevertheless more probable or not less probable than the conclusion; 2. Or if there are not many intermediate terms between the last and the middle; For in all such cases the effect is to bring us nearer to knowledge.

By way of explanation, Aristotle supplies two very instructive examples, one for each of the two varieties of abductive inference steps that he has just described in the abstract:

248 63.1. CLASSICAL SOURCES 249

A question mark

1. For example, let A stand for“that which can be taught”, B for“knowledge”, and C for “morality”. Then that knowledge can be taught is evident; but whether virtue is knowledge is not clear. Then if BC is not less probable or is more probable than AC, we have reduction; for we are nearer to knowledge for having introduced an additional term, whereas before we had no knowledge that AC is true. 2. Or again we have reduction if there are not many intermediate terms between B and C; for in this case too we are brought nearer to knowledge. For example, suppose that D is “to square”, E “rectilinear figure”, and F “circle”. Assuming that between E and F there is only one intermediate term —that the circle becomes equal to a rectilinear figure by means of lunules —we should approximate to knowledge. (Aristotle,"Prior Analytics", 2.25, with minor alterations)

Aristotle's latter variety of abductive reasoning, though it will take some explaining in the sequel, is well worth our contemplation, since it hints already at streams of inquiry that course well beyond the syllogistic source from which they spring, and into regions that Peirce will explore more broadly and deeply. 250 CHAPTER 63. INQUIRY

63.2 Inquiry in the pragmatic paradigm

In the pragmatic philosophies of Charles Sanders Peirce, , John Dewey, and others, inquiry is closely associated with the of logic. In its inception, the pragmatic model or theory of inquiry was extracted by Peirce from its raw materials in classical logic, with a little bit of help from Kant, and refined in parallel with the early development of symbolic logic by Boole, De Morgan, and Peirce himself to address problems about the nature and conduct of scientific reasoning. Borrowing a brace of concepts from Aristotle, Peirce examined three fundamental modes of reasoning that play a role in inquiry, commonly known as abductive, deductive, and inductive inference. In rough terms, abduction is what we use to generate a likely hypothesis or an initial diagnosis in response to a phenomenon of interest or a problem of concern, while deduction is used to clarify, to derive, and to explicate the relevant consequences of the selected hypothesis, and induction is used to test the sum of the predictions against the sum of the data. It needs to be observed that the classical and pragmatic treatments of the types of reasoning, dividing the generic territory of inference as they do into three special parts, arrive at a different characterization of the environs of reason than do those accounts that count only two. These three processes typically operate in a cyclic fashion, systematically operating to reduce the uncertainties and the difficulties that initiated the inquiry in question, and in this way, to the extent that inquiry is successful, leading to an increase in knowledge or in skills. In the pragmatic way of thinking everything has a purpose, and the purpose of each thing is the first thing we should try to note about it.*[1] The purpose of inquiry is to reduce doubt and lead to a state of belief, which a person in that state will usually call knowledge or certainty. As they contribute to the end of inquiry, we should appreciate that the three kinds of inference describe a cycle that can be understood only as a whole, and none of the three makes complete sense in isolation from the others. For instance, the purpose of abduction is to generate guesses of a kind that deduction can explicate and that induction can evaluate. This places a mild but meaningful constraint on the production of hypotheses, since it is not just any wild guess at explanation that submits itself to reason and bows out when defeated in a match with reality. In a similar fashion, each of the other types of inference realizes its purpose only in accord with its proper role in the whole cycle of inquiry. No matter how much it may be necessary to study these processes in abstraction from each other, the integrity of inquiry places strong limitations on the effective modularity of its principal components. In Logic: The Theory of Inquiry, John Dewey defined inquiry as “the controlled or directed transformation of an indeterminate situation into one that is so determinate in its constituent distinctions and relations as to convert the elements of the original situation into a unified whole”*[2] Dewey and Peirce's conception of inquiry extended beyond a system of thinking and incorporated the social nature of inquiry. These ideas are summarize in the notion Community of inquiry.*[3]*[4]*[5]

63.2.1 Art and science of inquiry

For our present purposes, the first feature to note in distinguishing the three principal modes of reasoning from each other is whether each of them is exact or approximate in character. In this light, deduction is the only one of the three types of reasoning that can be made exact, in essence, always deriving true conclusions from true premises, while abduction and induction are unavoidably approximate in their modes of operation, involving elements of fallible judgment in practice and inescapable error in their application. The reason for this is that deduction, in the ideal limit, can be rendered a purely internal process of the reasoning agent, while the other two modes of reasoning essentially demand a constant interaction with the outside world, a source of phenomena and problems that will no doubt continue to exceed the capacities of any finite resource, human or machine, to master. Situated in this larger reality, approximations can be judged appropriate only in relation to their context of use and can be judged fitting only with regard to a purpose in view. A parallel distinction that is often made in this connection is to call deduction a demonstrative form of inference, while abduction and induction are classed as non-demonstrative forms of reasoning. Strictly speaking, the latter two modes of reasoning are not properly called inferences at all. They are more like controlled associations of words or ideas that just happen to be successful often enough to be preserved as useful heuristic strategies in the repertoire of the agent. But non-demonstrative ways of thinking are inherently subject to error, and must be constantly checked out and corrected as needed in practice. In classical terminology, forms of judgment that require attention to the context and the purpose of the judgment are said to involve an element of “art”, in a sense that is judged to distinguish them from “science”, and in their 63.2. INQUIRY IN THE PRAGMATIC PARADIGM 251

renderings as expressive judgments to implicate arbiters in styles of rhetoric, as contrasted with logic. In a figurative sense, this means that only deductive logic can be reduced to an exact theoretical science, while the practice of any empirical science will always remain to some degree an art.

63.2.2 Zeroth order inquiry

Many aspects of inquiry can be recognized and usefully studied in very basic logical settings, even simpler than the level of syllogism, for example, in the realm of reasoning that is variously known as Boolean algebra, propositional calculus, sentential calculus, or zeroth-order logic. By way of approaching the learning curve on the gentlest availing slope, we may well begin at the level of zeroth-order inquiry, in effect, taking the syllogistic approach to inquiry only so far as the propositional or sentential aspects of the associated reasoning processes are concerned. One of the bonuses of doing this in the context of Peirce's logical work is that it provides us with doubly instructive exercises in the use of his logical graphs, taken at the level of his so-called "alpha graphs". In the case of propositional calculus or sentential logic, deduction comes down to applications of the transitive law for conditional implications and the approximate forms of inference hang on the properties that derive from these. In describing the various types of inference I will employ a few old “terms of art”from classical logic that are still of use in treating these kinds of simple problems in reasoning.

Deduction takes a Case, the minor premise X ⇒ Y and combines it with a Rule, the major premise Y ⇒ Z to arrive at a Act, the demonstrative conclusion X ⇒ Z.

Induction takes a Case of the form X ⇒ Y and matches it with a Fact of the form X ⇒ Z to infer a Rule of the form Y ⇒ Z.

Abduction takes a Fact of the form X ⇒ Z and matches it with a Rule of the form Y ⇒ Z to infer a Case of the form X ⇒ Y.

For ease of reference, Figure 1 and the Legend beneath it summarize the classical terminology for the three types of inference and the relationships among them. o------o | | | Z | | o | | |\ | | | \ | | | \ | | | \ | | | \ | | | \ R U L E | | | \ | | | \ | | F | \ | | | \ | | A | \ | | | o Y | | C | / | | | / | | T | / | | | / | | | / | | | / C A S E | | | / | | | / | | | / | | | / | | |/ | | o | | X | | | | Deduction takes a Case of the form X → Y, | | matches it with a Rule of the form Y → Z, | | then adverts to a Fact of the form X → Z. | | | | Induction takes a Case of the form X → Y, | | matches it with a Fact of the form X → Z, | | then adverts to a Rule of the form Y → Z. | | | | Abduction takes a Fact of the form X → Z, | | matches it with a Rule of the form Y → Z, | | then adverts to a Case of the form X → Y. | | | | Even more succinctly: | | | | Abduction Deduction Induction | | | | Premise: Fact Case Case | | Premise: Rule Rule Fact | | Outcome: Case Fact Rule | | | o------o Figure 1. Elementary Structure and Terminology In its original usage a statement of Fact has to do with a deed done or a record made, that is, a type of event that is openly observable and not riddled with speculation as to its very occurrence. In contrast, a statement of Case may refer to a hidden or a hypothetical cause, that is, a type of event that is not immediately observable to all concerned. Obviously, the distinction is a rough one and the question of which mode applies can depend on the points of view that different observers adopt over time. Finally, a statement of a Rule is called that because it states a regularity or a regulation that governs a whole class of situations, and not because of its syntactic form. So far in this discussion, all three types of constraint are expressed in the form of conditional propositions, but this is not a fixed requirement. In practice, these modes of statement are distinguished by the roles that they play within an argument, not by their style of expression. When the time comes to branch out from the syllogistic framework, we will find that propositional constraints can be discovered and represented in arbitrary syntactic forms. 252 CHAPTER 63. INQUIRY

63.3 Example of inquiry

Examples of inquiry, that illustrate the full cycle of its abductive, deductive, and inductive phases, and yet are both concrete and simple enough to be suitable for a first (or zeroth) exposition, are somewhat rare in Peirce's writings, and so let us draw one from the work of fellow pragmatician John Dewey, analyzing it according to the model of zeroth-order inquiry that we developed above.

A man is walking on a warm day. The sky was clear the last time he observed it; but presently he notes, while occupied primarily with other things, that the air is cooler. It occurs to him that it is probably going to rain; looking up, he sees a dark cloud between him and the sun, and he then quickens his steps. What, if anything, in such a situation can be called thought? Neither the act of walking nor the noting of the cold is a thought. Walking is one direction of activity; looking and noting are other modes of activity. The likelihood that it will rain is, however, something suggested. The pedestrian feels the cold; he thinks of clouds and a coming shower. (John Dewey, How We Think, 1910, pp. 6-7).

63.3.1 Once over quickly

Let's first give Dewey's example of inquiry in everyday life the quick once over, hitting just the high points of its analysis into Peirce's three kinds of reasoning.

Abductive phase

In Dewey's “Rainy Day”or “Sign of Rain”story, we find our peripatetic hero presented with a surprising Fact:

• Fact: C → A, In the Current situation the Air is cool.

Responding to an intellectual reflex of puzzlement about the situation, his resource of common knowledge about the world is impelled to seize on an approximate Rule:

• Rule: B → A, Just Before it rains, the Air is cool.

This Rule can be recognized as having a potential relevance to the situation because it matches the surprising Fact, C → A, in its consequential feature A. All of this suggests that the present Case may be one in which it is just about to rain:

• Case: C → B, The Current situation is just Before it rains.

The whole mental performance, however automatic and semi-conscious it may be, that leads up from a problematic Fact and a previously settled knowledge base of Rules to the plausible suggestion of a Case description, is what we are calling an abductive inference.

Deductive phase

The next phase of inquiry uses deductive inference to expand the implied consequences of the abductive hypothesis, with the aim of testing its truth. For this purpose, the inquirer needs to think of other things that would follow from the consequence of his precipitate explanation. Thus, he now reflects on the Case just assumed:

• Case: C → B, The Current situation is just Before it rains.

He looks up to scan the sky, perhaps in a random search for further information, but since the sky is a logical place to look for details of an imminent rainstorm, symbolized in our story by the letter B, we may safely suppose that our reasoner has already detached the consequence of the abduced Case, C → B, and has begun to expand on its further implications. So let us imagine that our up-looker has a more deliberate purpose in mind, and that his search for additional data is driven by the new-found, determinate Rule: 63.3. EXAMPLE OF INQUIRY 253

• Rule: B → D, Just Before it rains, Dark clouds appear.

Contemplating the assumed Case in combination with this new Rule leads him by an immediate deduction to predict an additional Fact:

• Fact: C → D, In the Current situation Dark clouds appear.

The reconstructed picture of reasoning assembled in this second phase of inquiry is true to the pattern of deductive inference.

Inductive phase

Whatever the case, our subject observes a Dark cloud, just as he would expect on the basis of the new hypothesis. The explanation of imminent rain removes the discrepancy between observations and expectations and thereby reduces the shock of surprise that made this process of inquiry necessary.

63.3.2 Looking more closely

Seeding hypotheses

Figure 4 gives a graphical illustration of Dewey's example of inquiry, isolating for the purposes of the present analysis the first two steps in the more extended proceedings that go to make up the whole inquiry. o------o | | | A D | | o o | | \ * * / | | \ * * / | | \ * * / | | \ * * / | | \ * * / | | \ R uleRule/||\**/||\**/||\**/||\*B*/||FactoFact||\*/||\*/||\*/||\*/||\Case/||\*/|| \ * / | | \ * / | | \ * / | | \ * / | | \*/ | | o | | C | | | | A = the Air is cool | | B = just Before it rains | | C = the Current situation | | D = a Dark cloud appears | | | | A is a major term | | B is a middle term | | C is a minor term | | D is a major term, associated with A | | | o------o Figure 4. Dewey's 'Rainy Day' Inquiry In this analysis of the first steps of Inquiry, we have a complex or a mixed form of inference that can be seen as taking place in two steps:

• The first step is an Abduction that abstracts a Case from the consideration of a Fact and a Rule.

Fact: C → A, In the Current situation the Air is cool. Rule: B → A, Just Before it rains, the Air is cool. Case: C → B, The Current situation is just Before it rains.

• The final step is a Deduction that admits this Case to another Rule and so arrives at a novel Fact.

Case: C → B, The Current situation is just Before it rains. Rule: B → D, Just Before it rains, a Dark cloud will appear. Fact: C → D, In the Current situation, a Dark cloud will appear.

This is nowhere near a complete analysis of the Rainy Day inquiry, even insofar as it might be carried out within the constraints of the syllogistic framework, and it covers only the first two steps of the relevant inquiry process, but maybe it will do for a start. One other thing needs to be noticed here, the formal duality between this expansion phase of inquiry and the . This can be seen most clearly in the propositional lattice diagrams shown in Figures 3 and 4, where analogy exhibits a rough“A”shape and the first two steps of inquiry exhibit a rough“V”shape, respectively. Since we find ourselves repeatedly referring to this expansion phase of inquiry as a unit, let's give it a name that suggests its duality with analogy—"catalogy" will do for the moment. This usage is apt enough if one thinks of a catalogue entry for an item as a text that lists its salient features. Notice that analogy has to do with the examples of a given quality, while catalogy has to do with the qualities of a given example. Peirce noted similar forms of duality in many of his early writings, leading to the consummate treatment in his 1867 paper “On a New List of Categories” (CP 1.545-559, W 2, 49-59). 254 CHAPTER 63. INQUIRY

Weeding hypotheses

In order to comprehend the bearing of inductive reasoning on the closing phases of inquiry there are a couple of observations that we need to make:

• First, we need to recognize that smaller inquiries are typically woven into larger inquiries, whether we view the whole pattern of inquiry as carried on by a single agent or by a complex community. • Further, we need to consider the different ways in which the particular instances of inquiry can be related to ongoing inquiries at larger scales. Three modes of inductive interaction between the micro-inquiries and the macro-inquiries that are salient here can be described under the headings of the “Learning”, the “Transfer”, and the “Testing”of rules.

Analogy of experience

Throughout inquiry the reasoner makes use of rules that have to be transported across intervals of experience, from the masses of experience where they are learned to the moments of experience where they are applied. Inductive reasoning is involved in the learning and the transfer of these rules, both in accumulating a knowledge base and in carrying it through the times between acquisition and application.

• Learning. The principal way that induction contributes to an ongoing inquiry is through the learning of rules, that is, by creating each of the rules that goes into the knowledge base, or ever gets used along the way. • Transfer. The continuing way that induction contributes to an ongoing inquiry is through the exploit of analogy, a two-step combination of induction and deduction that serves to transfer rules from one context to another. • Testing. Finally, every inquiry that makes use of a knowledge base constitutes a “field test”of its accumulated contents. If the knowledge base fails to serve any live inquiry in a satisfactory manner, then there is a prima facie reason to reconsider and possibly to amend some of its rules.

Let's now consider how these principles of learning, transfer, and testing apply to John Dewey's “Sign of Rain” example.

Learning Rules in a knowledge base, as far as their effective content goes, can be obtained by any mode of infer- ence. For example, a rule like:

• Rule: B → A, Just Before it rains, the Air is cool,

is usually induced from a consideration of many past events, in a manner that can be rationally reconstructed as follows:

• Case: C → B, In Certain events, it is just Before it rains, • Fact: C → A, In Certain events, the Air is cool, ------

• Rule: B → A, Just Before it rains, the Air is cool.

However, the very same proposition could also be abduced as an explanation of a singular occurrence or deduced as a conclusion of a presumptive theory. 63.3. EXAMPLE OF INQUIRY 255

Transfer What is it that gives a distinctively inductive character to the acquisition of a knowledge base? It is evidently the“analogy of experience”that underlies its useful application. Whenever we find ourselves prefacing an argument with the phrase“If past experience is any guide...”then we can be sure that this principle has come into play. We are invoking an analogy between past experience, considered as a totality, and present experience, considered as a point of application. What we mean in practice is this: “If past experience is a fair sample of possible experience, then the knowledge gained in it applies to present experience”. This is the mechanism that allows a knowledge base to be carried across gulfs of experience that are indifferent to the effective contents of its rules. Here are the details of how this notion of transfer works out in the case of the “Sign of Rain”example: Let K(pres) be a portion of the reasoner's knowledge base that is logically equivalent to the conjunction of two rules, as follows:

• K(pres) = (B → A) and (B → D).

K(pres) is the present knowledge base, expressed in the form of a logical constraint on the present universe of dis- course. It is convenient to have the option of expressing all logical statements in terms of their logical models, that is, in terms of the primitive circumstances or the elements of experience over which they hold true.

• Let E(past) be the chosen set of experiences, or the circumstances that we have in mind when we refer to “past experience”. • Let E(poss) be the collective set of experiences, or the projective total of possible circumstances. • Let E(pres) be the present experience, or the circumstances that are present to the reasoner at the current moment.

If we think of the knowledge base K(pres) as referring to the“regime of experience”over which it is valid, then all of these sets of models can be compared by the simple relations of set inclusion or logical implication. Figure 5 schematizes this way of viewing the “analogy of experience”. o------o | | | K(pres) | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / Rule \ | | / | \ | | / | \ | | / | \ | | / E(poss) \ | | Fact / o \ Fact | | / * * \ | | / * * \ | | / * * \ | | / * * \ | | / * * \ | | / * Case Case * \ | | / * * \ | | / * * \ | | /* *\ | | o<<<------<<<------<<

• Given Case: E(past) → E(poss), Chosen events fairly sample Collective events. • Given Fact: E(past) → K(pres), Chosen events support the Knowledge regime. ------• Induce Rule: E(poss) → K(pres), Collective events support the Knowledge regime.

Deductive Phase:

• Given Case: E(pres) → E(poss), Current events fairly sample Collective events. • Given Rule: E(poss) → K(pres), Collective events support the Knowledge regime. ------• Deduce Fact: E(pres) → K(pres), Current events support the Knowledge regime.

Testing If the observer looks up and does not see dark clouds, or if he runs for shelter but it does not rain, then there is fresh occasion to question the utility or the validity of his knowledge base. But we must leave our foulweather friend for now and defer the logical analysis of this testing phase to another occasion. 256 CHAPTER 63. INQUIRY

63.4 Citations

[1] Rescher, N. (2012). Pragmatism: The Restoration of its Scientific Roots. New Brunswick, NJ: Transaction Press.

[2] Dewey, John (1938). Logic: The Theory of Inquiry. New York:NY: D.C. Heath & Co.

[3] Wikisource:The Fixation of Belief

[4] Seixas, Peter.“The Community of Inquiry as a Basis for Knowledge and Learning: The Case of History”. Sage. Retrieved 4 June 2012.

[5] Shields, Patricia. “The Community of Inquiry”. Sage. Retrieved 4 June 2012.

63.5 Bibliography

• Angluin, Dana (1989), “Learning with Hints”, pp. 167–181 in David Haussler and Leonard Pitt (eds.), Proceedings of the 1988 Workshop on Computational Learning Theory, MIT, 3–5 August 1988, Morgan Kauf- mann, San Mateo, CA, 1989. • Aristotle,"Prior Analytics", Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, , William Heinemann, London, UK, 1938. • Awbrey, Jon, and Awbrey, Susan (1995),“Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15, 40–52. Eprint. • Delaney, C.F. (1993), Science, Knowledge, and Mind: A Study in the Philosophy of C.S. Peirce, University of Notre Dame Press, Notre Dame, IN. • Dewey, John (1910), How We Think, D.C. Heath, Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991. • Dewey, John (1938), Logic: The Theory of Inquiry, Henry Holt and Company, New York, NY, 1938. Reprinted as pp. 1–527 in John Dewey, The Later Works, 1925–1953, Volume 12: 1938, Jo Ann Boydston (ed.), Kath- leen Poulos (text. ed.), (intro.), Southern Illinois University Press, Carbondale and Edwardsville, IL, 1986. • Haack, Susan (1993), Evidence and Inquiry: Towards Reconstruction in Epistemology, Blackwell Publishers, Oxford, UK. • Hanson, Norwood Russell (1958), Patterns of Discovery, An Inquiry into the Conceptual Foundations of Science, Cambridge University Press, Cambridge, UK. • Hendricks, Vincent F. (2005), Thought 2 Talk: A Crash Course in Reflection and Expression, Automatic Press / VIP, New York, NY. ISBN 87-991013-7-8 • Misak, Cheryl J. (1991), Truth and the End of Inquiry, A Peircean Account of Truth, Oxford University Press, Oxford, UK. • Peirce, C.S., (1931–1935, 1958), Collected Papers of Charles Sanders Peirce, vols. 1–6, and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as CP volume.paragraph. • Stalnaker, Robert C. (1984), Inquiry, MIT Press, Cambridge, MA.

63.6 See also

• Charles Sanders Peirce bibliography • C. West Churchman • Curiosity 63.6. SEE ALSO 257

• Empirical limits in science

• Information • Information theory

• Logic of information • Pragmatic information

• Pragmatic theory of truth • Pragmaticism

• Uncertainty • Community of inquiry Chapter 64

Intension

Not to be confused with intention or intentionality. For the song “Intension”by Tool, see 10,000 Days.

In linguistics, logic, philosophy, and other fields, an intension is any property or quality connoted by a word, phrase, or another symbol. In the case of a word, the word's definition often implies an intension. For instance, intension of the word '[plant]' includes properties like “being composed of cellulose" and “alive”and “organism,”among others. Comprehension is the collection of all such . The meaning of a word can be thought of as the bond between the idea the word means and the physical form of the word. Swiss linguist Ferdinand de Saussure (1857–1913) contrasts three concepts:

1. the signifier – the “sound image”or the string of letters on a page that one recognizes as the form of a sign 2. the signified – the meaning, the concept or idea that a sign expresses or evokes 3. the referent – the actual thing or set of things a sign refers to. See Dyadic signs and Reference (semantics).

Without intension of some sort, a word has no meaning. For instance, the terms 'rantans' or 'brillig' have no intension and hence no meaning. Such terms may be suggestive, but a term can be suggestive without being meaningful. For instance, 'ran tan' is an archaic onomatopoeia for chaotic noise or din and may suggest to English speakers a din or meaningless noise; and 'brillig' though made up by Lewis Caroll may be suggestive of 'brilliant' or 'frigid.' Such terms, it may be argued, are always intensional since they connote the property 'meaningless term' but this paradox does not constitute a counterexample to the claim that without intension a word has no meaning. Intension is analogous to the signified in the Saussurean system, extension to the referent. In philosophical arguments about dualism versus monism, it is noted that thoughts have intensionality and physical objects do not (S. E. Palmer, 1999), but rather have extension in space and time. Note: Intension and intensionality (the state of having intension) should not be confused with intention and intentionality, which are pronounced the same and occasionally arise in the same philosophical context. Where this happens, the letter s or t is sometimes italicized to emphasize the distinction.

64.1 Intensional statement

An intensional statement-form is a statement-form with at least one instance such that substituting co-extensive ex- pressions into it does not always preserve logical value. An intensional statement is a statement that is an instance of an intensional statement-form. Here co-extensive expressions are expressions with the same extension. (A statement- form is simply a form obtained by putting blanks into a sentence where one or more expressions with extensions occur —for instance, “The quick brown ___ jumped over the lazy ___'s back.”An instance of the form is a statement obtained by filling the blanks in.) That is, a statement-form is intensional if it has, as one of its instances, a statement for which there are two co- extensive expressions (in the relevant language) such that one of them occurs in the statement, and if the other one

258 64.1. INTENSIONAL STATEMENT 259

is put in its place (uniformly, so that it replaces the former expression wherever it occurs in the statement), the result is a (different) statement with a different logical value. An intensional statement, then, is an instance of such a form; it has the same form as a statement in which substitution of co-extensive terms fails to preserve logical value. A non-intensional statement is also known as an extensional statement, since substitution of co-extensive expressions into it always preserves logical value. A language is intensional if it contains intensional statements, and extensional otherwise. English, in common with every other natural language, is an intensional language. The only extensional languages are artificially constructed languages used in mathematics or for other special purposes and small fragments of natural languages.

64.1.1 Examples of extensional statements

1. Mark Twain wrote Huckleberry Finn.

2. Aristotle had a sister.

Note that if “Samuel Clemens”is put into (1) in place of “Mark Twain”, the result is as true as the original statement. It should be clear that no matter what is put for “Mark Twain”, so long as it is a singular term picking out the same man, the statement remains true. Likewise, we can put in place of the predicate any other predicate belonging to Mark Twain and only to Mark Twain, without changing the logical value. For (2), likewise, consider the following substitutions: “Aristotle”→ “The tutor of "; “Aristotle”→ “The author of the 'Prior Analytics'"; “had a sister”→ “had a sibling with two X-chromosomes"; “had a sister”→ “had a parent who had a non-male child”.

64.1.2 Examples of intensional statements

1. Everyone who has read Huckleberry Finn knows that Mark Twain wrote it.

2. It is possible that Aristotle did not tutor Alexander the Great.

3. Aristotle was pleased that he had a sister.

To see that these are intensional, make the following substitutions: (1)“Mark Twain”→“The author of 'Corn-pone Opinions'"; (2) “Aristotle”→ “the tutor of Alexander the Great"; (3) can be seen to be intensional given “had a sister”→ “had a sibling with two X-chromosomes”. It will be noted that the intensional statements above feature expressions like“knows”,“possible”, and“pleased” . Such expressions always, or nearly always, produce intensional statements when added (in some intelligible manner) to an extensional statement, and thus they (or more complex expressions like “It is possible that”) are sometimes called intensional operators. A large class of intensional statements, but by no means all, can be spotted from the fact that they contain intensional operators.

64.1.3 Significance

Intensional languages cannot be given an adequate semantics in terms of the extensions of expressions in them, since the extensions themselves do not suffice to determine a logical value. (If they did, then one could not change the logical value by substituting co-extensive expressions.) On the other hand, for the first half of the 20th century the only known systems of formal semantics worked by assigning extensions to expressions and used a Tarski-style truth- definition of statements constructed from the primitive expressions of the language under analysis. Hence, these semantical methods were pathetically inadequate for understanding the semantics of any but a few small artificial languages or mutilated fragments of natural languages. This situation changed in the 1960s with the invention of possible-world or“intensional”semantics, the main form of which is due to Saul Kripke. Though this has enabled improvements in the semantic modelling of natural languages, much work remains to be done. 260 CHAPTER 64. INTENSION

64.2 See also

• Description logic

• • Intensional definition

• Intensional logic

64.3 References

• Ferdinand De Saussure: Course in General Linguistics. Open Court Classics, July 1986. ISBN 0-8126-9023-0 • S. E. Palmer, Vision Science: From Photons to Phenomenology, 1999. MIT Press, ISBN 0-2621-6183-4

64.4 External links

• Chalmers, David, “On Sense and Intension”. • Rapaport, William J., “Intensionality v. Intentionality”. Chapter 65

International Wittgenstein Symposium

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The International Wittgenstein Symposium is an international conference dedicated to the work of Ludwig Wittgen- stein and its relationship to philosophy and science. It is sponsored by the Austrian Ludwig Wittgenstein Soci- ety.*[1]*[2]*[3]

65.1 History

In 1976, the International Wittgenstein Symposium was founded by Elisabeth Leinfellner, Werner Leinfellner, Rudolf Haller, Paul Weingartner, and Adolf Hübner in Kirchberg am Wechsel, Lower Austria. The location was chosen be- cause in the 1920s, Ludwig Wittgenstein taught at elementary schools in the area surrounding Kirchberg am Wechsel. On the 24th to the 25th of April, 1976 (just prior to the 25th anniversary of Wittgenstein's death), the first conference took place. Only four of the five founders gave talks on his philosophical work at the first meeting, but at the second, 120 speakers attended from around the world.*[2]

65.2 Philosophical Topics

The general topic of each symposium centers around the philosophy and philosophy of science of Wittgenstein, but the specific topics change from year to year. For example, the topic of the second International Wittgenstein Symposium was“Wittgenstein and his impact on contemporary thought”and the topic of the third symposium was “Wittgenstein, the Vienna Circle, and (including a seminar on Popper's open society).”A survey of topics is available from the site of the Austrian Ludwig Wittgenstein Society.*[2]

65.3 Proceedings

Starting with the second symposium, the papers accepted for presentation have been published in edited proceedings. From 1978 to 2005 the proceedings of the International Wittgenstein Symposium were published by Hölder-Pichler- Tempsky and are now published by ontos verlag. The proceedings published at ontos verlag are available Open Access online at a site prepared by the Wittgenstein Archives at the University of Bergen. The Wittgenstein Archives have also prepared a site that contains an Open Access selection of symposium papers from the period 2001-10.*[2]

261 262 CHAPTER 65. INTERNATIONAL WITTGENSTEIN SYMPOSIUM

65.4 Sponsorship

The symposia are sponsored by the Austrian Ludwig Wittgenstein Society and they are largely funded by the govern- ment of Lower Austria and the Austrian Federal Ministry for Science and Research.*[2]

65.5 References

[1] “University of Graz, Department of Philosophy”. University of Graz.

[2] “Austrian Ludwig Wittgenstein Society”.

[3] “Wittgenstein News”.

65.6 External links

• Austrian Ludwig Wittgenstein Society

• From ontos verlag: Publications of the Austrian Ludwig Wittgenstein Society - New Series (Volumes 1-18) • From the ALWS archives: A selection of papers from the International Wittgenstein Symposia in Kirchberg am Wechsel • Wittgenstein News Chapter 66

Inverse consequences

The term "inverse consequences"*[1] or the "Law of Inverse Consequences"*[2] refers to results that are the opposite of the expected results as initially intended or planned.*[2] One consequence is in the“reverse predicament” of the other.*[1]

66.1 History

The term“inverse consequences”has been in use for over 175 years (since at least 1835).*[1] The term was also used by (1798–1857) in his book System of Positive Polity (published 1875), stating,“Inevitable increase in Complication, in proportion with the decrease of Generality, gives rise to two inverse consequences.”*[3]

66.2 Documented examples

The term “inverse consequences”has been applied in numerous situations, for example:

• In treatment of drug addiction, medications intended to reduce one type of addiction might trigger another addiction: long-term treatment with opiate medications (such as morphine) has inverse consequences.*[4]

• In management of work tasks, a total sequential execution, of work tasks, has inverse consequences, such as a decrease of the workload with an increase of the lead time.*[5]

• In asset management, plans for portfolio management might have inverse consequences to the potential bene- fits.*[6]

66.3 Related phrases

The concept of “inverse consequences”has a corollary in other phrases, as well:

•“the plan backfired" - meaning the opposite result occurred, as in a gun firing backward, rather than forward.

66.4 See also

66.5 Notes

[1] The Philosophy of Manufactures: Or, An Exposition (on factory systems), Andrew Ure, 1835, page 434 of 480 pages, Google Books link: booksGoogle-AU: states “the influence of which upon them will be manifested by inverse consequences; the one being in the reverse predicament of the other.”

263 264 CHAPTER 66. INVERSE CONSEQUENCES

[2]“Chatham County Center: Green Thumb Prints Newsletter 2007 Index”, NCSU.edu, May 2007, webpage: NCSU-law.

[3] System of Positive Polity: Social statics (on positivism), Auguste Comte, 1875, page 376, Google Books link (PDF 11.8mb): books-Google-AC: states“inevitable increase in Complication in proportion with the decrease of Generality, gives rise to two inverse consequences.”

[4]“Neuron : Experimental Genetic Approaches to Addiction”, A. Laakso, 2002, webpage: LinkingHub.elsevier.com-728: states “exposure to psychostimulants such as cocaine leads to sensitized response; long-term treatment with opiates (such as morphine) also has inverse consequences”.

[5]“Managing and Organizing the Cooperation in Design Processes”M. David, PDF file: Inria-fr-COOP2004-PDF.

[6]“The Infinite Asset: Managing Brands to Build New Value”(on business & economics), Sam Hill, Chris Lederer, 2001, 238 pages, Google Books link: books-Google-SH: states "...managers must understand and anticipate inverse consequences every bit as much as they preach the potential benefits of their action plans.”

66.6 References

• Andrew Ure, The Philosophy of Manufactures: Or, An Exposition (on factory systems), 1835, page 434 of 480 pages, Google Books link: booksGoogle-AU. Chapter 67

Language, Proof and Logic

Language, Proof and Logic is an educational software package, devised and written by Jon Barwise and John Etchemendy, geared to teaching formal logic through the use of a tight integration between a textbook (same name as the package) and four software programs, where three of them are logic related (Boole, Fitch and Tarski's World) and the other (Submit) is an internet-based grading service. The name is a pun derived from Language, Truth, and Logic, the philosophy book by A. J. Ayer. On September 2, 2014 launched also a massive open online course (MOOC) with the same name which utilizes this educational software package. A short description of the programs:

• Boole (named after George Boole) - a program that facilitates the construction and checking of truth tables and related notions (tautology, tautological consequence, etc.);

• Fitch (named after Frederic Brenton Fitch) - a natural deduction proof environment in Fitch-style calculus for giving and checking first-order proofs;

• Tarski's World (named after Alfred Tarski) - a program that teaches the basic first-order language and its semantics using a model theoretic-like approach, where the “world”consists of a little grid and some simple objects; • Submit - a program that allows students to submit exercises done with the above programs to the Grade Grinder, the online grading service.*[1]

67.1 References

[1] Grim, P. (2001). “Language, Proof and Logic”. The Bulletin of Symbolic Logic 7 (3): 377–379. doi:10.2307/2687756.

67.2 External links

• the home page of Language, Proof and Logic • 1st edition of Language, Proof and Logic Internet Archive

• massive open online course (MOOC) of Language, Proof and Logic

265 Chapter 68

Lexical definition

The lexical definition of a term, also known as the dictionary definition, is the meaning of the term in common usage. As its other name implies, this is the sort of definition one is likely to find in the dictionary. A lexical definition is usually the type expected from a request for definition, and it is generally expected that such a definition will be stated as simply as possible in order to convey information to the widest audience. Note that a lexical definition is descriptive, reporting actual usage within speakers of a language, and changes with changing usage of the term, rather than prescriptive, which would be to stick with a version regarded as “correct”, regardless of drift in accepted meaning. They tend to be inclusive, attempting to capture everything the term is used to refer to, and as such are often too vague for many purposes. When the breadth or vagueness of a lexical definition is unacceptable, a precising definition or a stipulative definition is often used. Words can be classified as lexical or nonlexical. Lexical words are those that have independent meaning (such as a Noun (N), verb (V), adjective (A), adverb (Adv), or preposition (P). The definition which reports the meaning of a word or a phrase as it is actually used by people is called a lexical definition. Meanings of words given in a dictionary are lexical definitions. As a word may have more than one meaning, it may also have more than one lexical definition. Lexical definitions are either true or false. If the definition is the same as the actual use of the word then it is true, otherwise it is false.

68.1 See also

• Definition

• Theoretical definition • Circular definition

266 Chapter 69

List of paradoxes

This is a list of paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. Because of varying definitions of the term paradox, some of the following are not considered to be paradoxes by everyone. This list collects only scenarios that have been called a paradox by at least one source and have their own article. Although considered paradoxes, some of these are based on fallacious reasoning, or incomplete/faulty analysis. In- formally, the term is often used to describe a counter-intuitive result.

69.1 Logic

• The supposition that if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved leads to paradoxical consequences. Not to be confused with the .

• What the Tortoise Said to Achilles “Whatever Logic is good enough to tell me is worth writing down...”, also known as Carroll's paradox, not to be confused with the physical paradox of the same name.

• Catch-22 A situation in which someone is in need of something that can only be had by not being in need of it. A soldier who wants to be declared insane in order to avoid combat is deemed not insane for that very reason, and will therefore not be declared insane.

• In any pub there is a customer of whom it is true to say: if that customer drinks, everybody in the pub drinks.

• Paradox of entailment Inconsistent premises always make an argument valid.

• If there is one winning ticket in a large lottery, it is reasonable to believe of any particular lottery ticket that it is not the winning ticket, but it is not reasonable to believe that no lottery ticket will win.

• (or Hempel's Ravens): Observing a green apple increases the likelihood of all ravens being black.

• Ross' paradox Disjunction introduction poses a problem for imperative inference by seemingly permitting arbitrary imperatives to be inferred.

• Unexpected hanging paradox The day of the hanging will be a surprise, so it cannot happen at all, so it will be a surprise. The surprise examination and Bottle Imp paradox use similar logic.

69.1.1 Self-reference

These paradoxes have in common a contradiction arising from either from self-reference or from circular reference, in which several statements refer to each other in a way that following some of the references leads back to the starting point.

267 268 CHAPTER 69. LIST OF PARADOXES

• Barber paradox A barber (who is a man) shaves all and only those men who do not shave themselves. Does he shave himself? (Russell's popularization of his set theoretic paradox.) • Bhartrhari's paradox The thesis that there are some things which are unnameable conflicts with the notion that something is named by calling it unnameable. • The phrase “the first number not nameable in under ten words”appears to name it in nine words. • Crocodile dilemma If a crocodile steals a child and promises its return if the father can correctly guess exactly what the crocodile will do, how should the crocodile respond in the case that the father guesses that the child will not be returned? • Paradox of the Court A law student agrees to pay his teacher after (and only after) winning his first case. The teacher then sues the student (who has not yet won a case) for payment. • Curry's paradox “If this sentence is true, then Santa Claus exists.” • paradox A Cretan says: “All Cretans are liars”. This paradox works in mainly the same way as the . • Grelling–Nelson paradox Is the word“heterological”, meaning“not applicable to itself”, a heterological word? (Another close relative of Russell's paradox.) • Kleene–Rosser paradox By formulating an equivalent to Richard's paradox, untyped is shown to be inconsistent. • Liar paradox “This sentence is false.”This is the canonical self-referential paradox. Also “Is the answer to this question no?", and “I'm lying.” • “The next statement is true. The previous statement is false.”A variant of the liar paradox that does not use self-reference. • Pinocchio paradox What would happen if Pinocchio said “My nose will be growing"?*[1] • Quine's paradox "'Yields a falsehood when appended to its own quotation' yields a falsehood when appended to its own quotation.”Shows that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals. • Yablo's paradox An ordered infinite sequence of sentences, each of which says that all following sen- tences are false. While constructed to avoid self-reference, there is no consensus whether it relies on self-reference or not. • “It is opposite day today.”Therefore it is not opposite day, but if you say it is a normal day it would be considered a normal day. • Petronius' paradox “Moderation in all things, including moderation”(unsourced quotation sometimes at- tributed to Petronius). • Richard's paradox We appear to be able to use simple English to define a decimal expansion in a way that is self-contradictory. • Russell's paradox Does the set of all those sets that do not contain themselves contain itself? • Socratic paradox “I know that I know nothing at all.”

69.1.2 Vagueness

• Ship of Theseus (a.k.a. George Washington's axe or Grandfather's old axe): It seems like you can replace any component of a ship, and it is still the same ship. So you can replace them all, one at a time, and it is still the same ship. However, you can then take all the original pieces, and assemble them into a ship. That, too, is the same ship you began with. • (also known as the paradox of the heap): If you remove a single grain of sand from a heap, you still have a heap. Keep removing single grains, and the heap will disappear. Can a single grain of sand make the difference between heap and non-heap? 69.2. MATHEMATICS 269

69.2 Mathematics

See also: Category:Mathematics paradoxes and

• All horses are the same color A proof by induction that all horses have the same color.

• Cramer's paradox The number of points of intersection of two higher-order curves can be greater than the number of arbitrary points needed to define one such curve.

• Elevator paradox Elevators can seem to be mostly going in one direction, as if they were being manufactured in the middle of the building and being disassembled on the roof and basement.

• Interesting number paradox The first number that can be considered “dull”rather than “interesting” becomes interesting because of that fact.

• Nontransitive dice You can have three dice, called A, B, and C, such that A is likely to win in a roll against B, B is likely to win in a roll against C, and C is likely to win in a roll against A.

• Potato paradox If you let potatoes consisting of 99% water dry so that they are 98% water, they lose 50% of their weight.

• Russell's paradox Does the set of all those sets that do not contain themselves contain itself?

69.2.1 Statistics

See also: Category:Statistical paradoxes

• Abelson's paradox Effect size may not be indicative of practical meaning.

• Accuracy paradox Predictive models with a given level of accuracy may have greater predictive power than models with higher accuracy.

• Benford's law Numbers starting with lower digits appear disproportionately often in seemingly random data sets.

• Berkson's paradox A complicating factor arising in statistical tests of proportions.

• Freedman's paradox Describes a problem in model selection where predictor variables with no explanatory power can appear artificially important.

• Friendship paradox For almost everyone, their friends have more friends than they do.

• Inspection paradox Why one will wait longer for a bus than one should.

• Lindley's paradox Tiny errors in the null hypothesis are magnified when large data sets are analyzed, leading to false but highly statistically significant results.

• Low birth weight paradox Low birth weight and mothers who smoke contribute to a higher mortality rate. Babies of smokers have lower average birth weight, but low birth weight babies born to smokers have a lower mortality rate than other low birth weight babies. This is a special case of Simpson's paradox.

• Simpson's paradox, or the Yule–Simpson effect: A trend that appears in different groups of data disappears when these groups are combined, and the reverse trend appears for the aggregate data.

• Will Rogers phenomenon The mathematical concept of an average, whether defined as the mean or median, leads to apparently paradoxical results—for example, it is possible that moving an entry from an encyclopedia to a dictionary would increase the average entry length on both books. 270 CHAPTER 69. LIST OF PARADOXES

1 2 ? ?

The Monty Hall problem: which door do you choose?

69.2.2 Probability

See also: Category:Probability theory paradoxes

• Bertrand's box paradox A paradox of conditional probability closely related to the Boy or Girl paradox. • Bertrand's paradox Different common-sense definitions of give quite different results. • Birthday paradox What is the chance that two people in a room have the same birthday? • Borel's paradox Conditional probability density functions are not invariant under coordinate transformations. • Boy or Girl paradox A two-child family has at least one boy. What is the probability that it has a girl? • False positive paradox A test that is accurate the vast majority of the time could show you have a disease, but the probability that you actually have it could still be tiny. • Grice's paradox Shows that the exact meaning of statements involving conditionals and probabilities is more complicated than may be obvious on casual examination. • Monty Hall problem An unintuitive consequence of conditional probability. • Necktie paradox A wager between two people seems to favour them both. Very similar in essence to the Two-envelope paradox. • Proebsting's paradox The Kelly criterion is an often optimal strategy for maximizing profit in the long run. Proebsting's paradox apparently shows that the Kelly criterion can lead to ruin. • Sleeping Beauty problem A probability problem that can be correctly answered as one half or one third depending on how the question is approached. • Three cards problem When pulling a random card, how do you determine the color of the underside? • Three Prisoners problem A variation of the Monty Hall problem. • Two-envelope paradox You are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it contains. You pick one envelope at random but before you open it you are given the chance to take the other envelope instead. 69.2. MATHEMATICS 271

69.2.3 Infinity and infinitesimals

• Burali-Forti paradox If the ordinal numbers formed a set, it would be an that is smaller than itself.

• Cantor's paradox The set of all sets would have its own as a subset, therefore its would be at least as great as that of its power set. But Cantor's theorem proves that power sets are strictly greater than the sets they are constructed from. Consequently, the set of all sets would contain a subset greater than itself.

• Galileo's paradox Though most numbers are not squares, there are no more numbers than squares. (See also Cantor's diagonal argument)

• Hilbert's paradox of the Grand Hotel If a hotel with infinitely many rooms is full, it can still take in more guests.

• Russell's paradox Does the set of all those sets that do not contain themselves contain itself?

• Skolem's paradox Countably infinite models of set theory contain uncountably infinite sets.

• Zeno's paradoxes “You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on.”(This is also a physical paradox.)

• Supertasks may result in paradoxes such as

• Benardete's paradox Apparently, a man can be“forced to stay where he is by the mere unfulfilled intentions of the gods”. • Ross–Littlewood paradox After alternatively adding and removing balls to a vase infinitely often, how many balls remain? • Thomson's lamp After flicking a lamp on and off infinitely often, is it on or off?

69.2.4 Geometry and topology

The Banach–Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original.

• Banach–Tarski paradox Cut a ball into a finite number of pieces, re-assemble the pieces to get two balls, both of equal size to the first. The von Neumann paradox is a two-dimensional analogue.

• Paradoxical set A set that can be partitioned into two sets, each of which is equivalent to the original.

• Coastline paradox the perimeter of a landmass is in general ill-defined.

• Coin rotation paradox a coin rotating along the edge of an identical coin will make a full revolution after traversing only half of the stationary coin's circumference.

• Gabriel's Horn or Torricelli's trumpet: A simple object with finite volume but infinite surface area. Also, the Mandelbrot set and various other fractals are covered by a finite area, but have an infinite perimeter (in fact, there are no two distinct points on the boundary of the Mandelbrot set that can be reached from one another by moving a finite distance along that boundary, which also implies that in a sense you go no further if you walk “the wrong way”around the set to reach a nearby point). This can be represented by a Klein bottle. 272 CHAPTER 69. LIST OF PARADOXES

• Hausdorff paradox There exists a countable subset C of the sphere S such that S\C is equidecomposable with two copies of itself.

• Nikodym set A set contained in and with the same as the unit square, yet for every one of its points there is a straight line intersecting the Nikodym set only in that point.

• Sphere eversion A sphere can, topologically, be turned inside out.

69.3 Decision theory

• People can make decisions based not on what they actually want to do, but on what they think that other people want to do, with the result that everybody decides to do something that nobody really wants to do, but only what they thought that everybody else wanted to do.

• Apportionment paradox Some systems of apportioning representation can have unintuitive results due to rounding

• Alabama paradox Increasing the total number of seats might shrink one block's seats. • New states paradox Adding a new state or voting block might increase the number of votes of another. • Population paradox A fast-growing state can lose votes to a slow-growing state.

• Arrow's paradox Given more than two choices, no system can have all the attributes of an ideal voting system at once.

• Buridan's ass How can a rational choice be made between two outcomes of equal value?

• Even those who know better play the so-called chain store game in an irrational manner.

• Decision-making paradox Selecting the best decision-making method is a decision problem in itself.

• Fenno's paradox The belief that people generally disapprove of the United States Congress as a whole, but support the Congressman from their own Congressional district.

• Fredkin's paradox The more similar two choices are, the more time a decision-making agent spends on deciding.

• Policies intending to reduce future CO2 emissions may lead to increased emissions in the present.

• Hedgehog's dilemma or Lover's paradox Despite goodwill, human intimacy cannot occur without substantial mutual harm.

• Inventor's paradox It is easier to solve a more general problem that covers the specifics of the sought-after solution.

• Kavka's toxin puzzle Can one intend to drink the non-deadly toxin, if the intention is the only thing needed to get the reward?

• Morton's fork Choosing between unpalatable alternatives.

• Navigation paradox Increased navigational precision may result in increased collision risk.

• Newcomb's paradox How do you play a game against an omniscient opponent?

• Should one tolerate intolerance if intolerance would destroy the possibility of tolerance?

• Also known as the Downs paradox. For a rational, self-interested voter the costs of voting will normally exceed the expected benefits, so why do people keep voting?

• Parrondo's paradox It is possible to play two losing games alternately to eventually win.

• Prevention paradox For one person to benefit, many people have to change their behavior —even though they receive no benefit, or even suffer, from the change. 69.4. PHYSICS 273

• Prisoner's dilemma Two people might not cooperate even if it is in both their best interests to do so. • Relevance paradox Sometimes relevant information is not sought out because its relevance only becomes clear after the information is available. • Voting paradox Also known as Condorcet's paradox and paradox of voting. A group of separately rational individuals may have preferences that are irrational in the aggregate. • Willpower paradox Those who kept their minds open were more goal-directed and more motivated than those who declared their objective to themselves.

69.4 Physics

For more details on this topic, see Physical paradox.

A demonstration of the tea leaf paradox.

• Cool tropics paradox A contradiction between modelled estimates of tropical temperatures during warm, ice-free periods of the Cretaceous and Eocene, and the lower temperatures that proxies suggest were present. • Irresistible force paradox What would happen if an unstoppable force hit an immovable object?

69.4.1 Astrophysics

• Algol paradox In some binaries the partners seem to have different ages, even though they are thought to have formed at the same time. • Faint young Sun paradox The contradiction between existence of liquid water early in the Earth's history and the expectation that the output of the young Sun would have been insufficient to melt ice on Earth. 274 CHAPTER 69. LIST OF PARADOXES

• GZK paradox Extreme-energy cosmic rays have been observed that seem to violate the Greisen-Zatsepin- Kuzmin limit, which is a consequence of .

• Paradox of youth Compared to theory, there is an overabundance of young stars close to the supermassive black hole in the Galactic Center.

69.4.2 Classical mechanics

• Archer's paradox An archer must, in order to hit his target, not aim directly at it, but slightly to the side.

• Hydrostatic paradox or Archimedes' paradox A massive battleship can float in a few litres of water.

• Aristotle's wheel paradox Rolling joined concentric wheels seem to trace the same distance with their cir- cumferences, even though the circumferences are different.

• Carroll's paradox The angular momentum of a stick should be zero, but is not.

• D'Alembert's paradox Flow of an inviscid fluid produces no net force on a solid body.

• Denny's paradox Surface-dwelling arthropods (such as the water strider) should not be able to propel them- selves horizontally.

• Elevator paradox Even though hydrometers are used to measure fluid density, a hydrometer will not indicate changes of fluid density caused by changing atmospheric pressure.

• Feynman sprinkler Which way does a sprinkler rotate when submerged in a tank and made to suck in the surrounding fluid?

• Painlevé paradox Rigid-body dynamics with contact and friction is inconsistent.

• Tea leaf paradox When a cup of tea is stirred, the leaves assemble in the center, even though centrifugal force pushes them outward.

• Upstream contamination When a fluid is poured from a higher container onto a lower one, particles can climb up the falling water.

69.4.3

• Bentley's paradox In a Newtonian universe, gravitation should pull all matter into a single point.

• Boltzmann brain If the universe we observe resulted from a random thermodynamic fluctuation, it would be vastly more likely to be a simple one than the complex one we observe. The simplest case would be just a brain floating in vacuum, having the thoughts and sensations you have.

• Fermi paradox If there are, as various arguments suggest, many other sentient species in the Universe, then where are they? Shouldn't their presence be obvious?

• Heat death paradox If the universe was infinitely old, it would be in thermodynamical equilibrium, which contradicts what we observe.

• Olbers' paradox Why is the night sky dark if there is an infinity of stars, covering every part of the celestial sphere?

69.4.4 Electromagnetism

• paradox An apparent violation of Faraday's law of electromagnetic induction. 69.4. PHYSICS 275

69.4.5 Quantum mechanics

• Aharonov–Bohm effect a charged particle is affected by an electromagnetic field even though it has no local contact with that field

• Bell's theorem Why do measured quantum particles not satisfy mathematical probability theory?

• Double-slit experiment Matter and energy can act as a wave or as a particle depending on the experiment.

• Einstein–Podolsky–Rosen paradox Can far away events influence each other in quantum mechanics?

• Extinction paradox In the small wavelength limit, the total scattering cross section of an impenetrable sphere is twice its geometrical cross-sectional area (which is the value obtained in classical mechanics).*[2]

• Hardy's paradox How can we make inferences about past events that we haven't observed while at the same time acknowledge that the act of observing it affects the reality we are inferring to?

• Klein paradox When the potential of a potential barrier becomes similar to the mass of the impinging particle, it becomes transparent.

• Mott problem Spherically symmetric wave functions, when observed, produce linear particle tracks.

• Quantum LC circuit paradox Energies stored on capacitance and inductance are not equal to the ground state energy of the quantum oscillator.

• Quantum pseudo-telepathy Two players who can not communicate accomplish tasks that seemingly require direct contact.

• Quantum Zeno effect (Turing paradox) echoing the Zeno paradox, a quantum particle that is continuously observed cannot change its state

• Schrödinger's cat paradox According to the Copenhagen interpretation of quantum mechanics, a cat could be simultaneously alive and dead, as long as we don't look.

• Uncertainty principle Attempts to determine position must disturb momentum, and vice versa.

69.4.6 Relativity

• Bell's spaceship paradox concerning relativity.

• Black hole information paradox Black holes violate a commonly assumed tenet of science —that information cannot be destroyed.

• Ehrenfest paradox On the kinematics of a rigid rotating disk.

• A classic relativity problem.

• Mocanu's velocity composition paradox a paradox in special relativity.

• Supplee's paradox the buoyancy of a relativistic object (such as a bullet) appears to change when the reference frame is changed from one in which the bullet is at rest to one in which the fluid is at rest.

• Trouton-Noble or Right-angle lever paradox Does a torque arise in static systems when changing frames?

• The predicts that a person making a round trip will return younger than his or her identical twin who stayed at home. 276 CHAPTER 69. LIST OF PARADOXES

69.4.7 Thermodynamics

• Gibbs paradox In an , is entropy an extensive variable?

• Loschmidt's paradox Why is there an inevitable increase in entropy when the laws of physics are invariant under time reversal? The time reversal symmetry of physical laws appears to contradict the second law of thermodynamics.

• Maxwell's demon The second law of thermodynamics seems to be violated by a cleverly operated trapdoor.*[3]

• Mpemba effect Hot water can, under certain conditions, freeze faster than cold water, even though it must pass the lower temperature on the way to freezing.

• Duncan's Paradox Gas-surface reactions create either steady-state temperature or pressure differentials that can be used in perpetual motion machines.*[4]*[5]

• Schmidt's Paradox (In Russian): - the contradiction (discovered in 1917) between the observed vertical turbu- lent heat flux in the surface layer and the theory of heat conduction in the atmosphere .

69.5 Biology

• Antarctic paradox In some areas of the oceans, phytoplankton concentrations are low despite there apparently being sufficient nutrients.

• C-value enigma Genome size does not correlate with organismal complexity. For example, some unicellular organisms have genomes much larger than that of humans.

• Cole's paradox Even a tiny fecundity advantage of one additional offspring would favor the evolution of semelparity.

• Gray's paradox Despite their relatively small muscle mass, dolphins can swim at high speeds and obtain large accelerations.

• Lek paradox Persistent female choice for particular male trait values should erode genetic variance in male traits and thereby remove the benefits of choice, yet choice persists.

• Lombard's paradox When rising to stand from a sitting or squatting position, both the hamstrings and quadri- ceps contract at the same time, despite their being antagonists to each other.

• Paradox of enrichment Increasing the food available to an ecosystem may lead to instability, and even to extinction.

• Paradox of the pesticides Applying pesticide to a pest may increase the pest's abundance.

• Paradox of the plankton Why are there so many different species of phytoplankton, even though competition for the same resources tends to reduce the number of species?

• Sherman paradox An anomalous pattern of inheritance in the fragile X syndrome.

• Temporal paradox (paleontology) When did the ancestors of birds live?

69.5.1 Health and nutrition

• French paradox The observation that the French suffer a relatively low incidence of coronary heart disease, despite having a diet relatively rich in saturated fats, which are assumed to be the leading dietary cause of such disease.

• Glucose paradox The large amount of glycogen in the liver cannot be explained by its small glucose absorption.

• Hispanic paradox The finding that Hispanics in the U.S. tend to have substantially better health than the average population in spite of what their aggregate socio-economic indicators predict. 69.6. CHEMISTRY 277

• Israeli paradox The observation that Israelis suffer a relatively high incidence of coronary heart disease, despite having a diet very low in saturated fats, which are assumed to be the leading dietary cause of such disease.

• Meditation paradox The amplitude of heart rate oscillations during meditation was significantly greater than in the pre-meditation control state and also in three non-meditation control groups*[6]

• Mexican paradox Mexican children tend to have higher birth weights than can be expected from their socio- economic status.

• Obesity survival paradox Although the negative health consequences of obesity in the general population are well supported by the available evidence, health outcomes in certain subgroups seem to be improved at an increased BMI.

• Peto's paradox Humans get cancer with high frequency, while larger mammals, like whales, do not. If cancer is essentially a negative outcome lottery at the cell level, and larger organisms have more cells, and thus more potentially cancerous cell divisions, one would expect larger organisms to be more predisposed to cancer.

• Pulsus paradoxus A pulsus paradoxus is a paradoxical decrease in systolic blood pressure during inspiration. It can indicate certain medical conditions in which there is reduced venous return of blood to the heart, such as cardiac tamponade or constrictive pericarditis. Also known as the Pulse Paradox.*[7]

69.6 Chemistry

• Faraday paradox () Diluted nitric acid will corrode steel, while concentrated nitric acid will not.

• Levinthal paradox The length of time that it takes for a protein chain to find its folded state is many orders of magnitude shorter than it would be if it freely searched all possible configurations.

• SAR paradox Exceptions to the principle that a small change in a molecule causes a small change in its chemical behaviour are frequently profound.

69.7

• Bootstrap paradox, also ontological paradox Can a time traveler send himself information with no outside source?

• Polchinski's paradox A billiard ball can be thrown into a wormhole in such a way that it would emerge in the past and knock its incoming past self away from the wormhole entrance, creating a variant of the .

• Predestination paradox A man travels back in time to discover the cause of a famous fire. While in the building where the fire started, he accidentally knocks over a kerosene lantern and causes a fire, the same fire that would inspire him, years later, to travel back in time. The bootstrap paradox is closely tied to this, in which, as a result of time travel, information or objects appear to have no beginning.

• Temporal paradox What happens when a time traveler does things in the past that prevent him from doing them in the first place?

• Grandfather paradox You travel back in time and kill your grandfather before he conceives one of your parents, which precludes your own conception and, therefore, you couldn't go back in time and kill your grandfather. • Hitler's murder paradox You travel back in time and kill a famous person in history before they become famous; but if the person had never been famous then he could not have been targeted as a famous person. 278 CHAPTER 69. LIST OF PARADOXES

69.8 Linguistics and artificial intelligence

• Bracketing paradox Is a“historical linguist”a linguist who is historical, or someone who studies“historical linguistics"?

• Code-talker paradox How can a language both enable communication and block communication?

• Moravec's paradox Logical thought is hard for humans and easy for computers, but picking a screw from a box of screws is an unsolved problem.

• Movement paradox In transformational linguistics, there are pairs of sentences in which the sentence without movement is ungrammatical while the sentence with movement is not.

• Sayre's paradox In automated handwriting recognition, a cursively written word cannot be recognized without being segmented and cannot be segmented without being recognized.

69.9 Philosophy

• It seems that no conceptual analysis can meet the requirements both of correctness and of informativeness.

• Buridan's bridge Will Plato throw Socrates into the water or not?

• Paradox of fiction How can people experience strong emotions from purely fictional things?

• Fitch's paradox If all truths are knowable, then all truths must in fact be known.

• Paradox of free will If God knew how we will decide when he created us, how can there be free will?

• Goodman's paradox Why can induction be used to confirm that things are“green”, but not to confirm that things are “grue"?

• When one pursues happiness itself, one is miserable; but, when one pursues something else, one achieves happiness.

• Hutton's Paradox If asking oneself “Am I dreaming?" in a dream proves that one is, what does it prove in waking life?

• “Minimal Liberty”is incompatible with Pareto optimality.

• Meno's paradox (Learner's paradox) A man cannot search either for what he knows or for what he does not know.

• (Parfit's paradox) Is a large population living a barely tolerable life better than a small, happy population?

• Moore's paradox “It's raining, but I don't believe that it is.”

• Newcomb's paradox A paradoxical game between two players, one of whom can predict the actions of the other.

• Paradox of Several distinct paradoxes share this name.

• paradox Can an omnipotent being create a rock too heavy for itself to lift?

• The author of a book may be justified in believing that all his statements in the book are correct, at the same time believing that at least one of them is incorrect.

• Problem of evil (Epicurean paradox) The existence of evil seems to be incompatible with the existence of an omnipotent, omniscient, and morally perfect God.

• Rule-following paradox Even though rules are intended to determine actions, “no course of action could be determined by a rule, because any course of action can be made out to accord with the rule”. 69.10. MYSTICISM 279

• When a white horse is not a horse White horses are not horses because white and horse refer to different things.

• Zeno's paradoxes “You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on ...”(This is also a paradox of the infinite)

69.10 Mysticism

• Tzimtzum In Kabbalah, how to reconcile self-awareness of finite Creation with Infinite Divine source, as an emanated causal chain would seemingly nullify existence. Luria's initial withdrawal of God in Hasidic involves simultaneous illusionism of Creation (Upper Unity) and self-aware existence (Lower Unity), God encompassing logical opposites.

69.11 Economics

See also: Category:Economics paradoxes

One class of paradoxes in economics are the paradoxes of competition, in which behavior that benefits a lone actor would leave everyone worse off if everyone did the same. These paradoxes are classified into circuit, classical and Marx paradoxes.

• A change in a possible outcome that is shared by different alternatives affects people's choices among those alternatives, in contradiction with expected utility theory.

• The Antitrust Paradox : A book arguing that antitrust enforcement artificially raised prices by protecting inefficient competitors from competition.

• Arrow information paradox To sell information you need to give it away before the sale.

• Bertrand paradox Two players reaching a state of both find themselves with no profits.

• Braess' paradox Adding extra capacity to a network can reduce overall performance.

• Deaton paradox Consumption varies surprisingly smoothly despite sharp variations in income.

• Demographic-economic paradox nations or subpopulations with higher GDP per capita are observed to have fewer children, even though a richer population can support more children.

• Downs–Thomson paradox Increasing road capacity at the expense of investments in public transport can make overall congestion on the road worse.

• For countries with income sufficient to meet basic needs, the reported level of happiness does not correlate with national income per person.

• With capacity constraints, there may not be an equilibrium.

• People exhibit (as distinct from risk aversion), in contradiction with expected utility theory.

• European paradox The perceived failure of European countries to translate scientific advances into marketable innovations.

• Gibson's paradox Why were interest rates and prices correlated?

• Giffen paradox Increasing the price of bread makes poor people eat more of it.

• Icarus paradox Some businesses bring about their own downfall through their own successes.

• Increases in efficiency lead to even larger increases in demand. 280 CHAPTER 69. LIST OF PARADOXES

• Some countries labor-intensive commodities and import capital-intensive commodi- ties, in contradiction with the Heckscher–Ohlin theorem.

• Lucas paradox Capital is not flowing from developed countries to developing countries despite the fact that developing countries have lower levels of capital per worker, and therefore higher returns to capital.

• Mandeville's paradox Actions that may be vicious to individuals may benefit society as a whole.

• Mayfield's paradox Keeping everyone out of an information system is impossible, but so is getting everybody in.

• The imposition of a tariff on imports may reduce the relative internal price of that good.

• Paradox of prosperity Why do generations that significantly improve the economic climate seem to generally rear a successor generation that consumes rather than produces?

• If everyone saves more money during times of recession, then aggregate demand will fall and will in turn lower total savings in the population.

• If everyone tries to work during times of recession, lower wages will reduce prices, leading to more deflationary expectations, leading to further thrift, reducing demand and thereby reducing employment.

• , also known as diamond-water paradox: Water is more useful than diamonds, yet is a lot cheaper.

• (also known as Solow computer paradox): Worker productivity may go down, despite technological improvements.

• Using the Kaldor–Hicks criterion, an allocation A may be more efficient than allocation B, while at the same time B is more efficient than A.

• Service recovery paradox Successfully fixing a problem with a defective product may lead to higher consumer satisfaction than in the case where no problem occurred at all.

• St. Petersburg paradox People will only offer a modest fee for a reward of infinite .

• Paradox of Plenty The Paradox of Plenty () refers to the paradox that countries and regions with an abundance of natural resources, specifically point-source non-renewable resources like minerals and fuels, tend to have less economic growth and worse development outcomes than countries with fewer natural resources.

• Throw away paradox A trader can gain by throwing away some of his/her initial .

• Tullock paradox Bribing politicians costs less than one would expect, considering how much profit it can yield.

69.12 Perception

For more details on this topic, see Perceptual paradox.

• Tritone paradox An auditory illusion in which a sequentially played pair of Shepard tones is heard as ascending by some people and as descending by others.

• Blub paradox Cognitive lock of some experienced programmers that prevents them from properly evaluating the quality of programming languages which they do not know.*[8]

• Optical illusion A visual illusion which suggests inconsistency, such as an impossible cube or the vertical- horizontal illusion, where the two lines are exactly the same length but appear to be of different lengths. 69.13. POLITICS 281

the vertical–horizontal illusion

69.13 Politics

• Stability–instability paradox When two countries each have nuclear weapons, the probability of a direct war between them greatly decreases, but the probability of minor or indirect conflicts between them increases. • Wollheim's paradox A person can simultaneously advocating two conflicting policy options A and B, provided that the person believes that democratic decisions should be followed.

69.14 Psychology and sociology

• Gender paradox Women conform more closely than men to sociolinguistics norms that are overtly prescribed, but conform less than men when they are not. • Ironic process theory Ironic processing is the psychological process whereby an individual's deliberate at- tempts to suppress or avoid certain thoughts (thought suppression) renders those thoughts more persistent. • Meat paradox People care about animals, but embrace diets that involve harming them. • Moral paradox A situation in which moral imperatives clash without clear resolution. • Outcomes paradox Schizophrenia patients in developing countries seem to fare better than their Western counterparts.*[9] • Region-beta paradox People can sometimes recover more quickly from more intense emotions or pain than from less distressing experiences. • Self-absorption paradox The contradictory association whereby higher levels of self-awareness are simulta- neously associated with higher levels of psychological distress and with psychological well-being.*[10] • Stapp's ironical paradox “The universal aptitude for ineptitude makes any human accomplishment an in- credible .” • Status paradox Several paradoxes involve the concept of medical or social status. • Stockdale paradox “You must never confuse faith that you will prevail in the end—which you can never afford to lose—with the discipline to confront the most brutal facts of your current reality, whatever they might be.” • The Paradox of Anti-Semitism A book arguing that the lack of external persecutions and antagonisms results in the dissolution of Jewish identity, a theory that resonates in works of Dershowitz and Sartre. 282 CHAPTER 69. LIST OF PARADOXES

69.15 Miscellaneous

• Absence paradox No one is ever “here”.

• Ant on a rubber rope An ant crawling on a rubber rope can reach the end even when the rope stretches much faster than the ant can crawl.

• Bonini's paradox Models or simulations that explain the workings of complex systems are seemingly impos- sible to construct. As a model of a complex system becomes more complete, it becomes less understandable, for it to be more understandable it must be less complete and therefore less accurate. When the model becomes accurate, it is just as difficult to understand as the real-world processes it represents.

• Buttered cat paradox Humorous example of a paradox from contradicting proverbs.

• Intentionally blank page Many documents contain pages on which the text “This page is intentionally left blank”is printed, thereby making the page not blank.

• Observer's paradox The outcome of an event or experiment is influenced by the presence of the observer.

69.16 See also

• Auto-antonym A word that is encoded with opposing meanings.

• Absurdity

• Excusable negligence If a behavior is excusable, it is not negligence.

• Gödel's incompleteness theorems and Tarski's undefinability theorem

• Ignore all rules To obey this rule, it is necessary to ignore it.

• Impossible object A type of optical illusion.

• Invalid proof An apparently correct mathematical derivation that leads to an obvious contradiction.

• Logical fallacy A misconception resulting from incorrect reasoning in argumentation.

• Paradox gun A gun that has characteristics of both (smoothbore) shotguns and rifles.

• Paradoxical laughter Inappropriate laughter, often recognized as such by the laughing person.

• Performative contradiction Some statements contradict the conditions that allow them to be stated.

• Proof that 0.999... equals 1

• Puzzle

• Self-refuting idea

• Theories of humor Incongruity theory and the Ridiculous.

69.17 Notes

[1] Eldridge-Smith, Peter; Eldridge-Smith, Veronique (13 January 2010). “The Pinocchio paradox”. Analysis 70 (2): 212– 215. doi:10.1093/analys/anp173. ISSN 1467-8284. Retrieved 23 July 2010. As of 2010, an image of Pinocchio with a speech bubble “My nose will grow now!" has become a minor Internet phe- nomenon (Google search, Google image search). It seems likely that this paradox has been independently conceived multiple times.

[2] Newton, Roger G. (2002). Scattering Theory of Waves and Particles, second edition. Dover Publications. p. 68. ISBN 0-486-42535-5. 69.17. NOTES 283

[3] Carnap is quoted as saying in 1977 "... the situation with respect to Maxwell's paradox”, in Leff, Harvey S.; Rex, A. F., eds. (2003). Maxwell's Demon 2: Entropy, Classical and Quantum Information, Computing (PDF). Institute of Physics. p. 19. ISBN 0-7503-0759-5. Archived from the original (PDF) on 2005-11-09. Retrieved 15 March 2010. On page 36, Leff and Rex also quote Goldstein and Goldstein as saying “Smoluchowski fully resolved the paradox of the demon in 1912”in Goldstein, Martin; Goldstein, Inge F. (1993). The Refrigerator and The Universe. Universities Press (India) Pvt. Ltd. p. 228. ISBN 978-81-7371-085-8. OCLC 477206415. Retrieved 15 March 2010.

[4] T.L. Duncan, Phys. Rev. E 61, 4661 (2000).

[5] Sheehan, D.P., D.J. Mallin, J.T. Garamella, and W.F. Sheehan, Found. Phys. 44 235 (2014).

[6] Peng, C.-K; Isaac C Henry; Joseph E Mietus; Jeffrey M Hausdorff; Gurucharan Khalsa; Herbert Benson; Ary L Goldberger (May 2004).“Heart rate dynamics during three forms of meditation”. International Journal of Cardiology 95 (1): 19–27. doi:10.1016/j.ijcard.2003.02.006. PMID 15159033. Retrieved 23 May 2012.

[7] Khasnis, A.; Lokhandwala, Y. (Jan–Mar 2002). “Clinical signs in medicine: pulsus paradoxus”. Journal of Postgraduate Medicine (Mumbai – 400 012, India: 49) 48 (1): 46–9. ISSN 0022-3859. PMID 12082330. Retrieved 21 March 2010. The “paradox”refers to the fact that heart sounds may be heard over the precordium when the radial pulse is not felt.

[8] Hidders, J. “Expressive Power of Recursion and Aggregates in XQuery” (PDF). Retrieved 23 May 2012. Chapter 1, Introduction.

[9] Developing countries: The outcomes paradox Nature.com

[10] Trapnell, P. D., & Campbell, J. D. (1999). “Private self-consciousness and the Five-Factor Model of Personality: Distin- guishing rumination from reflection”. Journal of Personality and Social Psychology, 76, 284–304. Chapter 70

Logic

This article is about reasoning and its study. For other uses, see Logic (disambiguation).

Logic (from the Ancient Greek: λογική, logike)*[1] is the branch of philosophy concerned with the use and study of valid reasoning.*[2]*[3] The study of logic also features prominently in mathematics and computer science. Logic was studied in several ancient civilizations, including Greece, India,*[4] and China.*[5] In the West, logic was established as a formal discipline by Aristotle, who gave it a fundamental place in philosophy. The study of logic was part of the classical trivium, which also included grammar and rhetoric. Logic was further extended by Al-Farabi who categorized it into two separate groups (idea and proof). Later, Avicenna revived the study of logic and developed relationship between temporalis and the implication. In the East, logic was developed by Hindus, Buddhists and Jains. Logic is often divided into three parts: inductive reasoning, abductive reasoning, and deductive reasoning.

70.1 The study of logic

The concept of logical form is central to logic. The validity of an argument is determined by its logical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic.

• Informal logic is the study of natural language arguments. The study of fallacies is an important branch of informal logic. The dialogues of Plato*[6] are good examples of informal logic. • Formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of Aristotle contain the earliest known formal study of logic. Modern formal logic follows and expands on Aristotle.*[7] In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language. • Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference.*[8]*[9] Symbolic logic is often divided into two branches: propositional logic and predicate logic. • Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.

However, agreement on what logic is has remained elusive, and although the field of universal logic has studied the common structure of logics, in 2007 Mossakowski et al. commented that “it is embarrassing that there is no widely acceptable formal definition of 'a logic'".*[10]

70.1.1 Logical form

Main article: Logical form

284 70.1. THE STUDY OF LOGIC 285

Logic is generally considered formal when it analyzes and represents the form of any valid argument type. The form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. If one considers the notion of form too philosophically loaded, one could say that formalizing simply means translating English sentences into the language of logic. This is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a considerable variety of form and complexity that makes their use in inference impractical. It requires, first, ignoring those grammatical features irrelevant to logic (such as gender and declension, if the argument is in Latin), replacing conjunctions irrelevant to logic (such as“but”) with logical conjunctions like“and”and replacing ambiguous, or alternative logical expressions (“any”, “every”, etc.) with expressions of a standard type (such as “all”, or the universal quantifier ∀). Second, certain parts of the sentence must be replaced with schematic letters. Thus, for example, the expression“all As are Bs”shows the logical form common to the sentences “all men are mortals”, “all cats are carnivores”, “all Greeks are philosophers”, and so on. That the concept of form is fundamental to logic was already recognized in ancient times. Aristotle uses variable letters to represent valid inferences in Prior Analytics, leading Jan Łukasiewicz to say that the introduction of variables was “one of Aristotle's greatest inventions”.*[11] According to the followers of Aristotle (such as Ammonius), only the logical principles stated in schematic terms belong to logic, not those given in concrete terms. The concrete terms “man”, “mortal”, etc., are analogous to the substitution values of the schematic placeholders A, B, C, which were called the “matter”(Greek hyle) of the inference. The fundamental difference between modern formal logic and traditional, or Aristotelian logic, lies in their differing analysis of the logical form of the sentences they treat.

• In the traditional view, the form of the sentence consists of (1) a subject (e.g.,“man”) plus a sign of quantity (“all”or “some”or “no”); (2) the copula, which is of the form “is”or “is not"; (3) a predicate (e.g., “mortal”). Thus: all men are mortal. The logical constants such as “all”, “no”and so on, plus sentential connectives such as “and”and “or”were called syncategorematic terms (from the Greek kategorei – to predicate, and syn – together with). This is a fixed scheme, where each judgment has an identified quantity and copula, determining the logical form of the sentence. • According to the modern view, the fundamental form of a simple sentence is given by a recursive schema, involving logical connectives, such as a quantifier with its bound variable, which are joined by juxtaposition to other sentences, which in turn may have logical structure. • The modern view is more complex, since a single judgement of Aristotle's system involves two or more logical connectives. For example, the sentence“All men are mortal”involves, in term logic, two non-logical terms“is a man”(here M) and“is mortal”(here D): the sentence is given by the judgement A(M,D). In predicate logic, the sentence involves the same two non-logical concepts, here analyzed as m(x) and d(x) , and the sentence is given by ∀x.(m(x) → d(x)) , involving the logical connectives for universal quantification and implication. • But equally, the modern view is more powerful. Medieval logicians recognized the problem of multiple gen- erality, where Aristotelian logic is unable to satisfactorily render such sentences as “Some guys have all the luck”, because both quantities“all”and“some”may be relevant in an inference, but the fixed scheme that Aristotle used allows only one to govern the inference. Just as linguists recognize recursive structure in natural languages, it appears that logic needs recursive structure.

70.1.2 Deductive and inductive reasoning, and abductive inference

Deductive reasoning concerns what follows necessarily from given premises. However, inductive reasoning—the process of deriving a reliable inference from observations—is often included in the study of logic. Similarly, it is important to distinguish deductive validity and inductive validity (called “strength”). An inference is deductively valid if and only if there is no possible situation in which all the premises are true but the conclusion false. An inference is inductively strong if and only if its premises give some degree of probability to its conclusion. The notion of deductive validity can be rigorously stated for systems of formal logic in terms of the well-understood notions of semantics. Inductive validity on the other hand requires us to define a reliable generalization of some set of observations. The task of providing this definition may be approached in various ways, some less formal than others; some of these definitions may use logical association rule induction, while others may use mathematical models of probability such as decision trees. For the most part this discussion of logic deals only with deductive logic. 286 CHAPTER 70. LOGIC

Abduction*[12] is a form of logical inference that goes from observation to a hypothesis that accounts for the reliable data (observation) and seeks to explain relevant evidence. The American philosopher Charles Sanders Peirce (1839– 1914) first introduced the term as“guessing”.*[13] Peirce said that to abduce a hypothetical explanation a from an observed surprising circumstance b is to surmise that a may be true because then b would be a matter of course.*[14] Thus, to abduce a from b involves determining that a is sufficient (or nearly sufficient), but not necessary, for b .

70.1.3 Consistency, validity, soundness, and completeness

Among the important properties that logical systems can have are:

• Consistency, which means that no theorem of the system contradicts another.*[15]

• Validity, which means that the system's rules of proof never allow a false inference from true premises. A logical system has the property of soundness when the logical system has the property of validity and uses only premises that prove true (or, in the case of axioms, are true by definition).*[15]

• Completeness, which means that if a formula is true, it can be proven (if it is true, it is a theorem of the system).

• Soundness, which has multiple separate meanings, creating a bit of confusion throughout the literature. Most commonly, soundness refers to logical systems, which means that if some formula can be proven in a system, then it is true in the relevant model/structure (if A is a theorem, it is true). This is the converse of completeness. A distinct, peripheral use of soundness refers to arguments, which means that the premises of a valid argument are true in the actual world.

Some logical systems do not have all four properties. As an example, Kurt Gödel's incompleteness theorems show that sufficiently complex formal systems of arithmetic cannot be consistent and complete;*[9] however, first-order predicate logics not extended by specific axioms to be arithmetic formal systems with equality can be complete and consistent.*[16]

70.1.4 Rival conceptions of logic

Main article: Definitions of logic

Logic arose (see below) from a concern with correctness of argumentation. Modern logicians usually wish to ensure that logic studies just those arguments that arise from appropriately general forms of inference. For example, Thomas Hofweber writes in the Stanford Encyclopedia of Philosophy that logic“does not, however, cover good reasoning as a whole. That is the job of the theory of rationality. Rather it deals with inferences whose validity can be traced back to the formal features of the representations that are involved in that inference, be they linguistic, mental, or other representations”.*[17] By contrast, Immanuel Kant argued that logic should be conceived as the science of judgement, an idea taken up in Gottlob Frege's logical and philosophical work. But Frege's work is ambiguous in the sense that it is both concerned with the “laws of thought”as well as with the “laws of truth”, i.e. it both treats logic in the context of a theory of the mind, and treats logic as the study of abstract formal structures.

70.2 History

Main article: History of logic In Europe, logic was first developed by Aristotle.*[18] Aristotelian logic became widely accepted in science and mathematics and remained in wide use in the West until the early 19th century.*[19] Aristotle's system of logic was responsible for the introduction of hypothetical syllogism,*[20] temporal modal logic,*[21]*[22] and inductive logic,*[23] as well as influential terms such as terms, predicables, syllogisms and propositions. In Europe during the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian faith. During the High Middle Ages, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments, often using variations of the methodology of . In 1323, William 70.2. HISTORY 287

Aristotle, 384–322 BCE.

of Ockham's influential Summa Logicae was released. By the 18th century, the structured approach to arguments had degenerated and fallen out of favour, as depicted in Holberg's satirical play Montanus. The Chinese logical philosopher Gongsun Long (c. 325–250 BCE) proposed the paradox “One and one cannot become two, since neither becomes two.”*[24] In China, the tradition of scholarly investigation into logic, however, was repressed by the Qin dynasty following the legalist philosophy of Han Feizi. In India, innovations in the scholastic school, called Nyaya, continued from ancient times into the early 18th century 288 CHAPTER 70. LOGIC

with the Navya-Nyaya school. By the 16th century, it developed theories resembling modern logic, such as Gottlob Frege's“distinction between sense and reference of proper names”and his“definition of number”, as well as the theory of “restrictive conditions for universals”anticipating some of the developments in modern set theory.*[25] Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such as Charles Babbage, Augustus De Morgan, and George Boole.*[26] In the 20th century, Western philosophers like Stanislaw Schayer and Klaus Glashoff have explored Indian logic more extensively. The syllogistic logic developed by Aristotle predominated in the West until the mid-19th century, when interest in the foundations of mathematics stimulated the development of symbolic logic (now called mathematical logic). In 1854, George Boole published An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, introducing symbolic logic and the principles of what is now known as Boolean logic. In 1879, Gottlob Frege published Begriffsschrift, which inaugurated modern logic with the invention of quantifier notation. From 1910 to 1913, Alfred North Whitehead and Bertrand Russell published Principia Mathematica*[8] on the foundations of mathematics, attempting to derive mathematical truths from axioms and inference rules in symbolic logic. In 1931, Gödel raised serious problems with the foundationalist program and logic ceased to focus on such issues. The development of logic since Frege, Russell, and Wittgenstein had a profound influence on the practice of philos- ophy and the perceived nature of philosophical problems (see Analytic philosophy), and Philosophy of mathematics. Logic, especially sentential logic, is implemented in computer logic circuits and is fundamental to computer science. Logic is commonly taught by university philosophy departments, often as a compulsory discipline.

70.3 Types of logic

70.3.1 Syllogistic logic

Main article: Aristotelian logic

The Organon was Aristotle's body of work on logic, with the Prior Analytics constituting the first explicit work in formal logic, introducing the syllogistic.*[27] The parts of syllogistic logic, also known by the name term logic, are the analysis of the judgements into propositions consisting of two terms that are related by one of a fixed number of relations, and the expression of inferences by means of syllogisms that consist of two propositions sharing a common term as premise, and a conclusion that is a proposition involving the two unrelated terms from the premises. Aristotle's work was regarded in classical times and from medieval times in Europe and the Middle East as the very picture of a fully worked out system. However, it was not alone: the Stoics proposed a system of propositional logic that was studied by medieval logicians. Also, the problem of multiple generality was recognized in medieval times. Nonetheless, problems with syllogistic logic were not seen as being in need of revolutionary solutions. Today, some academics claim that Aristotle's system is generally seen as having little more than historical value (though there is some current interest in extending term logics), regarded as made obsolete by the advent of propo- sitional logic and the predicate calculus. Others use Aristotle in argumentation theory to help develop and critically question argumentation schemes that are used in artificial intelligence and legal arguments.

70.3.2 Propositional logic (sentential logic)

Main article: Propositional calculus

A propositional calculus or logic (also a sentential calculus) is a formal system in which formulae representing propo- sitions can be formed by combining atomic propositions using logical connectives, and in which a system of rules establishes certain formulae as “theorems”.

70.3.3 Predicate logic

Main article: Predicate logic 70.3. TYPES OF LOGIC 289

Predicate logic is the generic term for symbolic formal systems such as first-order logic, second-order logic, many- sorted logic, and infinitary logic. Predicate logic provides an account of quantifiers general enough to express a wide set of arguments occurring in natural language. Aristotelian syllogistic logic specifies a small number of forms that the relevant part of the involved judgements may take. Predicate logic allows sentences to be analysed into subject and argument in several additional ways—allowing predicate logic to solve the problem of multiple generality that had perplexed medieval logicians. The development of predicate logic is usually attributed to Gottlob Frege, who is also credited as one of the founders of analytical philosophy, but the formulation of predicate logic most often used today is the first-order logic presented in Principles of Mathematical Logic by David Hilbert and Wilhelm Ackermann in 1928. The analytical generality of predicate logic allowed the formalization of mathematics, drove the investigation of set theory, and allowed the development of Alfred Tarski's approach to model theory. It provides the foundation of modern mathematical logic. Frege's original system of predicate logic was second-order, rather than first-order. Second-order logic is most prominently defended (against the criticism of Willard Van Orman Quine and others) by George Boolos and Stewart Shapiro.

70.3.4 Modal logic

Main article: Modal logic

In languages, modality deals with the phenomenon that sub-parts of a sentence may have their semantics modified by special verbs or modal particles. For example, "We go to the games" can be modified to give "We should go to the games", and "We can go to the games" and perhaps "We will go to the games". More abstractly, we might say that modality affects the circumstances in which we take an assertion to be satisfied. Aristotle's logic is in large parts concerned with the theory of non-modalized logic. Although, there are passages in his work, such as the famous sea-battle argument in De Interpretatione § 9, that are now seen as anticipations of modal logic and its connection with potentiality and time, the earliest formal system of modal logic was developed by Avicenna, whom ultimately developed a theory of "temporally modalized" syllogistic.*[28] While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of Clarence Irving Lewis in 1918, who formulated a family of rival axiomatizations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include deontic logic and epistemic logic. The seminal work of Arthur Prior applied the same formal language to treat temporal logic and paved the way for the marriage of the two subjects. Saul Kripke discovered (contemporaneously with rivals) his theory of frame semantics, which revolutionized the formal technology available to modal logicians and gave a new graph-theoretic way of looking at modality that has driven many applications in computational linguistics and computer science, such as dynamic logic.

70.3.5 Informal reasoning

Main article: Informal logic

The motivation for the study of logic in ancient times was clear: it is so that one may learn to distinguish good from bad arguments, and so become more effective in argument and oratory, and perhaps also to become a better person. Half of the works of Aristotle's Organon treat inference as it occurs in an informal setting, side by side with the development of the syllogistic, and in the Aristotelian school, these informal works on logic were seen as complementary to Aristotle's treatment of rhetoric. This ancient motivation is still alive, although it no longer takes centre stage in the picture of logic; typically dialectical logic forms the heart of a course in critical thinking, a compulsory course at many universities. Argumentation theory is the study and research of informal logic, fallacies, and critical questions as they relate to every day and practical situations. Specific types of dialogue can be analyzed and questioned to reveal premises, conclusions, and fallacies. Argumentation theory is now applied in artificial intelligence and law. 290 CHAPTER 70. LOGIC

70.3.6 Mathematical logic

Main article: Mathematical logic

Mathematical logic really refers to two distinct areas of research: the first is the application of the techniques of formal logic to mathematics and mathematical reasoning, and the second, in the other direction, the application of mathematical techniques to the representation and analysis of formal logic.*[29] The earliest use of mathematics and geometry in relation to logic and philosophy goes back to the ancient Greeks such as Euclid, Plato, and Aristotle.*[30] Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.*[31] One of the boldest attempts to apply logic to mathematics was undoubtedly the logicism pioneered by philosopher- logicians such as Gottlob Frege and Bertrand Russell: the idea was that mathematical theories were logical tautologies, and the programme was to show this by means to a reduction of mathematics to logic.*[8] The various attempts to carry this out met with a series of failures, from the crippling of Frege's project in his Grundgesetze by Russell's paradox, to the defeat of Hilbert's program by Gödel's incompleteness theorems. Both the statement of Hilbert's program and its refutation by Gödel depended upon their work establishing the second area of mathematical logic, the application of mathematics to logic in the form of proof theory.*[32] Despite the negative nature of the incompleteness theorems, Gödel's completeness theorem, a result in model theory and another application of mathematics to logic, can be understood as showing how close logicism came to being true: every rigorously defined mathematical theory can be exactly captured by a first-order logical theory; Frege's proof calculus is enough to describe the whole of mathematics, though not equivalent to it. If proof theory and model theory have been the foundation of mathematical logic, they have been but two of the four pillars of the subject. Set theory originated in the study of the infinite by Georg Cantor, and it has been the source of many of the most challenging and important issues in mathematical logic, from Cantor's theorem, through the status of the Axiom of Choice and the question of the independence of the continuum hypothesis, to the modern debate on large cardinal axioms. Recursion theory captures the idea of computation in logical and arithmetic terms; its most classical achievements are the undecidability of the Entscheidungsproblem by Alan Turing, and his presentation of the Church–Turing thesis.*[33] Today recursion theory is mostly concerned with the more refined problem of complexity classes—when is a problem efficiently solvable?—and the classification of degrees of unsolvability.*[34]

70.3.7 Philosophical logic

Main article: Philosophical logic

Philosophical logic deals with formal of ordinary, non-specialist“( natural”) language. Most philosophers assume that the bulk of everyday reasoning can be captured in logic if a method or methods to translate ordinary language into that logic can be found. Philosophical logic is essentially a continuation of the traditional discipline called “logic”before the invention of mathematical logic. Philosophical logic has a much greater concern with the connection between natural language and logic. As a result, philosophical logicians have contributed a great deal to the development of non-standard logics (e.g. free logics, tense logics) as well as various extensions of classical logic (e.g. modal logics) and non-standard semantics for such logics (e.g. Kripke's supervaluationism in the ). Logic and the philosophy of language are closely related. Philosophy of language has to do with the study of how our language engages and interacts with our thinking. Logic has an immediate impact on other areas of study. Studying logic and the relationship between logic and ordinary speech can help a person better structure his own arguments and critique the arguments of others. Many popular arguments are filled with errors because so many people are untrained in logic and unaware of how to formulate an argument correctly.*[35]*[36]

70.3.8 Computational logic

Main articles: Computational logic and Logic in computer science 70.3. TYPES OF LOGIC 291

Logic cut to the heart of computer science as it emerged as a discipline: Alan Turing's work on the Entscheidungsproblem followed from Kurt Gödel's work on the incompleteness theorems. The notion of the general purpose computer that came from this work was of fundamental importance to the designers of the computer machinery in the 1940s. In the 1950s and 1960s, researchers predicted that when human knowledge could be expressed using logic with mathematical notation, it would be possible to create a machine that reasons, or artificial intelligence. This was more difficult than expected because of the complexity of human reasoning. In logic programming, a program consists of a set of axioms and rules. Logic programming systems such as Prolog compute the consequences of the axioms and rules in order to answer a query. Today, logic is extensively applied in the fields of Artificial Intelligence and Computer Science, and these fields provide a rich source of problems in formal and informal logic. Argumentation theory is one good example of how logic is being applied to artificial intelligence. The ACM Computing Classification System in particular regards:

• Section F.3 on Logics and meanings of programs and F.4 on Mathematical logic and formal languages as part of the theory of computer science: this work covers formal semantics of programming languages, as well as work of formal methods such as Hoare logic;

• Boolean logic as fundamental to : particularly, the system's section B.2 on Arithmetic and logic structures, relating to operatives AND, NOT, and OR;

• Many fundamental logical formalisms are essential to section I.2 on artificial intelligence, for example modal logic and default logic in Knowledge representation formalisms and methods, Horn clauses in logic program- ming, and description logic.

Furthermore, computers can be used as tools for logicians. For example, in symbolic logic and mathematical logic, proofs by humans can be computer-assisted. Using automated theorem proving, the machines can find and check proofs, as well as work with proofs too lengthy to write out by hand.

70.3.9 Non-classical logics

Main article: Non-classical logic

The logics discussed above are all "bivalent" or “two-valued"; that is, they are most naturally understood as divid- ing propositions into true and false propositions. Non-classical logics are those systems that reject various rules of Classical logic. Hegel developed his own dialectic logic that extended Kant's transcendental logic but also brought it back to ground by assuring us that“neither in heaven nor in earth, neither in the world of mind nor of nature, is there anywhere such an abstract 'either–or' as the understanding maintains. Whatever exists is concrete, with difference and opposition in itself”.*[37] In 1910, Nicolai A. Vasiliev extended the law of excluded middle and the law of contradiction and proposed the law of excluded fourth and logic tolerant to contradiction.*[38] In the early 20th century Jan Łukasiewicz investigated the extension of the traditional true/false values to include a third value,“possible”, so inventing ternary logic, the first multi-valued logic.*[39] Logics such as fuzzy logic have since been devised with an infinite number of “degrees of truth”, represented by a real number between 0 and 1.*[40] Intuitionistic logic was proposed by L.E.J. Brouwer as the correct logic for reasoning about mathematics, based upon his rejection of the law of the excluded middle as part of his . Brouwer rejected formalization in math- ematics, but his student studied intuitionistic logic formally, as did Gerhard Gentzen. Intuitionistic logic is of great interest to computer scientists, as it is a constructive logic and can be applied for extracting verified programs from proofs. Newton da Costa has created systems of non-reflexive logics,“Schrödinger logics”, which reject the law of identity in whole or in part.*[41] Modal logic is not truth conditional, and so it has often been proposed as a non-classical logic. However, modal logic is normally formalized with the principle of the excluded middle, and its relational semantics is bivalent, so this inclusion is disputable. 292 CHAPTER 70. LOGIC

70.3.10 “Is logic empirical?"

Main article: Is logic empirical?

What is the epistemological status of the laws of logic? What sort of argument is appropriate for criticizing purported principles of logic? In an influential paper entitled“Is logic empirical?"*[42] Hilary Putnam, building on a suggestion of W. V. Quine, argued that in general the facts of propositional logic have a similar epistemological status as facts about the physical universe, for example as the laws of mechanics or of general relativity, and in particular that what physicists have learned about quantum mechanics provides a compelling case for abandoning certain familiar principles of classical logic: if we want to be realists about the physical phenomena described by quantum theory, then we should abandon the principle of distributivity, substituting for classical logic the quantum logic proposed by Garrett Birkhoff and John von Neumann.*[43] Another paper of the same name by Sir Michael Dummett argues that Putnam's desire for mandates the law of distributivity.*[44] Distributivity of logic is essential for the realist's understanding of how propositions are true of the world in just the same way as he has argued the principle of bivalence is. In this way, the question, “Is logic empirical?" can be seen to lead naturally into the fundamental controversy in metaphysics on realism versus anti-realism.

70.3.11 Implication: strict or material?

Main article: Paradox of entailment

The notion of implication formalized in classical logic does not comfortably translate into natural language by means of “if ... then ...”, due to a number of problems called the paradoxes of material implication. The first class of paradoxes involves counterfactuals, such as If the moon is made of green cheese, then 2+2=5, which are puzzling because natural language does not support the principle of explosion. Eliminating this class of paradoxes was the reason for C. I. Lewis's formulation of strict implication, which eventually led to more radically revisionist logics such as relevance logic. The second class of paradoxes involves redundant premises, falsely suggesting that we know the succedent because of the antecedent: thus “if that man gets elected, granny will die”is materially true since granny is mortal, regardless of the man's election prospects. Such sentences violate the Gricean maxim of relevance, and can be modelled by logics that reject the principle of monotonicity of entailment, such as relevance logic.

70.3.12 Tolerating the impossible

Main article: Paraconsistent logic

Hegel was deeply critical of any simplified notion of the Law of Non-Contradiction. It was based on Leibniz's idea that this law of logic also requires a sufficient ground to specify from what point of view (or time) one says that something cannot contradict itself. A building, for example, both moves and does not move; the ground for the first is our solar system and for the second the earth. In Hegelian dialectic, the law of non-contradiction, of identity, itself relies upon difference and so is not independently assertable. Closely related to questions arising from the paradoxes of implication comes the suggestion that logic ought to tolerate inconsistency. Relevance logic and paraconsistent logic are the most important approaches here, though the concerns are different: a key consequence of classical logic and some of its rivals, such as intuitionistic logic, is that they respect the principle of explosion, which means that the logic collapses if it is capable of deriving a contradiction. Graham Priest, the main proponent of dialetheism, has argued for paraconsistency on the grounds that there are in fact, true .*[45]

70.3.13 Rejection of logical truth

The philosophical vein of various kinds of skepticism contains many kinds of doubt and rejection of the various bases on which logic rests, such as the idea of logical form, correct inference, or meaning, typically leading to the conclusion 70.4. SEE ALSO 293

that there are no logical truths. Observe that this is opposite to the usual views in , where logic directs skeptical enquiry to doubt received wisdoms, as in the work of Sextus Empiricus. Friedrich Nietzsche provides a strong example of the rejection of the usual basis of logic: his radical rejection of idealization led him to reject truth as a "... mobile army of metaphors, metonyms, and anthropomorphisms—in short ... metaphors which are worn out and without sensuous power; coins which have lost their pictures and now matter only as metal, no longer as coins.”*[46] His rejection of truth did not lead him to reject the idea of either inference or logic completely, but rather suggested that “logic [came] into existence in man's head [out] of illogic, whose realm originally must have been immense. Innumerable beings who made inferences in a way different from ours perished” .*[47] Thus there is the idea that logical inference has a use as a tool for human survival, but that its existence does not support the existence of truth, nor does it have a reality beyond the instrumental: “Logic, too, also rests on assumptions that do not correspond to anything in the real world”.*[48] This position held by Nietzsche however, has come under extreme scrutiny for several reasons. He fails to demonstrate the validity of his claims and merely asserts them rhetorically. Although, since he is criticising the established criteria of validity, this does not undermine his position for one could argue that the demonstration of validity provided in the name of logic was just as rhetorically based. Some philosophers, such as Jürgen Habermas, claim his position is self-refuting—and accuse Nietzsche of not even having a coherent perspective, let alone a theory of knowledge.*[49] Again, it is unclear if this is a decisive critique for the criteria of coherency and consistent theory are exactly what is under question. Georg Lukács, in his book The Destruction of Reason, asserts that, “Were we to study Nietzsche's statements in this area from a logico-philosophical angle, we would be confronted by a dizzy chaos of the most lurid assertions, arbitrary and violently incompatible.”*[50] Still, in this respect his “theory”would be a much better depiction of a confused and chaotic reality than any consistent and compatible theory. Bertrand Russell described Nietzsche's irrational claims with “He is fond of expressing himself paradoxically and with a view to shocking conventional readers”in his book A History of Western Philosophy.*[51]

70.4 See also

• Digital electronics (also known as digital logic or logic gates)

• Fallacies

• List of logicians

• List of logic journals

• List of logic symbols

• Logic puzzle

• Mathematics

• List of mathematics articles • Outline of mathematics

• Metalogic

• Outline of logic

• Philosophy

• List of philosophy topics •

• Reason

• Straight and Crooked Thinking (book)

• Truth

• Vector logic 294 CHAPTER 70. LOGIC

70.5 Notes and references

[1]“possessed of reason, intellectual, dialectical, argumentative”, also related to λόγος (logos),“word, thought, idea, argument, account, reason, or principle”(Liddell & Scott 1999; Online Etymology Dictionary 2001).

[2] Richard Henry Popkin; Avrum Stroll (1 July 1993). Philosophy Made Simple. Random House Digital, Inc. p. 238. ISBN 978-0-385-42533-9. Retrieved 5 March 2012.

[3] Jacquette, D. (2002). A Companion to Philosophical Logic. Wiley Online Library. p. 2.

[4] For example, Nyaya (syllogistic recursion) dates back 1900 years.

[5] Mohists and the school of Names date back at 2200 years.

[6] Plato (1976). Buchanan, Scott, ed. The Portable Plato. Penguin. ISBN 0-14-015040-4.

[7] Aristotle (2001). "Posterior Analytics". In Mckeon, Richard. The Basic Works. Modern Library. ISBN 0-375-75799-6.

[8] Whitehead, Alfred North; Russell, Bertrand (1967). Principia Mathematica to *56. Cambridge University Press. ISBN 0-521-62606-4.

[9] For a more modern treatment, see Hamilton, A. G. (1980). Logic for Mathematicians. Cambridge University Press. ISBN 0-521-29291-3.

[10] T. Mossakowski, J. A. Goguen, R. Diaconescu, A. Tarlecki,“What is a Logic?", Logica Universalis 2007 Birkhauser, pp. 113–133.

[11] Łukasiewicz, Jan (1957). Aristotle's syllogistic from the standpoint of modern formal logic (2nd ed.). Oxford University Press. p. 7. ISBN 978-0-19-824144-7.

[12] • Magnani, L.“Abduction, Reason, and Science: Processes of Discovery and Explanation”. Kluwer Academic Plenum Publishers, New York, 2001. xvii. 205 pages. Hard cover, ISBN 0-306-46514-0. • R. Josephson, J. & G. Josephson, S.“Abductive Inference: Computation, Philosophy, Technology”Cambridge Uni- versity Press, New York & Cambridge (U.K.). viii. 306 pages. Hard cover (1994), ISBN 0-521-43461-0, Paperback (1996), ISBN 0-521-57545-1. • Bunt, H. & Black, W.“Abduction, Belief and Context in Dialogue: Studies in Computational Pragmatics”(Natural Language Processing, 1.) John Benjamins, Amsterdam & Philadelphia, 2000. vi. 471 pages. Hard cover, ISBN 90-272-4983-0 (Europe), 1-58619-794-2 (U.S.)

[13] Peirce, C. S.

•“On the Logic of drawing History from Ancient Documents especially from Testimonies”(1901), Collected Papers v. 7, paragraph 219. •“PAP”["Prolegomena to an Apology for Pragmatism"], MS 293 c. 1906, New Elements of Mathematics v. 4, pp. 319-320. • A Letter to F. A. Woods (1913), Collected Papers v. 8, paragraphs 385-388.

(See under "Abduction" and "Retroduction" at Commens Dictionary of Peirce's Terms.)

[14] Peirce, C. S. (1903), Harvard lectures on pragmatism, Collected Papers v. 5, paragraphs 188–189.

[15] Bergmann, Merrie; Moor, James; Nelson, Jack (2009). The Logic Book (Fifth ed.). New York, NY: McGraw-Hill. ISBN 978-0-07-353563-0.

[16] Mendelson, Elliott (1964). “Quantification Theory: Completeness Theorems”. Introduction to Mathematical Logic. Van Nostrand. ISBN 0-412-80830-7.

[17] Hofweber, T. (2004). “Logic and Ontology”. In Zalta, Edward N. Stanford Encyclopedia of Philosophy.

[18] E.g., Kline (1972, p.53) wrote “A major achievement of Aristotle was the founding of the science of logic”.

[19] "Aristotle", MTU Department of Chemistry.

[20] Jonathan Lear (1986). "Aristotle and Logical Theory". Cambridge University Press. p.34. ISBN 0-521-31178-0

[21] Simo Knuuttila (1981). "Reforging the great chain of being: studies of the history of modal theories". Springer Science & Business. p.71. ISBN 90-277-1125-9 70.5. NOTES AND REFERENCES 295

[22] Michael Fisher, Dov M. Gabbay, Lluís Vila (2005). "Handbook of temporal reasoning in artificial intelligence". Elsevier. p.119. ISBN 0-444-51493-7

[23] Harold Joseph Berman (1983). "Law and revolution: the formation of the Western legal tradition". Harvard University Press. p.133. ISBN 0-674-51776-8

[24] The four Catuṣkoṭi logical divisions are formally very close to the four opposed propositions of the Greek tetralemma, which in turn are analogous to the four truth values of modern relevance logic Cf. Belnap (1977); Jayatilleke, K. N., (1967, The logic of four alternatives, in Philosophy East and West, University of Hawaii Press).

[25] Kisor Kumar Chakrabarti (June 1976). “Some Comparisons Between Frege's Logic and Navya-Nyaya Logic”. Philos- ophy and Phenomenological Research (International Phenomenological Society) 36 (4): 554–563. doi:10.2307/2106873. JSTOR 2106873. This paper consists of three parts. The first part deals with Frege's distinction between sense and ref- erence of proper names and a similar distinction in Navya-Nyaya logic. In the second part we have compared Frege's definition of number to the Navya-Nyaya definition of number. In the third part we have shown how the study of the so-called 'restrictive conditions for universals' in Navya-Nyaya logic anticipated some of the developments of modern set theory.

[26] Jonardon Ganeri (2001). Indian logic: a reader. Routledge. pp. vii, 5, 7. ISBN 0-7007-1306-9.

[27] “Aristotle”. Encyclopædia Britannica.

[28] “History of logic: Arabic logic”. Encyclopædia Britannica.

[29] Stolyar, Abram A. (1983). Introduction to Elementary Mathematical Logic. Dover Publications. p. 3. ISBN 0-486-64561- 4.

[30] Barnes, Jonathan (1995). The Cambridge Companion to Aristotle. Cambridge University Press. p. 27. ISBN 0-521-42294- 9.

[31] Aristotle (1989). Prior Analytics. Hackett Publishing Co. p. 115. ISBN 978-0-87220-064-7.

[32] Mendelson, Elliott (1964). “Formal Number Theory: Gödel's Incompleteness Theorem”. Introduction to Mathematical Logic. Monterey, Calif.: Wadsworth & Brooks/Cole Advanced Books & Software. OCLC 13580200.

[33] Brookshear, J. Glenn (1989). “Computability: Foundations of Recursive Function Theory”. Theory of computation: formal languages, automata, and complexity. Redwood City, Calif.: Benjamin/Cummings Pub. Co. ISBN 0-8053-0143-7.

[34] Brookshear, J. Glenn (1989). “Complexity”. Theory of computation: formal languages, automata, and complexity. Redwood City, Calif.: Benjamin/Cummings Pub. Co. ISBN 0-8053-0143-7.

[35] Goldman, Alvin I. (1986), Epistemology and Cognition, Harvard University Press, p. 293, ISBN 9780674258969, untrained subjects are prone to commit various sorts of fallacies and mistakes.

[36] Demetriou, A.; Efklides, A., eds. (1994), Intelligence, Mind, and Reasoning: Structure and Development, Advances in Psychology 106, Elsevier, p. 194, ISBN 9780080867601.

[37] Hegel, G. W. F (1971) [1817]. . Encyclopedia of the Philosophical Sciences. trans. William Wallace. Oxford: Clarendon Press. p. 174. ISBN 0-19-875014-5.

[38] Joseph E. Brenner (3 August 2008). Logic in Reality. Springer. pp. 28–30. ISBN 978-1-4020-8374-7. Retrieved 9 April 2012.

[39] Zegarelli, Mark (2010), Logic For Dummies, John Wiley & Sons, p. 30, ISBN 9781118053072.

[40] Hájek, Petr (2006). “Fuzzy Logic”. In Zalta, Edward N. Stanford Encyclopedia of Philosophy.

[41] da Costa, Newton (1994), Schrödinger logics, Studia Logica, p. 533.

[42] Putnam, H. (1969). “Is Logic Empirical?". Boston Studies in the Philosophy of Science 5.

[43] Birkhoff, G.; von Neumann, J. (1936). “The Logic of Quantum Mechanics”. Annals of Mathematics (Annals of Math- ematics) 37 (4): 823–843. doi:10.2307/1968621. JSTOR 1968621.

[44] Dummett, M. (1978). “Is Logic Empirical?". Truth and Other Enigmas. ISBN 0-674-91076-1.

[45] Priest, Graham (2008). “Dialetheism”. In Zalta, Edward N. Stanford Encyclopedia of Philosophy.

[46] Nietzsche, 1873, On Truth and Lies in a Nonmoral Sense.

[47] Nietzsche, 1882, The Gay Science. 296 CHAPTER 70. LOGIC

[48] Nietzsche, 1878, Human, All Too Human

[49] Babette Babich, Habermas, Nietzsche, and Critical Theory

[50] Georg Lukács. “The Destruction of Reason by Georg Lukács 1952”. Marxists.org. Retrieved 2013-06-16.

[51] Russell, Bertrand (1945), A History of Western Philosophy And Its Connection with Political and Social Circumstances from the Earliest Times to the Present Day (PDF), Simon and Schuster, p. 762

70.6 Bibliography

• Nuel Belnap, (1977). “A useful four-valued logic”. In Dunn & Eppstein, Modern uses of multiple-valued logic. Reidel: Boston.

• Józef Maria Bocheński (1959). A précis of mathematical logic. Translated from the French and German editions by Otto Bird. D. Reidel, Dordrecht, South Holland.

• Józef Maria Bocheński, (1970). A history of formal logic. 2nd Edition. Translated and edited from the German edition by Ivo Thomas. Chelsea Publishing, New York.

• Brookshear, J. Glenn (1989). Theory of computation: formal languages, automata, and complexity. Redwood City, Calif.: Benjamin/Cummings Pub. Co. ISBN 0-8053-0143-7.

• Cohen, R.S, and Wartofsky, M.W. (1974). Logical and Epistemological Studies in Contemporary Physics. Boston Studies in the Philosophy of Science. D. Reidel Publishing Company: Dordrecht, Netherlands. ISBN 90-277-0377-9.

• Finkelstein, D. (1969). “Matter, Space, and Logic”. in R.S. Cohen and M.W. Wartofsky (eds. 1974).

• Gabbay, D.M., and Guenthner, F. (eds., 2001–2005). Handbook of Philosophical Logic. 13 vols., 2nd edition. Kluwer Publishers: Dordrecht.

• Hilbert, D., and Ackermann, W, (1928). Grundzüge der theoretischen Logik (Principles of Mathematical Logic). Springer-Verlag. OCLC 2085765

• Susan Haack, (1996). Deviant Logic, Fuzzy Logic: Beyond the Formalism, University of Chicago Press.

• Hodges, W., (2001). Logic. An introduction to Elementary Logic, Penguin Books.

• Hofweber, T., (2004), Logic and Ontology. Stanford Encyclopedia of Philosophy. Edward N. Zalta (ed.).

• Hughes, R.I.G., (1993, ed.). A Philosophical Companion to First-Order Logic. Hackett Publishing.

• Kline, Morris (1972). Mathematical Thought From Ancient to Modern Times. Oxford University Press. ISBN 0-19-506135-7.

• Kneale, William, and Kneale, Martha, (1962). The Development of Logic. Oxford University Press, London, UK.

• Liddell, Henry George; Scott, Robert. “Logikos”. A Greek-English Lexicon. Perseus Project. Retrieved 8 May 2009.

• Mendelson, Elliott, (1964). Introduction to Mathematical Logic. Wadsworth & Brooks/Cole Advanced Books & Software: Monterey, Calif. OCLC 13580200

• Harper, Robert (2001). “Logic”. Online Etymology Dictionary. Retrieved 8 May 2009.

• Smith, B., (1989). “Logic and the Sachverhalt”. The Monist 72(1):52–69.

• Whitehead, Alfred North and Bertrand Russell, (1910). Principia Mathematica. Cambridge University Press: Cambridge, England. OCLC 1041146 70.7. EXTERNAL LINKS 297

70.7 External links

• Logic at PhilPapers

• Logic at the Indiana Philosophy Ontology Project • Logic entry in the Internet Encyclopedia of Philosophy

• Hazewinkel, Michiel, ed. (2001), “Logical calculus”, Encyclopedia of Mathematics, Springer, ISBN 978-1- 55608-010-4 • An Outline for Verbal Logic

• Introductions and tutorials • An Introduction to Philosophical Logic, by Paul Newall, aimed at beginners. • forall x: an introduction to formal logic, by P.D. Magnus, covers sentential and quantified logic. • Logic Self-Taught: A Workbook (originally prepared for on-line logic instruction). • Nicholas Rescher. (1964). Introduction to Logic, St. Martin's Press.

• Essays • “Symbolic Logic” and “The Game of Logic”, Lewis Carroll, 1896. • Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas. In The Dictionary of the History of Ideas. • Online Tools

• Interactive Syllogistic Machine A web based syllogistic machine for exploring fallacies, figures, terms, and modes of syllogisms.

• Reference material • Translation Tips, by Peter Suber, for translating from English into logical notation. • Ontology and History of Logic. An Introduction with an annotated bibliography.

• Reading lists • The London Philosophy Study Guide offers many suggestions on what to read, depending on the student's familiarity with the subject: • Logic & Metaphysics • Set Theory and Further Logic • Mathematical Logic Chapter 71

Logic of Argumentation

The Logic of Argumentation (LA) is a formalised description of the ways in which humans reason and argue about propositions. It is used, for example, in computer artificial intelligence systems in the fields of medical diagnosis and prognosis, and research chemistry. Krause et al.*[1] appear to have been the first authors to use the term “logic of argumentation”in a paper about their model for using argumentation for qualitative reasoning under uncertainty, although the approach had been used earlier in prototype computer applications to support medical diagnosis.*[2]*[3] Their ideas have been de- veloped further,*[4]*[5] and used in applications for predicting chemical toxicity and xenobiotic metabolism, for example.*[6]*[7] In LA arguments for and arguments against a proposition are distinct; an argument for a proposi- tion contributes nothing to the case against it, and vice versa. Among other things, this means that LA can support contradiction – proof that an argument is true and that it is false. Arguments supporting the case for and arguments supporting the case against are aggregated separately, leading to a single assessment of confidence in the case for and a single assessment of confidence in the case against. Then the two are resolved to provide a single measure of confidence in the proposition. In most implementations of LA the default aggregated value is equal to the strongest value in the set of arguments for or against the proposition. Having more than one argument in agreement does not automatically increase confidence because it cannot be assumed that the arguments are independent when reasoning under uncertainty. If there is evi- dence that arguments are independent and there is a case for increased confidence when they agree, this is sometimes expressed in additional rules of the form “If A and B then ...”. The process of aggregation and resolution can be represented as follows: T = Resolve[Max{For(Ca,x, Cb,y, ...)}, Max{Against(Ca,x, Cb,y, ...)}] where T is the overall assessment of confidence in a proposition; Resolve[] is a function which returns the single confidence value which is the resolution of any pair of values; For and Against are the sets of arguments supporting and opposing the proposition, respectively; Ca,x, Cb,y, …, are the confidence values for those arguments; Max{...} is a function which returns the strongest member of the set upon which it operates (For or Against). Arguments may assign confidence to propositions that themselves influence confidence in other arguments, and one rule may be undercut by another. A computer implementation can recognize these inter-relationships to construct reasoning trees automatically.

71.1 References

[1] Paul J. Krause, Simon Ambler, Morten Elvang-Gøransson, and John Fox, A Logic of Argumentation for Reasoning Under Uncertainty, Computational Intelligence, 1995, 11(1), 113-131.

[2] Morten Elvang-Gøransson, Paul J. Krause, and John Fox, Dialectic Reasoning with Inconsistent Information. in Uncertainty in Artificial Intelligence: Proceedings of the Ninth Conference, eds. D. Heckerman and A. Mamdani, Morgan Kaufmann, San Francisco, 1993, pp. 114-121.

[3] John Fox, David W. Glasspool, and Jonathan Bury, Quantitative and Qualitative Approaches to Reasoning under Un- certainty in Medical Decision Making, in 8th Conference on Artificial Intelligence in Medicine in Europe, AIME 2001

298 71.1. REFERENCES 299

Cascais, Portugal, July 2001, Proceedings, eds. S. Quaglini, P. Barahone, and S. Andreassen, Springer, Berlin, 2001, pp 272-282.

[4] Philip N. Judson and Jonathan D. Vessey, A Comprehensive Approach to Argumentation, J. Chem. Inf. Comput. Sci., 2003, 43, 1356-1363.

[5] Leila Amgoud and Henri Prade, Towards a Logic of Argumentation, Lecture Notes in Comput. Sci., 2012, 7520, 558-565.

[6] Philip N. Judson; Carol A. Marchant; Jonathan D. Vessey. Using Argumentation for Absolute Reasoning about the Potential Toxicity of Chemicals. Journal of Chemical Information and Computer Science, 2003, 43, 1364-1370.

[7] William G. Button, Philip N. Judson, Anthony Long, and Jonathan D. Vessey. Using Absolute and Relative Reasoning in the Prediction of the Potential Metabolism of Xenobiotics, J. Chem. Inf. Comput. Sci., 2003, 43, 1371-1377. Chapter 72

Logic of class

The logic of class is a branch of logic that distinguishes valid from invalid syllogistic reasonings by the use of Venn Diagrams.*[1] In syllogistic reasoning each premise takes one of the following forms, referring to an individual or class of individuals. For example:

• Universal Affirmative (called type A) *[2] • For example, the proposition “All fish are aquatic”. This indicates that the class fish are included in full in the aquatic kind. This is a ratio of total inclusion and how to respond, or has or is expressed by: “All S is P”

• Universal Negative (called type E) *[2] • For example, the proposition “Any child is old”. This proposition indicates that any element of the class of “children”belongs to the class of “old.”This is a case of total exclusion and is expressed in the form “No S is P”

• Particular Affirmative (called type I) *[2] •“Some students are artists”is a proposition which states that at least one member of the class of students is included in the class of artists. This is a partial inclusion relation is expressed, answer or has the form “Some S are P”

• Particular Negative (called Type O) • The proposition “Some roses are not red”states that at least one of the roses is outside the class of the red. Here is a relation of partial exclusion, denoted as “Some S are not P”*[2]

Using Venn diagrams can be viewed as reasoning. If the argument is valid and the conclusion must be determined from the premises that are represented in the diagram *[3] Each form of reasoning has a convertient, a premise that is equivalent but with opposite *[4] Ex:

• All S is P. Convertiente: • Some P is S. P is a subset in S • Anything S is P Convertiente: • No P is S. P does not belong to S • Some S is P Convertiente: • Some P is S. There are elements belonging to P are S and vice versa • Some S is not P Convertiente: • (Not have)

300 72.1. REFERENCES 301

72.1 References

[1] N. Chavez, A. (2000) Introduction to Logic. Lima: Noriega.

[2] Garcia Zarate, Oscar. (2007) Logic. Lima: UNMSM.

[3] Ravello Rea, Bernardo. (2003) Introduction to Logic. Lima: Mantaro.

[4] Perez, M. (2006) Logic and Argumentation Daily Classic. Bogota: Editorial Pontificia Universidad Javeriana.

72.2 See also

• Logic

• Syllogism • Mathematical logic

• Propositional logic Chapter 73

Logic of information

The logic of information, or the logical theory of information, considers the information content of logical signs and expressions along the lines initially developed by Charles Sanders Peirce. In this line of work, the concept of information serves to integrate the aspects of signs and expressions that are separately covered, on the one hand, by the concepts of and extension, and on the other hand, by the concepts of connotation and comprehension. Peirce began to develop these ideas in his lectures “On the Logic of Science”at Harvard University (1865) and the Lowell Institute (1866).

73.1 See also

73.2 References

• Luciano Floridi, The Logic of Information, presentation, discussion, Télé-université (Université du Québec), 11 May 2005, Montréal, Canada.

• Luciano Floridi, The logic of being informed, Logique et Analyse. 2006, 49.196, 433-460.

73.3 External links

• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Eprint

302 Chapter 74

Logical abacus

A logical abacus is a mechanical digital computer. Also referred to as a “logical machine”, the logical abacus is analogous to the ordinary (mathematical) abacus. It is based on the principle of truth tables. It is constructed to show all the possible combinations of a set of logical terms with their negatives, and, further, the way in which these combinations are affected by the addition of attributes or other limiting words, i.e., to simplify mechanically the solution of logical problems. These instruments are all more or less elaborate developments of the “logical slate”, on which were written in vertical columns all the combinations of symbols or letters that could logically be made out of a definite number of terms. These were compared with any given premises, and incompatible ones crossed off. In the abacus the combinations are inscribed each on a single slip of wood or similar substance, which is moved by a key; incompatible combinations can thus be mechanically removed at will, in accordance with any given series of premises. The principal examples of such machines are those of William Stanley Jevons (logic piano), John Venn,*[1] and Allan Marquand.

74.1 References

[1] John Venn (1894). Symbolic logic (2nd ed.). London: Macmillan. Here: p.135f,

• Chisholm, Hugh, ed. (1911). "Abacus". Encyclopædia Britannica (11th ed.). Cambridge University Press.

• William Stanley Jevons, Element. Lessons in Logic, c. xxiii. • Allan Marquand, American Academy of Arts and Sciences, 1885, pp. 303–7.

• Allan Marquand, Johns Hopkins University Studies in Logic, 1883). • Barrett, Lindsay; Connell, Matthew (2005), “Jevons and the Logic 'Piano'", Rutherford Journal 1.

• William Stanley Jevons (1869). The substitution of similars, the true principle of reasoning, derived from a modification of Aristotle's dictum. London: MacMillan. —On p.55f, Jevons gives a description of his logical abacus.

This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "*article name needed". Encyclopædia Britannica (11th ed.). Cambridge University Press.

303 304 CHAPTER 74. LOGICAL ABACUS

Jevons' Logic Piano in the Powerhouse Museum in 2006 Chapter 75

Logical determinism

Logical determinism is the view that a proposition about the future is either necessarily true, or its negation is necessarily true. The argument for this is as follows. By excluded middle, the future tense proposition (‘There will be a sea-battle tomorrow’) is either true now, or its negation is true. But what makes it (or its negation true) is the present existence of a state of affairs – a truthmaker.*[1] If so, then the future is determined in the sense that the way things are now – namely the state of affairs that makes ‘There will be a sea-battle tomorrow’or its negation true – determines the way that things will be. Furthermore, if the past is necessary, in the sense that a state of affairs that existed yesterday cannot be altered, then the state of affairs that made the proposition ‘There will be a sea-battle tomorrow’true cannot be changed, and so the proposition or its negation is necessarily true, and it is either necessarily the case that there will be a sea-battle tomorrow, or necessarily not the case. The term ‘logical determinism’(Logische Determinismus) was introduced by .*[2] Logical determinism seems to present a problem for the conception of free will which requires that different courses of action are possible, for the sea-battle argument suggests that only one course is possible, because necessary. In trying to resolve the problem, the 13th century philosopher Duns Scotus argued in an early work that a future proposition can be understood in two ways: either as signifying something in reality that makes something be true in the future, or simply as signifying that something will be the case. The second sense is weaker in that it does not commit us to any present state of affairs that makes the future proposition true, only a future state of affairs.*[3]

75.1 References

• Schlick, M. ‘Das Kausalität in den gegenwärtigen Physik’, Naturwisseschaften 19 (1931),145-162; Eng. tr. (by P. Heath, 1979), in Philosophical Papers (Volume II). H. L. Mulder and B. F. van de Velde-Schlick (eds.), Dordrecht: D. Reidel. 176-209 • Woleński, J. 'An Analysis of Logical Determinism', 1996.

75.2 See also

• Master Argument • Determinism • Free will • Problem of future contingents

75.3 Notes

[1] Buckner and Zupko, Duns Scotus on Time and Existence: The Questions on Aristotle's 'De interpretatione' , translated with Introduction and Commentary by Edward Buckner and Jack Zupko, Washington, DC: Catholic University of America Press, 2014, p. 318

305 306 CHAPTER 75. LOGICAL DETERMINISM

[2] (Schlick 1979 p. 202), cited in Woleński 1996

[3] Buckner and Zupko, p. 318 Chapter 76

Logical extreme

A logical extreme is a useful, though often fallacious, rhetorical device for the disputation of propositions. Quite simply, a logical extreme is the relevant statement of an extreme or even preposterous position that is nonetheless consistent with the proposition in question. Thus, in so far as the logically extreme position is both relevant and untenable, it has succeeded in calling the proposition into question, at least in its stated form.

The worst effect of Clausewitz's views came through his metaphysical exposition of the idea of 'ab- solute' warfare. By taking the logical extreme as the theoretical ideal, he conveyed the impression, to superficial readers, that the road to success was through the unlimited application of force. [...] More- over, Clausewitz contributed to the subsequent decay of generalship when in an oft-quoted passage he wrote--'Philanthropists may easily imagine that there is a skilful method of disarming and overcoming the enemy without great bloodshed, and that this is the proper tendency of the Art of War....It is an error which must be extirpated.' [...] Unfortunately Clausewitz's corrective arguments would henceforth be cited by countless blunderers to excuse, and even to justify, their futile squandering of life in bull-headed assaults.*[1] —J.P.T. Bury, The New Cambridge Modern History

76.1 See also

• Reductio ad absurdum

76.2 References

[1] Bury, J.P.T. (1960). The New Cambridge Modern History 10. University Press. p. 319. ISBN 9780521045483. LCCN 89009938.

307 Chapter 77

Logical harmony

Logical harmony, a name coined by Sir Michael Dummett, is a supposed constraint on the rules of inference that can be used in a given logical system. The logician Gerhard Gentzen proposed that the meanings of logical connectives could be given by the rules for introducing them into discourse. For example, if one believes that the sky is blue and one also believes that grass is green, then one can introduce the connective and as follows: The sky is blue AND grass is green. Gentzen's idea was that having rules like this is what gives meaning to one's words, or at least to certain words. The idea has also been associated with Wittgenstein's dictum that in many cases we can say, the meaning is the use. Most contemporary logicians prefer to think that the introduction rules and the elimination rules for an expression are equally important. In this case, and is characterized by the following rules: An apparent problem with this was pointed out by Arthur Prior: Why can't we have an expression (call it "tonk") whose introduction rule is that of OR (from “p”to “p tonk q”) but whose elimination rule is that of AND (from “p tonk q”to “q”)? This lets us deduce anything at all from any starting point. Prior suggested that this meant that inferential rules could not determine meaning. He was answered by Nuel Belnap, that even though introduction and elimination rules can constitute meaning, not just any pair of such rules will determine a meaningful expression – they must meet certain constraints, such as not allowing us to deduce any new truths in the old vocabulary. These constraints are what Dummett was referring to. Harmony, then, refers to certain constraints that a proof theory must let hold between introduction and elimination rules for it to be meaningful, or in other words, for its inference rules to be meaning-constituting. The application of harmony to logic may be considered a special case; it makes sense to talk of harmony with respect to not only inferential systems, but also conceptual systems in human cognition, and to type systems in programming languages. Semantics of this form has not provided a very great challenge to that sketched in Tarski's Semantic theory of truth, but many philosophers interested in reconstituting the semantics of logic in a way that respects Ludwig Wittgenstein's meaning is use have felt that harmony holds the key.

77.1 References

• Prior, Arthur. “The runabout inference ticket.”Analysis, 21, pp38–39, 1960-61. • Belnap, Nuel D. Jr. “Tonk, Plonk, and Plink”, Analysis, 22, pp130–134, 1961-62.

77.2 External links

• Harmony at Greg Restall's Proof and Consequence wiki

308 Chapter 78

Logical pluralism

Logical pluralism is the philosophical view that there is more than one correct logic. It stands in contrast to logical monism which argues that there is a single unique logic. There are different standard both for what counts as a logic and what exactly it means for a logic to be “correct.”As a result, the debate about logical monism and pluralism is fractured into many different positions.

78.1 External links

Logical Pluralism at the Stanford Encyclopedia of Philosophy

309 Chapter 79

Loosely associated statements

A loosely associated statement is a type of simple non-inferential passage wherein statements about a general sub- ject are juxtaposed but make no inferential claim.*[1] As a rhetorical device, loosely associated statements may be intended by the speaker to infer a claim or conclusion, but because they lack a coherent logical structure any such interpretation is subjective as loosely associated statements prove nothing and attempt no obvious conclusion.*[2] Loosely associated statements can be said to serve no obvious purpose, such as illustration or explanation.*[3] Included statements can be premises, conclusions or both, and both true or false, but missing from the passage is a claim that any one statement supports another.

79.1 Examples

In A concise introduction to logic, Hurley demonstrates the concept with a quote by Lao-Tzu:

Not to honor men of worth will keep the people from contention; not to value goods which are hard to come by will keep them from theft; not to display what is desirable will keep them from being unset- tled of mind. —Lao-Tzu

While each clause in the quote may seem related to the others, each provides no reason to believe another.

79.2 See also

79.3 References

[1] Hurley, Patrick J. (2008). A Concise Introduction to Logic 10th ed. Thompson Wadsworth. p. 17. ISBN 0-495-50383-5.

[2] “The logic of arguments”. Retrieved April 28, 2012.

[3] “NONargument - Loosely associated statements”. Retrieved April 28, 2012.

310 Chapter 80

Markov's principle

An artistic representation of a Turing machine. Markov's principle says that if it is impossible that a Turing machine will not halt, then it must halt. Markov's principle, named after Andrey Markov Jr, is a specific statement in computability theory that is obvious true classically (i.e. it is a tautology), but must be proved when using constructive mathematics. There are many equivalent formulations of Markov's principle.

80.1 Statements of the principle

In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable, then the set is decidable. In predicate logic, if P is a predicate over the natural numbers, it is expressed as:

(∀n(P (n) ∨ ¬P (n)) ∧ (¬∀n¬P (n))) → (∃n P (n)). That is, if P is decidable, and it cannot be false for every natural number n, then it is true for some n. (In general, a predicate P over some domain is called decidable if for every x in the domain, either P(x) is true, or P(x) is not true, which is not always the case constructively.)

311 312 CHAPTER 80. MARKOV'S PRINCIPLE

It is equivalent in the language of arithmetic to:

¬¬∃n f(n) = 0 → ∃n f(n) = 0, for f a total recursive function on the natural numbers. It is equivalent, in the language of real analysis, to the following principles:

• For each real number x, if it is contradictory that x is equal to 0, then there exists y ∈ Q such that 0 < y < |x|, often expressed by saying that x is apart from, or constructively unequal to, 0. • For each real number x, if it is contradictory that x is equal to 0, then there exists y ∈ R such that xy = 1.

80.2 Realizability

If constructive arithmetic is translated using realizability into a classical meta-theory that proves the ω -consistency of the relevant classical theory (for example, Peano Arithmetic if we are studying ), then Markov's principle is justified: a realizer is the constant function that takes a realization that P is not everywhere false to the unbounded search that successively checks if P (0),P (1),P (2),... is true. If P is not everywhere false, then by ω -consistency there must be a term for which P holds, and each term will be checked by the search eventually. If however P does not hold anywhere, then the domain of the constant function must be empty, so although the search does not halt it still holds vacuously that the function is a realizer. By the Law of the Excluded Middle (in our classical metatheory), P must either hold nowhere or not hold nowhere, therefore this constant function is a realizer. If instead the realizability interpretation is used in a constructive meta-theory, then it is not justified. Indeed, for first-order arithmetic, Markov's principle exactly captures the difference between a constructive and classical meta- theory. Specifically, a statement is provable in Heyting arithmetic with Extended Church's thesis if and only if there is a number that provably realizes it in Heyting arithmetic; and it is provable in Heyting arithmetic with Extended Church's thesis and Markov's principle if and only if there is a number that provably realizes it in Peano arithmetic. Modified realizability does not justify Markov's principle, even if classical logic is used in the meta-theory: there is no realizer in the language of simply typed lambda calculus as this language is not Turing-complete and arbitrary loops cannot be defined in it.

80.3 Markov's rule

Markov's rule is the formulation of Markov's principle as a rule. It states that ∃n P (n) is derivable as soon as ¬¬∃n P (n) is, for P decidable. It was proved by Anne S. Troelstra*[1] that Markov's rule is an admissible rule in Heyting arithmetic. Later on, the logician Harvey Friedman showed that Markov's rule is an admissible rule in all of intuitionistic logic, Heyting arithmetic, and various other intuitionistic theories,*[2] using the Friedman translation.

80.4 Weak Markov's principle

A weaker form of Markov's principle may be stated in the language of analysis as

∀x ∈ R (∀y ∈ R ¬¬(0 < y) ∨ ¬¬(y < x)) → 0 < x. This form can be justified by Brouwer's continuity principles, whereas the stronger form contradicts them. Thus it can be derived from intuitionistic, realizability, and classical reasoning, in each case for different reasons, but this principle is not valid in the general constructive sense of Bishop.*[3]

80.5 See also

• Church's thesis (constructive mathematics) 80.6. REFERENCES 313

80.6 References

[1] Anne S. Troelstra. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Springer Verlag (1973), Theorem 4.2.4 of the 2nd edition.

[2] Harvey Friedman. Classically and Intuitionistically Provably Recursive Functions. In Scott, D. S. and Muller, G. H. Editors, Higher Set Theory, Volume 699 of Lecture Notes in Mathematics, Springer Verlag (1978), pp. 21–28.

[3] Ulrich Kohlenbach,"On weak Markov's principle". Mathematical Logic Quarterly (2002), vol 48, issue S1, pp. 59–65.

80.7 External links

• Constructive Mathematics (Stanford Encyclopedia of Philosophy) Chapter 81

Metamathematics

Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term itself) owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th Century. Metamathematics provides “a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic..”(Kleene 1952, p. 59). An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition “2+2=4”as belonging to mathematics while categorizing the proposition "'2+2=4' is valid”as belonging to metamathematics.

81.1 History

Metamathematical metatheorems about mathematics itself were originally differentiated from ordinary mathematical theorems in the 19th century to focus on what was then called the foundational crisis of mathematics. Richard's paradox (Richard 1905) concerning certain 'definitions' of real numbers in the English language is an example of the sort of contradictions that can easily occur if one fails to distinguish between mathematics and metamathematics. Something similar can be said around the well-known Russell's paradox (Does the set of all those sets that do not contain themselves contain itself?). Metamathematics was intimately connected to mathematical logic, so that the early histories of the two fields, during the late 19th and early 20th centuries, largely overlap. More recently, mathematical logic has often included the study of new pure mathematics, such as set theory, recursion theory and pure model theory, which is not directly related to metamathematics. Serious metamathematical reflection began with the work of Gottlob Frege, especially his Begriffsschrift. David Hilbert was the first to invoke the term “metamathematics”with regularity (see Hilbert's program). In his hands, it meant something akin to contemporary proof theory, in which finitary methods are used to study various axiomatized mathematical theorems (Kleene 1952, p. 55). Other prominent figures in the field include Bertrand Russell, Thoralf Skolem, Emil Post, Alonzo Church, Stephen Kleene, Willard Quine, Paul Benacerraf, Hilary Putnam, Gregory Chaitin, Alfred Tarski and Kurt Gödel. In par- ticular, arguably the greatest achievement of metamathematics and the philosophy of mathematics to date is Gödel's incompleteness theorem: proof that given any finite number of axioms for Peano arithmetic, there will be true state- ments about that arithmetic that cannot be proved from those axioms. Today, metalogic and metamathematics are largely synonymous with each other, and both have been substantially subsumed by mathematical logic in academia.

81.2 Milestones

314 81.2. MILESTONES 315

81.2.1 The discovery of

See also: Non- § History and Hyperbolic geometry § History

The discovery of hyperbolic geometry had important philosophical consequences for Metamathematics. Before its discovery there was just one geometry and mathematics even the idea of another geometry was improbable. When Gauss had discovered hyperbolic geometry, it is said that he did not publish anything about hyperbolic geom- etry out of fear of the “uproar of the Boeotians", which would ruin his status as princeps mathematicorum (Latin, “the Prince of Mathematicians”).*[1] The “uproar of the Boeotians”came and went, and gave an impetus to metamathematics and great improvements in mathematical rigour, analytical philosophy and logic.

81.2.2 Begriffsschrift

Main article: Begriffsschrift

Begriffsschrift (German for, roughly, “concept-script”) is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. Begriffsschrift is usually translated as concept writing or concept notation; the full title of the book identifies it as “a formula language, modeled on that of arithmetic, of pure thought.”Frege's motivation for developing his formal approach to logic resembled Leibniz's motivation for his calculus ratiocinator (despite that, in his Foreword Frege clearly denies that he reached this aim, and also that his main aim would be constructing an ideal language like Leibniz's, what Frege declares to be quite hard and idealistic, however, not impossible task). Frege went on to employ his logical calculus in his research on the foundations of mathematics, carried out over the next quarter century.

81.2.3 Principia Mathematica

Main article: Principia Mathematica

Principia Mathematica, or“PM”as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy,*[2] being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, there would in fact be some truths of mathematics which could not be deduced from them. One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.

81.2.4 Gödel's completeness theorem

Main article: Gödel's completeness theorem

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It makes a close link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proven in particular formal systems. More formally, the completeness theorem says that if a formula is logically valid then there is a finite deduction (a formal proof) of the formula. 316 CHAPTER 81. METAMATHEMATICS

It was first proved by Kurt Gödel in 1929. It was then simplified in 1947, when Leon Henkin observed in his Ph.D. thesis that the hard part of the proof can be presented as the Model Existence Theorem (published in 1949). Henkin's proof was simplified by Gisbert Hasenjaeger in 1953. Gödel's completeness theorem says that a deductive system of first-order predicate calculus is“complete”in the sense that no additional inference rules are required to prove all the logically valid formulas. A converse to completeness is soundness, the fact that only logically valid formulas are provable in the deductive system. Together with soundness (whose verification is easy), this theorem implies that a formula is logically valid if and only if it is the conclusion of a formal deduction.

81.2.5 Gödel's incompleteness theorem

Main article: Gödel's incompleteness theorem

Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.

81.2.6 Tarski's definition of model-theoretic satisfaction

Main article: T-schema

The T-schema or truth schema (not to be confused with 'Convention T') is used to give an inductive definition of truth which lies at the heart of any realisation of Alfred Tarski's semantic theory of truth. Some authors refer to it as the “Equivalence Schema”, a synonym introduced by Michael Dummett.*[3] The T-schema is often expressed in natural language, but it can be formalized in many-sorted predicate logic or modal logic; such a formalisation is called a T-theory. T-theories form the basis of much fundamental work in philosophical logic, where they are applied in several important controversies in analytic philosophy. As expressed in semi-natural language (where 'S' is the name of the sentence abbreviated to S): 'S' is true if and only if S Example: 'snow is white' is true if and only if snow is white.

81.2.7 The impossibility of the Entscheidungsproblem

Main article: Entscheidungsproblem

The Entscheidungsproblem (German for 'decision problem') is a challenge posed by David Hilbert in 1928.*[4] The Entscheidungsproblem asks for an algorithm that takes as input a statement of a first-order logic (possibly with a finite number of axioms beyond the usual axioms of first-order logic) and answers “Yes”or “No”according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic. In 1936, Alonzo Church and Alan Turing published independent papers*[5] showing that a general solution to the Entscheidungsproblem is impossible, assuming that the intuitive notation of "effectively calculable" is captured by 81.3. SEE ALSO 317 the functions computable by a Turing machine (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turing thesis.

81.3 See also

• Meta

• Metalogic

• Model theory • Philosophy of mathematics

• Proof theory

81.4 References

[1] Torretti, Roberto (1978). Philosophy of Geometry from Riemann to Poincare. Dordrecht Holland: Reidel. p. 255.

[2] Irvine, Andrew D. (1 May 2003).“Principia Mathematica (Stanford Encyclopedia of Philosophy)". Metaphysics Research Lab, CSLI, Stanford University. Retrieved 5 August 2009.

[3] Wolfgang Künne (2003). Conceptions of truth. Clarendon Press. p. 18. ISBN 978-0-19-928019-3.

[4] Hilbert and Ackermann

[5] Church's paper was presented to the American Mathematical Society on 19 April 1935 and published on 15 April 1936. Turing, who had made substantial progress in writing up his own results, was disappointed to learn of Church's proof upon its publication (see correspondence between Max Newman and Church in Alonzo Church papers). Turing quickly completed his paper and rushed it to publication; it was received by the Proceedings of the London Mathematical Society on 28 May 1936, read on 12 November 1936, and published in series 2, volume 42 (1936-7); it appeared in two sections: in Part 3 (pages 230-240), issued on Nov 30, 1936 and in Part 4 (pages 241-265), issued on Dec 23, 1936; Turing added corrections in volume 43(1937) pp. 544–546. See the footnote at the end of Soare:1996.

81.5 Further reading

• W. J. Blok and Don Pigozzi,“Alfred Tarski's Work on General Metamathematics”, The Journal of Symbolic Logic, v. 53, No. 1 (Mar., 1988), pp. 36–50. • I. J. Good. “A Note on Richard's Paradox”. Mind, New Series, Vol. 75, No. 299 (Jul., 1966), p. 431. JStor

• Douglas Hofstadter, 1980. Gödel, Escher, Bach. Vintage Books. Aimed at laypeople. • Stephen Cole Kleene, 1952. Introduction to Metamathematics. North Holland. Aimed at mathematicians.

• Jules Richard, Les Principes des Mathématiques et le Problème des Ensembles, Revue Générale des Sciences Pures et Appliquées (1905); translated in Heijenoort J. van (ed.), Source Book in Mathematical Logic 1879- 1931 (Cambridge, Mass., 1964). • Alfred North Whitehead, and Bertrand Russell. Principia Mathematica, 3 vols, Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3). Abridged as Principia Mathematica to *56, Cambridge University Press, 1962. 318 CHAPTER 81. METAMATHEMATICS

The title page of the Principia Mathematica (shortened version, including sections only up to *56), an important work of metamath- ematics. Chapter 82

Multiple-conclusion logic

A multiple-conclusion logic is one in which logical consequence is a relation, ⊢ , between two sets of sentences (or propositions). Γ ⊢ ∆ is typically interpreted as meaning that whenever each element of Γ is true, some element of ∆ is true; and whenever each element of ∆ is false, some element of Γ is false. This form of logic was developed in the 1970s by D. J. Shoesmith and Timothy Smiley*[1] but has not been widely adopted. Some logicians favor a multiple-conclusion consequence relation over the more traditional single-conclusion relation on the grounds that the latter is asymmetric (in the informal, non-mathematical sense) and favors truth over falsity (or assertion over denial).

82.1 See also

• Sequent calculus

82.2 References

[1] D. J. Shoesmith and T. J. Smiley, Multiple Conclusion Logic, Cambridge University Press, 1978

319 Chapter 83

Mutual exclusivity

This article is about logical exclusivity of events and propositions. For the concept in concurrent computing, see Mutual exclusion. For the concept in , see (psychology).

In logic and probability theory, two propositions (or events) are mutually exclusive or disjoint if they cannot both be true (occur). A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, but not both. In the coin-tossing example, both outcomes are, in theory, jointly exhaustive, which means that at least one of the outcomes must happen, so these two possibilities together exhaust all the possibilities.*[1] However, not all mutually exclusive events are collectively exhaustive. For example, the outcomes 1 and 4 of a single roll of a six-sided die are mutually exclusive (both cannot happen at the same time) but not collectively exhaustive (there are other possible outcomes; 2,3,5,6).

83.1 Logic

In logic, two mutually exclusive propositions are propositions that logically cannot be true in the same sense at the same time. To say that more than two propositions are mutually exclusive, depending on context, means that one cannot be true if the other one is true, or at least one of them cannot be true. The term pairwise mutually exclusive always means that two of them cannot be true simultaneously.

83.2 Probability

In probability theory, events E1, E2, ..., En are said to be mutually exclusive if the occurrence of any one of them implies the non-occurrence of the remaining n − 1 events. Therefore, two mutually exclusive events cannot both occur. Formally said, the intersection of each two of them is empty (the null event): A ∩ B = ∅. In consequence, mutually exclusive events have the property: P(A ∩ B) = 0.*[2] For example, it is impossible to draw a card that is both red and a club because clubs are always black. If just one card is drawn from the deck, either a red card (heart or diamond) or a black card (club or spade) will be drawn. When A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B).*[3] To find the probability of drawing a red card or a club, for example, add together the probability of drawing a red card and the probability of drawing a club. In a standard 52-card deck, there are twenty-six red cards and thirteen clubs: 26/52 + 13/52 = 39/52 or 3/4. One would have to draw at least two cards in order to draw both a red card and a club. The probability of doing so in two draws depends on whether the first card drawn were replaced before the second drawing, since without replacement there is one fewer card after the first card was drawn. The probabilities of the individual events (red, and club) are multiplied rather than added. The probability of drawing a red and a club in two drawings without replacement is then 26/52 × 13/51 × 2 = 676/2652, or 13/51. With replacement, the probability would be 26/52 × 13/52 × 2 = 676/2704, or 13/52. In probability theory, the word or allows for the possibility of both events happening. The probability of one or both

320 83.3. STATISTICS 321

events occurring is denoted P(A ∪ B) and in general it equals P(A) + P(B) – P(A ∩ B).*[3] Therefore, in the case of drawing a red card or a king, drawing any of a red king, a red non-king, or a black king is considered a success. In a standard 52-card deck, there are twenty-six red cards and four kings, two of which are red, so the probability of drawing a red or a king is 26/52 + 4/52 – 2/52 = 28/52. Events are collectively exhaustive if all the possibilities for outcomes are exhausted by those possible events, so at least one of those outcomes must occur. The probability that at least one of the events will occur is equal to one.*[4] For example, there are theoretically only two possibilities for flipping a coin. Flipping a head and flipping a tail are collectively exhaustive events, and there is a probability of one of flipping either a head or a tail. Events can be both mutually exclusive and collectively exhaustive.*[4] In the case of flipping a coin, flipping a head and flipping a tail are also mutually exclusive events. Both outcomes cannot occur for a single trial (i.e., when a coin is flipped only once). The probability of flipping a head and the probability of flipping a tail can be added to yield a probability of 1: 1/2 + 1/2 =1.*[5]

83.3 Statistics

In statistics and regression analysis, an independent variable that can take on only two possible values is called a dummy variable. For example, it may take on the value 0 if an observation is of a male subject or 1 if the observation is of a female subject. The two possible categories associated with the two possible values are mutually exclusive, so that no observation falls into more than one category, and the categories are exhaustive, so that every observation falls into some category. Sometimes there are three or more possible categories, which are pairwise mutually exclusive and are collectively exhaustive —for example, under 18 years of age, 18 to 64 years of age, and age 65 or above. In this case a set of dummy variables is constructed, each dummy variable having two mutually exclusive and jointly exhaustive categories —in this example, one dummy variable (called D1) would equal 1 if age is less than 18, and would equal 0 otherwise; a second dummy variable (called D2) would equal 1 if age is in the range 18-64, and 0 otherwise. In this set-up, the dummy variable pairs (D1,D2) can have the values (1,0) (under 18), (0,1) (between 18 and 64), or (0,0) (65 or older) (but not (1,1), which would nonsensically imply that an observed subject is both under 18 and between 18 and 64). Then the dummy variables can be included as independent (explanatory) variables in a regression. Note that the number of dummy variables is always one less than the number of categories: with the two categories male and female there is a single dummy variable to distinguish them, while with the three age categories two dummy variables are needed to distinguish them. Such qualitative data can also be used for dependent variables. For example, a researcher might want to predict whether someone goes to college or not, using family income, a gender dummy variable, and so forth as explanatory variables. Here the variable to be explained is a dummy variable that equals 0 if the observed subject does not go to college and equals 1 if the subject does go to college. In such a situation, ordinary least squares (the basic regression technique) is widely seen as inadequate; instead probit regression or logistic regression is used. Further, sometimes there are three or more categories for the dependent variable —for example, no college, community college, and four-year college. In this case, the multinomial probit or multinomial logit technique is used.

83.4 See also

• Dichotomy • Disjoint sets • Event structure • Oxymoron • Synchronicity • Double bind

83.5 Notes

[1] Miller, Scott; Childers, Donald (2012). Probability and Random Processes (Second ed.). Academic Press. p. 8. ISBN 978-0-12-386981-4. The sample space is the collection or set of 'all possible' distinct (collectively exhaustive and mutually 322 CHAPTER 83. MUTUAL EXCLUSIVITY

exclusive) outcomes of an experiment.

[2] intmath.com; Mutually Exclusive Events. Interactive Mathematics. December 28, 2008.

[3] Stats: Probability Rules.

[4] Scott Bierman. A Probability Primer. Carleton College. Pages 3-4.

[5] Non-Mutually Exclusive Outcomes. CliffsNotes.

83.6 References

• Whitlock, Michael C.; Schluter, Dolph (2008). The Analysis of Biological Data. Roberts and Co. ISBN 978-0-9815194-0-1.

• Lind, Douglas A.; Marchal, William G.; Wathen, Samuel A. (2003). Basic Statistics for Business & Economics (4th ed.). Boston: McGraw-Hill. ISBN 0-07-247104-2. Chapter 84

Mutual knowledge (logic)

Mutual knowledge is a fundamental concept about information in , (epistemic) logic and epistemology. An event is mutual knowledge if all agents know that the event occurred.*[1]*:73 However, mutual knowledge by itself implies nothing about what agents know about other agents' knowledge: i.e. it is possible that an event is mutual knowledge but that each agent is unaware that the other agents know it has occurred.*[2] Common knowledge is a related but stronger notion; any event that is common knowledge is also mutual knowledge. The philosopher Stephen Schiffer, in his book Meaning, developed a notion he called “mutual knowledge”which functions quite similarly to David K. Lewis's “common knowledge”.*[3]

84.1 See also

• Elephant in the room

• The Emperor's New Clothes

84.2 External links

• Steven Pinker. “The Stuff of Thought: Language as a window into human nature”. RSA.

84.3 References

[1] Osborne, Martin J., and Ariel Rubinstein. A Course in Game Theory. Cambridge, MA: MIT, 1994. Print.

[2] Peter Vanderschraaf, Giacomo Sillari (2007). Common Knowledge. Stanford Encyclopedia of Philosophy. Accessed 18 November 2011.

[3] Stephen Schiffer, Meaning, 2nd edition, Oxford University Press, 1988. The first edition was published by OUP in 1972. Also, David Lewis, Convention, Cambridge, MA: Harvard University Press, 1969. For a discussion of both Lewis's and Schiffer's notions, see Russell Dale, The Theory of Meaning (1996).

323 Chapter 85

Münchhausen trilemma

In epistemology, the Münchhausen trilemma is a used to demonstrate the impossibility to prove any truth even in the fields of logic and mathematics. The name Münchhausen-Trilemma was coined by the German philosopher Hans Albert in 1968 in reference to a Trilemma of "dogmatism vs. infinite regress vs. psychologism" used by ;*[1] it is a reference to the problem of "bootstrapping", after the story of Baron Munchausen (in German, “Münchhausen”), pulling himself and the horse on which he was sitting out of a mire by his own hair. It is also known as Agrippa's trilemma, after a similar argument by Sextus Empiricus, which was attributed to Agrippa the Skeptic by Diogenes Laertius; Sextus' argument, however, consists of five (not three)“modes”. Popper in his original 1935 publication mentions neither Sextus nor Agrippa, but attributes his trilemma to Jakob Fries.*[2]

85.1 Trilemma

If we ask of any knowledge: "How do I know that it's true?", we may provide proof; yet that same question can be asked of the proof, and any subsequent proof. The Münchhausen trilemma is that we have only three options when providing proof in this situation:

• The circular argument, in which theory and proof support each other (i.e. we repeat ourselves at some point) • The regressive argument, in which each proof requires a further proof, ad infinitum (i.e. we just keep giving proofs, presumably forever) • The axiomatic argument, which rests on accepted precepts (i.e. we reach some bedrock assumption or cer- tainty)

The first two methods of reasoning are fundamentally weak, and because the Greek skeptics advocated deep ques- tioning of all accepted values, they refused to accept proofs of the third sort. The trilemma, then, is the decision among the three equally unsatisfying options. In contemporary epistemology, advocates of coherentism are supposed to be accepting the “circular”horn of the trilemma; foundationalists are relying on the axiomatic argument. The view that accepts the infinite regress is called infinitism. Advocates of , though, point out that while it is indeed correct that a theory cannot be proven uni- versally true, it can be proven false (test method) or it can be deemed unnecessary (Occam's razor). Thus, conjectural theories can be held as long as they have not been refuted.

85.2 Agrippa and the Greek skeptics

The following tropes for Greek skepticism are given by Sextus Empiricus, in his Outlines of Pyrrhonism. According to Sextus, they are attributed only “to the more recent skeptics”and by Diogenes Laertius we attribute them to Agrippa.*[3] The tropes are:

1. Dissent – the uncertainty of the rules of common life, and of the opinions of philosophers

324 85.3. FRIES' TRILEMMA 325

2. Progress ad infinitum – All proof requires some further proof, and so on to infinity.

3. Relation – All things are changed as their relations become changed, or, as we look upon them from different points of view.

4. Assumption – The truth asserted is merely a hypothesis.

5. Circularity – The truth asserted involves a vicious circle (see , known in scholasticism as diallelus).

[165] According to the mode deriving from dispute, we find that undecidable dissension about the matter proposed has come about both in ordinary life and among philosophers. Because of this we are not able to choose or to rule out anything, and we end up with suspension of judgment. [166] In the mode deriving from infinite regress, we say that what is brought forward as a source of conviction for the matter proposed itself needs another such source, which itself needs another, and so ad infinitum, so that we have no point from which to begin to establish anything, and suspension of judgment follows. [167] In the mode deriving from relativity, as we said above, the existing object appears to be such-and-such relative to the subject judging and to the things observed together with it, but we suspend judgment on what it is like in its nature. [168] We have the mode from hypothesis when the Dogmatists, being thrown back ad infinitum, begin from something which they do not establish but claim to assume simply and without proof in virtue of a concession. [169] The reciprocal mode occurs when what ought to be confirmatory of the object under investigation needs to be made convincing by the object under investigation; then, being unable to take either in order to establish the other, we suspend judgment about both.*[4]

With reference to these five tropes, the first and third are a short summary of the ten original grounds of doubt which were the basis of the earlier skepticism.*[3] The three additional ones show a progress in the sceptical system, and a transition from the common objections derived from the fallibility of sense and opinion, to more abstract and metaphysical grounds of doubt. According to Victor Brochard, “the five tropes can be regarded as the most radical and most precise formulation of skepticism that has ever been given. In a sense, they are still irresistible today.”*[5]

85.3 Fries' trilemma

Jakob Friedrich Fries formulated a similar trilemma in which statements can be accepted either:*[6]

• dogmatically

• supported by infinite regress

• based on perceptual experience (psychologism)

The first two possibilities are rejected by Fries as unsatisfactory, requiring us to adopt the third option. Karl Popper argued that a way to avoid the trilemma was to use an intermediate approach incorporating some dogmatism, some infinite regress, and some perceptual experience.*[7]

85.4 Albert's formulation

The argument proposed by Hans Albert runs as follows: All of the only three (“tri"-lemma) possible attempts to get a certain justification must fail:

1. All justifications in pursuit of 'certain' knowledge have also to justify the means of their justification and doing so they have to justify anew the means of their justification. Therefore, there can be no end. We are faced with the hopeless situation of 'infinite regression'.

2. One can justify with a circular argument, but this sacrifices its validity. 326 CHAPTER 85. MÜNCHHAUSEN TRILEMMA

3. One can stop at self-evidence or common sense or fundamental principles or speaking ex cathedra or at any other evidence, but in doing so, the intention to install 'certain' justification is abandoned.

An English translation of a quote from the original German text by Albert is as follows:*[8]

Here, one has a mere choice between: 1. An infinite regression, which appears because of the necessity to go ever further back, but is not practically feasible and does not, therefore, provide a certain foundation. 2. A logical circle in the deduction, which is caused by the fact that one, in the need to found, falls back on statements which had already appeared before as requiring a foundation, and which circle does not lead to any certain foundation either. 3. A break of searching at a certain point, which indeed appears principally feasible, but would mean a random suspension of the principle of sufficient reason.

Albert stressed repeatedly that there is no limitation of the Münchhausen trilemma to deductive conclusions. The verdict concerns also inductive, causal, transcendental, and all otherwise structured justifications. They all will be in vain. Therefore, certain justification is impossible to attain. Once having given up the classical idea of certain knowledge, one can stop the process of justification where one wants to stop, presupposed one is ready to start critical thinking at this point always anew if necessary. This trilemma rounds off the classical problem of justification in the theory of knowledge. The failure of proving exactly any truth as expressed by the Münchhausen trilemma does not have to lead to dismissal of objectivity, as with relativism. One example of an alternative is the fallibilism of Karl Popper and Hans Albert, accepting that certainty is impossible, but that it is best to get as close as we can to truth, while remembering our uncertainty. In Albert's view, the impossibility to prove any certain truth is not in itself a certain truth. After all, one needs to assume some basic rules of logical inference to derive his result, and in doing so must either abandon the pursuit of “certain”justification, as above, or attempt to justify these rules, etc. He suggests that it has to be taken as true as long as nobody has come forward with a truth which is scrupulously justified as a certain truth. Several philosophers defied Albert's challenge; his responses to such criticisms can be found in his long addendum to his Treatise on Critical Reason and later articles.

85.5 See also

• Anti-foundationalism • Critical rationalism • Duhem–Quine thesis • Gödel's incompleteness theorems • Pyrrhonism • Tarski's undefinability theorem • What the Tortoise Said to Achilles

85.6 References

[1] Dogmatismus – unendlicher Regreß – Psychologismus Albert, Traktat über kritische Vernunft, 1968, p. 11, cited after Westermann, Argumentationen und Begründungen in der Ethik und Rechtslehre, 1977, p. 15.

[2] Robert Nola, “Conceptual and Non-Conceptual Content”, in : ^Karl Popper: A Centenary Assessment vol 2, 2006, p. 158. 85.7. FURTHER READING 327

[3] Diogenes Laërtius, ix.

[4] Sextus Empiricus, Pyrrhōneioi hypotypōseis i., from Annas, J., Outlines of Scepticism Cambridge University Press. (2000).

[5] Brochard, V., The Greek Skeptics.

[6] J. F. Fries, Neue oder anthropologische Kritik der Vernunft (1828 to 1831).

[7] Karl Popper, “The Logic of Scientific Discovery”, p. 87

[8] Albert, H., Traktat über kritische Vernunft, p. 15 (Tübingen: J.C.B. Mohr, 1991).

85.7 Further reading

• Hans Albert, Treatise on Critical Reason, Princeton University Press, 1985, chap. I, sect. 2.

• For Hans Albert's scientific articles see List of Publications in Hans Albert at opensociety.de

85.8 External links

• Epistemic regress at PhilPapers 328 CHAPTER 85. MÜNCHHAUSEN TRILEMMA

Baron Munchausen pulls himself out of a mire by his own hair (illustration by Oskar Herrfurth). Chapter 86

Natural kind

In analytic philosophy, the term natural kind is used to refer to a “natural”grouping, not an artificial one. Or, it is something that a set of things (objects, events, beings) has in common which distinguishes it from other things as a real set rather than as a group of things arbitrarily lumped together by a person or group of people. If any natural kinds exist at all, good candidates might include each of the chemical elements, like gold or potassium. Physical particles, like quarks, might also be natural kinds. That is, they would still be groups of things, distinct from other things as a group, even if there were no people around to say that they were members of the same group. The set of objects that weigh more than 50 pounds, on the other hand, almost certainly does not constitute a natural kind. A person might group those objects together for some purpose like shipping costs, but there is no particular reason that any other person should lump those objects together instead of placing them in some other grouping. A more formal definition has it that a natural kind is a family of "entities possessing properties bound by ; we know of natural kinds in the form of categories of minerals, plants, or animals, and we know that different human cultures classify natural that surround them in a completely analogous fashion”(Molino 2000, p.168). The term was brought into contemporary analytic philosophy by W.V.O. Quine in his essay“Natural Kinds”, where any set of objects forms a kind only if (and perhaps if) it is“projectible”, meaning judgments made about some members of that set can plausibly be extended by scientific induction to other members. Hence "raven" and "black" refer to concepts that match natural kinds, because any black raven constitutes at least some evidence that all ravens are black. But“nonblack”and“nonraven”are not, because a nonblack nonraven (an extremely wide category) is not evidence that all nonblack things are nonravens.*[1] 's problem predicate "grue", meaning“observed before 1 January 2050 and blue or observed after 1 January 2050 and green”, turns out to be inappropriate because it does not denote a natural kind, according to Quine. He argued that kind-hood was logically primitive: it could not be reduced non-trivially to any other relation among individuals. See also #Quine. Cultural artifacts are not generally considered natural kinds. As one author puts it, “they never stop changing, and terms that designate them constitute only what Wittgenstein called ' predicates'" (ibid, p.169). This point is more disputed; John McDowell has extensively argued that this opposition between “culture”and “nature”cannot be clearly formulated, and that in any case it ought to lead us to construing cultural products not as unnatural, but as, adopting Aristotle's terminology, a kind of “second nature.” There is considerable debate in analytic philosophy about whether there are any natural kinds at all, and if so, what they are. Philosophers of biology argue about whether biological species, like the bald eagle (Haliaeetus leucocephalus), are natural kinds; even such familiar species as bird, cat, and dog cannot be established as natural types, since any plausible definitions of those species leaves the classification of some animals ambiguous. Others debate whether races, sexes, or sexual orientations are natural kinds, or rather, to what extent they can be given the wide and continuous variety of race- and gender-related qualities. Meteorologists classify a number of different kinds of clouds, but it is not clear whether they are really different kinds, or whether those groups merely reflect the classifying interests of human beings – in order for them to classify as natural types, some clearly discrete circumstances would have to be shown to produce them in clearly distinguishable ways. The argument, however, is not so much about whether natural kinds exist or whether any of our current concepts succeed in defining true natural kinds, but rather why certain concepts result in more reliable theories and successful interactions with the world. One possible explanation is that it is because some concepts are better approximations of natural types than others, even if no perfectly discrete natural kinds exist and we can only hope to define concepts that classify reality with fewer marginal cases than others. In other words, the concept of the “natural type”may itself not refer to a natural type, but rather to an unattainable ideal that

329 330 CHAPTER 86. NATURAL KIND

Chemical elements like gold are good candidates for natural kinds. paradoxically leads to actions and decisions that achieve the best possible outcomes for human beings. Language is often drawn into the debate as a complicating factor. Those who contest that language heavily shapes abstract thought, or even that it makes abstract thought possible, tend to argue that the coincidental properties of 86.1. SEE ALSO 331

one's native language(s) will inevitably influence one's concepts and interfere with the recognition of natural kinds. Words that reference ideas, such as“love”and“honor”, or that reference metaphysical concepts, such as "" and "", may be conceived of as not even attempting to refer to natural kinds as consistent delineations of objective reality. At the other extreme, a word may claim to refer to a natural kind in a dogmatic and infallible manner, leaving those who understand the word as such unable to recognize or understand evidence that defies their linguistic categories; for example, members of a culture that defines democracy as inherently good may be unable or unwilling to understand the fact that Athens voted to commit genocide on several occasions. The linguistic argument is that it is the way in which cultures inevitably bias words such as “democracy”(or, for that matter, “genocide” ) that makes us ultimately unable to apprehend natural types even if they did exist. Such approaches are associated with Hermeneutics and the family of subjectivist philosophies generally.

86.1 See also

• Essentially contested concept

• Species problem

86.2 Footnotes

[1] web.telia.com Archived September 26, 2007, at the Wayback Machine.

86.3 References

• Andreasen, Robin O. 2005. The Meaning of 'Race': Folk Conceptions and the New Biology of Race. Journal of Philosophy 102(2): 94–106.

• Collins, Harry M. 1975. The Seven Sexes: A Study in the Sociology of a Phenomenon, or the Replication of Experiments in Physics. Sociology 9(2): 205–224.

• Dupré, John. 2001. In Defence of Classification. Studies in History and Philosophy of Biological and Biomed- ical Sciences 32(2): 203–219.

• Fausto-Sterling, Anne. 2000. Essay Review: The Sex/Gender Perplex. Studies in History and Philosophy of Biological and Biomedical Sciences 31(4): 637–646.

• Gadamer, Hans-Georg. “Truth and Method”. Continuum International Publishing Group, 2004. ISBN 082647697X, 9780826476975.

• Hacking, Ian. 1990. Natural Kinds. in Robert B. Barrett and , F., editors. Perspectives on Quine. Cambridge, Mass.: Blackwell.

• Hacking, Ian. 2002. How “Natural”Are “Kinds”of Sexual Orientation? Law and Philosophy 21(3): 335–347.

• Markman, Ellen. 1989. Categorization and Naming in Children. Cambridge, Mass.: MIT Press.

• McOuat, Gordon. 2001. From Cutting Nature at Its Joints to Measuring It: New Kinds and New Kinds of People in Biology. Studies in History and Philosophy of Science 32(4): 613–645.

• Molino, Jean (2000). “Toward an Evolutionary Theory of Music and Natural Language”, The Origins of Music. Cambridge, Mass: A Bradford Book, The MIT Press. ISBN 0-262-23206-5.

• Putnam, Hilary. 1975. The Meaning of 'Meaning'. in Keith Gunderson, editor. Minnesota Studies in the Philosophy of Science, vol. VII. Minneapolis: University of Minnesota Press.

• Willard Van Orman Quine (1970). “Natural Kinds” (PDF). In Nicholas Rescher; et al. Essays in Honor of Carl G. Hempel. Dordrecht: D. Reidel. pp. 41–56. Reprinted in: Quine (1969), Ontological Relativity and Other Essays, Ch. 5, Columbia Univ. Press. 332 CHAPTER 86. NATURAL KIND

• Sokal, Robert R. 1974. Classification: Purposes, Principles, Progress, Prospects. Science 185(4157): 1115– 1123. • Waters, C. Kenneth. 1998. Causal Regularities in the Biological World of Contingent Distributions. Biology and Philosophy 13(1): 5–36.

86.4 External links

• Natural Kinds, by Alexander Bird and Emma Tobin, in the Stanford Encyclopedia of Philosophy • Project Cosmology Modern effort to produce a comprehensive list of major categories for natural kinds Chapter 87

Neutrality (philosophy)

For other uses, see Neutral (disambiguation).

Neutrality is the tendency not to side in a conflict (physical or ideological),*[1]*[2]*[3] which may not suggest neutral parties do not have a side or are not a side themselves. In colloquial use“neutral”can be synonymous with“unbiased.” However, bias is a favoritism for some side,*[4]*[5] distinct of the tendency to act on that favoritism. Neutrality is distinct (though not exclusive) from apathy, ignorance, indifference, doublethink, equality,*[6] agreement, and objectivity. Objectivity suggests siding with the more reasonable position (except journalistic objectivity), where reasonableness is judged by some common basis between the sides, such as logic (thereby avoiding the problem of incommensurability). Neutrality implies tolerance regardless of how disagreeable, deplorable, or unusual a perspec- tive might be.*[6] Advocating neutrality is non-neutral. In moderation/mediation neutrality is often expected to make judgments or facilitate dialog independent of any bias, putting emphasis on the process rather than the outcome.*[6] For example, a neutral-party is seen as a party with no (or a fully disclosed) conflict of interest in a conflict,*[7] and is expected to operate as-if it has no bias. Neutral Parties are often perceived as more trustworthy, reliable, and safe.*[3]*[8] Alternative to acting without a bias, the bias of neutrality itself is the expectation upon the Swiss government (in Armed Neutrality),*[9] and the International Federation of Red Cross and Red Crescent Societies (in Non- interventionism).*[3]

87.1 Criticisms and views

Neutrality implies not judging the validity of an opinion. Thus, a neutral position will provide a platform for all opin- ions, including irrational or malicious ones. Such a platform might be perceived as supportive of positions normally prone to conflict. In classical periods of enlightenment, neutrality has been looked down upon as a character vice, an escape from one's duty to think and to act, as opposed to the modern trend of esteeming neutrality as a virtue. Other Views include:

• Woodrow Wilson: “Neutrality is a negative word. It does not express what America ought to feel. We are not trying to keep out of trouble; we are trying to preserve the foundations on which peace may be rebuilt.”*[10]

• In the Supreme Court decision “Southworth v. The Board of Regents of the University of Wisconsin Sys- tem”based on the United State's First Amendment, the court decided some funding decisions should be made through a neutral viewpoint.*[6]

87.2 In popular culture

• Prime Directive

333 334 CHAPTER 87. NEUTRALITY (PHILOSOPHY)

• Seerow's Kindness from Animorphs

87.3 See also

• Tatramajjhattatā • Alternative dispute resolution

• Moderation • Mediation

• Fairness • Toleration

87.4 References

[1] Dictionary.com, neutral

[2] Merriam-Webster Dictionary, Neutrality

[3] IFRC.org, Our Vision and Mission, The Seven Fundamental Principles, Neutrality

[4] Dictionary.com, bias”

[5] Merriam-Webster Dictionary, bias

[6] Associated Students of Madison, Viewpoint Neutrality in Funding Decisions

[7] Business Dictionary, neutral party

[8] Investopedia, Emotional Neutrality

[9] Swissinfo, Armed Neutrality

[10] Brainy Quote Chapter 88

Non-monotonic logic

A non-monotonic logic is a formal logic whose consequence relation is not monotonic. In other words, non- monotonic logics are devised to capture and represent defeasible inferences (c.f. defeasible reasoning), i.e., a kind of inference in which reasoners draw tentative conclusions, enabling reasoners to retract their conclusion(s) based on further evidence.*[1] Most studied formal logics have a monotonic consequence relation, meaning that adding a formula to a theory never produces a reduction of its set of consequences. Intuitively, monotonicity indicates that learning a new piece of knowledge cannot reduce the set of what is known. A monotonic logic cannot handle var- ious reasoning tasks such as reasoning by default (consequences may be derived only because of lack of evidence of the contrary), abductive reasoning (consequences are only deduced as most likely explanations), some important approaches to reasoning about knowledge (the ignorance of a consequence must be retracted when the consequence becomes known), and similarly, belief revision (new knowledge may contradict old beliefs).

88.1 Abductive reasoning

Abductive reasoning is the process of deriving the most likely explanations of the known facts. An abductive logic should not be monotonic because the most likely explanations are not necessarily correct. For example, the most likely explanation for seeing wet grass is that it rained; however, this explanation has to be retracted when learning that the real cause of the grass being wet was a sprinkler. Since the old explanation (it rained) is retracted because of the addition of a piece of knowledge (a sprinkler was active), any logic that models explanations is non-monotonic.

88.2 Reasoning about knowledge

If a logic includes formulae that mean that something is not known, this logic should not be monotonic. Indeed, learning something that was previously not known leads to the removal of the formula specifying that this piece of knowledge is not known. This second change (a removal caused by an addition) violates the condition of monotonicity. A logic for reasoning about knowledge is the autoepistemic logic.

88.3 Belief revision

Belief revision is the process of changing beliefs to accommodate a new belief that might be inconsistent with the old ones. In the assumption that the new belief is correct, some of the old ones have to be retracted in order to maintain consistency. This retraction in response to an addition of a new belief makes any logic for belief revision to be non-monotonic. The belief revision approach is alternative to paraconsistent logics, which tolerate inconsistency rather than attempting to remove it.

335 336 CHAPTER 88. NON-MONOTONIC LOGIC

88.4 Proof-theoretic versus model-theoretic formalizations of non-monotonic logics

Proof-theoretic formalization of a non-monotonic logic begins with adoption of certain non-monotonic rules of infer- ence, and then prescribes contexts in which these non-monotonic rules may be applied in admissible deductions. This typically is accomplished by means of fixed-point equations that relate the sets of premises and the sets of their non- monotonic conclusions. Defaults logics and autoepistemic logic are the most common examples of non-monotonic logics that have been formalized that way.*[2] Model-theoretic formalization of a non-monotonic logic begins with restriction of the semantics of a suitable mono- tonic logic to some special models, for instance, to minimal models, and then derives the set of non-monotonic rules of inference, possibly with some restrictions in which contexts these rules may be applied, so that the resulting deductive system is sound and complete with respect to the restricted semantics. Unlike some proof-theoretic formalizations that suffered from well-known paradoxes and were often hard to evaluate with respect of their consistency with the intuitions they were supposed to capture, model-theoretic formalizations were paradox-free and left little, if any, room for confusion about what non-monotonic patterns of reasoning they covered. Examples of proof-theoretic formaliza- tions of non-monotonic reasoning, which revealed some undesirable or paradoxical properties or did not capture the desired intuitive comprehensions, that have been successfully (consistent with respective intuitive comprehensions and with no paradoxical properties, that is) formalized by model-theoretic means include first-order circumscription, closed-world assumption, and autoepistemic logic.*[2]

88.5 See also

• Logic programming • Negation as failure • Stable model semantics • Rational consequence relation

88.6 References

• N. Bidoit and R. Hull (1989)“Minimalism, justification and non-monotonicity in deductive databases,”Journal of Computer and System Sciences 38: 290-325. • G. Brewka (1991). Nonmonotonic Reasoning: Logical Foundations of Commonsense. Cambridge University Press. • G. Brewka, J. Dix, K. Konolige (1997). Nonmonotonic Reasoning - An Overview. CSLI publications, Stanford. • M. Cadoli and M. Schaerf (1993)“A survey of complexity results for non-monotonic logics”Journal of Logic Programming 17: 127-60. • F. M. Donini, M. Lenzerini, D. Nardi, F. Pirri, and M. Schaerf (1990)“Nonmonotonic reasoning,”Artificial Intelligence Review 4: 163-210. • M. L. Ginsberg, ed. (1987) Readings in Nonmonotonic Reasoning. Los Altos CA: Morgan Kaufmann. • Horty, J. F., 2001, “Nonmonotonic Logic,”in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell. • W. Lukaszewicz (1990) Non-Monotonic Reasoning. Ellis-Horwood, Chichester, West Sussex, England. • C.G. Lundberg (2000) “Made sense and remembered sense: Sensemaking through abduction,”Journal of Economic Psychology: 21(6), 691-709. • D. Makinson (2005) Bridges from Classical to Nonmonotonic Logic, College Publications. • W. Marek and M. Truszczynski (1993) Nonmonotonic Logics: Context-Dependent Reasoning. Springer Verlag. 88.7. EXTERNAL LINKS 337

• A. Nait Abdallah (1995) The Logic of Partial Information. Springer Verlag.

[1] Strasser, Christian; Antonelli, G. Aldo. “Non-Monotonic Logic”. http://plato.stanford.edu/index.html. Stanford Ency- clopedia of Philosophy. Retrieved 19 March 2015. External link in |website= (help)

[2] Suchenek, Marek A. (2011), “Notes on Nonmonotonic Autoepistemic Propositional Logic” (PDF), Zeszyty Naukowe (Warsaw School of Computer Science) (6): 74–93.

88.7 External links

• Non-monotonic logic entry by G. Aldo Antonelli in the Stanford Encyclopedia of Philosophy • Non-monotonic logic at PhilPapers

• Non-monotonic logic at the Indiana Philosophy Ontology Project Chapter 89

Nonfirstorderizability

In formal logic, nonfirstorderizability is the inability of an expression to be adequately captured in particular theories in first-order logic. Nonfirstorderizable sentences are sometimes presented as evidence that first-order logic is not adequate to capture the nuances of meaning in natural language. The term was coined by George Boolos in his well-known paper “To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables).”Boolos argued that such sentences call for second-order symbolization, which can be interpreted as plural quantification over the same domain as first-order quantifiers use, without postulation of distinct “second-order objects”(properties, sets, etc.). A standard example, known as the Geach–Kaplan sentence, is:

Some critics admire only one another.

If Axy is understood to mean "x admires y,”and the universe of discourse is the set of all critics, then a reasonable translation of the sentence into second order logic is:

∃X(∃x, y(Xx ∧ Xy ∧ Axy) ∧ ∃x¬Xx ∧ ∀x ∀y(Xx ∧ Axy → Xy)) That this formula has no first-order equivalent can be seen as follows. Substitute the formula (y = x + 1 v x = y + 1) for Axy. The result,

∃X(∃x, y(Xx ∧ Xy ∧ (y = x + 1 ∨ x = y + 1)) ∧ ∃x¬Xx ∧ ∀x ∀y(Xx ∧ (y = x + 1 ∨ x = y + 1) → Xy)) states that there is a nonempty set which is closed under the predecessor and successor operations and yet does not contain all numbers. Thus, it is true in all nonstandard models of arithmetic but false in the standard model. Since no first-order sentence has this property, the result follows.

89.1 See also

• Plural quantification • Reification (linguistics) • Branching quantifier • Generalized quantifier

89.2 References

• George Boolos (1984). “To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables)". Journal of Philosophy (The Journal of Philosophy, Vol. 81, No. 8) 81 (8): 430–49. doi:10.2307/2026308.

338 89.2. REFERENCES 339

JSTOR 2026308. Reprinted in Boolos, George (1998). Logic, Logic, and Logic. Cambridge, MA: Harvard University Press. ISBN 0-674-53767-X. Chapter 90

Normal form (natural deduction)

An inference of natural deduction is a normal form, according to Dag Prawitz, if no formula occurrence is both the principal premise of an elimination rule and the conclusion of an introduction rule.

340 Chapter 91

Object of the mind

An object of the mind is an object which exists in the imagination, but which, in the real world, can only be repre- sented or modeled. Some such objects are mathematical abstractions, literary concepts, or fictional scenarios. Closely related are intentional objects, which are what thoughts and feelings are about, even if they are not about anything real (such as thoughts about unicorns, or feeling of apprehension about a dental appointment which is sub- sequently cancelled).*[1] However, intentional objects can coincide with real objects (as in thoughts about horses, or a feeling of regret about a missed appointment).

91.1 Mathematical objects

Mathematics and geometry describe abstract objects that sometimes correspond to familiar shapes, and sometimes do not. Circles, , rectangles, and so forth describe two-dimensional shapes that are often found in the real world. However, mathematical formulas do not describe individual physical circles, triangles, or rectangles. They describe ideal shapes that are objects of the mind. The incredible precision of mathematical expression permits a vast applicability of mental abstractions to real life situations. Many more mathematical formulas describe shapes that are unfamiliar, or do not necessarily correspond to objects in the real world. For example, the Klein bottle*[2] is a one-sided, sealed surface with no inside or outside (in other words, it is the three-dimensional equivalent of the Möbius strip). Such objects can be represented by twisting and cutting or taping pieces of paper together, as well as by computer simulations. To hold them in the imagination, abstractions such as extra or fewer dimensions are necessary.

91.2 Logical sequences

If-then arguments posit logical sequences that sometimes include objects of the mind. For example, a counterfactual argument proposes a hypothetical or subjunctive possibility which could or would be true, but might not be false. Conditional sequences involving subjunctives use intensional language, which is studied by modal logic,*[3] whereas classical logic studies the extensional language of necessary and sufficient conditions. In general, a logical antecedent is a necessary condition, and a logical consequent is a sufficient condition (or the contingency) in a logical conditional. But logical conditionals accounting only for necessity and sufficiency do not always reflect every day if-then reasoning, and for this reason they are sometimes known as material conditionals. In contrast, indicative conditionals, sometimes known as non-material conditionals,*[4] attempt to describe if-then reasoning involving hypotheticals, fictions, or counterfactuals. Truth tables for if-then statements identify four unique combinations of premises and conclusions: true premises and true conclusions; false premises and true conclusions; true premises and false conclusions; false premises and false conclusions. Strict conditionals assign a positive truth-value to every case except the case of a true premise and a false conclusion. This is sometimes regarded as counterintuitive, but makes more sense when false conditions are understood as objects of the mind.

341 342 CHAPTER 91. OBJECT OF THE MIND

91.2.1 False antecedent

A false antecedent is a premise known to be false, fictional, imaginary, or unnecessary. In a conditional sequence, a false antecedent may be the basis for any consequence, true or false. The subjects of literature are sometimes false antecedents. For example, the contents of false documents, the origins of stand-alone phenomena, or the implications of loaded words. Also, artificial sources, personalities, events, and histories. False antecedents are sometimes referred to as "nothing", or "nonexistent", whereas nonexistent are not referred to. Art and acting often portray scenarios without any antecedent except an artist's imagination. For example, mythical heroes, legendary creatures, gods, and .

91.2.2 False consequent

A false consequent, in contrast, is a conclusion known to be false, fictional, imaginary, or insufficient. In a conditional statement, a fictional conclusion is known as a non sequitur, which literally means out of sequence. A conclusion that is out of sequence is not contingent on any premises that precede it, and it does not follow from them, so such a sequence is not conditional. A conditional sequence is a connected series of statements. A false consequent cannot follow from true premises in a connected sequence. But, on the other hand, a false consequent can follow from a false antecedent. As an example, the name of a team, a genre, or a nation is a collective term applied ex post facto to a group of distinct individuals. None of the individuals on a sports team is the team itself, nor is any musical chord a genre, nor any person America. The name is an identity for a collection that is connected by consensus or reference, but not by sequence. A different name could equally follow, but it would have different social or political significance.

91.3 Philosophy of mind

Main article: Mind-body dualism

In philosophy, mind-body dualism is the doctrine that mental activities exist apart from the physical body, notably posited by René Descartes in Meditations on First Philosophy.

91.4 Invented sources

Main article: False document

Many objects in fiction follow the example of false antecedents or false consequents. For example, The Lord of the Rings by J.R.R. Tolkien is based on an imaginary book. In the Appendices to The Lord of the Rings, Tolkien's characters name the Red Book of Westmarch as the source material for The Lord of the Rings, which they describe as a translation. But the Red Book of Westmarch is a fictional document that chronicles events in an imaginary world. One might imagine a different translation, by another author.

91.5 Convenient fictions

Further information: Fictionalism

Social reality is composed of many standards and inventions that facilitate communication, but which are ultimately objects of the mind. For example, money is an object of the mind which currency represents. Similarly, languages signify ideas and thoughts. Objects of the mind are frequently involved in the roles that people play. For example, Acting is a profession which predicates real jobs on fictional premises. Charades is a game people play by guessing imaginary objects from short 91.6. SCIENCE 343

play-acts. Imaginary personalities and histories are sometimes invented to enhance the verisimilitude of fictional , and the immersion of role-playing games. In the sense that they exist independently of extant personalities and histories, they are believed to be fictional characters and fictional time frames. Science fiction is abundant with future times, alternate times, and past times that are objects of the mind. For example, in the novel Nineteen Eighty-Four by , the number 1984 represented a year that had not yet passed. Calendar dates also represent objects of the mind, specifically, past and future times. In The Transformers: The Movie, which was released in 1986, the narration opens with the statement, “It is the year 2005.”In 1986, that statement was futuristic. During the year 2005, that reference to the year 2005 was factual. Now, The Transformers: The Movie is retro-futuristic. The number 2005 did not change, but the object of the mind that it represents did change. Deliberate invention also may reference an object of the mind. The intentional invention of fiction for the purpose of deception is usually referred to as lying, in contrast to invention for entertainment or art. Invention is also often applied to problem solving. In this sense the physical invention of materials is associated with the mental invention of fictions. Convenient fictions also occur in science.

91.6 Science

The theoretical posits of one era's scientific theories may be demoted to mere objects of the mind by subsequent discoveries: some standard examples include phlogiston and ptolemaic epicycles. This raises questions, in the debate between scientific realism and about the status of current posits, such as black holes and quarks. Are they still merely intentional, even if the theory is correct? The situation is further complicated by the existence in scientific practice of entities which are explicitly held not to be real, but which nonetheless serve a purpose—convenient fictions. Examples include field lines, centers of gravity, and holes in semiconductor theory.

91.7 Self-reference

A reference that names an imaginary source is in some sense also a self-reference. A self-reference automatically makes a comment about itself. Premises that name themselves as premises are premises by self-reference; conclusions that name themselves as conclusions are conclusions by self-reference. In their respective imaginary worlds the Necronomicon, The Hitchhiker's Guide to the Galaxy, and the Red Book of Westmarch are realities, but only because they are referred to as real. Authors use this technique to invite readers to pretend or to make-believe that their imaginary world is real. In the sense that the stories that quote these books are true, the quoted books exist; in the sense that the stories are fiction, the quoted books do not exist.

91.8 Nonexistent objects

Austrian philosopher Alexius Meinong (1853–1920) advanced nonexistent objects in the 20th and 21st century within a “theory of objects”. He was interested in intentional states which are directed at nonexistent objects. Starting with the “principle of intentionality”, mental phenomena are intentionally directed towards an object. People may imagine, desire or fear something that does not exist. Other philosophers concluded that intentionality is not a real relation and therefore does not require the existence of an object, while Meinong concluded there is an object for every mental state whatsoever—if not an existent then at least a nonexistent one.*[5]

91.9 See also

• Abstraction 344 CHAPTER 91. OBJECT OF THE MIND

• Existence

• Intentionality • Noumenon

• List of thinking-related topics • The Concept of Mind

• Unobservable • Incompleteness theorems

91.10 References

[1] Tim Crane - Intentional Objects

[2] The Teaching Company, Course No. 1423. “Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas.” Lecture 14 - “A One-Sided, Sealed Surface - the Klein Bottle.”Professors Edward B. Burger, Michael Starbird.

[3] Modal Logic. Springer Online Reference Works.

[4] Payne, W. Russ The non-material conditional

[5] Stanford Encyclopedia of Philosophy, Nonexistent Objects, First published Tue Aug 22, 2006; substantive revision Thu Sep 7, 2006, Accessed May 18, 2010

91.11 External links

• Creatures of Imagination and Fiction - Olav Ashelm Chapter 92

Ontological commitment

An ontological commitment refers to a relation between a language and certain objects postulated to be extant by that language. The 'existence' referred to need not be 'real', but exist only in a universe of discourse. As an example, legal systems use vocabulary referring to 'legal persons' that are collective entities that have rights. One says the legal doctrine has an ontological commitment to non-singular individuals.*[1] In information systems and artificial intelligence, where an ontology refers to a specific vocabulary and a set of explicit assumptions about the meaning and usage of these words, then an ontological commitment is an agreement to use the shared vocabulary in a coherent and consistent manner within a specific context.*[2] In philosophy a“theory is ontologically committed to an object only if that object occurs in all the ontologies of that theory”*[3]

92.1 Background

The sentence “Napoleon is one of my ancestors”apparently commits us only to the existence of two individuals (i.e., Napoleon and the speaker) and a line of ancestry between them. The fact that no other people or objects are mentioned seems to limit the “commitment”of the sentence. However, it is well known that sentences of this kind cannot be interpreted in first-order logic, where individual variables stand for individual things. Instead, they must be represented in some second-order form. In ordinary language, such second-order forms use either grammatical plurals or terms such as “set of”or “group of”. For example, the sentence involving Napoleon can be rewritten as “any group of people that includes me and the parents of each person in the group must also include Napoleon,”which is easily interpreted as a statement in second- order logic (one would naturally start by assigning a name, such as G, to the group of people under consideration). Formally, collective noun forms such as “a group of people”are represented by second-order variables, or by first- order variables standing for sets (which are well-defined objects in mathematics and logic). Since these variables do not stand for individual objects, it seems we are“ontologically committed”to entities other than individuals —sets, classes, and so on. As Quine puts it,

the general adoption of class variables of quantification ushers in a theory whose laws were not in gen- eral expressible in the antecedent levels of logic. The price paid for this increased power is ontological: objects of a special and abstract kind, viz. classes, are now presupposed. Formally it is precisely in al- lowing quantification over class variables α, β, etc., that we assume a range of values for these variables to refer to. To be assumed as an entity is to be assumed as a value of a variable. (Methods of Logic p. 228)

Another statement about individuals that appears“ontologically innocent”is the well-known Geach–Kaplan sentence: Some critics admire only one another.

92.2 Quine's analysis

Willard Van Orman Quine provided an early and influential formulation of ontological commitment:*[4]

345 346 CHAPTER 92. ONTOLOGICAL COMMITMENT

If one affirms a statement using a name or other singular term, or an initial phrase of 'existential quantification', like 'There are some so-and-sos', then one must either (1) admit that one is committed to the existence of things answering to the singular term or satisfying the descriptions, or (2) provide a 'paraphrase' of the statement that eschews singular terms and quantification over so-and sos. Quine's criterion can be seen as a logical development of the methods of Bertrand Russell and G.E. Moore, who assumed that one must accept the existence of entities corresponding to the singular terms used in statements one accepts, unless and until one finds systematic methods of paraphrase that eliminate these terms.*[5] —Michael J. Loux & Dean W. Zimmerman, The Oxford Handbook of Metaphysics, p. 4

The purpose of Quine's strategy is to determine just how the ontological commitment of a theory is to be found. Quine argued that the only ontologically committing expressions are variables bound by a first-order existential quan- tifier, and natural language expressions which were formalized using variables bound by first-order existential quan- tifiers.*[6]*[7] Attempts have been made to argue that predicates are also ontologically committing, and thus that subject-predicate sentences bear additional ontological commitment to abstract objects such as universals, sets, or classes. It has been suggested that the use of meaningful names in nonexistence statements such as“Pegasus does not exist”brings with it an ontological commitment to fictional objects like Pegasus, a quandary referred to as Plato's beard and escaped by using quantifiers.*[8] This discussion has a connection to the Carnap-Quine argument over analytic and synthetic objects.*[9] Although Quine refers to 'ontological commitment' in this connection,*[10] in his rejection of the analytic/synthetic distinction he does not rely upon the formal translation of any particular theory along the lines he has suggested.*[11] Instead, Quine argues by using examples that although there are tautological statements in a formal theory, like “all squares are rectangles”, a formal theory necessarily contains references to objects that are not tautological, but have external connections. That is, there is an ontological commitment to such external objects. In addition, the terms used to interpret the application of the theory are not simply descriptions of sensory input, but are statements in a context. That is, inversely, there is an ontological commitment of these observational objects to the formal theory. As Ryan puts it: “Rather than being divided between contingent synthetic claims and indubitable analytic propositions, our beliefs constitute a continuous range from a periphery of sense-reports to interior concepts that are comparatively theory-laden and general.”*[9] Thus we end up with Quine's 'flat' ontology that does not see a distinction between analytic and synthetic objects.*[12]*[13] Whatever process one uses to determine the ontological commitments of a theory, that does not prescribe what ontological commitments one should have. Quine regarded this as a matter of epistemology, which theory one should accept. “Appeal is made to [concerns of] explanatory power, parsimony, , precision, and so on”.*[14]

92.2.1 Ontological parsimony

Ontological parsimony can be defined in various ways, and often is equated to versions of Occam's razor, a “rule of thumb, which obliges us to favor theories or hypotheses that make the fewest unwarranted, or ad hoc, assumptions about the data from which they are derived.”*[15] Glock, regards 'ontological parsimony' as one of the 'five main points' of Quine's conception of ontology.*[16] Following Quine,*[17] Baker states that a theory, T, is ontologically committed to items F if and only if T entails that F′s exist. If two theories, T1 and T2, have the same ontological commitments except that T2 is ontologically committed to F′s while T1 is not, then T1 is more parsimonious than T2. More generally, a sufficient condition for T1 * being more parsimonious than T2 is for the ontological commitments of T1 to be a proper subset of those of T2. [18]

These ideas lead to the following particular formulation of Occam's razor: 'Other things being equal, if T1 is more ontologically parsimonious than T2 then it is rational to prefer T1 to T2.' While a common formulation stipulates only that entities should not be multiplied beyond necessity, this version by contrast, states that entities should not be multiplied other things being equal, and this is compatible with parsimony being a comparatively weak theoretical virtue.*[18] 92.3. RECENT CONTROVERSIES 347

92.3 Recent controversies

The standard approach to ontological commitment has been that, once a theory has been regimented and/or “para- phrased”into an agreed “canonical”version,*[19] which may indeed be in formal logical notation rather than the original language of the theory, ontological commitments can be read off straightforwardly from the presence of certain ontologically committing expressions (e.g. bound variables of existential quantification). Although there is substantial debate about which expressions are ontologically committing,*[20]*[21] parties to that debate generally agree that the expressions they prefer are reliable bearers of ontological commitment, imparting ontological commit- ment to all regimented sentences in which they occur. This assumption has been challenged. Inwagen has taken issue with Quine's methodology,*[22] claiming that this process did not lead to a unique set of fundamental objects, but to several possible sets, and one never could be certain that all the possible sets had been found. He also took issue with Quine's notion of a theory, which he felt was tantamount to suggesting a 'theory' was just a collection of sentences. Inwagen suggested that Quine's approach provided useful tools for discovering what entities were ontological commitments, but that he had not been successful. His attempts are comparable to an “attempt to reach the moon by climbing ever higher trees...”*[23] It has been suggested that the ontological commitments of a theory cannot be discerned by analysis of the syntax of sentences, looking for ontologically committing expressions, because the true ontological commitments of a sentence (or theory) are restricted to the entities needed to serve as truthmakers for that sentence, and the syntax of even a regimented or formalized sentence is not a reliable guide to what entities are needed to make it true.*[24] However, this view has been attacked by Jonathan Shaffer, who has argued that truthmaking is not an adequate test for ontological commitment: at best, the search for the truthmakers of our theory will tell us what is “fundamental”, but not what our theory is ontologically committed to, and hence will not serve as a good way of deciding what exists.*[25] It also has been argued that the syntax of sentences is not a reliable guide to their ontological commitments because English has no form of words which reliably functions to make an existence-claim in every context in which it is used. For example, Jody Azzouni suggests that “There is”does not make any kind of genuine existence-claim when it is used in a sentence such as “There are mice that talk”. Since the meaning of the existential quantifier in formal notation is usually explained in terms of its equivalence to English expressions such as“there is”and“there exist”, and since these English expressions are not reliably ontologically committing, it comes to seem that we cannot be sure of our theory's ontological commitments even after it has been regimented into a canonical formulation.*[26] This argument has been attacked by Howard Peacock,*[27] who suggests that Azzouni's strategy conflates two different kinds of ontological commitment – one which is intended as a measure of what a theory explicitly claims to exist, and one which is intended as a measure of what is required for the theory to be true; what the ontological costs of the theory are. If ontological commitment is thought of as a matter of the ontological costs of a theory, then it is possible that a sentence may be ontologically committed to an entity even though competent speakers of the language do not recognize the sentence as asserting the existence of that entity. Ontological commitment is not a matter of what commitments one explicitly recognizes, but rather a matter of what commitments are actually incurred.

92.4 See also

• Conceptualization (information science)

• Holophrastic indeterminacy

• Indeterminacy of translation

92.5 References

[1] Burkhard Schäfer (1998). “Invariance principles and the community of heirs”. In N Guarino, ed. Formal Ontology in Information Systems: Proceedings of the 1st International Conference June 6-8, 1998, Trento, Italy. pp. 108 ff. ISBN 9051993994.

[2] Nicola Guarino (1998).“Formal ontology and information systems”. Formal Ontology in Information Systems: Proceedings of the First International Conference (FIOS'98), June 6-8, Trento, Italy. IOS Press. pp. 3 ff. ISBN 9051993994.

[3] , ed. (1999). “Ontological commitment”. The Cambridge Dictionary of Philosophy (Paperback 2nd ed.). p. 631. ISBN 0521637228. A shortened version of that definition is as follows: 348 CHAPTER 92. ONTOLOGICAL COMMITMENT

The ontological commitments of a theory are those things which occur in all the ontologies of that theory. To explain further, the ontology of a theory consists of the objects the theory makes use of. A dependence of a theory upon an object is indicated if the theory fails when the object is omitted. However, the ontology of a theory is not necessarily unique. A theory is ontologically committed to an object only if that object occurs in all the ontologies of that theory. A theory also can be ontologically committed to a class of objects if that class is populated (not necessarily by the same objects) in all its ontologies.

[4] Quine, W. V. (1948). “On What There Is”. Review of Metaphysics 2: 21–38.. Reprinted in From a Logical Point of View: Nine Logico-philosophical Essays (2nd ed.). Harvard University Press. 1980. pp. 1–19. ISBN 0674323513. See Wikisource.

[5] Michael J. Loux, Dean W. Zimmerman (2005). “Introduction”. In Michael J. Loux, Dean W. Zimmerman, eds. The Oxford Handbook of Metaphysics. Oxford Handbooks Online. ISBN 0199284229.

[6] Willard Van Orman Quine (1983). “Chapter 22: Ontology and ideology revisited”. Confessions of a Confirmed Exten- sionalist: And Other Essays. Harvard University Press. pp. 315 ff. ISBN 0674030842.

[7] Of course, this description is not understandable unless one knows what first-order existential quantifiers are and what is meant by saying they are bound. An approachable discussion of these matters is found in Jan Dejnožka (1996). “Chapter 1: Introduction”. The Ontology of the Analytic Tradition and Its Origins: Realism, Possibility, and Identity in Frege, Russell, Wittgenstein, and Quine. Rowman & Littlefield. pp. 1 ff. ISBN 0822630532.

[8] Robert J Fogelin (2004). The Cambridge Companion to Quine. Cambridge University Press. p. 36. ISBN 0521639492.

[9] Frank X Ryan (2004). “Analytic: Analytic/Synthetic”. In John Lachs, Robert B. Talisse, eds. American Philosophy: An Encyclopedia. Psychology Press. pp. 36–39. ISBN 020349279X.

[10] Quine, W. V. (1951). “On Carnap’s views on ontology”. Philosophical Studies 2: 65–72. doi:10.1007/bf02199422. Reprinted in Willard Van Orman Quine (1976). “Chapter 9: On Carnap's views on ontology”. The Ways of Paradox (2nd ed.). Harvard University Press. pp. 203–211. ISBN 0674948378.

[11] Willard Van Orman Quine (1980). “Chapter 2: Two of empiricism”. From a Logical Point of View: Nine Logico-philosophical Essays (2nd ed.). Harvard University Press. pp. 20 'ff. ISBN 0674323513. See this

[12] Jonathan Schaffer (2009). “On What Grounds What Metametaphysics”. In Chalmers, Manley, and Wasserman, eds. Metametaphysics (PDF). Oxford University Press. pp. 347–83. ISBN 0199546045. Reprinted by Philosopher’s Annual 29, eds. Grim, Charlow, Gallow, and Herold; also reprinted in Metaphysics: An Anthology, 2nd edition, eds. Kim, Korman, and Sosa (2011), 73-96: Blackwell.) Contains an analysis of Quine and proposes that questions of existence are not fundamental.

[13] See for example, Hilary Putnam (2001). “The analytic and the synthetic”. In Dagfinn Fllesdal, ed. Philosophy of Quine: General, reviews, and analytic/synthetic, Volume 1. Taylor & Francis. pp. 252 ff. ISBN 0815337388.

[14] Alex Orenstein (1998). “Quine, Willard Van Orman”. In Edward Craig, ed. Routledge Encyclopedia of Philosophy 8. pp. 8 ff. ISBN 0415073103. See also Choice of a theory.

[15] Kaila E Folinsbee; et al. (2007). “Quantitative approaches to phylogenetics; §5.2: Fount of stability and confusion: A synopsis of parsimony in systematics”. In Winfried Henke, ed. Handbook of Paleoanthropology: Primate evolution and human origins: Volume 2. Springer. p. 168. ISBN 3540324747.

[16] Hans-Johann Glock (2004). "§1: Ontological commitment and ontological parsimony”. Quine and Davidson on Language, Thought, and Reality. Cambridge University Press. pp. 41–47. ISBN 1139436732.

[17] Willard Van Quine (1981). Theories and Things (3rd ed.). Harvard University Press. pp. 144 ff. ISBN 0674879260. Cited by Alan Baker.

[18] This section is a slightly modified version of the discussion by Baker, Alan (Feb 25, 2010). Edward N. Zalta, ed, ed. “Simplicity”. The Stanford Encyclopedia of Philosophy (Summer 2011 Edition).

[19] Alex Ornstein (2008). “Quine vs. Quine: Canonical notation, paraphrase, and regimentation”. In Chase B Wrenn, ed. Naturalism, Reference and Ontology: Essays in Honor of Roger F. Gibson. Peter Lang Publishing, Inc. p. 171. ISBN 1433102293.

[20] Marion David (2008).“Quine's Ladder: Two and a half pages from the Philosophy of Logic". In Peter A. French, Howard Wettstein, eds. Midwest Studies in Philosophy, Truth and its Deformities (Volume XXXII). Wiley-Blackwell. pp. 274 ff. ISBN 1405191457.

[21] Mark Colyvan (2001). "§4.2 What is it to be indispensable?". The Indispensability of Mathematics. Oxford University Press. pp. 76 ff. ISBN 0198031440. 92.6. EXTERNAL LINKS 349

[22] (1998). “Meta-ontology” (PDF). Erkenntnis 48: 233–250. doi:10.1023/a:1005323618026.

[23] Peter van Inwagen (2008). “Chapter 6: Quine's 1946 lecture on ”. In Dean Zimmerman, ed. Oxford Studies in Metaphysics : Volume 4. Oxford University Press. pp. 125 ff. ISBN 0191562319. Quine has endorsed several closely related theses that I have referred to, collectively, as his “meta-ontolgy”. These are...those of his theses that pertain to the topic “ontological commitment”or “ontic commitment”.

[24] Heil, J. (2003). From an ontological point of view. Oxford: Oxford University Press.

[25] Shaffer, Jonathan. “Truthmaker Commitments” (PDF).

[26] Azzouni, Jody (2004). Deflating Existential Consequence: A Case for Nominalism. Oxford: Oxford University Press.

[27] Peacock, Howard (2011). “Two Kinds of Ontological Commitment”. The Philosophical Quarterly 61 (242): 79–104. doi:10.1111/j.1467-9213.2010.665.x.

92.6 External links

• Ontological Commitment entry by Phillip Bricker in the Stanford Encyclopedia of Philosophy

• Ontological commitment, a category in Philosophical Papers maintained by Henry Laycock Chapter 93

Original proof of Gödel's completeness theorem

The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 (and a rewritten version of the dissertation, published as an article in 1930) is not easy to read today; it uses concepts and formalism that are no longer used and terminology that is often obscure. The version given below attempts to represent all the steps in the proof and all the important ideas faithfully, while restating the proof in the modern language of mathematical logic. This outline should not be considered a rigorous proof of the theorem.

93.1 Definitions and assumptions

We work with first-order predicate calculus. Our languages allow constant, function and relation symbols. Structures consist of (non-empty) domains and interpretations of the relevant symbols as constant members, functions or relations over that domain. We fix some axiomatization of the predicate calculus: logical axioms and rules of inference. Any of the several well- known axiomatisations will do; we assume without proof all the basic well-known results about our formalism (such as the normal form theorem or the soundness theorem) that we need. We axiomatize predicate calculus without equality, i.e. there are no special axioms expressing the properties of equality as a special relation symbol. After the basic form of the theorem is proved, it will be easy to extend it to the case of predicate calculus with equality.

93.2 Statement of the theorem and its proof

In the following, we state two equivalent forms of the theorem, and show their equivalence. Later, we prove the theorem. This is done in the following steps:

1. Reducing the theorem to sentences (formulas with no free variables) in prenex form, i.e. with all quantifiers (∀ and ∃) at the beginning. Furthermore, we reduce it to formulas whose first quantifier is ∀. This is possible because for every sentence, there is an equivalent one in prenex form whose first quantifier is ∀.

2. Reducing the theorem to sentences of the form ∀x1∀x2...∀xk ∃y1∃y2...∃ym φ(x1...xk, y1...ym). While we cannot do this by simply rearranging the quantifiers, we show that it is yet enough to prove the theorem for sentences of that form. 3. Finally we prove the theorem for sentences of that form.

• This is done by first noting that a sentence such as B = ∃x1∃x2...∃xk ∃y1∃y2...∃ym φ(x1...xk, y1...ym) is either refutable or has some model in which it holds; this model is simply assigning truth values to the subpropositions from which B is built. The reason for that is the completeness of propositional logic, with the existential quantifiers playing no role.

350 93.2. STATEMENT OF THE THEOREM AND ITS PROOF 351

• We extend this result to more and more complex and lengthy sentences, Dn (n=1,2...), built out from B, so that either any of them is refutable and therefore so is φ, or all of them are not refutable and therefore each holds in some model.

• We finally use the models in which the Dn hold (in case all are not refutable) in order to build a model in which φ holds.

93.2.1 Theorem 1. Every valid formula (true in all structures) is provable.

This is the most basic form of the completeness theorem. We immediately restate it in a form more convenient for our purposes:

93.2.2 Theorem 2. Every formula φ is either refutable or satisfiable in some structure.

"φ is refutable”means by definition "¬φ is provable”.

93.2.3 Equivalence of both theorems

To see the equivalence, note first that if Theorem 1 holds, and φ is not satisfiable in any structure, then ¬φ is valid in all structures and therefore provable, thus φ is refutable and Theorem 2 holds. If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬φ is not satisfiable in any structure and therefore refutable; then ¬¬φ is provable and then so is φ, thus Theorem 1 holds.

93.2.4 Proof of theorem 2: first step

We approach the proof of Theorem 2 by successively restricting the class of all formulas φ for which we need to prove "φ is either refutable or satisfiable”. At the beginning we need to prove this for all possible formulas φ in our language. However, suppose that for every formula φ there is some formula ψ taken from a more restricted class of formulas C, such that "ψ is either refutable or satisfiable”→ "φ is either refutable or satisfiable”. Then, once this claim (expressed in the previous sentence) is proved, it will suffice to prove "φ is either refutable or satisfiable” only for φ's belonging to the class C. Note also that if φ is provably equivalent to ψ (i.e., (φ≡ψ) is provable), then it is indeed the case that "ψ is either refutable or satisfiable”→ "φ is either refutable or satisfiable”(the soundness theorem is needed to show this). There are standard techniques for rewriting an arbitrary formula into one that does not use function or constant symbols, at the cost of introducing additional quantifiers; we will therefore assume that all formulas are free of such symbols. Gödel's paper uses a version of first-order predicate calculus that has no function or constant symbols to begin with. Next we consider a generic formula φ (which no longer uses function or constant symbols) and apply the prenex form theorem to find a formula ψ in normal form such that φ≡ψ (ψ being in normal form means that all the quantifiers in ψ, if there are any, are found at the very beginning of ψ). It follows now that we need only prove Theorem 2 for formulas φ in normal form.

Next, we eliminate all free variables from φ by quantifying them existentially: if, say, x1...xn are free in φ, we form ψ = ∃x1...∃xnϕ . If ψ is satisfiable in a structure M, then certainly so is φ and if ψ is refutable, then ¬ψ = ∀x1...∀xn¬ϕ is provable, and then so is ¬φ, thus φ is refutable. We see that we can restrict φ to be a sentence, that is, a formula with no free variables. Finally, we would like, for reasons of technical convenience, that the prefix of φ (that is, the string of quantifiers at the beginning of φ, which is in normal form) begin with a universal quantifier and end with an existential quantifier. To achieve this for a generic φ (subject to restrictions we have already proved), we take some one-place relation symbol F unused in φ, and two new variables y and z.. If φ = (P)Φ, where (P) stands for the prefix of φ and Φ for the matrix (the remaining, quantifier-free part of φ) we form ψ = ∀y(P )∃z(Φ∧[F (y)∨¬F (z)]) . Since ∀y∃z(F (y)∨¬F (z)) is clearly provable, it is easy to see that ϕ = ψ is provable. 352 CHAPTER 93. ORIGINAL PROOF OF GÖDEL'S COMPLETENESS THEOREM

93.2.5 Reducing the theorem to formulas of degree 1

Our generic formula φ now is a sentence, in normal form, and its prefix starts with a universal quantifier and ends with an existential quantifier. Let us call the class of all such formulas R. We are faced with proving that every formula in R is either refutable or satisfiable. Given our formula φ, we group strings of quantifiers of one kind together in blocks:

∀ ∀ ∃ ∃ ∀ ∀ ∃ ∃ ϕ = ( x1... xk1 )( xk1+1... xk2 )...... ( xkn−2+1... xkn−1 )( xkn−1+1... xkn )(Φ) We define the degree of ϕ to be the number of universal quantifier blocks, separated by existential quantifier blocks as shown above, in the prefix of ϕ . The following lemma, which Gödel adapted from Skolem's proof of the Löwenheim- Skolem theorem, lets us sharply reduce the complexity of the generic formula ϕ we need to prove the theorem for: Lemma. Let k>=1. If every formula in R of degree k is either refutable or satisfiable, then so is every formula in R of degree k+1.

Comment: Take a formula φ of degree k+1 of the form ϕ = (∀x)(∃y)(∀u)(∃v)(P )ψ , where (P )ψ is the remainder of ϕ (it is thus of degree k-1). φ states that for every x there is a y such that... (something). It would have been nice to have a predicate Q' so that for every x, Q'(x,y) would be true if and only if y is the required one to make (something) true. Then we could have written a formula of degree k, which is equivalent to φ, namely (∀x′)(∀x)(∀y)(∀u)(∃v)(∃y′)(P )Q′(x′, y′) ∧ (Q′(x, y) → ψ) . This formula is indeed equivalent to φ because it states that for every x, if there is a y thatsatisfies Q'(x,y), then (something) holds, and furthermore, we know that there is such a y, because for every x', there is a y' that satisfies Q'(x',y'). Therefore φ follows from this formula. It is also easy to show that if the formula is false, then so is φ. Unfortunately, in general there is no such predicate Q'. However, this idea can be understood as a basis for the following proof of the Lemma.

Proof. Let φ be a formula of degree k+1; then we can write it as

ϕ = (∀x)(∃y)(∀u)(∃v)(P )ψ where (P) is the remainder of the prefix of ϕ (it is thus of degree k-1) and ψ is the quantifier-free matrix of ϕ . x, y, u and v denote here tuples of variables rather than single variables; e.g. (∀x) really stands for ∀x1∀x2...∀xn where x1...xn are some distinct variables. Let now x' and y' be tuples of previously unused variables of the same length as x and y respectively, and let Q be a previously unused relation symbol that takes as many arguments as the sum of lengths of x and y; we consider the formula

Φ = (∀x′)(∃y′)Q(x′, y′) ∧ (∀x)(∀y)(Q(x, y) → (∀u)(∃v)(P )ψ)

Clearly, Φ → ϕ is provable. Now since the string of quantifiers (∀u)(∃v)(P ) does not contain variables from x or y, the following equivalence is easily provable with the help of whatever formalism we're using:

(Q(x, y) → (∀u)(∃v)(P )ψ) ≡ (∀u)(∃v)(P )(Q(x, y) → ψ)

And since these two formulas are equivalent, if we replace the first with the second inside Φ, we obtain the formula Φ' such that Φ≡Φ':

Φ′ = (∀x′)(∃y′)Q(x′, y′) ∧ (∀x)(∀y)(∀u)(∃v)(P )(Q(x, y) → ψ)

Now Φ' has the form (S)ρ ∧ (S′)ρ′ , where (S) and (S') are some quantifier strings, ρ and ρ' are quantifier-free, and, furthermore, no variable of (S) occurs in ρ' and no variable of (S') occurs in ρ. Under such conditions every formula 93.2. STATEMENT OF THE THEOREM AND ITS PROOF 353

of the form (T )(ρ∧ρ′) , where (T) is a string of quantifiers containing all quantifiers in (S) and (S') interleaved among themselves in any fashion, but maintaining the relative order inside (S) and (S'), will be equivalent to the original formula Φ'(this is yet another basic result in first-order predicate calculus that we rely on). To wit, we form Ψ as follows:

Ψ = (∀x′)(∀x)(∀y)(∀u)(∃y′)(∃v)(P )Q(x′, y′) ∧ (Q(x, y) → ψ) and we have Φ′ ≡ Ψ . Now Ψ is a formula of degree k and therefore by assumption either refutable or satisfiable. If Ψ is satisfiable in a structure M, then, considering Ψ ≡ Φ′ ≡ Φ ∧ Φ → ϕ , we see that ϕ is satisfiable as well. If Ψ is refutable, then so is Φ , which is equivalent to it; thus ¬Φ is provable. Now we can replace all occurrences of Q inside the provable formula ¬Φ by some other formula dependent on the same variables, and we will still get a provable formula. (This is yet another basic result of first-order predicate calculus. Depending on the particular formalism adopted for the calculus, it may be seen as a simple application of a “functional substitution”rule of inference, as in Gödel's paper, or it may be proved by considering the formal proof of ¬Φ , replacing in it all occurrences of Q by some other formula with the same free variables, and noting that all logical axioms in the formal proof remain logical axioms after the substitution, and all rules of inference still apply in the same way.) In this particular case, we replace Q(x',y') in ¬Φ with the formula (∀u)(∃v)(P )ψ(x, y|x′, y′) . Here (x,y|x',y') means that instead of ψ we are writing a different formula, in which x and y are replaced with x' and y'. Note that Q(x,y) is simply replaced by (∀u)(∃v)(P )ψ . ¬Φ then becomes

¬((∀x′)(∃y′)(∀u)(∃v)(P )ψ(x, y|x′, y′) ∧ (∀x)(∀y)((∀u)(∃v)(P )ψ → (∀u)(∃v)(P )ψ)) and this formula is provable; since the part under negation and after the ∧ sign is obviously provable, and the part under negation and before the ∧ sign is obviously φ, just with x and y replaced by x' and y', we see that ¬ϕ is provable, and φ is refutable. We have proved that φ is either satisfiable or refutable, and this concludes the proof of the Lemma. Notice that we could not have used (∀u)(∃v)(P )ψ(x, y|x′, y′) instead of Q(x',y') from the beginning, because Ψ would not have been a well-formed formula in that case. This is why we cannot naively use the argument appearing at the comment that precedes the proof.

93.2.6 Proving the theorem for formulas of degree 1

As shown by the Lemma above, we only need to prove our theorem for formulas φ in R of degree 1. φ cannot be of degree 0, since formulas in R have no free variables and don't use constant symbols. So the formula φ has the general form:

(∀x1...xk)(∃y1...ym)ϕ(x1...xk, y1...ym).

Now we define an ordering of the k-tuples of natural numbers as follows: (x1...xk) < (y1...yk) should hold if either Σk(x1...xk) < Σk(y1...yk) , or Σk(x1...xk) = Σk(y1...yk) , and (x1...xk) precedes (y1...yk) in lexicographic n n order. [Here Σk(x1...xk) denotes the sum of the terms of the tuple.] Denote the nth tuple in this order by (a1 ...ak ) .

Set the formula B as ϕ(z n ...z n , z − , z − ...z ) . Then put D as n a1 ak (n 1)m+2 (n 1)m+3 nm+1 n

(∃z1...znm+1)(B1 ∧ B2... ∧ Bn).

Lemma: For every n, φ → Dn .

Proof: By induction on n; we have Dk ⇐ Dk−1 ∧ (∀z1...z(n−1)m+1)(∃z(n−1)m+2...znm+1)Bn ⇐ Dk−1 ∧ (∀z n ...z n )(∃y ...y )ϕ(z n ...z n , y ...y ) , where the latter implication holds by variable substitution, since the a1 ak 1 m a1 ak 1 m ∀ n n − ∧ ordering of the tuples is such that ( k)(a1 ...ak ) < (n 1)m + 2 . But the last formula is equivalent to Dk−1 φ. 354 CHAPTER 93. ORIGINAL PROOF OF GÖDEL'S COMPLETENESS THEOREM

For the base case, D1 ≡ (∃z1...zm+1)ϕ(z 1 ...z 1 , z2, z3...zm+1) ≡ (∃z1...zm+1)ϕ(z1...z1, z2, z3...zm+1) is ob- a1 ak viously a corollary of φ as well. So the Lemma is proven.

Now if Dn is refutable for some n, it follows that φ is refutable. On the other hand, suppose that Dn is not refutable for any n. Then for each n there is some way of assigning truth values to the distinct subpropositions Eh (ordered by their first appearance in Dn ; “distinct”here means either distinct predicates, or distinct bound variables) in Bk , such that Dn will be true when each proposition is evaluated in this fashion. This follows from the completeness of the underlying propositional logic.

We will now show that there is such an assignment of truth values to Eh , so that all Dn will be true: The Eh appear in the same order in every Dn ; we will inductively define a general assignment to them by a sort of“majority vote": Since there are infinitely many assignments (one for each Dn ) affecting E1 , either infinitely many make E1 true, or infinitely many make it false and only finitely many make it true. In the former case, we choose E1 to be true in general; in the latter we take it to be false in general. Then from the infinitely many n for which E1 through Eh−1 are assigned the same truth value as in the general assignment, we pick a general assignment to Eh in the same fashion.

This general assignment must lead to every one of the Bk and Dk being true, since if one of the Bk were false under the general assignment, Dn would also be false for every n > k. But this contradicts the fact that for the finite collection of general Eh assignments appearing in Dk , there are infinitely many n where the assignment making Dn true matches the general assignment.

From this general assignment, which makes all of the Dk true, we construct an interpretation of the language's predicates that makes φ true. The universe of the model will be the natural numbers. Each i-ary predicate Ψ should

be true of the naturals (u1...ui) precisely when the proposition Ψ(zu1 ...zui ) is either true in the general assignment, or not assigned by it (because it never appears in any of the Dk ). ∃ n n In this model, each of the formulas ( y1...ym)ϕ(a1 ...ak , y1...ym) is true by construction. But this implies that φ itself is true in the model, since the an range over all possible k-tuples of natural numbers. So φ is satisfiable, and we are done.

Intuitive explanation

We may write each Bᵢ as Φ(x1...xk,y1...ym) for some x-s, which we may call “first arguments”and y-s that we may call “last arguments”.

Take B1 for example. Its“last arguments”are z2,z3...zm+1, and for every possible combination of k of these variables there is some j so that they appear as “first arguments”in Bⱼ. Thus for large enough n1,Dn1 has the property that the“last arguments”of B1 appear, in every possible combinations of k of them, as“first arguments”in other Bⱼ-s within Dn. For every Bᵢ there is a Dnᵢ with the corresponding property.

Therefore in a model that satisfies all the Dn-s, there are objects corresponding to z1, z2... and each combination of k of these appear as“first arguments”in some Bⱼ, meaning that for every k of these objects zp1 ...zpk there are zq1 ...zqm , which makes Φ(zp1 ...zpk ,zq1 ...zqm ) satisfied. By taking a submodel with only these z1, z2... objects, we have a model satisfying φ.

93.3 Extensions

93.3.1 Extension to first-order predicate calculus with equality

Gödel reduced a formula containing instances of the equality predicate to ones without it in an extended language. His method involves replacing a formula φ containing some instances of equality with the formula

(∀x)Eq(x, x)∧(∀x, y, z)[Eq(x, y) → (Eq(x, z) → Eq(y, z))] ∧(∀x, y, z)[Eq(x, y) → (Eq(z, x) → Eq(z, y))] ∧(∀x1...xk, y1...xk)[(Eq(x1, y1) ∧ ... ∧ Eq(xk, yk)) → (A(x1...xk) ≡ A(y1...yk))] ′ ∧... ∧ (∀x1...xm, y1...xm)[(Eq(x1, y1) ∧ ... ∧ Eq(xm, ym)) → (Z(x1...xm) ≡ Z(y1...ym))] ∧φ .

Here A...Z denote the predicates appearing in φ (with k...m their respective arities), and φ' is the formula φ with all occurrences of equality replaced with the new predicate Eq. If this new formula is refutable, the original φ was as well; the same is true of satisfiability, since we may take a quotient of satisfying model of the new formula by the 93.4. REFERENCES 355

equivalence relation representing Eq. This quotient is well-defined with respect to the other predicates, and therefore will satisfy the original formula φ.

93.3.2 Extension to countable sets of formulas

Gödel also considered the case where there are a countably infinite collection of formulas. Using the same reductions as above, he was able to consider only those cases where each formula is of degree 1 and contains no uses of equality. i i For a countable collection of formulas ϕ of degree 1, we may define Bk as above; then define Dk to be the closure 1 1 k k of B1 ...Bk, ..., B1 ...Bk . The remainder of the proof then went through as before.

93.3.3 Extension to arbitrary sets of formulas

When there is an uncountably infinite collection of formulas, the Axiom of Choice (or at least some weak form of it) is needed. Using the full AC, one can well-order the formulas, and prove the uncountable case with the same argument as the countable one, except with transfinite induction. Other approaches can be used to prove that the completeness theorem in this case is equivalent to the Boolean prime ideal theorem, a weak form of AC.

93.4 References

• Gödel, K (1929). "Über die Vollständigkeit des Logikkalküls”. Doctoral dissertation. . The first proof of the completeness theorem.

• Gödel, K (1930). “Die Vollständigkeit der Axiome des logischen Funktionenkalküls”. Monatshefte für Mathematik (in German) 37 (1): 349–360. doi:10.1007/BF01696781. JFM 56.0046.04. The same material as the dissertation, except with briefer proofs, more succinct explanations, and omitting the lengthy introduction.

93.5 External links

• Stanford Encyclopedia of Philosophy:"Kurt Gödel"—by Juliette Kennedy.

• MacTutor biography: Kurt Gödel. Chapter 94

Ostensive definition

An ostensive definition conveys the meaning of a term by pointing out examples. This type of definition is often used where the term is difficult to define verbally, either because the words will not be understood (as with children and new speakers of a language) or because of the nature of the term (such as colors or sensations). It is usually accompanied with a gesture pointing out the object serving as an example, and for this reason is also often referred to as "definition by pointing". Ostensive definitions rely on an analogical or case-based reasoning by the subject they are intended to educate or inform. For example, defining red by pointing out red objects—apples, stop signs, roses—is giving ostensive definition, as is naming. Ostensive definition assumes the questioner has sufficient understanding to recognize the type of information being given. Ludwig Wittgenstein writes: So one might say: the ostensive definition explains the use—the meaning—of the word when the overall role of the word in language is clear. Thus if I know that someone means to explain a colour- word to me the ostensive definition “That is called 'sepia' " will help me to understand the word.... One has already to know (or be able to do) something in order to be capable of asking a thing's name. But what does one have to know?*[1] The limitations of ostensive definition are exploited in a famous argument from the Philosophical Investigations (which deal primarily with the philosophy of language), the private language argument, in which Wittgenstein asks if it is possible to have a private language that no one else can understand.*[2] John Passmore states that the term was first defined by the British logician William Ernest Johnson (1858–1931):

“His neologisms, as rarely happens, have won wide acceptance: such phrases as“ostensive definition” , such contrasts as those between ...“determinates”and“determinables”“, continuants”and“occurrents” , are now familiar in philosophical literature”(Passmore 1966, p. 344).

94.1 Ostension in folklore

The term ostension is also used by those who study folklore and urban legends to indicate real-life happenings that parallel the events told in pre-existing and well-established legends and lore. Semiotician Umberto Eco was the first to use the term to describe the way in which people communicate messages through miming actions, as by holding up a pack of cigarettes to say,“Would you like one?"*[3] The concept was applied to contemporary legends by folklorists Linda Dégh and Andrew Vázsonyi, who argued that the most direct form of ostension involved committing an actual crime mentioned in a well-known urban legend, such as microwaving someone's pet animal or placing poison in a child's Halloween candy. While such events are rare, the authors stressed that folklorists must recognize “that fact can become narrative and narrative can become fact.”*[4]*:29 Dégh and Vázsonyi, followed by other analysts, argued that there were two other forms of ostension that did not necessarily involve literal acting out of legends. Quasi-ostension involves interpretation of ambiguous events in terms of a legend, as when a murder is first believed to have been a “cult”sacrifice or “gang”murder when in fact the perpetrator had other motives. Many local media panics are based in this form of ostension.*[5] Pseudo-ostension involves legend-like events intentionally acted out by persons aware of the original narrative. For example, in 1991, Ebony published a letter written by “C.J.”a Dallas-area woman who said she was HIV-positive,

356 94.2. SEE ALSO 357

but intentionally having sex with as many men as possible. Soon after, a local radio talk-show broadcast a phone call from a woman who said she was the real“C.J.”“I blame it on men, period,”she said to the talk-show host. “I'm doing it to all the men because it was a man that gave it to me.”After a huge spike in males seeking HIV screening in the Dallas-Fort-Worth area, both the author of the letter and the talk-show caller were identified as hoaxers intending to raise consciousness of the disease.*[6] Ostension has become an important concept for folklorists studying the ways in which folklore affects everyday people's real lives, ranging from supernatural rituals such as legend tripping to the complex ways in which awareness of AIDS has affected people's sexual habits.*[7] Folklorist John McDowell, in an article preceding Dégh and Vázsonyi by a year, explored the relationship between iconicity—representation—and ostension—presentation—in mythic narrative, finding in episodes of ostention a vitual encounter with the experiential substrate, an experience that he termed “narrative epiphany.”*[8]

94.2 See also

• Archetype

• Definition

• Enumerative definition

• Extensional definition

• Intensional definition

• Comprehension

• Extension

• Intension

• Exemplification

• Legend tripping

94.3 Notes

[1] Wittgenstein, Ludwig. Philosophical Investigations, §30.

[2] Wittgenstein, Ludwig. Philosophical Investigations, §258.

[3] Eco, Umberto. A Theory of Semiotics. 224-26.

[4] Dégh, Linda; Vázsonyi, Andrew (1983). “Does the Word 'Dog' Bite? Ostensive Action: A Means of Legend Telling”. Journal of Folklore Research 20: 5–34.

[5] Ellis, Bill. Legend-Trips and Satanism: Adolescents' Ostensive Traditions as “Cult”Activity. In Richardson et al., The Satanism Scare, pp. 279-295.

[6] FOAFTale News 25 (March 1992): 11.

[7] Goldstein, Diane. Once upon a Virus (2004).

[8]“Beyond Iconicity: Ostension in Kamsa Mythic Narrative, Journal of the Folklore Institute 19 (1982): 119-139. 358 CHAPTER 94. OSTENSIVE

94.4 References

• Passmore, John (1966). A Hundred Years of Philosophy (2nd ed.). London: Penguin (1957).

• Dégh, Linda; Andrew Vázsonyi (1983). “Does the Word 'Dog' Bite? Ostensive Action: A Means of Legend Telling”. Journal of Folklore Research 20: 5–34.

• Eco, Umberto (1976). A Theory of Semiotics. Indiana University Press. ISBN 0-253-20217-5.

• Goldstein, Diane E (2004). Once upon a Virus: Aids Legends and Vernacular Risk Perception. Logan, UT: Utah State University Press. ISBN 0-87421-587-0.

• Richardson, James T.; Best, Joel; Bromley, David (1991). “The Satanism Scare”. New York: Aldine de Gruyter. ISBN 0-202-30379-9.

• Wittgenstein, Ludwig (2001) [1953]. Philosophical Investigations. Blackwell Publishing. ISBN 0-631-23127- 7.

• William Van Orman Quine (1974). The Roots of Reference. La Salle, Illinois: Open Court Publishing Co. (in particular Sect.11) Chapter 95

Outline of logic

The following outline is provided as an overview of and topical guide to logic: Logic – formal science of using reason and is considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and through the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving . One of the aims of logic is to identify the correct (or valid) and incorrect (or fallacious) inferences. Logicians study the criteria for the evaluation of arguments.

95.1 Foundations of logic

Main article: Philosophy of logic

• Analytic-synthetic distinction • Antinomy • A priori and a posteriori • Definition • Description • Entailment • Identity (philosophy) • Inference • Logical form • Logical implication • Logical truth • Logical consequence • Name • Necessity • Material conditional • Meaning (linguistic) • Meaning (non-linguistic)

359 360 CHAPTER 95. OUTLINE OF LOGIC

• Paradox (list)

• Possible world

• Presupposition • Probability

• Quantification • Reason

• Reasoning

• Reference • Semantics

• Strict conditional

• Syntax (logic) • Truth

• Truth value • Validity

95.2 Philosophical logic

Philosophical logic –

95.2.1 Informal logic and critical thinking

Informal logic – Critical thinking – Argumentation theory –

• Argument –

• Argument map – • Accuracy and precision –

• Ad hoc hypothesis –

• Ambiguity – • Analysis –

• Attacking Faulty Reasoning –

• Belief – • Belief bias –

• Bias – • Cogency –

• Cognitive bias –

• Confirmation bias – • Credibility –

• Critical pedagogy – 95.2. PHILOSOPHICAL LOGIC 361

• Critical reading –

• Decidophobia –

• Decision making –

• Dispositional and occurrent belief –

• Emotional reasoning –

• Evidence –

• Expert –

• Explanation –

• Explanatory power –

• Fact –

• Fallacy –

• Higher-order thinking –

• Inquiry –

• Interpretive discussion –

• Narrative logic –

• Occam's razor –

• Opinion –

• Practical syllogism –

• Precision questioning –

• Propaganda –

• Propaganda techniques –

• Prudence –

• Pseudophilosophy –

• Reasoning –

• Relevance –

• Rhetoric –

• Rigour –

• Socratic questioning –

• Source credibility –

• Source criticism –

• Theory of justification –

• Topical logic –

• Vagueness – 362 CHAPTER 95. OUTLINE OF LOGIC

95.2.2 Deductive reasoning

Theories of deduction

• Anti-psychologism

• Conceptualism

• Constructivism

• Conventionalism

• Counterpart theory

• Deflationary theory of truth

• Dialetheism

• Fictionalism

• Formalism (philosophy)

• Game theory

• Illuminationist philosophy

• Logical atomism

• Logical holism

• Logicism

• Modal fictionalism

• Nominalism

• Object theory

• Polylogism

• Pragmatism

• Preintuitionism

• Proof theory

• Psychologism

• Ramism

• Semantic theory of truth

• Sophism

• Trivialism

• Ultrafinitism

95.2.3 Fallacies

• Fallacy (list) – incorrect argumentation in reasoning resulting in a misconception or presumption. By or design, fallacies may exploit emotional triggers in the listener or interlocutor (), or take advantage of social relationships between people (e.g. argument from authority). Fallacious arguments are often structured using rhetorical patterns that obscure any logical argument. Fallacies can be used to win arguments regardless of the merits. There are dozens of types of fallacies. 95.3. FORMAL LOGIC 363

95.3 Formal logic

• Formal logic – Mathematical logic, symbolic logic and formal logic are largely, if not completely synonymous. The essential feature of this field is the use of formal languages to express the ideas whose logical validity is being studied. • List of mathematical logic topics

95.3.1 Symbols and strings of symbols

Logical symbols

Main articles: Table of logic symbols and Symbol (formal)

• Logical variables • Propositional variable • Predicate variable • Literal • Metavariable • Logical constants • • Quantifier • Identity • Brackets

Logical connectives Logical connective –

• Converse implication – • Converse nonimplication – • Exclusive or – • Logical NOR – • Logical biconditional – • Logical conjunction – • Logical disjunction – • Material implication – • Material nonimplication – • Negation – • Sheffer stroke –

Strings of symbols

Main article: Well-formed formula

• Atomic formula • Open sentence 364 CHAPTER 95. OUTLINE OF LOGIC

Types of propositions

Main article: Proposition

• Analytic proposition • Axiom • Atomic sentence • Clause (logic) • Contingent proposition • Contradiction • Logical truth • • Rule of inference • Sentence (mathematical logic) • Sequent • Statement (logic) • Tautology • Theorem

Rules of inference Rule of inference (list)

• Biconditional elimination • Biconditional introduction • Case analysis • Commutativity of conjunction • Conjunction introduction • Constructive dilemma • Contraposition (traditional logic) • Conversion (logic) • De Morgan's laws • • Disjunction elimination • Disjunction introduction • Disjunctive syllogism • Double negative elimination • Generalization (logic) • Hypothetical syllogism • Law of excluded middle 95.3. FORMAL LOGIC 365

• Law of identity

• Modus ponendo tollens

• Modus ponens

• Modus tollens

• Obversion

• Principle of contradiction

• Resolution (logic)

• Simplification

• (logic)

Formal theories

Main article: Theory (mathematical logic)

• Formal proof

• List of first-order theories

Expressions in an object language

Main article: Object language

• Symbol

• Formula

• Formal system

• Theorem

• Formal proof

• Theory

Expressions in a metalanguage

Main article: Metalanguage

• Metalinguistic variable

• Deductive system

• Metatheorem

• Metatheory

• Interpretation 366 CHAPTER 95. OUTLINE OF LOGIC

95.3.2 Propositional and boolean logic

Propositional logic

Main article: Propositional logic

• Absorption law • Clause (logic) • Deductive closure • Entailment • Formation rule • • Intermediate logic • Literal (mathematical logic) • Logical connective • Logical consequence • Negation normal form • Open sentence • Propositional calculus • Propositional formula • Propositional variable • Rule of inference • Strict conditional • Substitution instance • Truth table • Zeroth-order logic

Boolean logic

• Boolean algebra (list) • Boolean logic • Boolean algebra (structure) • Boolean algebras canonically defined • Introduction to Boolean algebra • Complete Boolean algebra • Free Boolean algebra • Monadic Boolean algebra • Residuated Boolean algebra • Two-element Boolean algebra 95.3. FORMAL LOGIC 367

• Modal algebra

• Derivative algebra (abstract algebra)

• Relation algebra

• Absorption law

• Laws of Form

• De Morgan's laws

• Algebraic normal form

• Canonical form (Boolean algebra)

• Boolean conjunctive query

• Boolean-valued model

• Boolean domain

• Boolean expression

• Boolean ring

• Boolean function

• Boolean-valued function

• Parity function

• Symmetric Boolean function

• Conditioned disjunction

• Field of sets

• Functional completeness

• Implicant

•

• Logic redundancy

• Logical connective

• Logical matrix

• Minimal negation operator

• Product term

• True quantified Boolean formula

• Truth table 368 CHAPTER 95. OUTLINE OF LOGIC

95.3.3 Predicate logic and relations

Predicate logic

Main article: Predicate logic

• Atomic formula • Atomic sentence • Domain of discourse • Empty domain • Extension (predicate logic) • First-order logic • First-order predicate • Formation rule • Free variables and bound variables • Generalization (logic) • Monadic predicate calculus • Predicate (mathematical logic) • Predicate logic • Predicate variable • Quantification • Second-order predicate • Sentence (mathematical logic) • Universal instantiation • (ε, δ)-definition of limit

Relations

Main article: Mathematical relation

• Finitary relation • Antisymmetric relation • Asymmetric relation • Bijection • Bijection, injection and surjection • Binary relation • Composition of relations • Concurrent relation • relation 95.4. MATHEMATICAL LOGIC 369

• Coreflexive relation

• Covering relation

• Cyclic order

• Dense relation

• Dependence relation

• Dependency relation

• Directed set

• Equivalence relation

• Euclidean relation

• Homogeneous relation

• Idempotence

• Intransitivity

• Inverse relation

• Involutive relation

• Partial equivalence relation

• Partial function

• Partially ordered set

• Preorder

• Prewellordering

• Propositional function

• Quasitransitive relation

• Reflexive relation

• Surjective function

• Symmetric relation

• Ternary relation

• Total relation

• Transitive relation

• Trichotomy (mathematics)

• Well-founded relation

95.4 Mathematical logic

Mathematical logic – 370 CHAPTER 95. OUTLINE OF LOGIC

95.4.1 Set theory

Set theory (list)–

• Aleph null • Bijection, injection and surjection • Binary set • Cantor's diagonal argument • Cantor's first uncountability proof • Cantor's theorem • Cardinality of the continuum • Cardinal number • Codomain • Complement (set theory) • Continuum hypothesis • • Decidable set • Denumerable set • Disjoint sets • Disjoint union • Domain of a function • Effective enumeration • Element (mathematics) • Empty function • Empty set • Enumeration • Extensionality • Finite set • Function (mathematics) • Function composition • Generalized continuum hypothesis • Index set • Infinite set • Intension • Intersection (set theory) • • Löwenheim–Skolem theorem 95.4. MATHEMATICAL LOGIC 371

• Map (mathematics)

• Multiset

• Naïve set theory

• Non-Cantorian set theory

• One to one correspondence

• Ordered pair

• Partition of a set

• Pointed set

• Power set

• Projection (set theory)

• Proper subset

• Proper superset

• Range (mathematics)

• Russell's paradox

• Sequence (mathematics)

• Set (mathematics)

• Set of all sets

• Simple theorems in the algebra of sets

• Singleton (mathematics)

• Skolem paradox

• Subset

• Superset

• Tuple

•

• Union (set theory)

• Zermelo–Fraenkel set theory

95.4.2 Metalogic

Metalogic – The study of the metatheory of logic.

• Completeness (logic)

• Syntax (logic)

• Consistency

• Decidability (logic)

• Deductive system

• Interpretation (logic) 372 CHAPTER 95. OUTLINE OF LOGIC

• Cantor's theorem

• Church's theorem

• Church's thesis

• Effective method

• Formal system

• Gödel's completeness theorem

• Gödel's first incompleteness theorem

• Gödel's second incompleteness theorem

• Independence (mathematical logic)

• Logical consequence

• Löwenheim-Skolem theorem

• Metalanguage

• Metasyntactic variable

• Metatheorem

• Object language

• Symbol (formal)

• Type-token distinction

• Use–mention distinction

• Well-formed formula

Proof theory

Proof theory – The study of deductive apparatus.

• Axiom

• Deductive system

• Formal proof

• Formal system

• Formal theorem

• Syntactic consequence

• Syntax (logic)

• Transformation rules 95.4. MATHEMATICAL LOGIC 373

Model theory

Model theory – The study of interpretation of formal systems.

• Interpretation (logic) • Logical validity • Non-standard model • Normal model • Model • Semantic consequence • Truth value

95.4.3 Computability theory

Computability theory – branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability. The basic questions addressed by recursion theory are “What does it mean for a function from the natural numbers to themselves to be computable?" and “How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?". The answers to these questions have led to a rich theory that is still being actively researched.

• Alpha recursion theory • Arithmetical set • Church–Turing thesis • Computability logic • Computable function • Computation • Decision problem • Effective method • Entscheidungsproblem • Enumeration • Forcing (recursion theory) • Halting problem • History of the Church–Turing thesis • Lambda calculus • List of undecidable problems • Post correspondence problem • Post's theorem • Primitive recursive function • Recursion (computer science) • Recursive language 374 CHAPTER 95. OUTLINE OF LOGIC

• Recursive languages and sets

• Recursive set

• Recursively enumerable language

• Recursively enumerable set

• Reduction (recursion theory)

• Turing machine

95.5 Classical logic

Classical logic –

• Properties of classical logics:

• Law of the excluded middle • Double negative elimination • Law of noncontradiction • Principle of explosion • Monotonicity of entailment • Idempotency of entailment • Commutativity of conjunction • De Morgan duality – every logical operator is dual to another

• Term logic

• General concepts in classical logic

• Baralipton • Baroco • Bivalence • Boolean logic • Boolean-valued function • Categorical proposition • Distribution of terms • End term • Enthymeme • Immediate inference • Law of contraries • Logical connective • Major term • Middle term • Minor term • Organon • Polysyllogism • Port-Royal Logic • Premise • Prior Analytics 95.6. NON-CLASSICAL LOGIC 375

• Relative term • Sorites paradox • Square of opposition • Sum of Logic • Syllogism • Tetralemma • Truth function

95.6 Non-classical logic

Non-classical logic – Deviant logic –

• Computability logic –

• Fuzzy logic –

• Linear logic –

• Decision theory –

• Game theory –

• Probability theory –

• Affine logic –

• Bunched logic –

• Description logic –

• Free logic –

• Intensional logic –

• Intuitionistic logic –

• Many-valued logic –

• Minimal logic –

• Noncommutative logic –

• Non-monotonic logic –

• Paraconsistent logic –

• Quantum logic –

• Relevance logic –

• Strict logic –

• Substructural logic – 376 CHAPTER 95. OUTLINE OF LOGIC

95.6.1 Modal logic

Modal logic –

• Alethic logic – • Axiological logic – • Deontic logic – • – • Epistemic logic – • Temporal logic –

95.7 Concepts of logic

• Deductive reasoning – • Inductive reasoning – • Abductive reasoning –

Mathematical logic –

• Proof theory – • Set theory – • Formal system – • Predicate logic – • Predicate – • Higher-order logic – • Propositional calculus – • Proposition – • Boolean algebra – • Boolean logic – • Truth value – • Venn diagram – • Pierce's law – • Aristotelian logic – • Non-Aristotelian logic – • Informal logic – • Fuzzy logic – • Infinitary logic – • Infinity – • Categorical logic – • College logic – • Linear logic – 95.7. CONCEPTS OF LOGIC 377

• Metalogic –

• Ordered logic –

• Temporal logic –

• Sequential logic –

• Provability logic –

• Interpretability logic – • Interpretability –

• Quantum logic –

• Relevant logic –

• Consequent –

• Affirming the consequent –

• Antecedent –

• Denying the antecedent –

• Theorem –

• Axiom –

• Axiomatic system –

• Axiomatization –

• Conditional proof –

• Invalid proof –

• Degree of truth –

• Truth –

• Truth condition –

• Truth function –

• Double negative –

• Double negative elimination –

• Fallacy –

• Existential fallacy – • Logical fallacy – • Syllogistic fallacy –

• Type theory –

• Game theory –

• –

• Rule of inference –

• Inference procedure –

• Inference rule –Quantification

• Introduction rule – 378 CHAPTER 95. OUTLINE OF LOGIC

• Law of excluded middle – • Law of non-contradiction – • Logical constant – • Logical connective – • Quantifier – • Logic gate – • Boolean Function – • Tautology – • Logical assertion – • Logical conditional – • Logical biconditional – • Logical equivalence – • Logical AND – • Negation – • Logical OR – • Logical NAND – • Logical NOR – • Contradiction – • Logicism – • Polysyllogism – • Syllogism – • Hypothetical syllogism – • Major premise – • Minor premise – • Term – • Singular term – • Major term – • Middle term – • Quantification – • Plural quantification – • Logical argument – • Validity – • Soundness – • Inverse (logic) – • Non sequitur – • Tolerance – 95.8. HISTORY OF LOGIC 379

• Satisfiability –

• Logical language –

• Paradox –

• Polish notation –

• Principia Mathematica –

• Quod erat demonstrandum –

• Reductio ad absurdum –

• Rhetoric –

• Self-reference –

• Necessary and sufficient –

• Sufficient condition –

• Nonfirstorderizability –

• Occam's Razor –

• Socratic dialoge –

• Socratic method –

• Argument form –

• Logic programming –

• Unification –

95.8 History of logic

Main article: History of logic

95.9 Literature about logic

95.9.1 Journals

• Journal of Logic, Language and Information

• Journal of Philosophical Logic

95.9.2 Books

• A System of Logic

• Attacking Faulty Reasoning

• Begriffsschrift

• Categories (Aristotle)

• Charles Sanders Peirce bibliography

• De Interpretatione 380 CHAPTER 95. OUTLINE OF LOGIC

• Gödel, Escher, Bach • Introduction to Mathematical Philosophy • Language, Truth, and Logic • Laws of Form • Linguistics and Philosophy • Logic Made Easy • Metamagical Themas • Minds, Machines and Gödel • Novum Organum • On Formally Undecidable Propositions of Principia Mathematica and Related Systems • Organon • Philosophical Investigations • Philosophy of Arithmetic • Polish Logic • Port-Royal Logic • Posterior Analytics • Principia Mathematica • Principles of Mathematical Logic • Prior Analytics • Rhetoric (Aristotle) • Sophistical Refutations • Sum of Logic • The Art of Being Right • The Foundations of Arithmetic • The Logic of Scientific Discovery • Topics (Aristotle) • Tractatus Logico-Philosophicus • What the Tortoise Said to Achilles • Where Mathematics Comes From

95.10 Logic organizations

• Association for Symbolic Logic

95.11 Logicians

• List of logicians • List of philosophers of language 95.12. SEE ALSO 381

95.12 See also

• Index of logic articles

• Mathematics • List of basic mathematics topics • List of mathematics articles

• Philosophy

• List of basic philosophy topics • List of philosophy topics • Outline of discrete mathematics – for introductory set theory and other supporting material see this

95.13 External links

• Taxonomy of Logical Fallacies • An Introduction to Philosophical Logic, by Paul Newall, aimed at beginners

• forall x: an introduction to formal logic, by P.D. Magnus, covers sentential and quantified logic • Translation Tips, by Peter Suber, for translating from English into logical notation

• Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas. In The Dictionary of the History of Ideas.

• Logic test Test your logic skills • Logic Self-Taught: A Workbook (originally prepared for on-line logic instruction) Chapter 96

Per fas et nefas

"Per faset nefas" (Latin for“through right and wrong”) refers to unfair eristic treatment. It occurs when Interlocutor A postulates a theory, and cites several reasons that justify it. Interlocutor B then refutes one of the arguments, and triumphantly declares that A's argument has no basis, even though he never said a word about the other arguments that A put forth.

96.1 See also

• Eristic • Dialectic

• Logic

382 Chapter 97

Persuasive definition

A persuasive definition is a form of definition which purports to describe the 'true' or 'commonly accepted' meaning of a term, while in reality stipulating an uncommon or altered use, usually to support an argument for some view, or to create or alter rights, duties or crimes.*[1]*[2] The terms thus defined will often involve emotionally charged but imprecise notions, such as“freedom”,“”,“democracy”, etc. In argumentation the use of a stipulative definition is sometimes called definist fallacy.*[3]*[4] Examples of persuasive definitions include:

• atheist – “someone who doesn't yet realize that God exists”*[4]

• Democrat – “a leftist who desires to overtax the corporations and abolish freedom in the economic sphere” *[4]

• Republican – “an old white man who feels threatened by change.”

• Loyalty – “a tool to get people to do things they don't want to do.”

• Sophistry – “a slogan used by ordinary common sense against educated reason”

Persuasive definitions commonly appear in controversial topics such as politics, sex, and religion, as participants in emotionally charged exchanges will sometimes become more concerned about swaying people to one side or another than expressing the unbiased facts. A persuasive definition of a term is favorable to one argument or unfavorable to the other argument, but is presented as if it were neutral and well-accepted, and the listener is expected to accept such a definition without question.*[1] The term “persuasive definition”was introduced by philosopher C.L. Stevenson as part of his emotive theory of meaning.*[5]

97.1 Overview and example

Language can simultaneously communicate information (informative) and feelings (expressive).*[6] Unlike other common types of definitions in logic, persuasive definitions focus on the expressive use of language to affect the feelings of readers and listeners ultimately with an aim to change their behavior.*[7] With this fundamentally different purpose, persuasive definitions are evaluated not on their truth or falsehood but rather on their effectiveness as a persuasive device.*[8] Stevenson *[9] showed how these two dimensions are combined when he investigated the terms he called“ethical”or emotive.*[10] He noted that some words, such as‘peace’or‘war,’, are not simply used to describe reality by modifying the cognitive response of the interlocutor. They have also the power of directing the interlocutor’s attitudes and suggesting a course of action. For this reason, they evoke a different kind of reaction, emotive in nature. As Stevenson *[11] put it “Instead of merely describing people’s interests, they change and intensify them. They recommend an interest in an object, rather than state that the interest already exists.”These words have the tendency to encourage future actions, to lead the hearer towards a decision by affecting his or her system of interests .*[12] Stevenson distinguished between the use of a word (a stimulus) and its possible psychological effects on the addressee’s cognitive and the emotive reactions by labeling them as “descriptive meaning”and “emotive

383 384 CHAPTER 97. PERSUASIVE DEFINITION

meaning”.*[13] Applying this distinction reveals how the redefinition of an ethical word is transformed into an instrument of persuasion, a tool for redirecting preferences and emotions:*[12]

Ethical definitions involve a wedding of descriptive and emotive meaning, and accordingly have a frequent use in redirecting and intensifying attitudes. To choose a definition is to plead a cause, so long as the word defined is strongly emotive.

In persuasive definitions the evaluative component associated with a concept is left unaltered while the descriptive meaning is modified. In this fashion, imprisonment can become“true freedom”,*[14] and massacres“pacification” .*[15] Persuasive definitions can change or distort the meaning while keeping the original evaluations that the use of a word evokes. Quasi-definitions consist in the modification of the emotive meaning of a word without altering the descriptive one. The speaker can quasi-define a word by qualifying the definiendum without setting forth what the term actually means. For instance, we can consider the following quasi-definition taken from Casanova’s Fuga dai Piombi. In this example (1), the speaker, Mr. Soradaci, tries to convince his interlocutor (Casanova) that being a “sneak”is an honorable behavior :*[16]

I have always despised the prejudice that attaches to the name“spy”a hateful meaning: this name sounds bad only to the ears of who hates the Government. A sneak is just a friend of the good of the State, the plague of the crooks, the faithful servant of his Prince.

This quasi-definition employed in case 1 underscores a fundamental dimension of the“emotive”meaning of a word, namely its relationship with the shared values, which are attacked as “prejudices.”This account given by the spy shows how describing the referent based on a different hierarchy of values can modify emotive meaning. The value of trust is not denied, but is placed in a hierarchy where the highest worth is given to the State .*[17] Patrick Hurley provides a number of examples of contentious terms with two opposing persuasive definitions, among them a favorable and unfavorable definition of taxation:*[8]

• definition supporting taxation: “the procedure by means of which our commonwealth is preserved and sus- tained” • definition opposing taxation: “the procedure used by bureaucrats to rip off the people who elected them”

Neither definition is particularly informative compared to a commonly accepted lexical definition. Note how the supporting view uses positive language where the opposing view uses negative language, such as the word bureaucrat which carries an unfavorable connotation alongside its informative meaning.*[18] Unclear, figurative language is often used in persuasive definitions.*[19] Although several techniques can be used to form such a definition ,*[10] the genus and difference technique is the usual one applied.*[20] Both definitions in the taxation example above agree that the genus is a procedure relating to governance but disagree on the difference. Persuasive definitions combine elements of stipulative definitions, lexical definitions, and sometimes theoretical def- initions.*[8] Persuasive definitions commonly appear in political speeches, editorials and other situations where the power to influence is most in demand.*[8] They have been dismissed as serving only to confuse readers and listeners without legitimate purpose.*[21] Critical scrutiny is often necessary to identify persuasive definitions in an argument as they are meant to appear as honest definitions.*[8]*[22]*[10]

97.2 See also

• Essentially contested concept • • Poisoning the well • Stipulative definition • Definition • The Devil's Dictionary 97.3. REFERENCES 385

97.3 References

[1] Bunnin, Nicholas; Jiyuan Yu (2004). “Persuasive definition”. The Blackwell Dictionary of Western Philosophy. Wiley- Blackwell. ISBN 978-1-4051-0679-5. Retrieved 2012-10-21.

[2] “Philosophy Pages”. Retrieved 2011-04-10.

[3] Bunnin, Nicholas; Jiyuan Yu (2008). “Definist fallacy”. The Blackwell Dictionary of Western Philosophy. John Wiley & Sons. p. 165. ISBN 978-0-470-99721-5. Retrieved 2014-10-15.

[4] Dowden, Bradley (December 31, 2010). “Fallacies”. Internet Encyclopedia of Philosophy. Retrieved 2011-04-10.

[5] Copi & Cohen 1990, p. 82.

[6] Copi & Cohen 1990, p. 67, 137.

[7] Copi & Cohen 1990, p. 137.

[8] Hurley 2008, p. 94.

[9] Stevenson 1937.

[10] Macagno & Walton 2014.

[11] Stevenson 1937, p. 18-19.

[12] Stevenson 1944, p. 210.

[13] Stevenson 1944, p. 54.

[14] Huxley 1944, p. 122.

[15] Orwell 1946.

[16] Casanova 1911, p. 112.

[17] Walton & Macagno 2015.

[18] “bureaucrat”. Macmillan Dictionary. Macmillan Publishers Limited. Retrieved 2011-04-09.

[19] Copi & Cohen 1990, p. 154.

[20] Hurley 2008, p. 103.

[21] Kemerling, Garth (2001-10-27). “Definition and Meaning”. Philosophy Pages. Retrieved 2011-04-09.

[22] Copi & Cohen 1990, pp. 137–138.

97.4 Sources

• Casanova, G. (1911). Historia della mia fuga dalle prigioni della republica di Venezia dette “li Piombi.”: . Milano: Alfieri e Lacroix. • Copi, Irving M.; Cohen, Carl (1990). Introduction to Logic (8th ed.). New York: Macmillan. ISBN 0-02- 946192-8. • Hurley, Patrick J. (2008). A Concise Introduction to Logic (10th ed.). Belmont, California: Thomson. ISBN 978-0-495-50383-5. • Huxley, A. (1955). Eyeless in Gaza. :. London: Chatto & Windus. • Macagno, Fabrizio; Walton, Douglas (2014). Emotive Language in Argumentation. New York: Cambdridge University Press. • Orwell, G. (1946). “Politics and the English Language. , .”. Horizon. April. • Stevenson, Charles (1937). “The Emotive Meaning of Ethical Terms.”. Mind 46: 14–31. • Stevenson, Charles (July 1938).“Persuasive Definitions”. Mind 47 (187): 331–350. doi:10.1093/mind/xlvii.187.331. 386 CHAPTER 97. PERSUASIVE DEFINITION

• Stevenson, Charles (1944). Ethics and Language. Connecticut: Yale University Press.

• Walton, Douglas; Macagno, Fabrizio (2015). "The Importance and Trickiness of Definition Strategies in Legal and Political Argumentation.”. Journal of Politics and Law 8 (1): 137–148. External link in |title= (help) Chapter 98

Philosophic burden of proof

This article is about burden of proof as a philosophical concept. For other uses, see Burden of proof (disambiguation).

In epistemology, the burden of proof (Latin: onus probandi) is the obligation on a party in a dispute to provide sufficient warrant for their position.

98.1 Holder of the burden

When two parties are in a discussion and one asserts a claim that the other disputes, the one who asserts has a burden of proof to justify or substantiate that claim.*[1] An occurs when either a proposition is assumed to be true because it has not yet been proved false or a proposition is assumed to be false because it has not yet been proved true.*[2]*[3] This has the effect of shifting the burden of proof to the person criticizing the proposition.*[4] While certain kinds of arguments, such as logical syllogisms, require mathematical or strictly logical proofs, the standard for evidence to meet the burden of proof is usually determined by context and community standards and conventions.*[5]*[6]

98.2 In public discourse

Burden of proof is also an important concept in the public arena of ideas. Once participants in discourse establish common assumptions, the mechanism of burden of proof helps to ensure that all parties contribute productively, using relevant arguments.*[7]*[8]*[9]*[10]

98.3 Proving a negative

A negative claim is a colloquialism for an affirmative claim that asserts the non-existence or exclusion of something. There are many proofs that substantiate negative claims in mathematics, science, and economics including Arrow's impossibility theorem. A negative claim may or may not exist as a counterpoint to a previous claim. A proof of impossibility or an evidence of absence argument are typical methods to fulfill the burden of proof for a negative claim.*[11]*[12]

98.4 Example

Matt Dillahunty gives the example of a large jar full of gumballs to illustrate the burden of proof.*[13]*[14] The number of whole gumballs in the jar is either even or odd, but the degree of personal acceptance or rejection of claims about that characteristic may vary. We can choose to consider two claims about the situation, given as:

387 388 CHAPTER 98. PHILOSOPHIC BURDEN OF PROOF

1. The number of gumballs is even.

2. The number of gumballs is odd.

Either claim could be explored separately, however, both claims represent the same proposition and do in fact ask the same question. Odd in this case means “not even”and could be described as a negative claim. Before we have any information about the number of gumballs, we have no means of checking either of the two claims. When we have no evidence to resolve the proposition, we may suspend judgment. From a cognitive sense, when no per- sonal preference toward opposing claims exists, one may be either skeptical of both claims or ambivalent of both claims.*[15]*[16]*[17] If there is a claim proposed and that claim is disputed, the burden of proof falls onto the proponent of the claim. If there is no agreeably adequate evidence to support a claim, the claim could be considered to be an argument from ignorance.

98.5 See also

• Argument from ignorance

• Evidentialism

• Justificationism

• Legal burden of proof

• Metaphysics

• Parsimony

• Pragma-dialectics

• Russell's teapot

• Scientific consensus

• Scientific method

• Statistical hypothesis testing

98.6 References

[1] Cargile, James (January 1997). “On the Burden of Proof”. Philosophy (Cambridge University Press) 72 (279): 59–83. doi:10.1017/s0031819100056655.

[2] “Argumentum ad Ignorantiam”. Philosophy 103: Introduction to Logic. Lander University. 2004. Archived from the original on 30 April 2009. Retrieved 2009-04-29.

[3] Dowden, Bradley. “Appeal to Ignorance”. Internet Encyclopedia of Philosophy. Retrieved 2016-02-24.

[4] Michalos, Alex (1969). Principles of Logic. Englewood Cliffs: Prentice-Hall. p. 370. usually one who makes an assertion must assume the responsibility of defending it. If this responsibility or burden of proof is shifted to a critic, the fallacy of appealing to ignorance is committed.

[5] Leite, Adam (2005). “A Localist Solution to the Regress of Justification”. Australasian Journal of Philosophy 83 (3): 395–421 [p. 418]. doi:10.1080/00048400500191974. [t]he point of articulating reasons in defense of one’s belief is to establish that one is justified in believing as one does.

[6] Leite, Adam (2005). “A Localist Solution to the Regress of Justification”. Australasian Journal of Philosophy 83 (3): 395–421 [p. 403]. doi:10.1080/00048400500191974. justificatory conversation...[is]...characterized by a person’s sincere attempt to vindicate his or her entitlement to a belief by providing adequate reasons in its defense and responding to objections.

[7] Goldman, Alvin (1994).“Argumentation and Social Epistemology”. Journal of Philosophy 91 (1): 27–49. doi:10.2307/2940949. JSTOR 2940949. 98.6. REFERENCES 389

[8] Eemeren, Frans van; Grootendorst, Rob (2004). A Systematic Theory of Argumentation. Cambridge: Cambridge University Press. p. 60. ISBN 0521830753. [t]here is no point in venturing to resolve a difference of opinion through an argumentative exchange of views if there is no mutual commitment to a common starting point.

[9] Brandom, Robert (1994). Making it Explicit. Cambridge: Harvard University Press. p. 222. ISBN 067454319X. [t]here are sentence types that would require a great deal of work for one to get into a position to challenge, such as 'Red is a color,' 'There have been black dogs,' 'Lighting frequently precedes thunder,' and similar commonplaces. These are treated as 'free moves' by members of our speech community—they are available to just about anyone any time to use as premises, to assert unchallenged.

[10] Adler, Jonathan (2002). Belief’s Own Ethics. Cambridge: MIT Press. pp. 164–167. ISBN 0262011921.

[11] Steven D. Hales (2005). “Thinking Tools: You Can Prove a Negative” (PDF). Bloomsburg University.

[12] T. Edward Dame (2009). Attacking Faulty Reasoning: A Practical Guide to Fallacy-Free Arguments. Cengage Learning. p. 17. ISBN 9780495095064. Retrieved 2016-02-24.

[13] “The Atheist Experience”. Episode 808. 7 April 2013. channelAustin 16. Missing or empty |series= (help)

[14] Matt Dillahunty (2013). Does God Exist? (Debate). Texas State University. Retrieved 2016-02-24.

[15] “Metacognitive Model of Ambivalence: The Role of Multiple Beliefs and Metacognitions in Creating Attitude Ambiva- lence”.

[16] “Reductionism, emergence, and burden of proof —part I”.

[17] “Reductionism, emergence, and burden of proof —part II”. Chapter 99

Philosophy of logic

Following the developments in formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to be termed either philosophy of logic or philosophical logic if no longer simply logic. Compared to the history of logic the demarcation between philosophy of logic and philosophical logic is of recent coinage and not always entirely clear. Characterisations include

• Philosophy of logic is the area of philosophy devoted to examining the scope and nature of logic.*[1] • Philosophy of logic is the investigation, critical analysis and intellectual reflection on issues arising in logic. The field is considered to be distinct from philosophical logic. • Philosophical logic is the branch of study that concerns questions about reference, predication, identity, truth, quantification, existence, entailment, modality, and necessity.*[2] • Philosophical logic is the application of formal logical techniques to philosophical problems.*[3]

This article outlines issues in philosophy of logic or provides links to relevant articles or both.

99.1 Introduction

This article makes use the following terms and concepts:

• Type–token distinction • Use–mention distinction

99.2 Truth

Parmenides said To say that that which is, is not or that which is not is, is a falsehood; and to say that which is, is and that which is not is not, is true*[4] This apparent truism has not proved unproblematic.

99.2.1 Truthbearers

Logic uses such terms as true, false, inconsistent, valid, and self-contradictory. Questions arise as Strawson (1952) writes*[5]

(a) when we use these words of logical appraisal, what is it exactly that we are appraising? and (b) how does logical appraisal become possible?

390 99.2. TRUTH 391

Main article: Truthbearer

See also: Sentence, Statement, Proposition.

99.2.2 Tarski's definition of Truth

See:

• Semantic theory of truth § Tarski's Theory • T-schema • Stanford Encyclopedia of Philosophy entry on Tarski's Truth Definitions • Self-reference:2.1 Consequences of the Semantic Paradoxes in Stanford Encyclopedia of Philosophy

99.2.3 Analytic Truths, Logical truth, Validity, Logical consequence and Entailment

Since the use, meaning, if not the meaningfulness, of the terms is part of the debate, it is possible only to give the following working definitions for the purposes of the discussion:

• A necessary truth is one that is true no matter what the state of the world or, as it is sometimes put, in all possible worlds.*[6] • Logical truths are those necessary truths that are necessarily true owing to the meaning of their logical constants only.*[7] • In formal logic a logical truth is just a“statement”(string of symbols in which no variable occurs free) which is true under all possible interpretations. • An analytic truth is one whose predicate concept is contained in its subject concept.

The concept of logical truth is intimately linked with those of validity, logical consequence and entailment (as well as self-contradiction, necessarily false etc.).

• If q is a logical truth, then p therefore q will be a valid argument. • If p1, p2, p3 ... pn therefore q is a valid argument then its corresponding conditional will be a logical truth. • If p1 & p2 & p3 ... pn entails q then If (p1 & p2 & p3 ... pn) then q is a logical truth. • If q is a logical consequence of p1 & p2 & p3 ... pn if and only if p1 & p2 & p3 ... pn entails q and if and only if If (p1 & p2 & p3..pn) then q is a logical truth

Issues that arise include:

• If there are truths that must be true, what makes them so? • Are there analytic truths that are not logical truths? • Are there necessary truths that are not analytic truths? • Are there necessary truths that are not logical truths? • Is the distinction between analytic truth and synthetic truth spurious?

Main articles: necessary truth, logical truth, validity (logic), Corresponding conditional (logic), logical consequence, entailment and Analytic-synthetic distinction

See also 392 CHAPTER 99. PHILOSOPHY OF LOGIC

99.2.4 Paradox

Main article: Paradox

99.3 Meaning and reference

See

• Sense and reference • Theory of reference • Mediated reference theory • • Causal theory of reference (section References) • Descriptivist theory of names (section References) • Saul Kripke (section References) • Frege's Puzzle (section New Theories of Reference and the Return of Frege's Puzzle) • Gottlob Frege (section References) • Failure of reference (section References) • Rigid designator (section Causal-Historical Theory of Reference) • Philosophy of language (section References) • Index of philosophy of language articles • Supposition theory (section References) • Referring expression • Meaning (philosophy of language) • Denotation and Connotation • Extension and Intension • Extensional definition • Intensional definition • Metacommunicative competence • Absent referent

99.4 Names and descriptions

• Failure to refer • Proper name (philosophy) • Definite description • Descriptivist theory of names • 99.5. LOGICAL CONSTANTS AND CONNECTIVES 393

• Singular term

• Term logic § Singular terms

• Empty name

• § Singular Terms, Truth-value Gaps, and Free Logic

• The Foundations of Arithmetic § Development of Frege's own view of a number

• Philosophy of language § references

• Direct reference

• Mediated reference theory

99.4.1 Formal and material consequence

• The problem of the material conditional: see Material conditional

99.5 Logical constants and connectives

Main articles: Logical constant and Logical connective

99.6 Quantifiers and quantificational theory

Main article: Quantifier (logic)

99.7 Modal logic

Main article: Modal logic

99.8 Deviant logics

Main article: Deviant logic

99.8.1 Classical v. non-classical logics

Main article: Classical logic

99.9 Philosophical theories of logic

• Conceptualism

• Constructivism

• Dialetheism 394 CHAPTER 99. PHILOSOPHY OF LOGIC

• Fictionalism

• Finitism

• Formalism

• Intuitionism

• Logical atomism

• Logicism

• Nominalism

• Realism

• Platonic realism

• Structuralism

99.10 Other Topics

• Leibniz's Law: see Identity of indiscernibles

• Vacuous names

• Do predicates have properties?: See Second-order logic

• Sense, Reference, Connotation, Denotation, Extension, Intension

• The status of the Laws of Logic

• Classical Logic

• Intuitionism

• Realism: see Platonic realism,

• The Law of Excluded Middle: see Law of excluded middle

• Modality, Intensionality and Propositional Attitude

• Counter-factuals

• Psychologism

99.11 See also

• Logic

• Is logic empirical?

• Type-token distinction

• Use–mention distinction

• Pierce's type-token distinction

• Concatenation theory 99.12. RESOURCES 395

99.12 Resources

• Haack, Susan. 1978. Philosophy of Logics. Cambridge University Press.(ISBN 0-521-29329-4) • Quine, W. V. O. 2004. Philosophy of Logic. 2nd ed. Harvard University Press.(ISBN 0-674-66563-5) • Alfred Tarski. 1983. The concept of truth in formalized languages, pp. 152–278, Logic,semantics, metamath- ematics, papers from 1923 to 1938, ed. John Corcoran (logician), Hackett,Indianapolis 1983.

99.13 References

[1] Audi, Robert, ed. (1999). The Cambridge Dictionary of Philosophy (2nd ed.). CUP. [2] Lowe, E. J.. Forms of Thought: A Study in Philosophical Logic. New York: Cambridge University Press, 2013. [3] Russell, Gillian Thoughts, Arguments, and Rants, Jc's Column. [4] Aristotle, Metaphysics,Books Γ, Δ, Ε 2nd edition 1011b25 (1993) trans Kirwan,: OUP [5] Strawson, P.F. (1952). Introduction to Logical Theory. Methuen: London. p. 3. [6] Wolfram (1989) p. 80 [7] Wolfram (1989), p. 273

99.14 Important figures

Important figures in the philosophy of logic include (but are not limited to):

99.15 Philosophers of logic

99.16 Literature

• Fisher Jennifer, On the Philosophy of Logic, Thomson Wadworth, 2008, ISBN 978-0-495-00888-0 • Goble, Lou, ed., 2001. (The Blackwell Guide to) Philosophical Logic. Oxford: Blackwell. ISBN 0-631-20693- 0. • Grayling, A. C., 1997. An Introduction to Philosophical Logic. 3rd ed. Oxford: Blackwell. ISBN 0-631- 19982-9. • Jacquette, Dale, ed., 2002. A Companion to Philosophical Logic. Oxford Blackwell. ISBN 1-4051-4575-7. • Kneale, W&M (1962). The development of logic. Oxford. • McGinn, Colin, 2000. Logical Properties: Identity, Existence, Predication, Necessity, Truth. Oxford: Oxford University Press. ISBN 0-19-926263-2. • Quine, Willard Van Orman (1970). Philosophy Of Logic. Prentice Hall: New JerseyUSA. • Sainsbury, Mark, 2001. Logical Forms: An Introduction to Philosophical Logic. 2nd ed. Oxford: Blackwell. ISBN 0-631-21679-0. • Strawson, PF (1967). Philosophical Logic. OUP. • Alfred Tarski,1983. The concept of truth in formalized languages, pp. 152–278, Logic,semantics, metamath- ematics, papers from 1923 to 1938, ed. John Corcoran (logician), Hackett,Indianapolis 1983. • Wolfram, Sybil, 1989. Philosophical Logic: An Introduction. London: Routledge. 290 pages. ISBN 0-415- 02318-1, ISBN 978-0-415-02318-4 • Journal of Philosophical Logic, Springer SBM 396 CHAPTER 99. PHILOSOPHY OF LOGIC

99.17 External links

• Routledge Encyclopedia of Philosophy entry Chapter 100

Polarity item

In linguistics, a polarity item is a lexical item that can appear only in environments associated with a particular grammatical polarity – affirmative or negative. A polarity item that appears in affirmative (positive) contexts is called a positive polarity item (PPI), and one that appears in negative contexts is a negative polarity item (NPI). The environment in which a polarity item is permitted to appear is called a "licensing context". In the simplest case, an affirmative statement provides a licensing context for a PPI, while negation provides a licensing context for an NPI. However, there are many complications, and not all polarity items of a given type need necessarily have exactly the same set of licensing contexts.

100.1 In English

As examples of polarity items, consider the English lexical items somewhat and at all, as used in the following sentences:

1. I liked the film somewhat. 2. I didn't like the film at all. 3. *I liked the film at all. 4. *I didn't like the film somewhat.

As can be seen, somewhat is licensed by the affirmative environment of sentence (1), but it is forbidden (anti-licensed) by the negative environment of sentence (4).*[1] It can therefore be considered to be a positive polarity item (PPI). On the other hand, at all is licensed by the negative environment of sentence (2), but anti-licensed by the positive environment of sentence (3), and is therefore considered a negative polarity item (NPI). Because standard English does not have negative concord, that is, double negatives are not used to intensify each other, the language makes frequent use of certain NPIs that correspond in meaning to negative items, and can be used in the environment of another negative. For example, anywhere is an NPI corresponding to the negative nowhere, as used in the following sentences:

1. I was going nowhere. (the negative nowhere is used when not preceded by another negative) 2. I was not going anywhere. (the NPI anywhere is used in the environment of the preceding negative not)

Note that the alternative form with the double negative, *I was not going nowhere, is ungrammatical in the standard language, although such forms are used in some colloquial English, and parallel the constructions used in certain other languages which have negative concord. (Note also that anywhere, like most of the other NPIs listed below, is also used in other senses where it is not an NPI, as in I would go anywhere with you.) Similar pairs of negatives and corresponding NPI are listed below.

• nobody/no one – anybody/anyone

397 398 CHAPTER 100. POLARITY ITEM

• nothing – anything

• no/none – any

• never – ever

• nowhere – anywhere

• no longer/no more – any longer/any more

See also English grammar: Negation, and Affirmative and negative: Double negatives.

100.2 Determination of licensing contexts

The actual set of contexts that license particular polarity items is not as easily defined as a simple distinction between affirmative and negative sentences. Baker*[2] noted that double negation may provide an acceptable context for positive polarity items:

I can't believe you don't fancy her somewhat.

John doesn't have any potatoes *John has any potatoes.

However, licensing contexts can take many forms besides simple negation/affirmation. To complicate matters, polarity items appear to be highly idiosyncratic, each with its own set of licensing contexts. Early discussion of polarity items can be found in the work of Otto Jespersen and Edward Klima. Much of the research on polarity items has centered around the question of what creates a negative context. In the late 1970s, William Ladusaw (building on work by Gilles Fauconnier) discovered that most English NPIs are licensed in downward en- tailing environments. This is known as the Fauconnier–Ladusaw hypothesis. A downward entailing environment, however, is not a necessary condition for an NPI to be licensed—they may be licensed by some non-monotone (and thus not downward entailing) contexts, like “exactly N”, as well.

*Some people have ever been on the moon. Exactly three people have ever been on the moon.

Nor is a downward entailing environment a sufficient condition for all negative polarity items, as first pointed out by Zwarts (1981) for Dutch “ook maar”. Licensing contexts across languages include the scope of n-words (negative particles, negative quantifiers), the antecedent of conditionals, questions, the restrictor of universal quantifiers, non-affirmative verbs (doubt), adversative predicates (be surprised), negative conjunctions (without), comparatives and superlatives, too-phrases, negative predicates (un- likely), some subjunctive complements, some disjunctions, imperatives, and others (finally, only). Given this wide range of mostly non-downward entailing environments, the Fauconnier-Ladusaw Hypothesis has gradually been re- placed in favor of theories based on the notion of nonveridicality (proposed by Zwarts and Giannakidou). Different NPIs may be licensed by different expressions. Thus, while the NPI anything is licensed by the downward entailing expression at most two of the visitors, the idiomatic NPI lift a finger (known as a “minimizer”) is not licensed by the same expression.

At most two of the visitors had seen anything. *At most two of the visitors lifted a finger to help.

While NPIs have been discovered in many languages, their distribution is subject to substantial cross-linguistic vari- ation; this aspect of NPIs is currently the subject of ongoing research in cross-linguistic semantics.*[3] 100.3. SEE ALSO 399

100.3 See also

• Grammatical polarity

100.4 Notes

[1] See Baker (1970).

[2] See Baker (1970).

[3] Giannakidou, Anastasia. “Negative and positive polarity items: licensing, compositionality and variation”. Prepared for Maienborn, Claudia, Klaus von Heusinger, and Paul Portner (eds). Semantics: An International Handbook of Natural Language Meaning. Berlin: Mouton de Gruyter. (January 2008).

100.5 References

• Baker, C. Lee (1970). “Double Negatives”. Linguistic Inquiry 1: 169–186. • Klima, Edward (1964).“Negation in English”. In Jerry A. Fodor & Jerrold J. Katz. The structure of language. Englewood Cliffs: Prentice Hall, 246-323. • Fauconnier, Gilles (1975). “Polarity and the scale principle”. Chicago Linguistic Society. pp. 188–199.

• Giannakidou, Anastasia (2001).“The Meaning of Free Choice”. Linguistics and Philosophy 24 (6): 659–735. doi:10.1023/A:1012758115458.

• Ladusaw, William A. (1979). Polarity Sensitivity as Inherent Scope Relations. Ph.D. Dissertation, University of Texas, Austin.

• Zwarts, Frans (1981). “Negatief Polaire Uitdrukkingen I”. GLOT 4: 35–102.

100.6 External links

• The Polarity Items Bibliography (Tübingen University) • The Collection of Distributionally Idiosyncratic Items, containing German and Rumanian NPIs (Tübingen University) Chapter 101

Portal:Logic

edit

101.1 Logic

Logic (from Classical Greek λόγος logos; meaning 'speech/word') is the study of the principles and criteria of valid inference and demonstration. The term“logos”was also believed by the Greeks to be the universal power by which all reality was sustained and made coherent and consistent. As a formal science, logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and through the study of arguments in natural language. The field of logic ranges from core topics such as the study of fallacies and paradoxes, to specialized analysis of reasoning using probability and to

400 101.2. SELECTED ARTICLE 401

arguments involving causality. Logic is also commonly used today in argumentation theory. *[1] Traditionally, logic is studied as a branch of philosophy, one part of the classical trivium, which consisted of grammar, logic, and rhetoric. Since the mid-nineteenth century formal logic has been studied in the context of the foundations of mathematics. In 1910 Bertrand Russell and Alfred North Whitehead attempted to establish logic as the cornerstone of mathematics formally with the publication of Principia Mathematica. However, the system of Principia is no longer much used, having been largely supplanted by set theory. The development of formal logic and its implementation in computing machinery is the foundation of computer science. Read more... edit

101.2 Selected article

The history of logic is the study of the development of the science of valid inference (logic). While many cultures have employed intricate systems of reasoning, and logical methods are evident in all human thought, an explicit analysis of the principles of reasoning was developed only in three traditions: those of China, India, and Greece. Of these, only the treatment of logic descending from the Greek tradition, particularly Aristotelian logic, found wide application and acceptance in science and mathematics. The Greek tradition was further developed by Islamic logicians and then medieval European logicians. Not until the 19th century does the next great advance in logic arise, with the development of symbolic logic by George Boole and its subsequent development into formal calculable logical systems by Gottlob Frege and set theorists such as Georg Cantor and Giuseppe Peano, ushering in the Information Age. Logic was known as 'dialectic' or 'analytic' in Ancient Greece. The word 'logic' (from the Greek logos, meaning discourse or sentence) does not appear in the modern sense until the commentaries of Alexander of Aphrodisias, writing in the third century A.D. edit

101.3 Selected biography

Aristotle (Greek: Ἀριστοτέλης, Aristotélēs) (384 BC – 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. He wrote on many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, politics, government, ethics, biology and zoology. 402 CHAPTER 101. PORTAL:LOGIC

Together with Plato and Socrates (Plato's teacher), Aristotle is one of the most important founding figures in Western philosophy. He was the first to create a comprehensive system of Western philosophy, encompassing morality and aesthetics, logic and science, politics and metaphysics. Aristotle's views on the physical sciences profoundly shaped medieval scholarship, and their influence extended well into the Renaissance, although they were ultimately replaced by Newtonian Physics. In the biological sciences, some of his observations were confirmed to be accurate only in the nineteenth century. His works contain the earliest known formal study of logic, which was incorporated in the late nineteenth century into modern formal logic. In metaphysics, had a profound influence on philosophical and theological thinking in the Islamic and Jewish traditions in the Middle Ages, and it continues to influence Christian theology, especially Eastern Orthodox theology, and the scholastic tradition of the Roman Catholic Church. All aspects of Aristotle's philosophy continue to be the object of active academic study today. edit

101.4 Did you know?

• ...that the Law of noncontradiction is the same as the Law of contradiction? • ...that NAND alone or NOR alone make a functionally complete set of logical operators? • ...that according to the continuum hypothesis, there is no cardinal number in between aleph null and the cardinality of the continuum? • ...that all 24 valid syllogistic forms have names like 'Camestros' and 'Felapton'? • ...that if a statement P implies another statement Q, and a third statement R also implies Q, and either P or R is true; then Q has to be true? • ...that the dunce cap was named after a logician? • ...that the collective noun for a group of logicians is a "sequitur of logicians” edit

101.5 Categories

Logic Belief revision Classical logic Concepts in logic Critical thinking Deductive reasoning History of logic Inductive reasoning Logic and statistics Logic literature Logic organizations Logical fallacies Logicians Mathematical logic Metalogic Non-classical logic Philosophical logic Philosophy of logic Logic puzzles Reasoning Syntax (logic) 101.6. TOPICS 403

Theories of deduction ► Wikipedia books on logic ► Logic stubs edit

101.6 Topics edit

101.7 Related portals

• Philosophy • Mathematics • Set theory 404 CHAPTER 101. PORTAL:LOGIC

• Philosophy of science

• Thinking • Mind and Brain

• Epistemology • Ethics

• Constructed Languages • Linguistics

• Computer Science edit

101.8 WikiProjects

• Logic

• Philosophy • Mathematics

• Philosophy of Language • Computer Science

• Theoretical Linguistics •

• Rational Skepticism • Systems edit

101.9 Things to do

101.10 References

[1] J. Robert Cox and Charles Arthur Willard, eds. Advances in Argumentation Theory and Research, Southern Illinois Uni- versity Press, 1983 ISBN 0809310503, ISBN-13 978-0809310500

• What are portals?

• List of portals • Featured portals Chapter 102

Post's lattice

In logic and universal algebra, Post's lattice denotes the lattice of all clones on a two-element set {0, 1}, ordered by inclusion. It is named for Emil Post, who published a complete description of the lattice in 1941.*[1] The relative simplicity of Post's lattice is in stark contrast to the lattice of clones on a three-element (or larger) set, which has the cardinality of the continuum, and a complicated inner structure. A modern exposition of Post's result can be found in Lau (2006).*[2]

102.1 Basic concepts

A Boolean function, or logical connective, is an n-ary operation f: 2*n → 2 for some n ≥ 1, where 2 denotes the two-element set {0, 1}. Particular Boolean functions are the projections

n πk (x1, . . . , xn) = xk,

and given an m-ary function f, and n-ary functions g1, ..., gm, we can construct another n-ary function

h(x1, . . . , xn) = f(g1(x1, . . . , xn), . . . , gm(x1, . . . , xn)), called their composition. A set of functions closed under composition, and containing all projections, is called a clone. Let B be a set of connectives. The functions which can be defined by a formula using propositional variables and connectives from B form a clone [B], indeed it is the smallest clone which includes B. We call [B] the clone generated by B, and say that B is the basis of [B]. For example, [¬, ∧] are all Boolean functions, and [0, 1, ∧, ∨] are the monotone functions. We use the operations ¬ (negation), ∧ (conjunction or meet), ∨ (disjunction or join), → (implication), ↔ (biconditional), + (exclusive disjunction or Boolean ring addition), ↛ (nonimplication), ?: (the ternary conditional operator) and the constant unary functions 0 and 1. Moreover, we need the threshold functions

{ { | } ≥ n 1 if i xi = 1 k, th (x1, . . . , xn) = k 0 otherwise.

* * For example, th1 n is the large disjunction of all the variables xi, and thn n is the large conjunction. Of particular importance is the majority function

3 ∧ ∨ ∧ ∨ ∧ maj = th2 = (x y) (x z) (y z). * * We denote elements of 2 n (i.e., truth-assignments) as vectors: a = (a1, ..., an). The set 2 n carries a natural product Boolean algebra structure. That is, ordering, meets, joins, and other operations on n-ary truth assignments are defined pointwise:

405 406 CHAPTER 102. POST'S LATTICE

(a1, . . . , an) ≤ (b1, . . . , bn) ⇐⇒ ai ≤ bi every for i = 1, . . . , n,

(a1, . . . , an) ∧ (b1, . . . , bn) = (a1 ∧ b1, . . . , an ∧ bn).

102.2 Naming of clones

Intersection of an arbitrary number of clones is again a clone. It is convenient to denote intersection of clones by simple juxtaposition, i.e., the clone C1 ∩ C2 ∩ ... ∩ Ck is denoted by C1C2...Ck. Some special clones are introduced below:

• M is the set of monotone functions: f(a) ≤ f(b) for every a ≤ b. • D is the set of self-dual functions: ¬f(a) = f(¬a). • A is the set of affine functions: the functions satisfying

f(a1, . . . , ai−1, c, ai+1, . . . , an) = f(a1, . . . , d, ai+1,... ) ⇒ f(b1, . . . , c, bi+1,... ) = f(b1, . . . , d, bi+1,... )

* for every i ≤ n, a, b ∈ 2 n, and c, d ∈ 2. Equivalently, the functions expressible as f(x1, ..., xn) = a0 + a1x1 + ... + anxn for some a0, a.

• U is the set of essentially unary functions, i.e., functions which depend on at most one input variable: there exists an i = 1, ..., n such that f(a) = f(b) whenever ai = bi.

• Λ is the set of conjunctive∧ functions: f(a ∧ b) = f(a) ∧ f(b). The clone Λ consists of the conjunctions f(x1, . . . , xn) = i∈I xi for all subsets I of {1, ..., n} (including the empty conjunction, i.e., the constant 1), and the constant 0.

• V is the set of disjunctive∨ functions: f(a ∨ b) = f(a) ∨ f(b). Equivalently, V consists of the disjunctions f(x1, . . . , xn) = i∈I xi for all subsets I of {1, ..., n} (including the empty disjunction 0), and the constant 1.

* • For any k ≥ 1, T0 k is the set of functions f such that

a1 ∧ · · · ∧ ak = 0 ⇒ f(a1) ∧ · · · ∧ f(ak) = 0. ∩ ∞ ∞ k Moreover, T0 = k=1 T0 is the set of functions bounded above by a variable: there exists i = 1, ..., n such that f(a) ≤ ai for all a. 1 As a special case, P0 = T0 is the set of 0-preserving functions: f(0) = 0.

* • For any k ≥ 1, T1 k is the set of functions f such that

a1 ∨ · · · ∨ ak = 1 ⇒ f(a1) ∨ · · · ∨ f(ak) = 1, ∩ ∞ ∞ k and T1 = k=1 T1 is the set of functions bounded below by a variable: there exists i = 1, ..., n such that f(a) ≥ ai for all a. 1 The special case P1 = T1 consists of the 1-preserving functions: f(1) = 1.

• The largest clone of all functions is denoted ⊤, the smallest clone (which contains only projections) is denoted ⊥, and P = P0P1 is the clone of constant-preserving functions. 102.3. DESCRIPTION OF THE LATTICE 407

Hasse diagram of Post's lattice

102.3 Description of the lattice

The set of all clones is a closure system, hence it forms a complete lattice. The lattice is countably infinite, and all its members are finitely generated. All the clones are listed in the table below. 1 The eight infinite families have actually also members with k = 1, but these appear separately in the table: T0 = P0, 1 1 1 1 1 1 1 T1 = P1, PT0 = PT1 = P, MT0 = MP0, MT1 = MP1, MPT0 = MPT1 = MP. * * * The lattice has a natural symmetry mapping each clone C to its dual clone C d = {f d | f ∈ C}, where f d(x1, ..., xn) * * * * * = ¬f(¬x1, ..., ¬xn) is the de Morgan dual of a Boolean function f. For example, Λ d = V, (T0 k) d = T1 k, and M d = M. 408 CHAPTER 102. POST'S LATTICE

Central part of the lattice

102.4 Applications

The complete classification of Boolean clones given by Post helps to resolve various questions about classes of Boolean functions. For example:

• An inspection of the lattice shows that the maximal clones different from ⊤ (often called Post's classes) are M, D, A, P0,P1, and every proper subclone of ⊤ is contained in one of them. As a set B of connectives is functionally complete if and only if it generates ⊤, we obtain the following characterization: B is functionally complete iff it is not included in one of the five Post's classes. • The satisfiability problem for Boolean formulas is NP-complete by Cook's theorem. Consider a restricted version of the problem: for a fixed finite set B of connectives, let B-SAT be the algorithmic problem of checking whether a given B-formula is satisfiable. Lewis*[3] used the description of Post's lattice to show that B-SAT is * NP-complete if the function ↛ can be generated from B (i.e., [B] ⊇ T0 ∞), and in all the other cases B-SAT is polynomial-time decidable. 102.5. VARIANTS 409

102.5 Variants

Post originally did not work with the modern definition of clones, but with the so-called iterative systems, which are sets of operations closed under substitution

h(x1, . . . , xn+m−1) = f(x1, . . . , xn−1, g(xn, . . . , xn+m−1)), as well as permutation and identification of variables. The main difference is that iterative systems do not necessarily contain all projections. Every clone is an iterative system, and there are 20 non-empty iterative systems which are not clones. (Post also excluded the empty iterative system from the classification, hence his diagram has no least element and fails to be a lattice.) As another alternative, some authors work with the notion of a closed class, which is an iterative system closed under introduction of dummy variables. There are four closed classes which are not clones: the empty set, the set of constant 0 functions, the set of constant 1 functions, and the set of all constant functions. Composition alone does not allow to generate a nullary function from the corresponding unary constant function, this is the technical reason why nullary functions are excluded from clones in Post's classification. If we lift the restriction, we get more clones. Namely, each clone C in Post's lattice which contains at least one constant function corresponds to two clones under the less restrictive definition: C, and C together with all nullary functions whose unary versions are in C.

102.6 References

[1] E. L. Post, The two-valued iterative systems of mathematical logic, Annals of Mathematics studies, no. 5, Princeton Uni- versity Press, Princeton 1941, 122 pp.

[2] D. Lau, Function algebras on finite sets: Basic course on many-valued logic and clone theory, Springer, New York, 2006, 668 pp. ISBN 978-3-540-36022-3

[3] H. R. Lewis, Satisfiability problems for propositional calculi, Mathematical Systems Theory 13 (1979), pp. 45–53. Chapter 103

Pragmatic mapping

Pragmatic mapping —a term in current use in linguistics, computing, cognitive psychology, and related fields —is the process by which a given abstract predicate (a symbol) comes to be associated through action (a dynamic index) with some particular logical object (an icon). The logical object may be a thing, person, relation, event, situation, or a string of these at any conceivable level of complexity. A relatively simple example is the conventional —successful, appropriate, and mundanely“true”—linking of a proper name to the person of whom it is a conventional designation. There are three parts to this process when it succeeds. There is the abstract symbol which is used to represent some- thing else (the name or the entire signifying predication, for instance); there is the something else that is represented by that symbol (whatever is signified); and there is the act of using the symbol in a conventional way to represent whatever it usually represents (the act of signifying). Pragmatic mapping is the process by which any material argu- ment, or any imagined one, comes to be associated with a predicate that purports to be and succeeds in being about it. That is the predicate must be appropriate (“true”in the most mundane sense relative) to its logical object. The pred- ication may be as simple as a naming act or as complex as a representation consisting of many distinct propositions with many associated clauses. For instance, if we say“Jesse James was an American outlaw" the name "Jesse James" purports to be about a certain historical person whom we may know to have been shot by another individual named Robert Ford. We may know that a movie featuring Brad Pitt as Jesse James was released in September 2007 in select theaters across America. If the pragmatic mapping of the name “Jesse James”is complete, i.e., if it succeeds, it is mapped onto that certain individual that was actually shot by Robert Ford. Nothing of importance changes in the pragmatic mapping process if it turns out that Jesse James and Robert Ford are figments of someone’s imagination, excepting, of course, the truth value of the propositions that include the logical object of the name, Jesse James. In ordinary conversation and human communication in general, it has been demonstrated logically and mathematically that meaning is utterly dependent on the true and appropriate pragmatic mapping of symbols to their conventional logical objects. Infants depend on exemplification of such mapping relations to acquire languages and all meaningful linguistic representations have been proved to depend on such mappings.

103.1 See also

• Wörter und Sachen

103.2 References

• Frege, G. (1967). Begriffsschrift, a formula language modeled upon that of arithmetic for pure thought. In J. van Heijenoort (Ed. and Trans.), Frege and Gödel: Two fundamental texts in mathematical logic (pp. 5–82). Harvard University Press, Cambridge, Massachusetts. (Original work published 1879)

• Krashen, S. D. 1982. Principles and Practices in Second Language Acquisition. New York: Pergamon.

• Oller, J. W., Jr. (1975). Pragmatic mappings. Lingua, 35, 333-344.

410 103.2. REFERENCES 411

• Oller, J. W., Jr. (2005). Common ground between form and content: The pragmatic solution to the bootstrap- ping problem. Modern Language Journal, 89, 92-114. • Oller, J. W., Jr., Oller, S. D., & Badon, L. C. (2006). Milestones: Normal speech and language development across the life span. San Diego, CA: Plural Publishing, Inc. • Pearson, L. (2007). Patterns of development in Spanish L2 pragmatic acquisition: An analysis of novice learners' production of directives. The Modern Language Journal 90 (4), 473–495. • Peirce, C. S. (1897). The logic of relatives. The Monist, 7, 161 – 217. Also in C. Hartshorne & P.Weiss (Eds), (1932), Collected Papers of C. S. Peirce, Volume 2 (pp. 288 – 345). Cambridge, MA: Harvard University Press. See http://www.cs.cmu.edu/afs/cs/project/jair/pub/volume20/fox03a-html/node16.html for the use of the term “pragmatic mapping”in modern computing. • Tarski, A. (1949). The semantic conception of truth. In H. Feigl & W. Sellars (Eds. and Trans.), Readings in philosophical analysis (pp. 341–374). New York: Appleton. (Original work published 1944)

• Tarski, A. (1956). The concept of truth in formalized languages. In J. J. Woodger (Ed. and Trans.), Logic, semantics, and metamathematics (pp. 152–278). Oxford: Oxford University. (Original work published 1936) Chapter 104

Pragmatic maxim

The pragmatic maxim, also known as the maxim of pragmatism or the maxim of pragmaticism, is a maxim of logic formulated by Charles Sanders Peirce. Serving as a normative recommendation or a regulative principle in the normative science of logic, its function is to guide the conduct of thought toward the achievement of its purpose, advising on an optimal way of“attaining clearness of apprehension". Here is its original 1878 statement in English*[2] when it was not yet named:

It appears, then, that the rule for attaining the third grade of clearness of apprehension is as follows: Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object. (Peirce on p. 293 of "How to Make Our Ideas Clear", Popular Science Monthly, v. 12, pp. 286–302. Reprinted widely, including Collected Papers of Charles Sanders Peirce (CP) v. 5, paragraphs 388–410.)

104.1 Seven ways of looking at the pragmatic maxim

Peirce stated the pragmatic maxim in many different ways over the years, each of which adds its own bit of clarity or correction to their collective corpus.

• The first excerpt appears in the form of a dictionary entry, intended as a definition of pragmatism as an opinion favoring application of the pragmatic maxim as a recommendation about how to clarify meaning.

Pragmatism. The opinion that metaphysics is to be largely cleared up by the application of the following maxim for attaining clearness of apprehension: Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object. (Peirce, 1902, "Pragmatic and Pragmatism" in the Dictionary of Philosophy and Psychology, including quote of himself from 1878, "How to Make Our Ideas Clear" in Popular Science Monthly v. 12, pp. 286-302. Reprinted in CP 5.2).

• The second excerpt presents the pragmatic maxim (with added emphases on forms of the word “conceive” ) as a recommendation to you, the addressee, on how you can clarify your conception, then restates it in the indicative, in a way that emphasizes the generalism of pragmatism:

Pragmaticism was originally enounced in the form of a maxim, as follows: Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object.

I will restate this in other words, since ofttimes one can thus eliminate some unsuspected source of perplexity to the reader. This time it shall be in the indicative mood, as follows:

412 104.1. SEVEN WAYS OF LOOKING AT THE PRAGMATIC MAXIM 413

The entire intellectual purport of any symbol consists in the total of all general modes of ra- tional conduct which, conditionally upon all the possible different circumstances and desires, would ensue upon the acceptance of the symbol.

(Peirce, 1905, from “Issues of Pragmaticism”in The Monist v. XV, n. 4, pp. 481-499, see p. 481 via Google Books and via Internet Archive. Reprinted in CP 5.438.).

• The third excerpt puts a gloss on the meaning of a practical bearing and provides an alternative statement of the maxim. Such reasoning, and all reasonings turn upon the idea that one who exerts certain kinds of volition will undergo, in return, certain compulsory perceptions. Now this sort of consideration—that certain lines of conduct entail certain kinds of inevitable experiences—is called a practical consideration. This justifies the maxim as a practical belief, that:

To ascertain the meaning of an intellectual conception one should consider what prac- tical consequences might result from the truth of that conception—and the sum of these consequences constitute the entire meaning of the conception.

(Peirce, 1905, CP 5.9.)

• The fourth excerpt illustrates one of Peirce's many attempts to get the sense of the pragmatic philosophy across by rephrasing the pragmatic maxim. Introducing this version, he addresses prospective critics who do not believe a simple heuristic maxim, much less one that concerns itself with a routine matter of logical procedure, forms a sufficient basis for a whole philosophy. He suggests they might feel he makes pragmatism “a mere maxim of logic instead of a sublime principle of speculative philosophy.”For better philosophical standing, he endeavors to put pragmatism into the same form of a philosophical theorem:

Pragmatism is the principle that every theoretical judgment expressible in a sentence in the indicative mood is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a having its apodosis in the imperative mood.

(Peirce, 1903, from the lectures on Pragmatism, CP 5.18, also in Pragmatism as a Principle and Method of Right Thinking: The 1903 Harvard 'Lectures on Pragmatism', p. 110, and in Essential Peirce v. 2, pp. 134-5.)

• The fifth excerpt is useful by way of additional clarification, and is aimed to correct a variety of historical misunderstandings that arose with regard to the intended meaning of the pragmatic maxim. For a source of such misunderstanding, Peirce points to his younger self (but will retract the confession as itself mistaken— see the seventh excerpt).

The doctrine appears to assume that the end of man is action - a stoical axiom which, to the present writer at the age of sixty, does not recommend itself so forcibly as it did at thirty. If it be admitted, on the contrary, that action wants an end, and that that end must be something of a general description, then the spirit of the maxim itself, which is that we must look to the upshot of our concepts in order rightly to apprehend them, would direct us towards something different from practical facts, namely, to general ideas, as the true interpreters of our thought. (Peirce, 1902, from "Pragmatic and Pragmatism" in the Dictionary of Philosophy and Psychology. Reprinted CP 5.3, 1902).

• A sixth excerpt is useful in stating the bearing of the pragmatic maxim on the topic of reflection, namely, that it makes all of pragmatism boil down to nothing more or less than a method of reflection.

The study of philosophy consists, therefore, in reflexion, and pragmatism is that method of reflexion which is guided by constantly holding in view its purpose and the purpose of the ideas it analyzes, whether these ends be of the nature and uses of action or of thought. . It will be seen that pragmatism is not a Weltanschauung but is a method of reflexion having for its purpose to render ideas clear. 414 CHAPTER 104. PRAGMATIC MAXIM

(Peirce, 1902, CP 5.13 note 1).

• The seventh excerpt is a late reflection on the reception of pragmatism. With a sense of exasperation that is almost palpable, Peirce tries to justify the maxim of pragmatism and to correct its misreadings by pinpointing a number of false impressions that the intervening years have piled on it, and he attempts once more to prescribe against the deleterious effects of these mistakes. Recalling the very conception and birth of pragmatism, he reviews its initial promise and its intended lot in the light of its subsequent vicissitudes and its apparent fate. Adopting the style of a post mortem analysis, he presents a veritable autopsy of the ways that the main idea of pragmatism, for all its practicality, can be murdered by a host of misdissecting disciplinarians, by what are ostensibly its most devoted followers. He proceeds here (1906) to retract a philosophical confession—in the fifth excerpt (above)—which he wrote in 1902 about his 1878 original presentation of pragmatism.

This employment five times over of derivates of concipere must then have had a purpose. In point of fact it had two. One was to show that I was speaking of meaning in no other sense than that of intellectual purport. The other was to avoid all danger of being understood as attempting to explain a concept by percepts, images, schemata, or by anything but concepts. I did not, therefore, mean to say that acts, which are more strictly singular than anything, could constitute the purport, or adequate proper interpretation, of any symbol. I compared action to the finale of the symphony of thought, belief being a demicadence. Nobody conceives that the few bars at the end of a musical movement are the purpose of the movement. They may be called its upshot. But the figure obviously would not bear detailed application. I only mention it to show that the suspicion I myself expressed after a too hasty rereading of the forgotten magazine paper, that it expressed a stoic, that is, a nominalistic, materialistic, and utterly philistine state of thought, was quite mistaken. (Peirce, 1906, CP 5.402 note 3).

104.2 References

• Peirce, C.S., Collected Papers of Charles Sanders Peirce, Volumes 1–6, Charles Hartshorne and Paul Weiss (eds.), Vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Cited as CP x.y for volume x, paragraph y.

[1] Brent, Joseph (1998), Charles Sanders Peirce: A Life, 2nd edition, Bloomington and Indianapolis: Indiana University Press (catalog page); also NetLibrary.

[2] The article containing it was originally written in French as "Comment rendre nos idées claires" in 1877 for Revue Philo- sophique, which published it in its Volume VII in January 1879. There the maxim appeared on p. 48 as "Considérer quels sont les effets pratiques que nous pensons pouvoir être produits par l'objet de notre conception. La conception de tous ces effets est la conception complète de l'objet.". Curiously, the Revue Philosophique version omits the article's last one and a half paragraphs. Chapter 105

Pragmatic theory of truth

A pragmatic theory of truth is a theory of truth within the philosophies of pragmatism and pragmaticism. Pragmatic theories of truth were first posited by Charles Sanders Peirce, William James, and John Dewey. The common features of these theories are a reliance on the pragmatic maxim as a means of clarifying the meanings of difficult concepts such as truth; and an emphasis on the fact that belief, certainty, knowledge, or truth is the result of an inquiry.

105.1 Background

Pragmatic theories of truth developed from the earlier ideas of , the Scholastics, and Immanuel Kant. Pragmatic ideas about truth are often confused with the quite distinct notions of“logic and inquiry”,“judging what is true”, and “truth predicates”.

105.1.1 Logic and inquiry

In one classical formulation, truth is defined as the good of logic, where logic is a normative science, that is, an inquiry into a good or a value that seeks knowledge of it and the means to achieve it. In this view, truth cannot be discussed to much effect outside the context of inquiry, knowledge, and logic, all very broadly considered. Most inquiries into the character of truth begin with a notion of an informative, meaningful, or significant element, the truth of whose information, meaning, or significance may be put into question and needs to be evaluated. Depending on the context, this element might be called an artefact, expression, image, impression, lyric, mark, performance, picture, sentence, sign, string, symbol, text, thought, token, utterance, word, work, and so on. Whatever the case, one has the task of judging whether the bearers of information, meaning, or significance are indeed truth-bearers. This judgment is typically expressed in the form of a specific truth predicate, whose positive application to a sign, or so on, asserts that the sign is true.

105.1.2 Judging what is true

Considered within the broadest horizon, there is little reason to imagine that the process of judging a work, that leads to a predication of false or true, is necessarily amenable to formalization, and it may always remain what is commonly called a judgment call. But there are indeed many well-circumscribed domains where it is useful to consider disciplined forms of evaluation, and the observation of these limits allows for the institution of what is called a method of judging truth and falsity. One of the first questions that can be asked in this setting is about the relationship between the significant performance and its reflective critique. If one expresses oneself in a particular fashion, and someone says “that's true”, is there anything useful at all that can be said in general terms about the relationship between these two acts? For instance, does the critique add value to the expression criticized, does it say something significant in its own right, or is it just an insubstantial echo of the original sign?

415 416 CHAPTER 105. PRAGMATIC THEORY OF TRUTH

105.1.3 Truth predicates

Theories of truth may be described according to several dimensions of description that affect the character of the predicate “true”. The truth predicates that are used in different theories may be classified by the number of things that have to be mentioned in order to assess the truth of a sign, counting the sign itself as the first thing. In formal logic, this number is called the arity of the predicate. The kinds of truth predicates may then be subdivided according to any number of more specific characters that various theorists recognize as important.

1. A monadic truth predicate is one that applies to its main subject —typically a concrete representation or its abstract content —independently of reference to anything else. In this case one can say that a truthbearer is true in and of itself.

2. A dyadic truth predicate is one that applies to its main subject only in reference to something else, a second subject. Most commonly, the auxiliary subject is either an object, an interpreter, or a language to which the representation bears some relation.

3. A triadic truth predicate is one that applies to its main subject only in reference to a second and a third subject. For example, in a pragmatic theory of truth, one has to specify both the object of the sign, and either its interpreter or another sign called the interpretant before one can say that the sign is true of its object to its interpreting agent or sign.

Several qualifications must be kept in mind with respect to any such radically simple scheme of classification, as real practice seldom presents any pure types, and there are settings in which it is useful to speak of a theory of truth that is“almost”k-adic, or that“would be”k-adic if certain details can be abstracted away and neglected in a particular context of discussion. That said, given the generic division of truth predicates according to their arity, further species can be differentiated within each genus according to a number of more refined features. The truth predicate of interest in a typical correspondence theory of truth tells of a relation between representations and objective states of affairs, and is therefore expressed, for the most part, by a dyadic predicate. In general terms, one says that a representation is true of an objective situation, more briefly, that a sign is true of an object. The nature of the correspondence may vary from theory to theory in this family. The correspondence can be fairly arbitrary or it can take on the character of an analogy, an icon, or a morphism, whereby a representation is rendered true of its object by the existence of corresponding elements and a similar structure.

105.2 Peirce

Main article: Charles Sanders Peirce

Very little in Peirce's thought can be understood in its proper light without understanding that he thinks all thoughts are signs, and thus, according to his theory of thought, no thought is understandable outside the context of a sign relation. Sign relations taken collectively are the subject matter of a theory of signs. So Peirce's semiotic, his theory of sign relations, is key to understanding his entire philosophy of pragmatic thinking and thought. In his contribution to the article“Truth and Falsity and Error”for Baldwin's Dictionary of Philosophy and Psychology (1901),*[1] Peirce defines truth in the following way:

Truth is that concordance of an abstract statement with the ideal limit towards which endless inves- tigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth. (Peirce 1901, see Collected Papers (CP) 5.565).

This statement emphasizes Peirce's view that ideas of approximation, incompleteness, and partiality, what he describes elsewhere as fallibilism and“reference to the future”, are essential to a proper conception of truth. Although Peirce occasionally uses words like concordance and correspondence to describe one aspect of the pragmatic sign relation, he is also quite explicit in saying that definitions of truth based on mere correspondence are no more than nominal definitions, which he follows long tradition in relegating to a lower status than real definitions. 105.2. PEIRCE 417

That truth is the correspondence of a representation with its object is, as Kant says, merely the nominal definition of it. Truth belongs exclusively to propositions. A proposition has a subject (or set of subjects) and a predicate. The subject is a sign; the predicate is a sign; and the proposition is a sign that the predicate is a sign of that of which the subject is a sign. If it be so, it is true. But what does this correspondence or reference of the sign, to its object, consist in? (Peirce 1906, CP 5.553).

Here Peirce makes a statement that is decisive for understanding the relationship between his pragmatic definition of truth and any theory of truth that leaves it solely and simply a matter of representations corresponding with their objects. Peirce, like Kant before him, recognizes Aristotle's distinction between a nominal definition, a definition in name only, and a real definition, one that states the function of the concept, the reason for conceiving it, and so indicates the essence, the underlying substance of its object. This tells us the sense in which Peirce entertained a correspondence theory of truth, namely, a purely nominal sense. To get beneath the superficiality of the nominal definition it is necessary to analyze the notion of correspondence in greater depth. In preparing for this task, Peirce makes use of an allegorical story, omitted here, the moral of which is that there is no use seeking a conception of truth that we cannot conceive ourselves being able to capture in a humanly conceivable concept. So we might as well proceed on the assumption that we have a real hope of comprehending the answer, of being able to “handle the truth”when the time comes. Bearing that in mind, the problem of defining truth reduces to the following form:

Now thought is of the nature of a sign. In that case, then, if we can find out the right method of thinking and can follow it out —the right method of transforming signs —then truth can be nothing more nor less than the last result to which the following out of this method would ultimately carry us. In that case, that to which the representation should conform, is itself something in the nature of a representation, or sign —something noumenal, intelligible, conceivable, and utterly unlike a thing-in- itself. (Peirce 1906, CP 5.553).

Peirce's theory of truth depends on two other, intimately related subject matters, his theory of sign relations and his theory of inquiry. Inquiry is a special case of semiosis, a process that transforms signs into signs while maintaining a specific relationship to an object, which object may be located outside the trajectory of signs or else be found at the end of it. Inquiry includes all forms of belief revision and logical inference, including scientific method, what Peirce here means by“the right method of transforming signs”. A sign-to-sign transaction relating to an object is a transaction that involves three parties, or a relation that involves three roles. This is called a ternary or triadic relation in logic. Consequently, pragmatic theories of truth are largely expressed in terms of triadic truth predicates. The statement above tells us one more thing: Peirce, having started out in accord with Kant, is here giving notice that he is parting ways with the Kantian idea that the ultimate object of a representation is an unknowable thing-in-itself. Peirce would say that the object is knowable, in fact, it is known in the form of its representation, however imperfectly or partially. Reality and truth are coordinate concepts in pragmatic thinking, each being defined in relation to the other, and both together as they participate in the time evolution of inquiry. Inquiry is not a disembodied process, nor the occupation of a singular individual, but the common life of an unbounded community.

The real, then, is that which, sooner or later, information and reasoning would finally result in, and which is therefore independent of the vagaries of me and you. Thus, the very origin of the conception of reality shows that this conception essentially involves the notion of a COMMUNITY, without definite limits, and capable of an indefinite increase of knowledge. (Peirce 1868, CP 5.311).

Different minds may set out with the most antagonistic views, but the progress of investigation carries them by a force outside of themselves to one and the same conclusion. This activity of thought by which we are carried, not where we wish, but to a foreordained goal, is like the operation of destiny. No modification of the point of view taken, no selection of other facts for study, no natural bent of mind even, can enable a man to escape the predestinate opinion. This great law is embodied in the conception of truth and reality. The opinion which is fated to be ultimately agreed to by all who investigate, is what we mean by the truth, and the object represented in this opinion is the real. That is the way I would explain reality. (Peirce 1878, CP 5.407). 418 CHAPTER 105. PRAGMATIC THEORY OF TRUTH

105.3 James

Main article: William James

William James's version of the pragmatic theory is often summarized by his statement that “the 'true' is only the expedient in our way of thinking, just as the 'right' is only the expedient in our way of behaving.”*[2] By this, James meant that truth is a quality the value of which is confirmed by its effectiveness when applying concepts to actual practice (thus, “pragmatic”). James's pragmatic theory is a synthesis of correspondence theory of truth and coherence theory of truth, with an added dimension. Truth is verifiable to the extent that thoughts and statements correspond with actual things, as well as“hangs together,”or coheres, fits as pieces of a puzzle might fit together, and these are in turn verified by the observed results of the application of an idea to actual practice.*[2]*[3]*[4]*[5]*[6] James said that“all true processes must lead to the face of directly verifying sensible experiences somewhere.”*[7] He also extended his pragmatic theory well beyond the scope of scientific verifiability, and even into the realm of the mystical: “On pragmatic principles, if the hypothesis of God works satisfactorily in the widest sense of the word, then it is 'true.' "*[3]

Truth, as any dictionary will tell you, is a property of certain of our ideas. It means their 'agree- ment', as falsity means their disagreement, with 'reality'. Pragmatists and intellectualists both accept this definition as a matter of course. They begin to quarrel only after the question is raised as to what may precisely be meant by the term 'agreement', and what by the term 'reality', when reality is taken as something for our ideas to agree with. (James 1907, 198).

William James (1907) begins his chapter on“Pragmatism's Conception of Truth”in much the same letter and spirit as the above selection from Peirce (1906), noting the nominal definition of truth as a plausible point of departure, but immediately observing that the pragmatist's quest for the meaning of truth can only begin, not end there.

The popular notion is that a true idea must copy its reality. Like other popular views, this one follows the analogy of the most usual experience. Our true ideas of sensible things do indeed copy them. Shut your eyes and think of yonder clock on the wall, and you get just such a true picture or copy of its dial. But your idea of its 'works' (unless you are a clockmaker) is much less of a copy, yet it passes muster, for it in no way clashes with reality. Even though it should shrink to the mere word 'works', that word still serves you truly; and when you speak of the 'time-keeping function' of the clock, or of its spring's 'elasticity', it is hard to see exactly what your ideas can copy. (James 1907, 199).

James exhibits a knack for popular expression that Peirce seldom sought, and here his analysis of correspondence by way of a simple thought experiment cuts right to the quick of the first major question to ask about it, namely: To what extent is the notion of correspondence involved in truth covered by the ideas of analogues, copies, or iconic images of the thing represented? The answer is that the iconic aspect of correspondence can be taken literally only in regard to sensory experiences of the more precisely eidetic sort. When it comes to the kind of correspondence that might be said to exist between a symbol, a word like “works”, and its object, the springs and catches of the clock on the wall, then the pragmatist recognizes that a more than nominal account of the matter still has a lot more explaining to do.

105.3.1 Making truth

Instead of truth being ready-made for us, James asserts we and reality jointly“make”truth. This idea has two senses: (1) truth is mutable, (often attributed to William James and F.C.S. Schiller); and (2) truth is relative to a conceptual scheme (more widely accepted in Pragmatism). (1) Mutability of truth “Truth”is not readily defined in Pragmatism. Can beliefs pass from being true to being untrue and back? For James, beliefs are not true until they have been made true by verification. James believed propositions become true over the long term through proving their utility in a person's specific situation. The opposite of this process is not falsification, but rather the belief ceases to be a “live option.”F.C.S. Schiller, on the other hand, clearly asserted beliefs could pass into and out of truth on a situational basis. Schiller held that truth was relative to specific problems. If I want to know how to return home safely, the true answer will be whatever is useful to solving that problem. Later on, when 105.4. DEWEY 419 faced with a different problem, what I came to believe with the earlier problem may now be false. As my problems change, and as the most useful way to solve a problem shifts, so does the property of truth. C.S. Peirce considered the idea that beliefs are true at one time but false at another (or true for one person but false for another) to be one of the “seeds of death”*[8] by which James allowed his pragmatism to become“infected.” For Peirce the pragmatic view implies theoretical claims should be tied to verification processes (i.e. they should be subject to test). They shouldn't be tied to our specific problems or life needs. Truth is defined, for Peirce, as what would be the ultimate outcome (not any outcome in real time) of inquiry by a (usually scientific) community of investigators. John Dewey, while agreeing with this definition, also characterized truthfulness as a species of the good: if something is true it is trustworthy and reliable and will remain so in every conceivable situation. Both Peirce and Dewey connect the definitions of truth and warranted assertability. Hilary Putnam also developed his internal realism around the idea a belief is true if it is ideally justified in epistemic terms. About James' and Schiller's view, Putnam says:

Truth cannot simply be rational acceptability for one fundamental reason; truth is supposed to be a property of a statement that cannot be lost, whereas justification can be lost. The statement 'The earth is flat' was, very likely, rationally acceptable 3000 years ago; but it is not rationally acceptable today. Yet it would be wrong to say that 'the earth is flat' was true 3,000 years ago; for that would mean that the earth has changed its shape. (Putnam 1981, p. 55)

Rorty has also weighed in against James and Schiller:

Truth is, to be sure, an absolute notion, in the following sense: “true for me but not for you”and “true in my culture but not in yours”are weird, pointless locutions. So is “true then, but not now.” [...] James would, indeed, have done better to say that phrases like“the good in the way of belief”and “what it is better for us to believe”are interchangeable with“justified”rather than with“true.”(Rorty 1998, p. 2)

(2) Conceptual relativity With James and Schiller we make things true by verifying them—a view rejected by most pragmatists. However, nearly all pragmatists do accept the idea there can be no truths without a conceptual scheme to express those truths. That is,

Unless we decide upon how we are going to use concepts like 'object', 'existence' etc., the question 'how many objects exist' does not really make any sense. But once we decide the use of these concepts, the answer to the above-mentioned question within that use or 'version', to put in Nelson Goodman's phrase, is no more a matter of 'convention'. (Maitra 2003 p. 40)

F.C.S. Schiller used the analogy of a chair to make clear what he meant by the phrase that truth is made: just as a carpenter makes a chair out of existing materials and doesn't create it out of nothing, truth is a transformation of our experience—but this doesn't imply reality is something we're free to construct or imagine as we please.

105.4 Dewey

Main article: John Dewey

John Dewey, less broadly than William James but much more broadly than Charles Peirce, held that inquiry, whether scientific, technical, sociological, philosophical or cultural, is self-corrective over time if openly submitted for testing by a community of inquirers in order to clarify, justify, refine and/or refute proposed truths.*[9] In his Logic: The Theory of Inquiry (1938), Dewey gave the following definition of inquiry:

Inquiry is the controlled or directed transformation of an indeterminate situation into one that is so determinate in its constituent distinctions and relations as to convert the elements of the original situation into a unified whole. (Dewey, p. 108).

The index of the same book has exactly one entry under the heading truth, and it refers to the following footnote: 420 CHAPTER 105. PRAGMATIC THEORY OF TRUTH

The best definition of truth from the logical standpoint which is known to me is that by Peirce:“The opinion which is fated to be ultimately agreed to by all who investigate is what we mean by the truth, and the object represented in this opinion is the real [CP 5.407]. (Dewey, 343 n).

Dewey says more of what he understands by truth in terms of his preferred concept of warranted assertibility as the end-in-view and conclusion of inquiry (Dewey, 14–15).

105.5 Mead

Main article:

105.6 Criticisms

Several objections are commonly made to pragmatist account of truth, of either sort. First, due originally to Bertrand Russell (1907) in a discussion of James's theory, is that pragmatism mixes up the notion of truth with epistemology. Pragmatism describes an indicator or a sign of truth. It really cannot be regarded as a theory of the meaning of the word “true”. There's a difference between stating an indicator and giving the meaning. For example, when the streetlights turn on at the end of a day, that's an indicator, a sign, that evening is coming on. It would be an obvious mistake to say that the word“evening”just means“the time that the streetlights turn on”. In the same way, while it might be an indicator of truth, that a proposition is part of that perfect science at the ideal limit of inquiry, that just isn't what “true”means. Russell's objection is that pragmatism mixes up an indicator of truth with the meaning of the predicate 'true'. There is a difference between the two and pragmatism confuses them. In this pragmatism is akin to Berkeley's view that to be is to be perceived, which similarly confuses an indication or proof of that something exists with the meaning of the word 'exists', or with what it is for something to exist. Other objections to pragmatism include how we define what it means to say a belief “works”, or that it is “useful to believe”. The vague usage of these terms, first popularized by James, has led to much debate. A final objection is that pragmatism of James's variety entails relativism. What is useful for you to believe might not be useful for me to believe. It follows that “truth”for you is different from “truth”for me (and that the relevant facts don't matter). This is relativism. A viable, more sophisticated consensus theory of truth, a mixture of Peircean theory with speech-act theory and social theory, is that presented and defended by Jürgen Habermas, which sets out the universal pragmatic conditions of ideal consensus and responds to many objections to earlier versions of a pragmatic, consensus theory of truth. Habermas distinguishes explicitly between factual consensus, i.e. the beliefs that happen to hold in a particular community, and rational consensus, i.e. consensus attained in conditions approximating an "ideal speech situation", in which inquirers or members of a community suspend or bracket prevailing beliefs and engage in rational discourse aimed at truth and governed by the force of the better argument, under conditions in which all participants in discourse have equal opportunities to engage in constative (assertions of fact), normative, and expressive speech acts, and in which discourse is not distorted by the intervention of power or the internalization of systematic blocks to communication. Recent Peirceans, Cheryl Misak, and Robert B. Talisse have attempted to formulate Peirce's theory of truth in a way that improves on Habermas and provides an epistemological conception of deliberative democracy.

105.7 Notes and references

[1] Peirce, C.S. (1901), “Truth and Falsity and Error”(in part), pp. 716–720 in James Mark Baldwin, ed., Dictionary of Philosophy and Psychology, v. 2. Peirce's section is entitled "Logical", beginning on p. 718, column 1, and ending on p. 720 with the initials "(C.S.P.)", see Google Books Eprint. Reprinted, Collected Papers v. 5, pp. 565–573.

[2] James, William. The Meaning of Truth (1909).

[3] James, William. Pragmatism, 1907 105.7. NOTES AND REFERENCES 421

[4] James, William. A World of Pure Experience (1904).

[5] James, William. Essays in Radical Empiricism, Ch.3: “The Thing and its Relations”(1912): 92-122.

[6] Encyclopedia of Philosophy, Vol.6, “Pragmatic Theory of Truth”, p427-428 (Macmillan, 1969)

[7] James, William, Pragmatism: A New Name for Some Old Ways of Thinking Lect. 6,“Pragmatism's Conception of Truth,” (1907)

[8] See Peirce's 1908 "A Neglected Argument for the Reality of God", final paragraph.

[9] Encyclopedia of Philosophy, Vol.2, “Dewey, John”p383 (Macmillan, 1969)

• Allen, James Sloan, ed. William James on Habit, Will, Truth, and the . Frederic C. Beil, Publisher, Savannah, GA.

• Awbrey, Jon, and Awbrey, Susan (1995), “Interpretation as Action: The Risk of Inquiry”, Inquiry: Critical Thinking Across the Disciplines 15, 40–52. Eprint

• Baldwin, J.M. (1901–1905), Dictionary of Philosophy and Psychology, 3 volumes in 4, New York, NY.

• Dewey, John (1929), The Quest for Certainty: A Study of the Relation of Knowledge and Action, Minton, Balch, and Company, New York, NY. Reprinted, pp. 1–254 in John Dewey, The Later Works, 1925–1953, Volume 4: 1929, Jo Ann Boydston (ed.), Harriet Furst Simon (text. ed.), Stephen Toulmin (intro.), Southern Illinois University Press, Carbondale and Edwardsville, IL, 1984.

• Dewey, John (1938), Logic: The Theory of Inquiry, Henry Holt and Company, New York, NY, 1938. Reprinted, pp. 1–527 in John Dewey, The Later Works, 1925–1953, Volume 12: 1938, Jo Ann Boydston (ed.), Kathleen Poulos (text. ed.), Ernest Nagel (intro.), Southern Illinois University Press, Carbondale and Edwardsville, IL, 1986.

• Ferm, Vergilius (1962), “Consensus Gentium”, p. 64 in Runes (1962).

• Haack, Susan (1993), Evidence and Inquiry: Towards Reconstruction in Epistemology, Blackwell Publishers, Oxford, UK.

• Habermas, Jürgen (1976),“What Is Universal Pragmatics?", 1st published,“Was heißt Universalpragmatik?", Sprachpragmatik und Philosophie, Karl-Otto Apel (ed.), Suhrkamp Verlag, Frankfurt am Main. Reprinted, pp. 1–68 in Jürgen Habermas, Communication and the Evolution of Society, Thomas McCarthy (trans.), Beacon Press, Boston, MA, 1979.

• Habermas, Jürgen (1979), Communication and the Evolution of Society, Thomas McCarthy (trans.), Beacon Press, Boston, MA.

• Habermas, Jürgen (1990), Moral Consciousness and Communicative Action, Christian Lenhardt and Shierry Weber Nicholsen (trans.), Thomas McCarthy (intro.), MIT Press, Cambridge, MA.

• Habermas, Jürgen (2003), Truth and Justification, Barbara Fultner (trans.), MIT Press, Cambridge, MA.

• James, William (1907), Pragmatism, A New Name for Some Old Ways of Thinking, Popular Lectures on Phi- losophy, Longmans, Green, and Company, New York, NY.

• James, William (1909), The Meaning of Truth, A Sequel to 'Pragmatism', Longmans, Green, and Company, New York, NY.

• Kant, Immanuel (1800), Introduction to Logic. Reprinted, Thomas Kingsmill Abbott (trans.), Dennis Sweet (intro.), Barnes and Noble, New York, NY, 2005. 422 CHAPTER 105. PRAGMATIC THEORY OF TRUTH

• Peirce, C.S., Writings of Charles S. Peirce, A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianoplis, IN, 1981–. Volume 1 (1857–1866), 1981. Volume 2 (1867– 1871), 1984. Volume 3 (1872–1878), 1986. Cited as W volume:page.

• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Cited as CP vol.para.

• Peirce, C.S., The Essential Peirce, Selected Philosophical Writings, Volume 1 (1867–1893), Nathan Houser and Christian Kloesel (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1992. Cited as EP 1:page.

• Peirce, C.S., The Essential Peirce, Selected Philosophical Writings, Volume 2 (1893–1913), Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1998. Cited as EP 2:page.

• Peirce, C.S. (1868),“Some Consequences of Four Incapacities”, Journal of Speculative Philosophy 2 (1868), 140–157. Reprinted (CP 5.264–317), (W 2:211–242), (EP 1:28–55). Eprint. NB. Misprints in CP and Eprint copy.

• Peirce, C.S. (1877), "The Fixation of Belief", Popular Science Monthly 12 (1877), 1–15. Reprinted (CP 5.358– 387), (W 3:242–257), (EP 1:109–123). Eprint.

• Peirce, C.S. (1878), "How to Make Our Ideas Clear", Popular Science Monthly 12 (1878), 286–302. Reprinted (CP 5.388–410), (W 3:257–276)), (EP 1:124–141).

• Peirce, C.S. (1901), section entitled "Logical", pp. 718–720 in “Truth and Falsity and Error”, pp. 716–720 in J.M. Baldwin (ed.), Dictionary of Philosophy and Psychology, vol. 2. Google Books Eprint. Reprinted (CP 5.565–573).

• Peirce, C.S. (1905), “What Pragmatism Is”, The Monist 15, 161–181. Reprinted (CP 5.411–437), (EP 2:331–345). Internet Archive Eprint.

• Peirce, C.S. (1906),“Basis of Pragmaticism”, first published in Collected Papers, CP 1.573–574 and 5.549– 554.

• Rescher, Nicholas (1995), Pluralism: Against the Demand for Consensus, Oxford University Press, Oxford, UK.

• Rorty, R. (1979), Philosophy and the Mirror of Nature, Princeton University Press, Princeton, NJ.

• Runes, Dagobert D. (ed., 1962), Dictionary of Philosophy, Littlefield, Adams, and Company, Totowa, NJ. Cited as DOP.

105.8 See also

105.8.1 See also Chapter 106

Principle of bivalence

This article is about logical principle. For chemical meaning (an atom with 2 bonds), see Bivalent (chemistry).

In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false.*[1]*[2] A logic satisfying this principle is called a two-valued logic*[3] or bivalent logic.*[2]*[4] In formal logic, the principle of bivalence becomes a property that a semantics may or may not possess. It is not the same as the law of excluded middle, however, and a semantics may satisfy that law without being bivalent.*[2] It may be written in the second-order sentence as: ∀P ∀x(x ∈ P ∨ x ∈/ P ) , demonstrating similarity yet differing mainly by quantified set elements. The principle of bivalence is studied in philosophical logic to address the question of which natural-language state- ments have a well-defined truth value. Sentences which predict events in the future, and sentences which seem open to interpretation, are particularly difficult for philosophers who hold that the principle of bivalence applies to all declarative natural-language statements.*[2] Many-valued logics formalize ideas that a realistic characterization of the notion of consequence requires the admissibility of premises which, owing to vagueness, temporal or quantum in- determinacy, or reference-failure, cannot be considered classically bivalent. Reference failures can also be addressed by free logics.*[5]

106.1 Relationship with the law of the excluded middle

The principle of bivalence is related to the law of excluded middle though the latter is a syntactic expression of the language of a logic of the form “P ∨ ¬P”. The difference between the principle and the law is important because there are logics which validate the law but which do not validate the principle.*[2] For example, the three-valued Logic of Paradox (LP) validates the law of excluded middle, but not the law of non-contradiction, ¬(P ∧ ¬P), and its intended semantics is not bivalent.*[6] In classical two-valued logic both the law of excluded middle and the law of non-contradiction hold.*[1] Many modern logic programming systems replace the law of the excluded middle with the concept of negation as failure. The programmer may wish to add the law of the excluded middle by explicitly asserting it as true; however, it is not assumed a priori.

106.2 Classical logic

The intended semantics of classical logic is bivalent, but this is not true of every semantics for classical logic. In Boolean-valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra,“true”corresponds to the maximal element of the algebra, and“false”corresponds to the minimal element. Intermediate elements of the algebra correspond to truth values other than “true”and “false”. The principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements.

423 424 CHAPTER 106. PRINCIPLE OF BIVALENCE

Assigning Boolean semantics to classical predicate calculus requires that the model be a complete Boolean algebra be- cause the universal quantifier maps to the infimum operation, and the existential quantifier maps to the supremum;*[7] this is called a Boolean-valued model. All finite Boolean algebras are complete.

106.3 Suszko's thesis

In order to justify his claim that true and false are the only logical values, Suszko (1977) observes that every structural Tarskian many-valued propositional logic can be provided with a bivalent semantics.*[8]

106.4 Criticisms

106.4.1 Future contingents

Main article: Problem of future contingents

A famous example*[2] is the contingent sea battle case found in Aristotle's work, De Interpretatione, chapter 9:

Imagine P refers to the statement “There will be a sea battle tomorrow.”

The principle of bivalence here asserts:

Either it is true that there will be a sea battle tomorrow, or it is false that there will be a sea battle tomorrow.

Aristotle hesitated to embrace bivalence for such future contingents; Chrysippus, the Stoic logician, did embrace bivalence for this and all other propositions. The controversy continues to be of central importance in both the philosophy of time and the philosophy of logic. One of the early motivations for the study of many-valued logics has been precisely this issue. In the early 20th century, the Polish formal logician Jan Łukasiewicz proposed three truth-values: the true, the false and the as-yet- undetermined. This approach was later developed by Arend Heyting and L. E. J. Brouwer;*[2] see Łukasiewicz logic. Issues such as this have also been addressed in various temporal logics, where one can assert that "Eventually, either there will be a sea battle tomorrow, or there won't be.”(Which is true if “tomorrow”eventually occurs.)

106.4.2 Vagueness

Such puzzles as the Sorites paradox and the related continuum fallacy have raised doubt as to the applicability of classical logic and the principle of bivalence to concepts that may be vague in their application. Fuzzy logic and some other multi-valued logics have been proposed as alternatives that handle vague concepts better. Truth (and falsity) in fuzzy logic, for example, comes in varying degrees. Consider the following statement in the circumstance of sorting apples on a moving belt:

This apple is red.*[9]

Upon observation, the apple is an undetermined color between yellow and red, or it is motled both colors. Thus the color falls into neither category " red " nor " yellow ", but these are the only categories available to us as we sort the apples. We might say it is “50% red”. This could be rephrased: it is 50% true that the apple is red. Therefore, P is 50% true, and 50% false. Now consider:

This apple is red and it is not-red. 106.5. SEE ALSO 425

In other words, P and not-P. This violates the law of noncontradiction and, by extension, bivalence. However, this is only a partial rejection of these laws because P is only partially true. If P were 100% true, not-P would be 100% false, and there is no contradiction because P and not-P no longer holds. However, the law of the excluded middle is retained, because P and not-P implies P or not-P, since“or”is inclusive. The only two cases where P and not-P is false (when P is 100% true or false) are the same cases considered by two-valued logic, and the same rules apply. Example of a 3-valued logic applied to vague (undetermined) cases: Kleene 1952*[10] (§64, pp. 332–340) offers a 3-valued logic for the cases when algorithms involving partial recursive functions may not return values, but rather end up with circumstances“u”= undecided. He lets“t”=“true”,“f”=“false”,“u”=“undecided” and redesigns all the propositional connectives. He observes that:

We were justified intuitionistically in using the classical 2-valued logic, when we were using the connectives in building primitive and general recursive predicates, since there is a decision procedure for each general recursive predicate; i.e. the law of the excluded middle is proved intuitionistically to apply to general recursive predicates. Now if Q(x) is a partial recursive predicate, there is a decision procedure for Q(x) on its range of definition, so the law of the excluded middle or excluded “third”(saying that, Q(x) is either t or f) applies intuitionistically on the range of definition. But there may be no algorithm for deciding, given x, whether Q(x) is defined or not. […] Hence it is only classically and not intuitionistically that we have a law of the excluded fourth (saying that, for each x, Q(x) is either t, f, or u). The third “truth value”u is thus not on par with the other two t and f in our theory. Consideration of its status will show that we are limited to a special kind of truth table”.

The following are his “strong tables":*[11] For example, if a determination cannot be made as to whether an apple is red or not-red, then the truth value of the assertion Q: " This apple is red " is " u ". Likewise, the truth value of the assertion R " This apple is not-red " is " u ". Thus the AND of these into the assertion Q AND R, i.e. " This apple is red AND this apple is not-red " will, per the tables, yield " u ". And, the assertion Q OR R, i.e. " This apple is red OR this apple is not-red " will likewise yield " u ".

106.5 See also

• Dualism

• Exclusive disjunction

• Degrees of truth

•

• Extensionality

• False dilemma

• Fuzzy logic

• Logical disjunction

• Logical equality

• Logical value

• Multi-valued logic

• Propositional logic

• Relativism

• Supervaluationism 426 CHAPTER 106. PRINCIPLE OF BIVALENCE

• Truthbearer • Truthmaker • Truth-value link • Quantum logic • Perspectivism • Rhizome (philosophy) • True and false

106.6 References

[1] Lou Goble (2001). The Blackwell guide to philosophical logic. Wiley-Blackwell. p. 309. ISBN 978-0-631-20693-4.

[2] Paul Tomassi (1999). Logic. Routledge. p. 124. ISBN 978-0-415-16696-6.

[3] Lou Goble (2001). The Blackwell guide to philosophical logic. Wiley-Blackwell. p. 4. ISBN 978-0-631-20693-4.

[4] Mark Hürlimann (2009). Dealing with Real-World Complexity: Limits, Enhancements and New Approaches for Policy Makers. Gabler Verlag. p. 42. ISBN 978-3-8349-1493-4.

[5] Dov M. Gabbay; John Woods (2007). The Many Valued and Nonmonotonic Turn in Logic. The handbook of the history of logic 8. Elsevier. p. vii. ISBN 978-0-444-51623-7. Retrieved 4 April 2011.

[6] Graham Priest (2008). An introduction to non-classical logic: from if to is. Cambridge University Press. pp. 124–125. ISBN 978-0-521-85433-7.

[7] Morten Heine Sørensen; Paweł Urzyczyn (2006). Lectures on the Curry-Howard isomorphism. Elsevier. pp. 206–207. ISBN 978-0-444-52077-7.

[8] Shramko, Y.; Wansing, H. (2015). "Truth Values, Stanford Encyclopedia of Philosophy”.

[9] Note the use of the (extremely) definite article: " This " as opposed to a more-vague " The ". " The " would have to be accompanied with a pointing-gesture to make it definitive. Ff Principia Mathematica (2nd edition), p. 91. Russell & Whitehead observe that this " this " indicates“something given in sensation”and as such it shall be considered“elementary” .

[10] Stephen C. Kleene 1952 Introduction to Metamathematics, 6th Reprint 1971, North-Holland Publishing Company, Ams- terdam NY, ISBN 0-7294-2130-9.

[11]“Strong tables”is Kleene's choice of words. Note that even though " u " may appear for the value of Q or R, " t " or " f " may, in those occasions, appear as a value in " Q V R ", " Q & R " and " Q → R ". “Weak tables”on the other hand, are “regular”, meaning they have " u " appear in all cases when the value " u " is applied to either Q or R or both. Kleene notes that these tables are not the same as the original values of the tables of Łukasiewicz 1920. (Kleene gives these differences on page 335). He also concludes that " u " can mean any or all of the following: “undefined”, “unknown (or value immaterial)", “value disregarded for the moment”, i.e. it is a third category that does not (ultimately) exclude " t " and " f " (page 335).

106.7 Further reading

• Devidi, D.; Solomon, G. (1999). “On Confusions About Bivalence and Excluded Middle”. Dialogue (in French) 38 (4): 785–799. doi:10.1017/S0012217300006715.. • Betti Arianna (2002) The Incomplete Story of Łukasiewicz and Bivalence in T. Childers (ed.) The Logica 2002 Yearbook, Prague: The Czech Academy of Sciences—Filosofia, pp. 21–26 • Jean-Yves Béziau (2003) "Bivalence, excluded middle and non contradiction", in The Logica Yearbook 2003, L.Behounek (ed), Academy of Sciences, Prague, pp. 73–84. • Font, J. M. (2009).“Taking Degrees of Truth Seriously”. Studia Logica 91 (3): 383–406. doi:10.1007/s11225- 009-9180-7. 106.8. EXTERNAL LINKS 427

106.8 External links

• Truth Values entry by Yaroslav Shramko, Heinrich Wansing in the Stanford Encyclopedia of Philosophy Chapter 107

Principle of nonvacuous contrast

The principle of nonvacuous contrast is a logical or methodological principle which requires that a genuine predicate never refer to everything, or to nothing, within its universe of discourse.

107.1 References

• William Dray (1964). . Englewood Cliffs, NJ: Prentice-Hall, Inc. pp. 29 pp.

428 Chapter 108

Propositional representation

Propositional representation is the psychological theory, first developed in 1973 by Dr. Zenon Pylyshyn, that mental relationships between objects are represented by symbols and not by mental images of the scene.*[1]

108.1 Examples

A propositional network describing the sentence “John believes that Anna will pass her exam”is illustrated below.

Figure 1: A Propositional Network

Each circle represents a single proposition, and the connections between the circles describe a network of propositions. Another example is the sentence “Debby donated a big amount of money to Greenpeace, an organisation which protects the environment”, which contains the propositions“Debby donated money to Greenpeace”,“The amount of money was big”and “Greenpeace protects the environment”. If one or more of the propositions is false, the whole sentence is false. This is illustrated in Figure 2: Propositional representations are also:

• Language-like only in the sense that they manipulate symbols as a language does. The language of thought cannot be thought of as a natural language; it can only be a formal language that applies across different linguistic subjects, it therefore must be a language common to mind rather than culture, must be organizational rather than communicative. Thus Mentalese is best expressed through predicate and propositional calculus.

429 430 CHAPTER 108. PROPOSITIONAL REPRESENTATION

Figure 2: A more complex propositional network

• Made up of discrete symbols; each symbol has a smallest constituent part; i.e. limit to how far units of rep. can be broken down.

• Explicit; each symbol represents something (object, action, relation) specifically and thus explicitly.

• Grammatical; symbolic manipulation follows (requires?) syntactical rules and semantical rules.

• Abstract and amodal; symbols may represent any ideational content irrespective of which sensory modality was involved in its perception. (Unlike a pictorial representation which must be modality specific to the visual sensory mode).

Each proposition consists of a set of predicates and arguments which are represented in the form of predicate calculus. For instance: An event; (X) John hit Chris with a unicycle, the unicycle broke, because of this John started to cry, which caused Chris to be happy. A propositional representation

• P [hit (John, Chris, unicycle)]

• Q [broke (unicycle)]

• R [cry (John)]

• S [happy (Chris)]

• Cause (Q,R)

• Cause (R,S) 108.2. REFERENCES 431

Each set of predicates (words like hit, broke, cry, happy are first order-predicates; Cause is a second-order predicate) and arguments (often consisting of an agent/subject (e.g. John in‘P’), a recipient/object (e.g. Chris in‘P’) and an instrument (e.g. the unicycle in ‘P’)) are in turn manipulated as propositions: event/statement“John hit Chris with the unicycle”is represented as proposition ‘P’. Also, features of particular objects may be characterized through attribute lists.‘John’as a singular object may have the attributes‘plays guitar’,‘juggles’,‘eats a lot’,‘rides a unicycle’etc. Thus reference to‘John’identifies him as the object of thought in virtue of his having certain of these attributes. So in predicate calculus, if “John (F) has the property of being‘rides a unicycle’(x)”we may say salva veritate: (x)(Fx). These elements have been called semantic primitives or semantic markers/features. Each primitive may in turn form part of a propositional statement, which in turn could be represented by an abstract figure e.g. ‘P’. The primitives themselves play a crucial role in categorizing and classifying objects and concepts. The meaningful relations between ideas and concepts expressed between and within the propositions are in part dealt with through the general laws of inference. One of the most common of these is Modus Ponens Ponendum (MPP), which is a simple inference of relation between two objects, the latter supervening on the former (P-›Q). Thus if we have two propositions (P, Q) and we assume a law of inference that relates to them both (P-›Q), then if we have P we must necessarily have Q. Relations of causation and may be expressed in this fashion, i.e. one state (P) causing (-›) another (Q) So a purely formal characterization of the event (X) written above in natural language would be something like:

• P, Q (A)

• Q -› R (A)

• Q (A1)

• R (2,3 MPP)

• R -› S (A)

• S (4,5 MPP)

108.2 References

[1] Elport, Daniel “Cognitive Psychology and Cognitive Neuroscience”, Wikibooks, July 2007, accessed March 07, 2011. Chapter 109

Quantifier (logic)

In logic, quantification is a construct that specifies the quantity of specimens in the domain of discourse that satisfy an open formula. For example, in arithmetic, it allows the expression of the statement that every natural number has a successor. A language element which generates a quantification (such as “every”) is called a quantifier. The resulting expression is a quantified expression, it is said to be quantified over the predicate (such as “the natural number x has a successor”) whose free variable is bound by the quantifier. In formal languages, quantification is a formula constructor that produces new formulas from old ones. The semantics of the language specifies how the constructor is interpreted. Two fundamental kinds of quantification in predicate logic are universal quantification and existential quantification. The traditional symbol for the universal quantifier “all”is "∀", a rotated letter "A", and for the existential quantifier “exists”is "∃", a rotated letter "E". These quantifiers have been generalized beginning with the work of Mostowski and Lindström. Quantification is used as well in natural languages; examples of quantifiers in English are for all, for some, many, few, a lot, and no; see Quantifier (linguistics) for details.

109.1 Mathematics

Consider the following statement:

1 ·2 = 1 + 1, and 2 ·2 = 2 + 2, and 3 ·2 = 3 + 3, ..., and 100 ·2 = 100 + 100, and ..., etc.

This has the appearance of an infinite conjunction of propositions. From the point of view of formal languages this is immediately a problem, since syntax rules are expected to generate finite objects. The example above is fortunate in that there is a procedure to generate all the conjuncts. However, if an assertion were to be made about every irrational number, there would be no way to enumerate all the conjuncts, since irrationals cannot be enumerated. A succinct formulation which avoids these problems uses universal quantification:

For each natural number n, n ·2 = n + n.

A similar analysis applies to the disjunction,

1 is equal to 5 + 5, or 2 is equal to 5 + 5, or 3 is equal to 5 + 5, ... , or 100 is equal to 5 + 5, or ..., etc.

which can be rephrased using existential quantification:

For some natural number n, n is equal to 5+5.

109.2 Algebraic approaches to quantification

It is possible to devise abstract algebras whose models include formal languages with quantification, but progress has been slow and interest in such algebra has been limited. Three approaches have been devised to date:

432 109.3. NOTATION 433

• Relation algebra, invented by Augustus De Morgan, and developed by Charles Sanders Peirce, Ernst Schröder, Alfred Tarski, and Tarski's students. Relation algebra cannot represent any formula with quantifiers nested more than three deep. Surprisingly, the models of relation algebra include the axiomatic set theory ZFC and Peano arithmetic; • Cylindric algebra, devised by Alfred Tarski, Leon Henkin, and others; • The polyadic algebra of Paul Halmos.

109.3 Notation

The two most common quantifiers are the universal quantifier and the existential quantifier. The traditional symbol for the universal quantifier is "∀", an inverted letter "A", which stands for “for all”or “all”. The corresponding symbol for the existential quantifier is "∃", a rotated letter "E", which stands for “there exists”or “exists”. An example of translating a quantified English statement would be as follows. Given the statement,“Each of Peter's friends either likes to dance or likes to go to the beach”, we can identify key aspects and rewrite using symbols including quantifiers. So, let X be the set of all Peter's friends, P(x) the predicate "x likes to dance”, and lastly Q(x) the predicate "x likes to go to the beach”. Then the above sentence can be written in formal notation as ∀x∈X,P (x) ∨ Q(x) , which is read, “for every x that is a member of X, P applies to x or Q applies to x.” Some other quantified expressions are constructed as follows,

∃x P ∀x P for a formula P. These two expressions (using the definitions above) are read as “there exists a friend of Peter who likes to dance”and “all friends of Peter like to dance”respectively. Variant notations include, for set X and set members x:

(∃x)P (∃x . P ) ∃x · P (∃x : P ) ∃x(P ) ∃x P ∃x, P ∃x∈XP ∃ x:XP All of these variations also apply to universal quantification. Other variations for the universal quantifier are

∧ (x) P P x Some versions of the notation explicitly mention the range of quantification. The range of quantification must always be specified; for a given mathematical theory, this can be done in several ways:

• Assume a fixed domain of discourse for every quantification, as is done in Zermelo–Fraenkel set theory, • Fix several domains of discourse in advance and require that each variable have a declared domain, which is the type of that variable. This is analogous to the situation in statically typed computer programming languages, where variables have declared types. • Mention explicitly the range of quantification, perhaps using a symbol for the set of all objects in that domain or the type of the objects in that domain.

One can use any variable as a quantified variable in place of any other, under certain restrictions in which variable capture does not occur. Even if the notation uses typed variables, variables of that type may be used. Informally or in natural language, the "∀x" or "∃x" might appear after or in the middle of P(x). Formally, however, the phrase that introduces the dummy variable is placed in front. Mathematical formulas mix symbolic expressions for quantifiers, with natural language quantifiers such as

For every natural number x, .... There exists an x such that .... For at least one x. 434 CHAPTER 109. (LOGIC)

Keywords for uniqueness quantification include:

For exactly one natural number x, .... There is one and only one x such that ....

Further, x may be replaced by a pronoun. For example,

For every natural number, its product with 2 equals to its sum with itself Some natural number is prime.

109.4 Nesting

The order of quantifiers is critical to meaning, as is illustrated by the following two propositions:

For every natural number n, there exists a natural number s such that s = n2.

This is clearly true; it just asserts that every natural number has a square. The meaning of the assertion in which the quantifiers are turned around is different:

There exists a natural number s such that for every natural number n, s = n2.

This is clearly false; it asserts that there is a single natural number s that is at the same time the square of every natural number. This is because the syntax directs that any variable cannot be a function of subsequently introduced variables. A less trivial example from mathematical analysis are the concepts of uniform and pointwise continuity, whose def- initions differ only by an exchange in the positions of two quantifiers. A function f from R to R is called

• pointwise continuous if ∀ε>0 ∀x∈R ∃δ>0 ∀h∈R (|h| < δ → |f(x) – f(x + h)| < ε) • uniformly continuous if ∀ε>0 ∃δ>0 ∀x∈R ∀h∈R (|h| < δ → |f(x) – f(x + h)| < ε)

In the former case, the particular value chosen for δ can be a function of both ε and x, the variables that precede it. In the latter case, δ can be a function only of ε, i.e. it has to be chosen independent of x. For example, f(x) = x2 satisfies pointwise, but not uniform continuity. In contrast, interchanging the two initial universal quantifiers in the definition of pointwise continuity does not change the meaning. The maximum depth of nesting of quantifiers inside a formula is called its quantifier rank.

109.5 Equivalent expressions

If D is a domain of x and P(x) is a predicate dependent on x, then the universal proposition can be expressed as

∀x∈DP (x)

This notation is known as restricted or relativized or bounded quantification. Equivalently,

∀x (x∈D → P (x))

The existential proposition can be expressed with bounded quantification as

∃x∈DP (x) 109.6. RANGE OF QUANTIFICATION 435

or equivalently

∃x (x∈D ∧ P (x))

Together with negation, only one of either the universal or existential quantifier is needed to perform both tasks:

¬(∀x∈DP (x)) ≡ ∃x∈D ¬P (x),

which shows that to disprove a “for all x" proposition, one needs no more than to find an x for which the predicate is false. Similarly,

¬(∃x∈DP (x)) ≡ ∀x∈D ¬P (x), to disprove a “there exists an x" proposition, one needs to show that the predicate is false for all x.

109.6 Range of quantification

Every quantification involves one specific variable and a domain of discourse or range of quantification of that variable. The range of quantification specifies the set of values that the variable takes. In the examples above, the range of quantification is the set of natural numbers. Specification of the range of quantification allows us to express the difference between, asserting that a predicate holds for some natural number or for some real number. Expository conventions often reserve some variable names such as "n" for natural numbers and "x" for real numbers, although relying exclusively on naming conventions cannot work in general since ranges of variables can change in the course of a mathematical argument. A more natural way to restrict the domain of discourse uses guarded quantification. For example, the guarded quan- tification

For some natural number n, n is even and n is prime means

For some even number n, n is prime.

In some mathematical theories a single domain of discourse fixed in advance is assumed. For example, in Zermelo– Fraenkel set theory, variables range over all sets. In this case, guarded quantifiers can be used to mimic a smaller range of quantification. Thus in the example above to express

For every natural number n, n·2 = n + n in Zermelo–Fraenkel set theory, it can be said

For every n, if n belongs to N, then n·2 = n + n,

where N is the set of all natural numbers.

109.7 Formal semantics

Mathematical Semantics is the application of mathematics to study the meaning of expressions in a formal language. It has three elements: A mathematical specification of a class of objects via syntax, a mathematical specification of various semantic domains and the relation between the two, which is usually expressed as a function from syntactic objects to semantic ones. This article only addresses the issue of how quantifier elements are interpreted. 436 CHAPTER 109. QUANTIFIER (LOGIC)

Given a model theoretical logical framework, the syntax of a formula can be given by a syntax tree. Quantifiers have scope and a variable x is free if it is not within the scope of a quantification for that variable. Thus in

∀x(∃yB(x, y)) ∨ C(y, x)

the occurrence of both x and y in C(y,x) is free.

Syntactic tree illustrating scope and variable capture

An interpretation for first-order predicate calculus assumes as given a domain of individuals X. A formula A whose free variables are x1, ..., xn is interpreted as a boolean-valued function F(v1, ..., vn) of n arguments, where each argument ranges over the domain X. Boolean-valued means that the function assumes one of the values T (interpreted as truth) or F (interpreted as falsehood). The interpretation of the formula

∀xnA(x1, . . . , xn) is the function G of n−1 arguments such that G(v1, ...,vn−1) = T if and only if F(v1, ..., vn−1, w) = T for every w in X. If F(v1, ..., vn−1, w) = F for at least one value of w, then G(v1, ...,vn−1) = F. Similarly the interpretation of the formula

∃xnA(x1, . . . , xn)

is the function H of n−1 arguments such that H(v1, ...,vn−1) = T if and only if F(v1, ...,vn−1, w) = T for at least one w and H(v1, ..., vn−1) = F otherwise. The semantics for uniqueness quantification requires first-order predicate calculus with equality. This means there is given a distinguished two-placed predicate "="; the semantics is also modified accordingly so that "=" is always interpreted as the two-place equality relation on X. The interpretation of

∃!xnA(x1, . . . , xn) 109.8. PAUCAL, MULTAL AND OTHER DEGREE QUANTIFIERS 437

then is the function of n−1 arguments, which is the logical and of the interpretations of

∃xnA(x1, . . . , xn)

∀y, z {A(x1, . . . , xn−1, y) ∧ A(x1, . . . , xn−1, z) =⇒ y = z} .

109.8 Paucal, multal and other degree quantifiers

See also: Fubini's theorem and measurable

None of the quantifiers previously discussed apply to a quantification such as

There are many integers n < 100, such that n is divisible by 2 or 3 or 5.

One possible interpretation mechanism can be obtained as follows: Suppose that in addition to a semantic domain X, we have given a probability measure P defined on X and cutoff numbers 0 < a ≤ b ≤ 1. If A is a formula with free variables x1,...,xn whose interpretation is the function F of variables v1,...,vn then the interpretation of

many ∃ xnA(x1, . . . , xn−1, xn) is the function of v1,...,vn−1 which is T if and only if

P{w : F (v1, . . . , vn−1, w) = T} ≥ b and F otherwise. Similarly, the interpretation of

few ∃ xnA(x1, . . . , xn−1, xn) is the function of v1,...,vn−1 which is F if and only if

0 < P{w : F (v1, . . . , vn−1, w) = T} ≤ a and T otherwise.

109.9 Other quantifiers

A few other quantifiers have been proposed over time. In particular, the solution quantifier,*[1] noted § (section sign) and read “those”. For example:

[ ] §n ∈ N n2 ≤ 4 = {0, 1, 2}

is read “those n in N such that n2 ≤ 4 are in {0,1,2}.”The same construct is expressible in set-builder notation:

{n ∈ N : n2 ≤ 4} = {0, 1, 2}

Some other quantifiers sometimes used in mathematics include:

• There are infinitely many elements such that... 438 CHAPTER 109. QUANTIFIER (LOGIC)

• For all but finitely many elements... (sometimes expressed as “for almost all elements...”).

• There are uncountable many elements such that...

• For all but countably many elements...

• For all elements in a set of positive measure...

• For all elements except those in a set of measure zero...

109.10 History

Term logic, also called Aristotelian logic, treats quantification in a manner that is closer to natural language, and also less suited to formal analysis. Term logic treated All, Some and No in the 4th century BC, in an account also touching on the alethic modalities. Gottlob Frege, in his 1879 Begriffsschrift, was the first to employ a quantifier to bind a variable ranging over a domain of discourse and appearing in predicates. He would universally quantify a variable (or relation) by writing the variable over a dimple in an otherwise straight line appearing in his diagrammatic formulas. Frege did not devise an explicit notation for existential quantification, instead employing his equivalent of ~∀x~, or contraposition. Frege's treatment of quantification went largely unremarked until Bertrand Russell's 1903 Principles of Mathematics. In work that culminated in Peirce (1885), Charles Sanders Peirce and his student Oscar Howard Mitchell indepen- dently invented universal and existential quantifiers, and bound variables. Peirce and Mitchell wrote Πₓ and Σₓ where we now write ∀x and ∃x. Peirce's notation can be found in the writings of Ernst Schröder, Leopold Loewenheim, Thoralf Skolem, and Polish logicians into the 1950s. Most notably, it is the notation of Kurt Gödel's landmark 1930 paper on the completeness of first-order logic, and 1931 paper on the incompleteness of Peano arithmetic. Peirce's approach to quantification also influenced William Ernest Johnson and Giuseppe Peano, who invented yet another notation, namely (x) for the universal quantification of x and (in 1897) ∃x for the existential quantification of x. Hence for decades, the canonical notation in philosophy and mathematical logic was (x)P to express“all individuals in the domain of discourse have the property P,”and "(∃x)P" for“there exists at least one individual in the domain of discourse having the property P.”Peano, who was much better known than Peirce, in effect diffused the latter's thinking throughout Europe. Peano's notation was adopted by the Principia Mathematica of Whitehead and Russell, Quine, and Alonzo Church. In 1935, Gentzen introduced the ∀ symbol, by analogy with Peano's ∃ symbol. ∀ did not become canonical until the 1960s. Around 1895, Peirce began developing his existential graphs, whose variables can be seen as tacitly quantified. Whether the shallowest instance of a variable is even or odd determines whether that variable's quantification is universal or existential. (Shallowness is the contrary of depth, which is determined by the nesting of negations.) Peirce's graphical logic has attracted some attention in recent years by those researching heterogeneous reasoning and diagrammatic inference.

109.11 See also

• Generalized quantifier —a higher-order property used as standard semantics of quantified noun phrases

• Lindström quantifier —a generalized polyadic quantifier

• Quantifier elimination

109.12 References

[1] Hehner, Eric C. R., 2004, Practical Theory of Programming, 2nd edition, p. 28

• Barwise, Jon; and Etchemendy, John, 2000. Language Proof and Logic. CSLI (University of Chicago Press) and New York: Seven Bridges Press. A gentle introduction to first-order logic by two first-rate logicians. 109.13. EXTERNAL LINKS 439

• Frege, Gottlob, 1879. Begriffsschrift. Translated in Jean van Heijenoort, 1967. From Frege to Gödel: A Source Book on Mathematical Logic, 1879-1931. Harvard University Press. The first appearance of quantification. • Hilbert, David; and Ackermann, Wilhelm, 1950 (1928). Principles of Mathematical Logic. Chelsea. Transla- tion of Grundzüge der theoretischen Logik. Springer-Verlag. The 1928 first edition is the first time quantification was consciously employed in the now-standard manner, namely as binding variables ranging over some fixed domain of discourse. This is the defining aspect of first-order logic. • Peirce, C. S., 1885, “On the Algebra of Logic: A Contribution to the Philosophy of Notation, American Journal of Mathematics, Vol. 7, pp. 180–202. Reprinted in Kloesel, N. et al., eds., 1993. Writings of C. S. Peirce, Vol. 5. Indiana University Press. The first appearance of quantification in anything like its present form.

• Reichenbach, Hans, 1975 (1947). Elements of Symbolic Logic, Dover Publications. The quantifiers are dis- cussed in chapters §18 “Binding of variables”through §30 “Derivations from Synthetic Premises”.

• Westerståhl, Dag, 2001,“Quantifiers,”in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Black- well. • Wiese, Heike, 2003. Numbers, language, and the human mind. Cambridge University Press. ISBN 0-521- 83182-2.

109.13 External links

• Hazewinkel, Michiel, ed. (2001),“Quantifier”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608- 010-4 • Stanford Encyclopedia of Philosophy:

• "Classical Logic —by Stewart Shapiro. Covers syntax, model theory, and metatheory for first order logic in the natural deduction style. • "Generalized quantifiers" —by Dag Westerståhl.

• Peters, Stanley; Westerståhl, Dag (2002). "Quantifiers." Chapter 110

Quantization (linguistics)

In linguistics, a quantized expression is such that, whenever it is true of some entity, it is not true of any proper subparts of that entity. Example: If something is an“apple”, then no proper subpart of that thing is an“apple”. If something is“water”, then many of its subparts will also be“water”. Hence,“apple”is quantized, while“water” is not. Quantization has proven relevant to the proper characterization of grammatical telicity (roughly, sentences that present events as bounded/unbounded in time) and the mass/count distinction for nouns. The notion was first applied to linguistic semantics by the linguist Manfred Krifka. Formally, a quantization predicate QUA can be defined as follows, where U is the universe of discourse, and F is a variable over sets, and p is a mereological part structure on U with

(∀F ⊆ Up)(QUA(F ) ⇐⇒ (∀x, y)(F (x) ∧ F (y) ⇒ ¬x

110.1 See also

• Fewer vs. less

110.2 References

Krifka, Manfred 1989. Nominal reference, temporal constitution and quantification in event semantics. In Renate Bartsch, Johan van Benthem and Peter van Emde Boas (eds.), Semantics and Contextual Expressions 75-115. Dor- drecht: Foris.

440 Chapter 111

Regular modal logic

In modal logic, a regular modal logic L is a modal logic closed under the duality of the modal operators: ♢A ≡ ¬□¬A and the rule (A ∧ B) → C ⊢ (□A ∧ □B) → □C. Every regular modal logic is classical, and every normal modal logic is regular and hence classical.

111.1 References

Chellas, Brian. Modal Logic: An Introduction. Cambridge University Press, 1980.

441 Chapter 112

Relevance

Relevance is the concept of one topic being connected to another topic in a way that makes it useful to consider the first topic when considering the second. The concept of relevance is studied in many different fields, including cognitive sciences, logic, and library and information science. Most fundamentally, however, it is studied in epistemology (the theory of knowledge). Different theories of knowledge have different implications for what is considered relevant and these fundamental views have implications for all other fields as well.

112.1 Definition

“Something (A) is relevant to a task (T) if it increases the likelihood of accomplishing the goal (G), which is implied by T.”(Hjørland & Sejer Christensen,2002).*[1] A thing might be relevant, a document or a piece of information may be relevant. The basic understanding of relevance does not depend on whether we speak of “things”or “information”. For example, the Gandhian principles are of great relevance in today's world.

112.2 Epistemology

If you believe that schizophrenia is caused by bad communication between mother and child, then family interaction studies become relevant. If, on the other hand, you subscribe to a genetic theory of relevance then the study of genes becomes relevant. If you subscribe to the epistemology of empiricism, then only intersubjectively controlled observations are relevant. If, on the other hand, you subscribe to feminist epistemology, then the sex of the observer becomes relevant. Epistemology is not just one domain among others. Epistemological views are always at play in any domain. Those views determine or influence what is regarded relevant.

112.3 Relevance logic

In formal reasoning, relevance has proved an important but elusive concept. It is important because the solution of any problem requires the prior identification of the relevant elements from which a solution can be constructed. It is elusive, because the meaning of relevance appears to be difficult or impossible to capture within conventional logical systems. The obvious suggestion that q is relevant to p if q is implied by p breaks down because under standard definitions of material implication, a false proposition implies all other propositions. However though 'iron is a metal' may be implied by 'cats lay eggs' it doesn't seem to be relevant to it the way in which 'cats are mammals' and 'mammals give birth to living young' are relevant to each other. If one states“I love ice cream,”and another person responds“I have a friend named Brad Cook,”then these statements are not relevant. However, if one states“I love ice cream,” and another person responds “I have a friend named Brad Cook who also likes ice cream,”this statement now becomes relevant because it relates to the first person's idea.

442 112.4. APPLICATION 443

Graphic of relevance in digital ecosystems

More recently a number of theorists have sought to account for relevance in terms of "possible world logics”in intensional logic. Roughly, the idea is that necessary truths are true in all possible worlds, contradictions (logical falsehoods) are true in no possible worlds, and contingent propositions can be ordered in terms of the number of possible worlds in which they are true. Relevance is argued to depend upon the “remoteness relationship”between an actual world in which relevance is being evaluated and the set of possible worlds within which it is true.

112.4 Application

112.4.1 Politics

During the 1960s, relevance became a fashionable buzzword, meaning roughly 'relevance to social concerns', such as racial equality, poverty, social justice, world , world economic development, and so on. The implication was that some subjects, e.g., the study of medieval poetry and the practice of , were not worthwhile because they did not address pressing social issues.

112.4.2 Economics

The economist John Maynard Keynes saw the importance of defining relevance to the problem of calculating risk in economic decision-making. He suggested that the relevance of a piece of evidence, such as a true proposition, should be defined in terms of the changes it produces of estimations of the probability of future events. Specifically, Keynes proposed that new evidence e is irrelevant to a proposition, given old evidence q, if and only if p/q & e = p/q and relevant otherwise. 444 CHAPTER 112. RELEVANCE

There are technical problems with this definition, for example, the relevance of a piece of evidence can be sensitive to the order in which other pieces of evidence are received.

112.4.3 Cognitive science and pragmatics

Further information:

In 1986, Dan Sperber and Deirdre Wilson drew attention to the central importance of relevance decisions in reasoning and communication. They proposed an account of the process of inferring relevant information from any given utterance. To do this work, they used what they called the “Principle of Relevance": namely, the position that any utterance addressed to someone automatically conveys the presumption of its own optimal relevance. The central idea of Sperber and Wilson's theory is that all utterances are encountered in some context, and the correct interpretation of a particular utterance is the one that allows most new implications to be made in that context on the basis of the least amount of information necessary to convey it. For Sperber and Wilson, relevance is conceived as relative or subjective, as it depends upon the state of knowledge of a hearer when they encounter an utterance. Sperber and Wilson stress that this theory is not intended to account for every intuitive application of the English word “relevance”. Relevance, as a technical term, is restricted to relationships between utterances and interpretations, and so the theory cannot account for intuitions such as the one that relevance relationships obtain in problems involving physical objects. If a plumber needs to fix a leaky faucet, for example, some objects and tools are relevant (i.e. a wrench) and others are not (i.e. a waffle iron). And, moreover, the latter seems to be irrelevant in a manner which does not depend upon the plumber's knowledge, or the utterances used to describe the problem. A theory of relevance that seems to be more readily applicable to such instances of physical problem solving has been suggested by Gorayska and Lindsay in a series of articles published during the 1990s. The key feature of their theory is the idea that relevance is goal-dependent. An item (e.g., an utterance or object) is relevant to a goal if and only if it can be an essential element of some plan capable of achieving the desired goal. This theory embraces both propositional reasoning and the problem-solving activities of people such as plumbers, and defines relevance in such a way that what is relevant is determined by the real world (because what plans will work is a matter of empirical fact) rather than the state of knowledge or belief of a particular problem solver.

112.4.4 Law

Main article: Relevance (law)

The meaning of“relevance”in U.S. law is reflected in Rule 401 of the Federal Rules of Evidence. That rule defines relevance as “having any tendency to make the existence of any fact that is of consequence to the determinations of the action more probable or less probable than it would be without the evidence.”In other words, if a fact were to have no bearing on the truth or falsity of a conclusion, it would be legally irrelevant.

112.4.5 Library and information science

Main article: Relevance (information retrieval)

This field has considered when documents (or document representations) retrieved from databases are relevant or non-relevant. Given a conception of relevance, two measures have been applied: Precision and recall: Recall = a : (a + c) X 100%, where a = number of retrieved, relevant documents, c = number of non-retrieved, relevant documents (sometimes termed “silence”). Recall is thus an expression of how exhaustive a search for documents is. Precision = a : (a + b) X 100%, where a = number of retrieved, relevant documents, b = number of retrieved, non-relevant documents (often termed “noise”). Precision is thus a measure of the amount of noise in document-retrieval. Relevance itself has in the literature often been based on what is termed“the system's view”and“the user's view” . Hjørland (2010) criticize these two views and defends a “subject knowledge view of relevance”. 112.5. SEE ALSO 445

112.5 See also

• Source criticism

• Description • Distraction

• Information-action ratio

• Information overload • Intention

• Relevance paradox • Relevance theory

112.6 References

[1] Hjørland, B. & Sejer Christensen, F. (2002). Work tasks and socio-cognitive relevance: a specific example. Journal of the American Society for Information Science and Technology, 53(11), 960-965.

• Gorayska B. & R. O. Lindsay (1993). The Roots of Relevance. Journal of Pragmatics 19, 301–323. Los Alamitos: IEEE Computer Society Press.

• Hjørland, Birger (2010). The foundation of the concept of relevance. Journal of the American Society for Information Science and Technology, 61(2), 217-237. • Keynes, J. M. (1921). Treatise on Probability. London: MacMillan

• Lindsay, R. & Gorayska, B. (2002) Relevance, Goals and Cognitive Technology. International Journal of Cognitive Technology, 1, (2), 187–232

• Sperber, D. & D. Wilson (1986/1995) Relevance: Communication and Cognition. 2nd edition. Oxford: Black- well.

• Sperber, D. & D. Wilson (1987). Précis of Relevance: Communication and Cognition. Behavioral and Brain Science, 10, 697–754.

• Sperber, D. & D. Wilson (2004). Relevance Theory. In Horn, L.R. & Ward, G. (eds.) 2004 The Handbook of Pragmatics. Oxford: Blackwell, 607-632. http://www.dan.sperber.fr/?p=93

• Zhang, X, H. (1993). A Goal-Based Relevance Model and its Application to Intelligent Systems. Ph.D. Thesis, Oxford Brookes University, Department of Mathematics and Computer Science, October, 1993.

112.7 External links

• Malcolm Gladwell - Blink - full show: TVOntario interview regarding “snap judgements”and Blink Chapter 113

Segment addition postulate

In geometry, the segment addition postulate states that given two points A and C, a third point B lies on the line segment AC if and only if the distances between the points satisfy the equation AB + BC = AC. This is related to the triangle inequality, which states that AB + BC ≥ AC with equality if and only if A, B, and C are collinear (on the same line). This in turn is equivalent to the proposition that the shortest distance between two points lies on a straight line. The segment addition postulate is often useful in proving results on the congruence of segments.

113.1 External links

• http://www.course-notes.org/Geometry/Segments_and_Rays/Segment_Addition_Postulate

446 Chapter 114

Self-reference

Self-reference occurs in natural or formal languages when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding. In philosophy, it also refers to the ability of a subject to speak of or refer to itself: to have the kind of thought expressed by the first person nominative singular pronoun, the word “I” in English. Self-reference is studied and has applications in mathematics, philosophy, computer programming, and linguistics. Self-referential statements are sometimes paradoxical.

114.1 Usage

An example of a self-referential situation is the one of self-creation, as the logical organization produces itself the physical structure which creates itself. Self-reference also occurs in literature and film when an author refers to his or her own work in the context of the work itself. Famous examples include Cervantes's Don Quixote, Shakespeare's A Midsummer Night's Dream, Denis Diderot's Jacques le fataliste et son maître, Italo Calvino's If on a winter's night a traveler, many stories by Nikolai Gogol, Lost in the Funhouse by John Barth, Luigi Pirandello's Six Characters in Search of an Author and Federico Fellini's 8½. This is closely related to the concepts of breaking the fourth wall and meta-reference, which often involve self-reference. The surrealist painter René Magritte is famous for his self-referential works. His painting The Treachery of Images, includes the words this is not a pipe, the truth of which depends entirely on whether the word “ceci”(in English, “this”) refers to the pipe depicted—or to the painting or the word or sentence itself.*[2] In computer science, self-reference occurs in reflection, where a program can read or modify its own instructions like any other data.*[3] Numerous programming languages support reflection to some extent with varying degrees of expressiveness. Additionally, self-reference is seen in recursion (related to the mathematical recurrence relation), where a code structure refers back to itself during computation.*[4]

114.2 Examples

114.2.1 In language

See also: Appendix:Autological words

A word that describes itself is called an autological word (or autonym). This generally applies to adjectives, for example sesquipedalian (i.e.“sesquipedalian”is a sesquipedalian word), but can also apply to other parts of speech, such as TLA, as a three-letter abbreviation for "three-letter abbreviation", and PHP which is a recursive acronym for “PHP: Hypertext Preprocessor”.*[5] A sentence which inventories its own letters and punctuation marks is called an autogram.

447 448 CHAPTER 114. SELF-REFERENCE

The Ouroboros, a dragon that continually consumes itself, is used as a symbol for self-reference.*[1]

• There is a special case of meta-sentence in which the content of the sentence in the metalanguage and the content of the sentence in the object language are the same. Such a sentence is referring to itself. However some meta-sentences of this type can lead to paradoxes. “This is a sentence.”can be considered to be a self-referential meta-sentence which is obviously true. However “This sentence is false”is a meta-sentence which leads to a self-referential paradox.

Hofstadter's law, which specifies that “It always takes longer than you expect, even when you take into account Hofstadter's Law”*[6] is an example of a self-referencing adage. Fumblerules are a list of rules of good grammar and writing, demonstrated through sentences that violate those very rules, such as “Avoid cliches like the plague”and “Don't use no double negatives”. The term was coined in a published list of such rules by William Safire.*[7]*[8]

114.2.2 In mathematics

• Gödel sentence 114.3. SEE ALSO 449

• Impredicativity

• Loop (graph theory) • Tupper's self-referential formula

114.2.3 In literature, film, and popular culture

Main article: Metafiction

• The subgenre of "recursive science fiction" is now so extensive that it has fostered a fan-maintained bibliography at the New England Science Fiction Association's website; some of it is about science fiction fandom, some about science fiction and its authors.*[9]

114.3 See also

114.4 References

[1] Soto-Andrade, Jorge; Jaramillo, Sebastian; Gutierrez, Claudio; Letelier, Juan-Carlos.“Ouroboros avatars: A mathematical exploration of Self-reference and Metabolic Closure” (PDF). MIT Press. Retrieved 16 May 2015.

[2] Nöth, Winfried; Bishara, Nina (2007). Self-reference in the Media. Walter de Gruyter. p. 75. ISBN 978-3-11-019464-7.

[3] Malenfant, J.; Demers, F-N. “A Tutorial on Behavioral Reflection and its Implementation” (PDF). PARC. Retrieved 17 May 2015.

[4] Drucker, Thomas (4 January 2008). Perspectives on the History of Mathematical Logic. Springer Science & Business Media. p. 110. ISBN 978-0-8176-4768-1.

[5] PHP Manual: Preface, www.php.net

[6] Hofstadter, Douglas. Gödel, Escher, Bach: An Eternal Golden Braid. 20th anniversary ed., 1999, p. 152. ISBN 0-465- 02656-7

[7] alt.usage.english.org's Humorous Rules for Writing

[8] Safire, William (1979-11-04). “On Language; The Fumblerules of Grammar”. New York Times. p. SM4.

[9] “Recursive Science Fiction” New England Science Fiction Association website, last updated 3 August 2008

114.5 Sources

• Bartlett, Steven J. [James] (Ed.) (1992). Reflexivity: A Source-book in Self-reference. Amsterdam, North- Holland. (PDF). RePub, Erasmus University

• Hofstadter, D. R. (1980). Gödel, Escher, Bach: an Eternal Golden Braid. New York, Vintage Books. • Smullyan, Raymond (1994), Diagonalization and Self-Reference, Oxford Science Publications, ISBN 0-19- 853450-7 Chapter 115

Self-refuting idea

Self-refuting ideas or self-defeating ideas are ideas or statements whose falsehood is a logical consequence of the act or situation of holding them to be true. Many ideas are called self-refuting by their detractors, and such accusations are therefore almost always controversial, with defenders stating that the idea is being misunderstood or that the argument is invalid. For these reasons, none of the ideas below are unambiguously or incontrovertibly self- refuting. These ideas are often used as axioms, which are definitions taken to be true (tautological assumptions), and cannot be used to test themselves, for doing so would lead to only two consequences: consistency (circular reasoning) or exception (self contradiction). It is important to know that the conclusion of an argument that is self-refuting is not necessarily false, since it could be supported by another, more valid, argument.

115.1 Variations

115.1.1 Directly self-denying statements

The is a statement of the form“this statement is false”. Such statements troubled philosophers, especially when there was a serious attempt to formalize the foundations of logic. Bertrand Russell developed his "Theory of Types" to formalize a set of rules that would prevent such statements (more formally Russell's paradox) being made in symbolic logic.*[1] This work has led to the modern formulation of axiomatic set theory. While Rus- sell's formalization did not contain such paradoxes, Kurt Gödel showed that it must contain independent statements. Any logical system that is rich enough to contain elementary arithmetic contains at least one proposition whose inter- pretation is this proposition is unprovable (from within the logical system concerned), and hence no such system can be both complete and consistent.

115.1.2 Indirectly self-denying statements or “fallacy of the stolen concept”

Objectivists define the fallacy of the stolen concept: the act of using a concept while ignoring, contradicting or denying the validity of the concepts on which it logically and genetically depends. An example of the stolen concept fallacy is anarchist Pierre-Joseph Proudhon's statement, “All property is theft”.

While discussing the hierarchical nature of knowledge, Nathaniel Branden states,“Theft”is a concept that logically and genetically depends on the antecedent concept of “rightfully owned property”—and refers to the act of taking that property without the owner's . If no property is rightfully owned, that is, if nothing is property, there can be no such concept as “theft.”Thus, the statement “All property is theft”has an internal contradiction: to use the concept “theft”while denying the validity of the concept of “property,”is to use “theft”as a concept to which one has no logical right—that is, as a stolen concept.*[2]

Others have said the statement is fallacious only on a superficial reading of Proudhon, devoid of context. Proudhon used the term “property”with reference to claimed ownership in land, factories, etc. He believed such claims were illegitimate, and thus a form of theft from the commons.*[3] Proudhon explicitly states that the phrase “property

450 115.2. EXAMPLES 451

is theft”is analogous to the phrase “slavery is murder”. According to Proudhon, the slave, though biologically alive, is clearly in a sense “murdered”. The “theft”in his terminology does not refer to ownership any more than the “murder”refers directly to physiological death, but rather both are meant as terms to represent a denial of specific rights.*[4] Others point out that the difference between the two examples is that “slavery is murder,” unlike “property is theft,”does not make a statement that denies the validity of one of the concepts it utilizes. We should note as well that Proudhon does not actually say all property is theft—he is referring to a very specific kind of property rights. Proudhon favored another kind, which he called possession, based on occupancy and use, a sort of usufruct rights idea. In What is Property? he therefore says with the apparent contradiction “property is theft”to denote one sort he feels is this,“property is liberty”, referring to the kind he favored, and“property is impossible” to make it clear any sort of property rights cannot be absolute. Separate concepts are therefore laid out in a way that can be confusing, especially if one is not familiar with them.

115.1.3 In logic

Self-refutation plays an important role in some inconsistency tolerant logics (e.g. paraconsistent logics and direct logic*[5]) that lack proof by contradiction. For example, the negation of a proposition can be proved by showing that the proposition implies its own negation. Likewise, it can be inferred that a proposition cannot be proved by (1) showing that a proof would imply the negation of the proposition or by (2) showing a proof would imply that the negation of the proposition can be proved.

115.2 Examples

115.2.1 Brain in a vat

Brain in a vat is a thought experiment in philosophy which is premised upon the skeptical hypothesis that one could actually be a brain in a vat receiving electrical input identical to that which would be coming from the nervous system. Similar premises are found in Descartes's evil demon and . Philosopher Hilary Putnam argues that some versions of the thought experiment would be inconsistent due to . For a brain in a vat that had only ever experienced the simulated world, the statement “I'm not a brain in a vat”is true. The only possible brains and vats it could be referring to are simulated, and it is true that it is not a simulated brain in a simulated vat. By the same argument, saying “I'm a brain in a vat”would be false.*[6]

115.2.2 Determinism

It has been argued, particularly by Christian apologists, that to call determinism a rational statement is doubly self- defeating.*[7]

1. To count as rational, a belief must be freely chosen, which according to the determinist is impossible

2. Any kind of debate seems to be posited on the idea that the parties involved are trying to change each other's minds.

The argument does not succeed against the compatibilistic view, since in the latter there is no conflict between de- terminism and free will. Moreover, the argument fails if one denies either of the above or its implicit implications. That is, one could avoid the argument by maintaining that free will is not required for rationality or for trying to change one's mind. The latter is a sensible position insofar as one could be determined to try to persuade someone of something, and the listener could be determined to accept it. There is no internal contradiction in that view. One can also consider a deterministic computer algorithm which is able to make a correct conclusion, such as a mathematical calculation or fingerprint identification. However, on some notions of “rationality”, such programs are themselves not rational because they simply follow a certain deterministic pre-programmed path and nothing more. This does not apply if one takes on a position with regards to rationality analogous to , namely, one could simply view rationality as the property of correctly executing the laws of logic, in which case there simply is no contradiction with determinism. The contradiction would arise if one defines “rationality”in a manner that is incompatiblist. Some argue that machines cannot“think”, and if rationality is defined so that it requires human-like 452 CHAPTER 115. SELF-REFUTING IDEA

thought, this might pose a problem. But the view that machines cannot “think”in principle is rejected by most philosophers who accept a computational theory of mind.

115.2.3

It has been argued that extreme ethical egoism is self-defeating. Faced with a situation of limited resources, egoists would consume as much of the resource as they could, making the overall situation worse for everybody. Egoists may rejoin that if the situation becomes worse for everybody, that would include the egoist, so it is not in fact in his or her rational self-interest to take things to such extremes.*[8] However, the (unregulated) and the (one off) prisoner's dilemma are cases in which on the one hand it is rational for an individual to seek to take as much as possible even though that makes things worse for everybody, and on the other hand those cases are not self- refuting since that behaviour remains rational even though it is ultimately self-defeating, i.e. self-defeating does not imply self-refuting. A tragedy of the commons, however, assumes some degree of public land. That is, a commons forbidding homesteading requires regulation. Thus, an argument against the tragedy of the commons is fundamentally an argument for private property rights and the system that recognizes both property rights and rational self-interest —.*[9] A prisoner's dilemma assumes a zero-sum game imposed by a warden, which is not part of the everyday life of nearly all of humanity—especially people living in rights-respecting capitalism. More generally, an increasing respect for individual rights uniquely allows for increasing wealth creation and increasing usable resources despite a fixed amount of raw materials (e.g. the West pre-1776 versus post-1776, East versus , Hong Kong versus mainland China, North versus , etc.).*[10]

115.2.4

The philosopher Mary Midgley states that the idea that nothing exists except matter is also self-refuting because if it were true neither it, nor any other idea, would exist, and similarly that an argument to that effect would be self-refuting because it would deny its own existence.*[11] Several other philosophers also argue that eliminative materialism is self-refuting.*[12]*[13]*[14] However, other forms of materialism may escape this kind of argument because, rather than eliminating the mental, they seek to identify it with, or reduce it to, the material.*[15] For instance, identity theorists such as J. J. C. Smart, Ullin Place and E. G. Boring state that ideas exist materially as patterns of neural structure and activity.*[16]*[17] Christian apologist J.P. Moreland states that such arguments are based on semantics.*[18]

115.2.5 Epimenides paradox

The first notable self-refuting idea is the Epimenides paradox, a statement attributed to Epimenides, a Cretan philoso- pher, that “All Cretans are liars”. This cannot be true if uttered by a Cretan. A more common example is the self-refuting statement“I am lying”(because the first statement allows the possibility “some Cretans do not speak the truth,”the speaker being one of them). The second statement has no third alternative —the speaker's statement is either true or false.

115.2.6 Evolutionary naturalism

Alvin Plantinga argues in his evolutionary argument against naturalism that the combination of naturalism and evo- lution is “in a certain interesting way self-defeating”because if it were true there would be insufficient grounds to believe that human cognitive faculties are reliable.*[19] Consequently, if human cognitive abilities are unreliable, then any human construct, which by implication utilizes cognitive faculties, such as evolutionary theory, would be undermined. In this particular case, it is the confluence of evolutionary theory and naturalism that, according to the argument, undermine the reason for believing themselves to be true. Since Plantinga originally formulated the argu- ment, a few theistic philosophers and Christian apologists have agreed.*[20]*[21] There has also been a considerable backlash of papers arguing that the argument is flawed in a number of ways, one of the more recent ones published in 2011 by Feng Ye*[22] (see also the references in the Evolutionary argument against naturalism article). 115.2. EXAMPLES 453

115.2.7 Foundationalism

The philosopher argues that the idea, “common to theists like Aquinas and Descartes and to an atheist like Russell" that “Rational belief [is] either self-evident or based directly or indirectly on what is evident” (which he termed “foundationalism”following Plantinga) is self-refuting on the basis that this idea is itself neither self-evident nor based directly or indirectly on what is evident and that the same applies to other formulations of such foundationalism.*[23] However, the self-evident impossibility of infinite regress can be offered as a justification for foundationalism.*[24] Following the identification of problems with“naive foundationalism”, the term is now often used to focus on incorrigible beliefs (modern foundationalism), or basic beliefs (reformed foundationalism).

115.2.8 Philosophical skepticism

Philosophical skeptics state that“nothing can be known.”*[25] This has caused wonder as to whether that statement itself be known, or be self-refuting.*[26] One very old response to this problem is :*[27] an exception is made for the skeptic's own statement. This leads to further debate about consistency and . Another response is to accept that nothing can be known cannot itself be known, so that it is not known whether anything is knowable or not. This is Pyrrhonic skepticism. However, the issue can be pressed further. To state that nothing can be known is itself unknowable, is to state that it is a fact that nothing can be known is unknowable. This admits at least one form of knowledge, and the metaphysical proposition I can discern truth. It is therefore self- refuting; only justified through an infinite regress of not knowing knowledge claims and not knowing that one can know them ad infinitum.

115.2.9 Relativism

It is often stated that relativism about truth must be applied to itself.*[28]*[29] The cruder form of the argument concludes that since the relativist is calling relativism an absolute truth, it leads to a contradiction. Relativists often rejoin that in fact relativism is only relatively true, leading to a subtler problem: the absolutist, the relativist's opponent, is perfectly entitled, by the relativist's own standards, to reject relativism. That is, the relativist's arguments can have no normative force over someone who has different basic beliefs.*[30]

115.2.10

On the face of it, a statement of solipsism is —at least performatively —self-defeating, because a statement assumes another person to whom the statement is made. (That is to say, an unexpressed private belief in solipsism is not self-refuting). This, of course, assumes the solipsist would not communicate with a hallucination, even if just for self-amusement. One response is that the solipsist's interlocutor is in fact a figment of their imagination, but since their interlocutor knows they are not, they are not going to be convinced.*[31]

115.2.11 Verification- and falsification-principles

The statements“statements are meaningless unless they can be empirically verified”and“statements are meaningless unless they can be empirically falsified”have both been called self-refuting on the basis that they can neither be empirically verified nor falsified.*[32] Similar arguments have been made for statements such as “no statements are true unless they can be shown empirically to be true,”which was a problem for .*[33]

115.2.12 Wittgenstein's Tractatus

The Tractatus Logico-Philosophicus is an unusual example of a self-refuting argument, in that Ludwig Wittgenstein explicitly admits to the issue at the end of the work:

My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it.) 454 CHAPTER 115. SELF-REFUTING IDEA

(6.54)

However, this idea can be solved in the sense that, even if the argument itself is self-refuting, the effects of the argument elicit understandings that go beyond the argument itself. Søren Kierkegaard describes it as such:

[The reader] can understand that the understanding is a revocation--the understanding with him as the sole reader is indeed the revocation of the book. He can understand that to write a book and to revoke it is not the same as refraining from writing it, that to write a book that does not demand to be important for anyone is still not the same as letting it be unwritten. —Concluding Unscientific Postscript

115.3 See also

• Contradiction

• Paradox

• Performative contradiction

• Self-defeating prophecy

115.4 References

[1] Russell B, Whitehead A.N., Principia Mathematica

[2] The Stolen Concept by Nathaniel Branden - originally published in The Objectivist Newsletter in January 1963.

[3] Rockwell, L. Performative Contradictions and Subtle Misunderstandings

[4] http://www.marxists.org/reference/subject/economics/proudhon/property/ch01.htm

[5] Hewitt, C. “Large-scale Organizational Computing requires Unstratified Reflection and Strong Paraconsistency”Coor- dination, Organizations, Institutions, and Norms in Agent Systems III Jaime Sichman, Pablo Noriega, Julian Padget and Sascha Ossowski (ed.). Springer-Verlag. 2008.

[6] Brains in a vat, Reason, Truth, and History ch. 1, Hilary Putnam

[7] “Determinism”. Determinism is self-defeating. A determinist insists that both determinists and non-determinists are determined to believe what they believe. However, determinists believe self-determinists are wrong and ought to change their view. But “ought to change”implies they are free to change, which, within the incompatibilist view, is contrary to determinism.

[8] “Ethics” Britannica

[9] Walter Block. “ and Economic Freedom: The Case for Private Property Rights (Journal of Business Ethics , Vol. 17, No. 16 (Dec., 1998), pp. 1887-1899)". Retrieved 2014-03-14.

[10] Julian Simon. “The Ultimate Resource II: People, Materials, and Environment (1996)". Retrieved 2014-03-14.

[11] see Mary Midgley The Myths we Live by

[12] Baker, L. (1987). Saving Belief. Princeton: Princeton University Press. ISBN 0-691-07320-1.

[13] Reppert, V. (1992). “Eliminative Materialism, Cognitive Suicide, and Begging the Question”. 23 (4): 378–392. doi:10.1111/j.1467-9973.1992.tb00550.x.

[14] Boghossian, P. (1990). “The Status of Content”. Philosophical Review 99 (2): 157–184. doi:10.2307/2185488. And — (1991). “The Status of Content Revisited”. Pacific Philosophical Quarterly 71: 264–278.

[15] Hill, C. “Identity Theory” (PDF). 115.4. REFERENCES 455

[16] Place, U. T. “Identity Theories”. In Nanni, Marco. A Field Guide to the Philosophy of Mind. Società italiana per la filosofia analitica. To the author a perfect correlation is identity. Two events that always occur together at the same time in the same place, without any temporal or spatial differentiation at all, are not two events but the same event. The mind-body correlations as formulated at present, do not admit of spatial correlation, so they reduce to matters of simple correlation in time. The need for identification is no less urgent in this case.

[17] “Dictionary of the Philosophy of Mind”.

[18] Moreland, J.P., The Recalcitrant Imago Dei: Human Persons and the Failure of Naturalism

[19] in Naturalism Defeated?, Ed. James Beilby Cornell University Press, 2002

[20] John Polkinghorne is an example of a scientist-theologian who is supportive of Plantinga's position

[21] Richard Swimburne is a philosopher that supports and utilizes Plantinga's argument effectively in his book“The ”

[22] http://sites.google.com/site/fengye63/naturalizedtruthandplantinga

[23] Kenny, Anthony (1992). What is Faith?. Oxford: OUP. pp. 9–10. ISBN 0-19-283067-8. This particular chapter is based on a 1982 lecture which may explain the shift in the meaning of the term “foundationalism”since then

[24] Stanford Encyclopedia of Philosophy on foundationalism

[25] The Gallilean Library

[26] Suber, P. Classical Skepticism

[27] Internet Encyclopedia of Philosophy

[28] Cognitive Relativism, Internet Encyclopedia of Philosophy

[29] The problem of self-refutation is quite general. It arises whether truth is relativized to a framework of concepts, of beliefs, of standards, of practices. Stanford Encyclopedia of Philosophy

[30]“If truth is relative, then non-relativist points of view can legitimately claim to be true relative to some standpoints.” Westacott, E. On the Motivations for Relativism

[31] Russell, B. (1948). Human Knowledge: Its Scope and Limits. New York: Simon and Schuster. p. 180. As against solipsism it is to be said, in the first place, that it is psychologically impossible to believe, and is rejected in fact even by those who mean to accept it. I once received a letter from an eminent logician, Mrs. Christine Ladd Franklin, saying that she was a solipsist, and was surprised that there were no others. Coming from a logician and a solipsist, her surprise surprised me.

[32] See e.g. the discussion by Alston, William P. (2003). “Religious language and verificationism”. In Moser, Paul K.; Copan, Paul. The Rationality of Theism. New York: Routledge. pp. 26–34. ISBN 0-415-26332-8.

[33] see e.g. Keith Ward, Is Religion Dangerous? Chapter 116

Ship of Theseus

This article is about Theseus' Paradox. For the 2013 Indian film, see Ship of Theseus (film).

The ship of Theseus, also known as Theseus' paradox, is a thought experiment that raises the question of whether an object that has had all of its components replaced remains fundamentally the same object. The paradox is most notably recorded by Plutarch in Life of Theseus from the late first century. Plutarch asked whether a ship that had been restored by replacing every single wooden part remained the same ship. The paradox had been discussed by more ancient philosophers such as Heraclitus, Socrates, and Plato prior to Plutarch's writings;*[1] and more recently by Thomas Hobbes and John Locke. Several variants are known, including the grandfather's axe, which has had both head and handle replaced.

116.1 Variations of the paradox

116.1.1 Ancient philosophy

This particular version of the paradox was first introduced in Greek legend as reported by the historian, biographer, and essayist Plutarch,

The ship wherein Theseus and the youth of Athens returned from had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their places, in so much that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same. —Plutarch, Theseus*[2]

Plutarch thus questions whether the ship would remain the same if it were entirely replaced, piece by piece. Centuries later, the philosopher Thomas Hobbes introduced a further puzzle, wondering what would happen if the original planks were gathered up after they were replaced, and used to build a second ship.*[3] Hobbes asked which ship, if either, would be the original Ship of Theseus.

116.1.2 Modern era

John Locke proposed a scenario regarding a favorite sock that develops a hole. He pondered whether the sock would still be the same after a patch was applied to the hole, and if it would be the same sock, would it still be the same sock after a second patch was applied, and a third, etc., until all of the material of the original sock has been replaced with patches.*[4] George Washington's axe (sometimes “my grandfather's axe”) is the subject of an apocryphal story of unknown origin in which the famous artifact is “still George Washington's axe”despite having had both its head and handle replaced.*[5]

456 116.2. PROPOSED RESOLUTIONS 457

This has also been recited as "Abe Lincoln's axe";*[6] Lincoln was well known for his ability with an axe, and axes associated with his life are held in various museums.*[7] The French equivalent is the story of Jeannot's knife, where the eponymous knife has had its blade changed fifteen times and its handle fifteen times, but is still the same knife.*[8] In some Spanish-speaking countries, Jeannot's knife is present as a proverb, though referred to simply as “the family knife". The principle, however, remains the same. Hungarian version of the story features "Lajos Kossuth's pocket knife”, having its blade and handle continuously replaced but still being referred to as the very knife of the famous statesman. As a proverbial expression it is used for objects or solutions being repeatedly renewed and gradually replaced to an extent that it hasn't left any actual original parts. In Great Britain, it has become known as “Trigger's Broom”. This is from the popular road sweeper “Trigger” in the sit com “Only Fools and Horses”who boasted that in 20 years on the job he had had the same broom. But during that time it had had 7 new brushes and 6 new poles.

116.1.3 In popular culture

Main article: List of Ship of Theseus examples

The paradox appears in various forms in fictional contexts, particularly in fantasy or science-fiction, for example where a character has body parts swapped for artificial replacements until the person has been entirely replaced. There are many other variations with reference to the same concept in popular culture for example axes and brooms.

116.2 Proposed resolutions

116.2.1 Heraclitus

The Greek philosopher Heraclitus attempted to solve the paradox by introducing the idea of a river where water replenishes it. Arius Didymus quoted him as saying “upon those who step into the same rivers, different and again different waters flow”.*[9] Plutarch disputed Heraclitus' claim about stepping twice into the same river, citing that it cannot be done because “it scatters and again comes together, and approaches and recedes”.*[10]

116.2.2 Aristotle's causes

According to the philosophical system of Aristotle and his followers, four causes or reasons describe a thing; these causes can be analyzed to get to a solution to the paradox. The formal cause or 'form' (perhaps best parsed as the cause of an object's form or of its having that form) is the design of a thing, while the material cause is the matter of which the thing is made. Another of Aristotle's causes is the 'end' or final cause, which is the intended purpose of a thing. The ship of Theseus would have the same ends, those being, mythically, transporting Theseus, and politically, convincing the Athenians that Theseus was once a living person, though its material cause would change with time. The efficient cause is how and by whom a thing is made, for example, how artisans fabricate and assemble something; in the case of the ship of Theseus, the workers who built the ship in the first place could have used the same tools and techniques to replace the planks in the ship. According to Aristotle, the “what-it-is”of a thing is its formal cause, so the ship of Theseus is the 'same' ship, because the formal cause, or design, does not change, even though the matter used to construct it may vary with time. In the same manner, for Heraclitus's paradox, a river has the same formal cause, although the material cause (the particular water in it) changes with time, and likewise for the person who steps in the river. This argument's validity and soundness as applied to the paradox depend on the accuracy not only of Aristotle's expressed premise that an object's formal cause is not only the primary or even sole determiner of its defining char- acteristic(s) or essence (“what-it-is”) but also of the unstated, stronger premise that an object's formal cause is the sole determiner of its identity or "which-it-is”(i.e., whether the previous and the later ships or rivers are the“same” ship or river). This latter premise is subject to attack by indirect proof using arguments such as “Suppose two ships are built using the same design and exist at the same time until one sinks the other in battle. Clearly the two ships are not the same ship even before, let alone after, one sinks the other, and yet the two have the same formal cause; 458 CHAPTER 116. SHIP OF THESEUS

therefore, formal cause cannot by itself suffice to determine an object's identity”or " [...] therefore, two objects' or object-instances' having the same formal cause does not by itself suffice to make them the same object or prove that they are the same object.”

116.2.3 Definitions of “the same”

One common argument found in the philosophical literature is that in the case of Heraclitus' river one is tripped up by two different definitions of “the same”. In one sense, things can be “qualitatively identical”, by sharing some properties. In another sense, they might be“numerically identical”by being“one”. As an example, consider two different marbles that look identical. They would be qualitatively, but not numerically, identical. A marble can be numerically identical only to itself. Note that some languages differentiate between these two forms of identity. In German, for example, "gleich"( “equal”) and "selbe"(“self-same”) are the pertinent terms, respectively. At least in formal speech, the former refers to qualitative identity (e.g. die gleiche Murmel, “the same [qualitative] marble”) and the latter to numerical identity (e.g. dieselbe Murmel, “the same [numerical] marble”). Colloquially, "gleich" is also used in place of "selbe", however.

116.2.4 Four-dimensionalism

Main article: Perdurantism

Ted Sider and others have proposed that considering objects to extend across time as four-dimensional causal series of three-dimensional “time-slices”could solve the ship of Theseus problem because, in taking such an approach, each time-slice and all four dimensional objects remain numerically identical to themselves while allowing individual time-slices to differ from each other. The aforementioned river, therefore, comprises different three-dimensional time-slices of itself while remaining numerically identical to itself across time; one can never step into the same river-time-slice twice, but one can step into the same (four-dimensional) river twice.*[11]

116.2.5 In Japan

In Japan, Shinto shrines are rebuilt every twenty years with entirely“new wood”. The continuity over the centuries is spiritual and comes from the source of the wood in the case of the Ise Jingu's Naiku shrine, which is harvested from an adjoining forest that is considered sacred. The shrine has currently been rebuilt sixty-two times.*[12]

116.3 See also

• Haecceity

• Identity and change

• Mereological essentialism

• Neurathian bootstrap

• S. (Dorst novel)

• Ship of Theseus film

• Sorites paradox

• Śūnyatā

•

• Teletransportation paradox

• USS Niagara (1813) 116.4. REFERENCES 459

• USS Constellation (1854)

• Vehicle restoration • One-horse shay

116.4 References

[1] Plato (1925). Parmenides 9. Translated by N. Fowler, Harold. London: Harvard University Press. p. 139.

[2] Plutarch. “Theseus”. The Internet Classics Archive. Retrieved 2008-07-15.

[3] Page 89:The Ship of Theseus, Person and Object: A Metaphysical Study, By Roderick M. Chisholm - Google Books

[4] Cohen, M. (2010). Philosophy for Dummies. Chichester: John Wiley & Sons.

[5] Browne, Ray Broadus (1982). Objects of Special Devotion: Fetishism in Popular Culture. Popular Press. p. 134. ISBN 0-87972-191-X.

[6] “Atomic Tune-Up: How the Body Rejuvenates Itself”. National Public Radio. 2007-07-14. Retrieved 2009-11-11.

[7] Bruce Rushton (2008-02-22).“Ax turns out to be Lincoln's last swing”. Rockford Register-Star. Retrieved 2009-11-11.

[8] “Dumas in his Curricle”. Blackwood's Edinburgh Magazine LV (CCCXLI): 351. January–June 1844.

[9] Didymus, Fr 39.2, Dox. gr. 471.4

[10] Plutarch. “On the 'E' at Delphi”. Retrieved 2008-07-15.

[11] David Lewis, “Survival and Identity”in Amelie O. Rorty [ed.] The Identities of Persons (1976; U. of California P.) Reprinted in his Philosophical Papers I.

[12] Olson, Brad (30 August 2013). “Japan's most sacred site rebuilt, for the 62nd time”. CNN News. Chapter 117

Simple non-inferential passage

A simple non-inferential passage is a type of nonargument characterized by the lack of a claim that anything is being proved.*[1] Simple non-inferential passages include warnings, pieces of advice, statements of belief or opinion, loosely associated statements, and reports. Simple non-inferential passages are nonarguments because while the statements involved may be premises, conclusions or both, the statements do not serve to infer a conclusion or support one another. This is distinct from a logical fallacy, which indicates an error in reasoning.

117.1 Types

117.1.1 Warnings

A warning is a type of simple non-inferential passage that serves to alert a person to any sort of potential danger. This can be as simple as a road sign indicating falling rock or a janitorial sign indicating a wet, slippery floor.

117.1.2 Piece of advice

A piece of advice is a type of simple non-inferential passage that recommends some future action or course of conduct. A mechanic recommending regular oil changes or a doctor recommending that a patient refrain from smoking are examples of pieces of advice.

117.1.3 Statements of belief or opinion

Main article: Opinion

A statement of belief or opinion is a type of simple non-inferential passage containing an expression of belief or opinion lacking an inferential claim. In A concise introduction to logic, Hurley uses the following example to illustrate:

We believe that our company must develop and produce outstanding products that will perform a great service or fulfill a need for our customers. We believe that our business must be run at an adequate profit and that the services and products we offer must be better than those offered by competitors —A concise introduction to logic, 10th edition

117.1.4 Loosely associated statements

Main article: Loosely associated statements

460 117.2. REFERENCES 461

A loosely associated statement is a type of simple non-inferential passage wherein statements about a general sub- ject are juxtaposed but make no inferential claim.*[1] As a rhetorical device, loosely associated statements may be intended by the speaker to infer a claim or conclusion, but because they lack a coherent logical structure any such interpretation is subjective as loosely associated statements prove nothing and attempt no obvious conclusion.*[2] Loosely associated statements can be said to serve no obvious purpose, such as illustration or explanation.*[3]

117.1.5 Reports

A report is a type of simple non-inferential passage wherein the statements serve to convey knowledge.

Even though more of the world is immunized than ever before, many old diseases have proven quote resilient in the face of changing population and environmental conditions, especially in the developing world. New diseases, such as AIDS, have taken their toll in both the North and the South —Steven L. Spiedel, World politics in a new era

The above is considered a report because it informs the reader without making any sort of claim, ethical or otherwise. However, the statements being made could be seen as a set of premises, and with the addition of a conclusion it would be considered an argument.

117.2 References

[1] Hurley, Patrick J. (2008). A Concise Introduction to Logic 10th ed. Thompson Wadsworth. p. 16. ISBN 0-495-50383-5.

[2] “The logic of arguments”. Retrieved April 28, 2012.

[3] “NONargument - Loosely associated statements”. Retrieved April 28, 2012. Chapter 118

Situational analysis

Situational analysis (or Situational logic) is a concept advanced by Popper in his The Poverty of . Situational analysis is a process by which a social scientist tries to reconstruct the problem situation confronting an agent in order to understand that agent's choice. Koertge (1975) provides a helpful clarificatory summary.*[note 1]

First provide a description of the situation: ''Agent A was in a situation of type C''.

This situation is then analysed ''In a situation of type C, the appropriate thing to do is X.''

The rationality principle may then be called upon: ''agents always act appropriately to their situation'' Finally we have the explanadum: ''(therefore) A did X.''*[1]*[2]*[3]

118.1 Notes

[1] This use of this summary is from Boumans and Davis (2010)

118.2 References

[1] Boumans, M and Davis, John B. (2010) Economic Methodology: Understanding Economics as a science, Palgrave Macmillan (p129-133)

[2] Popper, Karl (1957) The Poverty of Historicism, Routledge

[3] Koertge, N (1975) Popper's Metaphysical Research Program for the Human Sciences,Inquiry, 18 (1975), 437-62.

462 Chapter 119

Social software (social procedure)

This article is about the field of research. For computer software used for social interaction, see Social software.

In philosophy and the social sciences, social software is an interdisciplinary research program that borrows mathe- matical tools and techniques from game theory and computer science in order to analyze and design social procedures. The goals of research in this field are modeling social situations, developing theories of correctness, and designing social procedures.*[1] Work under the term social software has been going on since about 1996, and conferences in Copenhagen, London, Utrecht and New York, have been partly or wholly devoted to it. Much of the work is carried out at the City University of New York under the leadership of Rohit Jivanlal Parikh, who was influential in the development of the field.

119.1 Goals and tools of social software

Current research in the area of social software include the analysis of social procedures and examination of them for fairness, appropriateness, correctness and efficiency. For example, an election procedure could be a simple majority vote, Borda count, a Single Transferable vote (STV), or Approval voting. All of these procedures can be examined for various properties like monotonicity. Monotonicity has the property that voting for a candidate should not harm that candidate. This may seem obvious, true under any system, but it is something which can happen in STV. Another question would be the ability to elect a Condorcet winner in case there is one. Other principles which are considered by researchers in social software include the concept that a procedure for fair division should be Pareto optimal, equitable and envy free. A procedure for auctions should be one which would encourage bidders to bid their actual valuation – a property which holds with the Vickrey auction. What is new in social software compared to older fields is the use of tools from computer science like program logic, analysis of algorithms and epistemic logic. Like programs, social procedures dovetail into each other. For instance an airport provides runways for planes to land, but it also provides security checks, and it must provide for ways in which buses and taxis can take arriving passengers to their local destinations. The entire mechanism can be analyzed in the way in which a complex computer program can be analyzed. The Banach-Knaster procedure for dividing a cake fairly, or the Brams and Taylor procedure for fair division have been analyzed in this way. To point to the need for epistemic logic, a building not only needs restrooms, for obvious reasons, it also needs signs indicating where they are. Thus epistemic considerations enter in addition to structural ones. For a more urgent example, in addition to medicines, physicians also need tests to indicate what a patient’s problem is.

119.2 See also

• Social procedure

• Social technology

• Epistemic Logic

463 464 CHAPTER 119. SOCIAL SOFTWARE (SOCIAL PROCEDURE)

• Game Theory

• Mechanism Design

• Fair Division

• No-trade Theorem

• Dynamic Logic

119.3 Notes

[1] Pacuit (2005), p.10

119.4 Further reading

• , The Construction of (1995) New York : Free Press, c1995.

• Rohit Parikh, “Social Software,”Synthese, 132, Sep 2002, 187-211.

• Eric Pacuit and Rohit Parikh, “Social Interaction, Knowledge, and Social Software”, in Interactive Compu- tation: The New Paradigm, ed. Dina Goldin, Sott Smolka, Peter Wegner, Springer 2007, 441-461.

• Ludwig Wittgenstein, Philosophical Investigations, Macmillan, 1953.

• , Knowledge and Belief: an introduction to the logic of the two notions, Cornell University press, 1962, ISBN 9781904987086

• D. Lewis, Convention, a Philosophical Study, Harvard U. Press, 1969.

• R. Aumann, Agreeing to disagree, Annals of Statistics, 4 (1976) 1236-1239.

• and Nancy Stokey (1982). “Information, trade and common knowledge” (PDF). Journal of Economic Theory 26 (1): 17–27. doi:10.1016/0022-0531(82)90046-1.

• J. Geanakoplos and H. Polemarchakis, We Can't Disagree Forever, J. Economic Theory, 28 (1982), 192-200.

• R. Parikh and P. Krasucki, Communication, Consensus and Knowledge, J. Economic Theory 52 (1990) pp. 178–189.

• W. Brian Arthur. Inductive reasoning and . Complexity in Economic Theory, 84(2):406- 411, 1994.

• Ronald Fagin, Joseph Halpern, Yoram Moses and Moshe Vardi, Reasoning about Knowledge, MIT Press 1995.

• Steven Brams and Alan Taylor, The Win-Win Solution: guaranteeing fair shares to everybody, Norton 1999.

• David Harel, Dexter Kozen and Jerzy Tiuryn, Dynamic Logic, MIT Press, 2000.

• Michael Chwe, Rational ritual : culture, coordination, and common knowledge, Princeton University Press, 2001.

• Marc Pauly, Logic for Social Software, Ph.D. Thesis, University of Amsterdam. ILLC Dissertation Series 2001-10, ISBN 90-6196-510-1.

• Rohit Parikh, Language as social software, in Future Pasts: the Analytic Tradition in Twentieth Century Philos- ophy, Ed. J. Floyd and S. Shieh, Oxford U. Press, 2001, 339-350.

• Parikh, R. and Ramanujam, R., A knowledge based semantics of messages, in J. Logic, Language, and Infor- mation, 12, pp. 453 – 467, 2003.

• Eric Pacuit, Topics in Social Software: Information in Strategic Situations, Doctoral dissertation, City University of New York (2005). 119.5. EXTERNAL LINKS 465

• Eric Pacuit, Rohit Parikh and Eva Cogan, The Logic of Knowledge Based Obligation, Knowledge, Rationality and Action, a subjournal of Synthese, 149(2), 311 – 341, 2006. • Eric Pacuit and Rohit Parikh, Reasoning about Communication Graphs, in Interactive Logic, Edited by Johan van Benthem, Dov Gabbay and Benedikt Lowe (2007). • Mike Wooldridge, Thomas Ågotnes, Paul E. Dunne, and Wiebe van der Hoek. Logic for Automated Mech- anism Design - A Progress Report. In Proceedings of the Twenty-Second Conference on Artificial Intelligence (AAAI-07), Vancouver, Canada, July 2007.

119.5 External links

• Knowledge, Games and Beliefs Group. City University of New York, Graduate Center.

• Social Software conference. Carlsberg Academy, Copenhagen. May 27–29, 2004. Retrieved on 2009-06-26. • Interactive Logic: Games and Social Software workshop. King's College, London. November 4–7, 2005. Retrieved on 2009-06-26. • Games, action and social software workshop. Lorentz Center, Leiden University, Netherlands. 30 Oct 2006–3 Nov 2006. Retrieved on 2009-06-26. • Social Software Mini-conference. Knowledge, Games and Beliefs Group, City University of New York. May 18–19, 2007. Retrieved on 2009-06-26. Chapter 120

Soku hi

Soku-hi (Japanese: 即⾮) means “is and is not”. The term is primarily used by the representatives of the of . The logic of soku-hi or “is and is not”represents a balanced logic of symbolization reflecting sensitivity to the mutual determination of universality and particularity in nature, and a corresponding emphasis on nonattachment to linguistic predicates and subjects as representations of the real.*[1]

120.1 See also

• Emptiness, a concept in Kyoto School philosophy

• Nishida Kitaro

120.2 Notes

[1] G. S. Axtell. Comparative Dialectics: Nishida Kitaro's Logic of Place and Western Dialectical Thought, Philosophy East and West. Vol. 41, No. 2 (April 1991). pp. 163-184. University of Hawaii Press, Hawaii, USA.

120.3 References and external links

• Logic of soku-hi by D.T. Suzuki (poetry) • Rude awakenings: Zen, the Kyoto school, & the question of nationalism, by James W. Heisig & John C. Maraldo. p. 24. • 'The Kyoto School' on the Stanford Encyclopedia of Philosophy

466 Chapter 121

Specialization (logic)

Specialisation,(or specialization) is an important way to generate propositional knowledge, by applying general knowledge, such as the theory of gravity, to specific instances, such as “when I release this apple, it will fall to the floor”. Specialisation is the opposite of generalisation. Concept B is a specialisation of concept A if and only if:

• every instance of concept B is also an instance of concept A; and

• there are instances of concept A which are not instances of concept B.

121.1 See also

• Generalisation

467 Chapter 122

Tacit assumption

A tacit assumption or implicit assumption is an assumption that includes the underlying agreements or statements made in the development of a logical argument, course of action, decision, or judgment that are not explicitly voiced nor necessarily understood by the decision maker or judge. Often, these assumptions are made based on personal life experiences, and are not consciously apparent in the decision making environment. These assumptions can be the source of apparent paradoxes, misunderstandings and resistance to change in human organizational behavior.

122.1 See also

• Implied consent

122.2 References

• Edgar H. Schein, Organizational Culture and Leadership, Jossey-Bass, 2004, ISBN 0-7879-7597-4

468 Chapter 123

Term logic

In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century. This entry is an introduction to the term logic needed to understand philosophy texts written before predicate logic came to be seen as the only formal logic of interest. Readers lacking a grasp of the basic terminology and ideas of term logic can have difficulty understanding such texts, because their authors typically assumed an acquaintance with term logic.

123.1 Aristotle's system

Aristotle's logical work is collected in the six texts that are collectively known as the Organon. Two of these texts in particular, namely the Prior Analytics and De Interpretatione, contain the heart of Aristotle's treatment of judgements and formal inference, and it is principally this part of Aristotle's works that is about term logic. Modern work on Aristotle's logic builds on the tradition started in 1951 with the establishment by Jan Lukasiewicz of a revolutionary paradigm.*[1] The Jan Lukasiewicz approach was reinvigorated in the early 1970s by John Corcoran and Timothy Smiley - which informs modern translations of Prior Analytics by Robin Smith in 1989 and Gisela Striker in 2009.*[2]

123.2 Basics

The fundamental assumption behind the theory is that propositions are composed of two terms - hence the name "two-term theory" or “term logic”- and that the reasoning process is in turn built from propositions:

• The term is a part of speech representing something, but which is not true or false in its own right, such as “man”or “mortal”.

• The proposition consists of two terms, in which one term (the "predicate") is “affirmed”or “denied”of the other (the "subject"), and which is capable of truth or falsity.

• The syllogism is an inference in which one proposition (the "conclusion") follows of necessity from two others (the "premises").

A proposition may be universal or particular, and it may be affirmative or negative. Traditionally, the four kinds of propositions are:

• A-type: Universal and affirmative (“Every philosopher is mortal”) • I-type: Particular and affirmative (“Some philosopher is mortal”) • E-type: Universal and negative (“Every philosopher is not immortal”) • O-type: Particular and negative (“Some philosopher is not immortal”)

469 470 CHAPTER 123. TERM LOGIC

This was called the fourfold scheme of propositions (see types of syllogism for an explanation of the letters A, I, E, and O in the traditional square). Aristotle's original square of opposition, however, does not lack existential import:

• A-type: Universal and affirmative (“Every philosopher is mortal”) • I-type: Particular and affirmative (“Some philosopher is mortal”) • E-type: Universal and negative (“No philosopher is mortal”) • O-type: Particular and negative (“Not every philosopher is mortal”)

In the Stanford Encyclopedia of Philosophy article, “The Traditional Square of Opposition”, Terence Parsons explains:

One central concern of the Aristotelian tradition in logic is the theory of the categorical syllogism. This is the theory of two-premised arguments in which the premises and conclusion share three terms among them, with each proposition containing two of them. It is distinctive of this enterprise that everybody agrees on which syllogisms are valid. The theory of the syllogism partly constrains the inter- pretation of the forms. For example, it determines that the A form has existential import, at least if the I form does. For one of the valid patterns (Darapti) is: Every C is B Every C is A So, some A is B This is invalid if the A form lacks existential import, and valid if it has existential import. It is held to be valid, and so we know how the A form is to be interpreted. One then naturally asks about the O form; what do the syllogisms tell us about it? The answer is that they tell us nothing. This is because Aristotle did not discuss weakened forms of syllogisms, in which one concludes a particular proposition when one could already conclude the corresponding universal. For example, he does not mention the form: No C is B Every A is C So, some A is not B If people had thoughtfully taken sides for or against the validity of this form, that would clearly be relevant to the understanding of the O form. But the weakened forms were typically ignored... One other piece of subject-matter bears on the interpretation of the O form. People were interested in Aristotle's discussion of “infinite”negation, which is the use of negation to form a term from a term instead of a proposition from a proposition. In modern English we use “non”for this; we make “non-horse,”which is true for exactly those things that are not horses. In medieval Latin “non”and “not”are the same word, and so the distinction required special discussion. It became common to use infinite negation, and logicians pondered its logic. Some writers in the twelfth century and thirteenth centuries adopted a principle called "conversion by contraposition.”It states that • 'Every S is P ' is equivalent to 'Every non-P is non-S ' • 'Some S is not P ' is equivalent to 'Some non-P is not non-S ' Unfortunately, this principle (which is not endorsed by Aristotle) conflicts with the idea that there may be empty or universal terms. For in the universal case it leads directly from the truth: Every man is a being to the falsehood: Every non-being is a non-man (which is false because the universal affirmative has existential import, and there are no non-beings). And in the particular case it leads from the truth (remember that the O form has no existential import): A chimera is not a man to the falsehood: A non-man is not a non-chimera 123.3. TERM 471

These are [Jean] Buridan's examples, used in the fourteenth century to show the invalidity of contraposi- tion. Unfortunately, by Buridan's time the principle of contraposition had been advocated by a number of authors. The doctrine is already present in several twelfth century tracts, and it is endorsed in the thirteenth century by Peter of Spain, whose work was republished for centuries, by William Sherwood, and by . By the fourteenth century, problems associated with contraposition seem to be well-known, and authors generally cite the principle and note that it is not valid, but that it becomes valid with an additional assumption of existence of things falling under the subject term. For example, Paul of Venice in his eclectic and widely published Logica Parva from the end of the fourteenth century gives the traditional square with simple conversion but rejects conversion by contraposition, essentially for Buridan's reason.*[3] —Terence Parsons, The Stanford Encyclopedia of Philosophy

123.3 Term

A term (Greek horos) is the basic component of the proposition. The original meaning of the horos (and also of the Latin terminus) is“extreme”or“boundary”. The two terms lie on the outside of the proposition, joined by the act of affirmation or denial. For early modern logicians like Arnauld (whose Port-Royal Logic was the best-known text of his day), it is a psychological entity like an “idea”or "concept". Mill considers it a word. To assert “all Greeks are men”is not to say that the concept of Greeks is the concept of men, or that word“Greeks”is the word“men” .A proposition cannot be built from real things or ideas, but it is not just meaningless words either.

123.4 Proposition

In term logic, a “proposition”is simply a form of language: a particular kind of sentence, in which the subject and predicate are combined, so as to assert something true or false. It is not a thought, or an abstract entity. The word “propositio” is from the Latin, meaning the first premise of a syllogism. Aristotle uses the word premise (protasis) as a sentence affirming or denying one thing or another (Posterior Analytics 1. 1 24a 16), so a premise is also a form of words. However, as in modern philosophical logic, it means that which is asserted by the sentence. Writers before Frege and Russell, such as Bradley, sometimes spoke of the “judgment”as something distinct from a sentence, but this is not quite the same. As a further confusion the word “sentence”derives from the Latin, meaning an opinion or judgment, and so is equivalent to "proposition". The logical quality of a proposition is whether it is affirmative (the predicate is affirmed of the subject) or negative (the predicate is denied of the subject). Thus every philosopher is mortal is affirmative, since the mortality of philosophers is affirmed universally, whereas no philosopher is mortal is negative by denying such mortality in particular. The quantity of a proposition is whether it is universal (the predicate is affirmed or denied of all subjects or of “the whole”) or particular (the predicate is affirmed or denied of some subject or a “part”thereof). In case where existential import is assumed, quantification implies the existence of at least one subject, unless disclaimed.

123.5 Singular terms

For Aristotle, the distinction between singular and universal is a fundamental metaphysical one, and not merely grammatical. A singular term for Aristotle is primary substance, which can only be predicated of itself: (this) “Callias”or (this) “Socrates”are not predicable of any other thing, thus one does not say every Socrates one says every human (De Int. 7; Meta. D9, 1018a4). It may feature as a grammatical predicate, as in the sentence “the person coming this way is Callias”. But it is still a logical subject. He contrasts “universal”(katholou, “whole”) secondary substance, genera, with primary substance, particular specimens. The formal nature of universals, in so far as they can be generalized “always, or for the most part”, are the subject matter of both scientific study and formal logic.*[4] The essential feature of the syllogistic is that, of the four terms in the two premises, one must occur twice. Thus

All Greeks are men 472 CHAPTER 123. TERM LOGIC

All men are mortal.

The subject of one premise, must be the predicate of the other, and so it is necessary to eliminate from the logic any terms which cannot function both as subject and predicate, namely singular terms. However, in a popular 17th century version of the syllogistic, Port-Royal Logic, singular terms were treated as uni- versals:*[5]

All men are mortals All Socrates are men All Socrates are mortals

This is clearly awkward, a weakness exploited by Frege in his devastating attack on the system (from which, ultimately, it never recovered, see concept and object). The famous syllogism “Socrates is a man ...”, is frequently quoted as though from Aristotle,*[6] but fact, it is nowhere in the Organon. It is first mentioned by Sextus Empiricus in his Hyp. Pyrrh. ii. 164.

123.6 Influence on philosophy

123.7 Decline of term logic

Term logic began to decline in Europe during the Renaissance, when logicians like Rodolphus Agricola Phrisius (1444-1485) and Ramus (1515-1572) began to promote place logics. The logical tradition called Port-Royal Logic, or sometimes "traditional logic", saw propositions as combinations of ideas rather than of terms, but otherwise followed many of the conventions of term logic. It remained influential, especially in England, until the 19th century. Leibniz created a distinctive logical calculus, but nearly all of his work on logic remained unpublished and unremarked until went through the Leibniz Nachlass around 1900, publishing his pioneering studies in logic. 19th-century attempts to algebraize logic, such as the work of Boole (1815-1864) and Venn (1834-1923), typically yielded systems highly influenced by the term-logic tradition. The first predicate logic was that of Frege's landmark Begriffsschrift (1879), little read before 1950, in part because of its eccentric notation. Modern predicate logic as we know it began in the 1880s with the writings of Charles Sanders Peirce, who influenced Peano (1858-1932) and even more, Ernst Schröder (1841-1902). It reached fruition in the hands of Bertrand Russell and A. N. Whitehead, whose Principia Mathematica (1910–13) made use of a variant of Peano's predicate logic. Term logic also survived to some extent in traditional Roman Catholic education, especially in seminaries. Medieval Catholic theology, especially the writings of , had a powerfully Aristotelean cast, and thus term logic became a part of Catholic theological reasoning. For example, Joyce's Principles of Logic (1908; 3rd edition 1949), written for use in Catholic seminaries, made no mention of Frege or of Bertrand Russell.*[7]

123.8 Revival

Some philosophers have complained that predicate logic:

• Is unnatural in a sense, in that its syntax does not follow the syntax of the sentences that figure in our everyday reasoning. It is, as Quine acknowledged, "Procrustean,”employing an artificial language of function and argument, quantifier, and bound variable.

• Suffers from theoretical problems, probably the most serious being empty names and identity statements.

Even academic philosophers entirely in the mainstream, such as Gareth Evans, have written as follows:

“I come to semantic investigations with a preference for homophonic theories; theories which try to take serious account of the syntactic and semantic devices which actually exist in the language ...I would 123.9. SEE ALSO 473

prefer [such] a theory ... over a theory which is only able to deal with [sentences of the form “all A's are B's"] by “discovering”hidden logical constants ... The objection would not be that such [Fregean] truth conditions are not correct, but that, in a sense which we would all dearly love to have more exactly explained, the syntactic shape of the sentence is treated as so much misleading surface structure”(Evans 1977)

123.9 See also

123.10 Notes

[1] Degnan, M. 1994. Recent Work in Aristotle's Logic. Philosophical Books 35.2 (April, 1994): 81-89.

[2] • Review of “Aristotle, Prior Analytics: Book I, Gisela Striker (translation and commentary), Oxford UP, 2009, 268pp., $39.95 (pbk), ISBN 978-0-19-925041-7.”in the Notre Dame Philosophical Reviews, 2010.02.02.

[3] Parsons, Terence (2012). “The Traditional Square of Opposition”. In Edward N. Zalta. The Stanford Encyclopedia of Philosophy (Fall 2012 ed.). 3-4.

[4] They are mentioned briefly in the De Interpretatione. Afterwards, in the chapters of the Prior Analytics where Aristotle methodically sets out his theory of the syllogism, they are entirely ignored.

[5] Arnauld, Antoine and Nicole, Pierre; (1662) La logique, ou l'art de penser. Part 2, chapter 3

[6] For example: Kapp, Greek Foundations of Traditional Logic, New York 1942, p. 17, Copleston A History of Philosophy Vol. I., p. 277, Russell, A History of Western Philosophy London 1946 p. 218.

[7] Copleston's A History of Philosophy

123.11 References

• Bochenski, I. M., 1951. Ancient Formal Logic. North-Holland. • Louis Couturat, 1961 (1901). La Logique de Leibniz. Hildesheim: Georg Olms Verlagsbuchhandlung. • Gareth Evans, 1977, “Pronouns, Quantifiers and Relative Clauses,”Canadian Journal of Philosophy. • , 1976. Reason and Argument. University of California Press. • Hammond and Scullard, 1992. The Oxford Classical Dictionary. Oxford University Press, ISBN 0-19-869117- 3. • Joyce, George Hayward, 1949 (1908). Principles of Logic, 3rd ed. Longmans. A manual written for use in Catholic seminaries. Authoritative on traditional logic, with many references to medieval and ancient sources. Contains no hint of modern formal logic. The author lived 1864-1943. • Jan Lukasiewicz, 1951. Aristotle's Syllogistic, from the Standpoint of Modern Formal Logic. Oxford Univ. Press. • John Stuart Mill, 1904. A System of Logic, 8th ed. London. • Parry and Hacker, 1991. Aristotelian Logic. State University of New York Press. • Arthur Prior 1962: Formal Logic, 2nd ed. Oxford Univ. Press. While primarily devoted to modern formal logic, contains much on term and medieval logic. 1976: The Doctrine of Propositions and Terms. Peter Geach and A. J. P. Kenny, eds. London: Duckworth. • Willard Quine, 1986. Philosophy of Logic 2nd ed. Harvard Univ. Press. • Rose, Lynn E., 1968. Aristotle's Syllogistic. Springfield: Clarence C. Thomas. 474 CHAPTER 123. TERM LOGIC

• Sommers, Fred

1970: “The Calculus of Terms,”Mind 79: 1-39. Reprinted in Englebretsen, G., ed., 1987. The new syllogistic New York: Peter Lang. ISBN 0-8204-0448-9 1982: The logic of natural language. Oxford University Press. 1990: "Predication in the Logic of Terms," Notre Dame Journal of Formal Logic 31: 106-26. and Englebretsen, George, 2000: An invitation to formal reasoning. The logic of terms. Aldershot UK: Ashgate. ISBN 0-7546-1366-6.

• Szabolcsi Lorne, 2008. Numerical Term Logic. Lewiston: Edwin Mellen Press.

123.12 External links

• Term logic at PhilPapers

• Aristotle's Logic entry by Robin Smith in the Stanford Encyclopedia of Philosophy • Term logic entry in the Internet Encyclopedia of Philosophy

• Aristotle's term logic online-This online program provides a platform for experimentation and research on Aristotelian logic.

• Annotated bibliographies: Fred Sommers. George Englebretsen.

• PlanetMath: Aristotelian Logic. • Interactive Syllogistic Machine for Term Logic A web based syllogistic machine for exploring fallacies, figures, terms, and modes of syllogisms. Chapter 124

Testability

This article is about hypothesis testing. For the ability of equipment to be tested, see Non-functional requirement and Software testability.

Testability, a property applying to an empirical hypothesis, involves two components:

1. The logical property that is variously described as contingency, defeasibility, or falsifiability, which means that counterexamples to the hypothesis are logically possible. 2. The practical feasibility of observing a reproducible series of such counterexamples if they do exist.

In short, a hypothesis is testable if there is some real hope of deciding whether it is true or false of real experience. Upon this property of its constituent hypotheses rests the ability to decide whether a theory can be supported or falsified by the data of actual experience. If hypotheses are tested, initial results may also be labeled inconclusive.

124.1 See also

• Confirmability • Contingency

• Controllability • Observability

• Scientific method

124.2 Further reading

• Popper, Karl (2008). The logic of scientific discovery (Reprint ed.). London: Routledge. ISBN 0415278449.

475 Chapter 125

Tetralemma

The tetralemma is a figure that features prominently in the classical logic of India. It states that with reference to any a logical proposition X, there are four possibilities:

X

¬X X ∧ ¬X ¬(X ∨ ¬X)

125.1 Catuskoti

The history of fourfold negation, the Catuskoti (Sanskrit), is evident in the logico-epistemological tradition of India, given the categorical nomenclature 'Indian logic' in Western discourse. Subsumed within the auspice of Indian logic, 'Buddhist logic' has been particularly focused in its employment of the fourfold negation, as evidenced by the traditions of Nagarjuna and the , particularly the school of Madhyamaka given the retroactive nomenclature of Prasangika by the Tibetan Buddhist logico-epistemological tradition.

125.2 See also

• Paraconsistent logic • Prasangika

• Two-truths doctrine • Catuṣkoṭi, a similar concept in Indian philosophy

125.3 External links

• Wiktionary definition of tetralemma

• Twelve links blog Notes on the tetralemma

476 Chapter 126

The Game of Logic

The Game of Logic is a book written by Lewis Carroll, published in 1886.*[1] *[2] *[3] *[4] *[5]

126.1 References

[1] “Scanned copy of the book at archive.org”.

[2] “The Game Of Logic”. goodreads.com. Retrieved 23 December 2013.

[3] “The game of logic”. amazon.com. Retrieved 23 December 2013.

[4] “The Game of Logic”. barnesandnoble.com. Retrieved 23 December 2013.

[5] “The Game of Logic: By Lewis Carroll”. books.google.com. Retrieved 23 December 2013.

126.2 External links

• Scanned copy at archive.org • Entry at gutenberg.org

477 Chapter 127

Theoretical definition

A theoreticals (or conceptual) definition is an abstract concept that defines a term in an academic discipline. With- out a falsifiable operational definition, conceptual definitions assume both knowledge and acceptance of the theories that it depends on.*[1] A hypothetical construct may serve as a theoretical definition, as can a stipulative definition. A theoretical definition is a proposed way of thinking about potentially related events *[1]*[2] Indeed, theoretical definitions contain built-in theories; they cannot be simply reduced to describing a set of observations. The definition may contain implicit inductions and deductive consequences that are part of the theory.*[3] A theoretical definition of a term can change, overtime, based on the methods in the field that created it.

127.1 In Different Fields

127.1.1 Sciences

The term scientific theory is reserved for concepts that are widely acepted. A scientific law often refers to regularities that can be expressed by a mathematical statement. However, there is no consensus about the distinction between these terms.*[4] Every scientific concept must have an operational definition, however the operational definition can use both direct observations and latent variables.*[5]

Natural Sciences

In the natural sciences, a concept is an abstract conclusion drawn from observations.*[5]

Social and Health Sciences

Social and health sciences interact with non-empirical fields and use both observation based and pre-existing concepts such as intelligence, race, and gender.

• In psychology the term “conceptual definition”is used for a concept variable.*[6]

127.1.2 Interdisciplinary

Most interdisciplinary fields are designed to address specific real world concerns and theoretical definitions in inter- disciplinary fields is still evolving.*[7]

478 127.2. EXAMPLES 479

In psychology and neuroscience, the concept of“intelligence”must be understood in terms of a combination of falsifiable operational concepts.

127.2 Examples

127.2.1 In

The definitions of substances as various configurations of atoms are theoretical definitions, as are definitions of colors as specific wavelengths of reflected light. 480 CHAPTER 127. THEORETICAL DEFINITION

Physics

The first postulate of special relativity theory that the speed of light in vacuum is the same to all inertial observers (i.e. it is a constant, and therefore a good measure of length). Of interest, this theoretical concept is the basis of an operational definition for the length of a metre is “the distance traveled by light in a vacuum during a time interval of 1/299,792,458 of a second". Thus we have defined 'metre' according to other ideas contained in modern scientific theory. Rejection of the theory underlying a theoretical definition leaves the definition invalid for use in argument with those who reject it —neither side will advance its position by using terms the others do not accept . Heat explains a collection of various laws of nature and that predict certain results.*[2]

127.2.2 In Medicine

127.2.3 In

Psychology

In psychology, the concept of intelligence is meant to explain correlations in performance on certain cognitive tasks.*[8] Recent models suggest several cognitive processes may be involved in tasks that have been associated with intelligence.*[9] However, overall the "g" or general intelligence factor is relatively supported by research*[10]

127.2.4 In Philosophy

John Searle's Chinese room thought experiment illustrates how differing theoretical definitions of “thinking”have caused conflict amongst artificial intelligence philosophers. Some philosophers might call“thought”merely“having the ability to convince another person that you can think”. An accompanying operational definition for this theoretical definition could be a simple conversation test (e.g. Turing test). In contrast, Searle believes that better theoretical and operational definitions are required.

127.3 See also

• Latent variable

• Operational definition

• Construct (philosophy)

• Stipulative definitions

• General Conference on Weights and Measures

• International Committee for Weights and Measures

127.4 References

[1] About.com, Logical Arguments, “Theoretical Definitions”

[2] A Concise Introduction to Logic by Patrick J. Hurley. 2007. Cengage learning. Entry on “Theoretical Definitions”may even be available through google books

[3] http://plato.stanford.edu/entries/science-theory-observation

[4] http://science.kennesaw.edu/~{}rmatson/3380theory.html

[5] Watt, James H.; van den Berg, Sjef (2002). Philosophy of Science, Empiricism, and the Scientific Method. http://www.cios. org. p. 11. Retrieved 24 March 2015. External link in |publisher= (help)

[6] http://www.rit.edu/cla/gssp400/lectures/l5.html 127.4. REFERENCES 481

[7] http://www.sagepub.com/upm-data/43242_1.pdf

[8] “Intelligent intelligence testing”. American Psychological Association. Retrieved 24 March 2015.

[9] Conway, Andrew R.A; Cowan, Nelson; Bunting, Michael F; Therriault, David J; Minkoff, Scott R.B (1999).“A latent vari- able analysis of working memory capacity, short-term memory capacity, processing speed, and general fluid intelligence” . Intelligence 30 (2): 163–183. doi:10.1016/S0160-2896(01)00096-4. Retrieved 24 March 2015.

[10] http://tip.duke.edu/node/1280 Chapter 128

Train of thought

For other uses, see Train of thought (disambiguation).

The train of thought or track of thought refers to the interconnection in the sequence of ideas expressed during a connected discourse or thought, as well as the sequence itself, especially in discussion how this sequence leads from one idea to another. When a reader or listener “loses the train of thought”(i.e., loses the relation between consecutive sentences or phrases, or the relation between non-verbal concepts in an argument or presentation), comprehension is lost of the expressed or unexpressed thought.*[1] The term“train of thoughts”was introduced and elaborated as early as in 1651 by Thomas Hobbes in his Leviathan, though with a somewhat different meaning (similar to the meaning used by the British associationists):

By Consequence, or train of thoughts, I understand that succession of one thought to another which is called, to distinguish it from discourse in words, mental discourse.

When a man thinketh on anything whatsoever, his next thought after is not altogether so casual as it seems to be. Not every thought to every thought succeeds indifferently. —Thomas Hobbes, Leviathan, The First Part: Of Man, Chapter III: Of the Consequence or Train of Imagination

128.1 See also

• Derailment (thought disorder) • Absent-mindedness

• Internal monologue • Mind-wandering

• Association of Ideas • Associationism

• Stream of consciousness

128.2 References

[1] Edward Parmelee Morris, “On Principles and Methods in Latin Syntax”(1901), Chapter VI: Parataxis

482 Chapter 129

Transferable belief model

The transferable belief model (TBM) is an elaboration on the Dempster–Shafer theory of evidence.

129.1 Context

Consider the following classical problem of information fusion. A patient has an illness that can be caused by three different factors A, B and C. Doctor 1 says that the patient's illness is very likely to be caused by A (very likely, meaning probability p = 0.95), but B is also possible but not likely (p = 0.05). Doctor 2 says that the cause is very likely C (p = 0.95), but B is also possible but not likely (p = 0.05). How is one to make one's own opinion from this ? Bayesian updating the first opinion with the second (or the other way round) implies certainty that the cause is B. Dempster's rule of combination lead to the same result. This can be seen as paradoxical, since although the two doctors point at different causes, A and C, they both agree that B is not likely. (For this reason the standard Bayesian approach is to adopt Cromwell's rule and avoid the use of 0 or 1 as probabilities.) The transferable belief model (TBM) is an elaboration on the Dempster–Shafer theory of evidence developed by Philippe Smets, based on the intuition that in the situation above, the result should allocate most of the belief weight to the empty set (i.e. neither A, B, nor C). Technically, this would be done by using the TBM conjunction rule for non-interactive sources of information, which is the same as Dempster's rule of combination without renormalization. While most other theories adhere to the axiom the probability (or belief mass) of the empty set is always zero, there is another intuitive reason to drop this axiom: the open-world assumption. It applies when the frame of reference is not exhaustive, so there are reason to believe that an event not described in this frame of reference will occur. For example, when tossing a coin one usually assumes that Head or Tail will occur, so that Pr(Head)+Pr(Tail)=1 . The open-world assumption is that the coin can be stolen in mid-air, disappear, break apart or otherwise fall sideway so that neither Head nor Tail occurs, so that the power set of {Head,Tail} is considered and there is a decompostion of the overall probability (i.e. 1) of the following form:

Pr(∅) + Pr(Head) + Pr(Tail) + Pr(Head, Tail) = 1.

129.2 See also

• Dempster–Shafer theory

129.3 References

• Smets Ph. (1988) “Belief function”. In: Non Standard Logics for Automated Reasoning, ed. Smets Ph., Mamdani A, Dubois D. and Prade H. Academic Press, London

483 484 CHAPTER 129. TRANSFERABLE BELIEF MODEL

• Smets Ph. (1990) “The combination of evidence in the transferable belief model”, IEEE Pattern Analysis and Machine Intelligence, 12, 447–458 • Smets Ph. (1993) “An axiomatic justification for the use of belief function to quantify beliefs”, IJCAI'93 (Inter. Joint Conf. on AI), Chambery, 598–603 • Smets Ph. and Kennes R. (1994) “The transferable belief model”, Artificial Intelligence, 66,191–234

• Smets Ph. and Kruse R. (1995)“The transferable belief model for belief representation”In: Smets and Motro A. (eds.) Uncertainty Management in Information Systems: from Needs to solutions. Kluwer, Boston

• Ramasso, E., Rombaut, M., Pellerin D. (2007) “Forward-Backward-Viterbi procedures in the Transferable Belief Model for state sequence analysis using belief functions”, ECSQARU, Hammamet : Tunisie (2007).

• Touil, K., Zribi, M., Benjelloun, M. (2007) “Application of transferable belief model to navigation system” , Integrated Computer-Aided Engineering, 14 (1), 93–105

129.4 External links

• The Transferable Belief Model

• Publications on TBM • Software for TBM in Matlab Chapter 130

Trikonic

Trikonic, is a technique of triadic analysis-synthesis which has been developed by Gary Richmond based on the original idea of a possible applied science making three categorial distinctions, which philosopher Charles Sanders Peirce, its creator, called“Trichotomic.”Peirce introduces trichotomic as the“art of making three-fold divisions” and trikonic is the name given to the electronic trichotomic approach to this science of the trichotomic or triadic analysis of anything which can be so analysed.

Figure 1.0 The Trikonic Symbol

Trikonic is a method that has been developed to assist in certain researches and decision making processes which are of most interest to the virtual community. In the future, it will be developed into an electronic tool used for analysis & discussion, the terminology related to the components of various triads, the relations that triads hold with other triads, and much more. The symbol shown in Figure 1.0 is referred to as the‘trikon’symbol by Richmond (2005). It displays the divisions of three categorial elements and their relationship for the subject in consideration

485 486 CHAPTER 130. TRIKONIC

130.1 Trikonic Analysis/Peircean Category Theory

A major part of trikonic analysis is the three Peircean categories; these consist of firstness, secondness and thirdness:

•“Firstness is the mode of being that of which is such as it is, positively and without reference to anything else.

• Secondness is the mode of being that which is such as it is, with respect to a second but regardless of any third.

• Thirdness is the mode of being that which is such as it is, in bringing a second and third into relation to each other.”

Figure 2.0 The Three Peircean Categories

Another way of describing the basic understanding of these notions is from Sowa:

“First is the conception of being or existing independent of anything else. Second is the conception of being relative to, the conception of reaction, with something else. Third is the conception of mediation, whereby a first and a second are brought into relation. (1891)”

Trikonically represented, the categories are (Figure 2.0): There are many ways in which these notions can be interpreted. These are can be represented as: Peirce's Three Universes of Experience (Figure 3.0): These in turn represent Peirce's three Universal Categories (Figure 4.0 The Universal Categories): In addition to the universal categories, there are equivalent Existential Categories (Figure 5.0 The Existential Cate- gories): We can also trikonically analyse thought, by identifying three Logical Modalities (Figure 6.0 The Logical Modalities): With these representations in mind, firstness, secondness and thirdness can be defined as the following: 130.2. VECTOR ANALYSIS 487

Figure 3.0 Peirce's Three Universes of Experience:

• Firstness is concerned with ideas, character, qualities, feeling, image and possibility.

• Secondness signifies events, brute actions, reactions, existence and character.

• Thirdness brings together firstness in relation to secondness.

130.2 Vector Analysis

There are six vectors that can be used in trikonic vector analysis; these are shown in Figure 7.0. These six vectors have also been referred to as “directions of movement through the trikon”(Richmond, 2005). They are authentic permutations of the triad, i.e. they are different arrangements of the order of firstness, secondness and thirdness. They represent the relationships between firstness (1ns), secondness (2ns) and thirdness (3ns) for the object that is being analysed.

• Vector of Process – (1ns) chance sporting, then follows patterns of habit formation (3ns) which leads to some actual structural change in an organism (2ns).

• Vector of Order – thesis (1ns), subsequently antithesis (2ns) leading to synthesis (3ns).

• Vector of Representation – An engineer (3ns) create a CAD drawing (1ns) of a signalling design (2ns).

• Vector of Analysis – 3ns which involves 2ns which in turn involves 1ns.

• Vector of Determination – The object determines (2ns) a sign (1ns) for an interpretant (3ns). 488 CHAPTER 130. TRIKONIC

Figure 4.0 The Universal Categories:

• Vector of Aspiration - (2ns), (3ns), (1ns), represents the unique character specifically in human community development.

Within trikonic are six directions, which are all permutations of logical paths of relations keeping the notion of 1ns, 2ns and 3ns throughout the object under consideration. You can interpret these in many ways, for example, the Determination Vector and Representation Vector (Please refer to Figure 8.0 and Figure 9.0). The reasoning behind the exploration of the six vectors within the triad aims at bringing a new view on analysis and a more systematic treatment to some of the difficult issues which arise especially in semiotic analysis, i.e. the vector process is a graphically logical analysis with aspects of dependence and constraint, and correlation, which tie in with the “living”reflection of the categories by the semiotic triad. The reasoning behind the vector analysis is to use the permutations within the triad for the theory and any actual semiosis, mostly the analysis of complex virtual communicative projects. Semiosis is defined as: “Semiosis is the making or production of meaning. The term was introduced by Charles Peirce (1839-1914) to describe the process of signification within the science of signs now termed semiology”. Trikonic analysis could lead to a kind of trikonic synthesis (the combination of separate elements or substances to form a coherent whole producing fundamental change in the further evolution of virtual community development.

130.3 Trikonic Analysis in HCI

Richmond suggests that “not all things can be trikonically analysed” (2005). However, there has been investigation into whether trikonic can be applied to HCI related issues. Especially, those issues concerned with websites and culture, trikonic analysis using the Peircean Category Theory has been used to analyse how trust is invoked in a user when using websites. It may be possible to use 1ns, 2ns and 3ns to categorise features of a website, for example the 130.4. TRIKONIC ANALYSIS ON THE INTEROPERABILITY OF CONCEPTUAL GRAPH (CG) TOOLS 489

Figure 5.0 The Existential Categories:

visual elements, colours or text used, layout, the content of the website, credibility, how the elements relate to the users actual purpose of using the website etc. One of the purposes of trikonic is to explore whether it can help to identify what elements of a website actually influence the user to perceive it to be credible or not.

130.4 Trikonic Analysis on the Interoperability of Conceptual Graph (CG) Tools

When approaching software problems like the interoperability of CG tools, it is often useful to look at a situation from a software engineering perspective to find a solution. However, with the use of trikonic it is possible to approach the software engineering problem in this way. Trikonic analysis is a very sophisticated method, and as shown in the examples above it can be used in many different ways and scenarios. Again, using the notions of 1ns, 2ns and 3ns it may be possible to see how certain elements of the CG tools relate to each other in this way and produce a good solution from this. Examples of the three categories could be the programming language, the CG language and the architecture of the software. From this you could say, that: “this CG language is a possibility”(1ns), “the programming language needs to be…”(2ns), “the architecture of the software is…”(3ns).

130.5 See also

• Categories (Peirce) • Semiosis 490 CHAPTER 130. TRIKONIC

Figure 6.0 The Logical Modalities:

• Thesis, antithesis, synthesis

130.6 References

130.7 Further reading

• HOUSER, N. & KLOESEL, C., ed. 1998. The Essential Peirce: Selected Philosophical Writings in 2 volumes, Indiana University Press.

• RICHMOND, G., 2005. Outline of trikonic Diagrammatic Trichotomic. In: F. DAU, ML. MUGNIER, G. STUMME, ed. Conceptual Structures: Common Semantics for Sharing Knowledge: 13th International Con- ference on Conceptual Structures, ICCS 2005, Kassel, Germany, 17-22 July, 2005. Springer-Verlag GmbH, pp. 453 – 466. • SOWA, J.F., 2000. Knowledge Representation – Logical, Philosophical and Computational Foundations. 130.7. FURTHER READING 491

Figure 7.0 The Vectors

Figure 8.0 The Determination Vector 492 CHAPTER 130. TRIKONIC

Figure 9.0 The Representation Vector Chapter 131

Truth-bearer

This article is about a term used in philosophy, logic and philosophy of logic.

A truth-bearer is an entity that is said to be either true or false and nothing else. The thesis that some things are true while others are false has led to different theories about the nature of these entities. Since there is divergence of opinion on the matter, the term truth-bearer is used to be neutral among the various theories. Truth-bearer candidates include propositions, sentences, sentence-tokens, statements, concepts, beliefs, thoughts, intuitions, utterances, and judgements but different authors exclude one or more of these, deny their existence, argue that they are true only in a derivative sense, assert or assume that the terms are synonymous,*[1] or seek to avoid addressing their distinction or do not clarify it.*[2]

131.1 Introduction

Some distinctions and terminology as used in this article, based on Wolfram 1989*[3] (Chapter 2 Section1) follow. It should be understood that the terminology described is not always used in the ways set out, and it is introduced solely for the purposes of discussion in this article. Use is made of the type–token and use–mention distinctions. Reflection on occurrences of numerals might be helpful.*[4] In grammar a sentence can be a declaration, an explanation, a question, a command. In logic a declarative sentence is considered to be a sentence that can be used to communicate truth. Some sentences which are grammatically declarative are not logically so. A character*[nb 1] is a typographic character (printed or written) etc. A word token*[nb 2] is a pattern of characters. A word-type*[nb 3] is an identical pattern of characters. A meaningful-word-token*[nb 4] is a meaningful word-token. Two word-tokens which mean the same are of the same word-meaning*[nb 5] A sentence-token*[nb 6] is a pattern of word-tokens. A meaningful-sentence-token*[nb 7] is a meaningful sentence- token or a meaningful pattern of meaningful-word-tokens. Two sentence-tokens are of the same sentence-type if they are identical patterns of word-tokens characters*[nb 8] A declarative-sentence-token is a sentence-token which that can be used to communicate truth or convey information.*[5] A meaningful-declarative-sentence-token is a mean- ingful declarative-sentence-token*[nb 9] Two meaningful-declarative-sentence-tokens are of the same meaningful- declarative-sentence-type*[nb 10] if they are identical patterns of word-tokens. A nonsense-declarative-sentence- token*[nb 11] is a declarative-sentence-token which is not a meaningful-declarative-sentence-token. A meaningful- declarative-sentence-token-use*[nb 12] occurs when and only when a meaningful-declarative-sentence-token is used declaratively. A referring-expression*[nb 13] is expression that can be used to pick out or refer to particular entity. A referential success*[nb 14] is a referring-expression’s success in identifying a particular entity. A referential failure*[nb 15] is a referring-expression’s failure to identify a particular entity. A referentially-successful-meaningful-declarative- sentence-token-use*[nb 16] is a meaningful-declarative-sentence-token-use containing no referring-expression that fails to identify a particular entity.

493 494 CHAPTER 131. TRUTH-BEARER

131.2 Sentences in natural languages

As Aristotle pointed out, since some sentences are questions, commands, or meaningless, not all can be truth-bearers. If in the proposal “What makes the sentence Snow is white true is the fact that snow is white”it is assumed that sentences like Snow is white are truth-bearers, then it would be more clearly stated as“What makes the meaningful- declarative-sentence Snow is white true is the fact that snow is white”. Theory 1a:

All and only meaningful-declarative-sentence-types*[nb 17]) are truth-bearers

Criticisms of Theory 1a Some meaningful-declarative-sentence-types will be both truth and false, contrary to our definition of truth-bearer, e.g. (i) the liar-paradox sentences such as“This sentence is false”. (see Fisher 2008*[6]) (ii) Time, place and person dependent sentences e.g. “It is noon”. “This is London”, “I'm Spartacus”. Anyone may ..ascribe truth and falsity to the deterministic propositional signs we here call utterances. But if he takes this line, he must, like Leibniz, recognise that truth cannot be an affair solely of actual utterances, since it makes sense to talk of the discovery of previously un-formulated truths. (Kneale, W&M (1962)*[7]) Revision to Theory 1a, by making a distinction between type and token. To escape the time, place and person dependent criticism the theory can be revised, making use or the type–token distinction,*[8] as follows Theory 1b:

All and only meaningful-declarative-sentence-tokens are truth-bearers

Quine argued that the primary truth-bearers are utterances *[nb 18]

Having now recognised in a general way that what are true are sentences, we must turn to certain refinements. What are best seen as primarily true or false are not sentences but events of utterances. If a man utters the words 'It is raining' in the rain, or the words 'I am hungry' while hungry, his verbal performance counts as true. Obviously one utterance of a sentence may be true and another utterance of the same sentence be false.

QUINE 1970*[9] page 13 Criticisms of Theory 1b (i) Theory 1b prevents sentences which are meaningful-declarative-sentence-types from being truth-bearers. If all meaningful-declarative-sentence-types typographically identical to “The whole is greater than the part”are true then it surely follow that the meaningful-declarative-sentence-type“The whole is greater than the part”is true (just as all meaningful-declarative-sentence-tokens typographically identical to “The whole is greater than the part”are English entails the meaningful-declarative-sentence-types“The whole is greater than the part”is English) (ii) Some meaningful-declarative-sentences-tokens will be both truth and false, or neither, contrary to our definition of truth- bearer. E.g. A token, t, of the meaningful-declarative-sentence-type ‘P: I'm Spartacus’, written on a placard. The token t would be true when used by Spartacus, false when used by Bertrand Russell, neither true nor false when mentioned by Spartacus or when being neither used nor mentioned. Theory 1b.1

All meaningful-declarative-sentence-token-uses are truth-bearers; some meaningful-declarative-sentence- types are truth-bearers

To allow that at least some meaningful-declarative-sentence-types can be truth-bearers Quine allowed so-called eter- nal sentences*[nb 19] to be truth-bearers.

In Peirces's terminology, utterances and inscriptions are tokens of the sentence or other linguistic expression concerned; and this linguistic expression is the type of those utterances and inscriptions. In 131.2. SENTENCES IN NATURAL LANGUAGES 495

Frege's terminology, truth and falsity are the two truth values. Succinctly then, an eternal sentence is a sentence whose tokens have the same truth values.... What are best regarded as true and false are not propositions but sentence tokens, or sentences if they are eternal

Quine 1970*[9] pages 13–14 Theory 1c

All and only meaningful-declarative-sentence-token-uses are truth-bearers

Arguments for Theory 1c By respecting the use–mention Theory 1c avoids criticism (ii) of Theory 1b. Criticisms of Theory 1c (i) Theory 1c does not avoid criticism (i) of Theory 1b. (ii) meaningful-declarative-sentence-token-uses are events (located in particular positions in time and space) and entail a user. This implies that (a) nothing (no truth-bearer) exists and hence nothing (no truth-bearer) is true (of false) anytime anywhere (b) nothing (no truth-bearer) exists and hence nothing (no truth-bearer) is true (of false) in the absence of a user. This implies that (a) nothing was true before the evolution of users capable of using meaningful-declarative-sentence-tokens and (b) nothing is true (or false) accept when being used (asserted) by a user. Intuitively the truth (or falsity) of ‘The tree continues to be in the quad’continues in the absence of an agent to asset it. Referential Failure A problem of some antiquity is the status of sentences such as U: The King of France is bald V: The highest prime has no factors W: Pegasus did not exist Such sentences purport to refer to entitles which do not exist (or do not always exist). They are said to suffer from referential failure. We are obliged to choose either (a) That they are not truth-bearers and consequently neither true nor false or (b) That they are truth-bearers and per se are either true of false. Theory 1d

All and only referentially-successful-meaningful-declarative-sentence-token-uses are truth-bearers.

Theory 1d takes option (a) above by declaring that meaningful-declarative-sentence-token-uses that fail referentially are not truth-bearers. Theory 1e

All referentially-successful-meaningful-declarative-sentence-token-uses are truth-bearers; some meaningful- declarative-sentence-types are truth-bearers

Arguments for Theory 1e Theory 1e has the same advantages as Theory 1d. Theory 1e allows for the existence of truth-bearers (i.e., meaningful- declarative-sentence-types) in the absence of users and between uses. If for any x, where x is a use of a referentially successful token of a meaningful-declarative-sentence-type y x is a truth-bearer then y is a truth-bearer otherwise y is not a truth bearer. E.g. If all uses of all referentially successful tokens of the meaningful-declarative-sentence-type ‘The whole is greater than the part’are truth-bearers (i.e. true or false) then the meaningful-declarative-sentence-type ‘The whole is greater than the part’is a truth-bearer. If some but not all uses of some referentially successful tokens of the meaningful-declarative-sentence-type ‘I am Spartacus’are true then the meaningful-declarative-sentence-type ‘I am Spartacus’is not a truth-bearer. Criticisms of Theory 1e Theory 1e makes implicit use of the concept of an agent or user capable of using (i.e. asserting) a referentially- successful-meaningful-declarative-sentence-token. Although Theory 1e does not depend on the actual existence (now, in the past or in the future) of such users, it does depend on the possibility and cogency of their existence. Consequently the concept of truth-bearer under Theory 1e is dependent upon giving an account of the concept of a ‘user’. In so far as referentially-successful-meaningful-declarative-sentence-tokens are particulars (locatable in time and space) the definition of truth-bearer just in terms of referentially-successful-meaningful-declarative-sentence is attractive to those who are (or would like to be) nominalists. The introduction of ‘use’and ‘users’threatens the introduction of intentions, attitudes, minds &c. as less-than=welcome ontological baggage 496 CHAPTER 131. TRUTH-BEARER

131.3 Sentences in languages of classical logic

In classical logic a sentence in a language is true or false under (and only under) an interpretation and is therefore a truth-bearer. For example a language in the first-order predicate calculus might include one of more predicate symbols and one or more individual constants and one or more variables. The interpretation of such a language would define a domain (universe of discourse); assign an element of the domain to each individual constant; assign the donation in the domain of some property to each unary (one-place) predicate symbol.*[10] For example if a language L consisted in the individual constant a, two unary predicate letters F and G and the variable x, then an interpretation I of L might define the Domain D as animals, assign Socrates to a, the denotation of the property being a man to F and the denotation of the property being mortal to G. Under the interpretation I of L then Fa would be true if, and only if Socrates is a man, and the sentence (Fx → Gx) would be true if, and only if all men (in the domain) are mortal. In some texts an interpretation is said to give“meaning”to the symbols of the language. Since Fa has the value true under some (but not all interpretations) it is not the sentence-type Fa which is said to be true but only some sentence-tokens of Fa under particular interpretations. A token of Fa without an interpretation is neither true nor false. Some sentences of a Language like L are said to be true under all interpretations of the sentence, e.g. (Fx ¬Fx), such sentences being termed logical truths, but again such sentences are neither true nor false in the absence of an interpretation.

131.4 Propositions

Many authors*[11] use the term proposition as truth-bearers. There is no single definition or usage.*[12]*[13] Sometimes it is used to mean a meaningful declarative sentence itself; sometimes it is used to mean the meaning of a meaningful declarative sentence.*[14] This provides two possible definitions for the purposes of discussion as below Theory 2a:

All and only meaningful-declarative-sentences are propositions

Theory 2b:

A meaningful-declarative-sentence-token expresses a proposition; two meaningful-declarative-sentence- tokens which have the same meaning express the same proposition; two meaningful-declarative-sentence- tokens with different meanings express different propositions.

(cf Wolfram 1989,*[3] p. 21) Proposition is not always used in one or other of these ways. Criticisms of Theory 2a.

• If all and only meaningful-declarative-sentences are propositions, as advanced by Theory 2a, then the terms are synonymous and we can just as well speak of the meaningful-declarative-sentences themselves as the trutbearers - there is no distinct concept of proposition to consider, and the term proposition is literally redundant.

Criticisms of Theory 2b

• Theory 2b entails that if all meaningful-declarative-sentence-tokens typographically identical to say, “I am Spartacus”have the same meaning then they (i) express the same proposition (ii) that proposition is both true and false,*[15] contrary to the definition of truth-bearer.

• The concept of a proposition in this theory rests upon the concept of meaning as applied to meaningful- declarative-sentences, in a word synonymy among meaningful-declarative-sentence s. Quine 1970 argues that the concept of synonymy among meaningful-declarative-sentences cannot be sustained or made clear, consequently the concepts of “propositions”and “meanings of sentences”are, in effect, vacuous and su- perfluous*[16] 131.5. STATEMENTS 497

see also Willard Van Orman Quine, Proposition, The Russell-Myhill Antinomy, also known as the Principles of Math- ematics Appendix B Paradox see also Internet Encycypedia of Philosophy Propositions are abstract entities; they do not exist in space and time. They are sometimes said to be“timeless”,“eternal”, or“omnitemporal”entities. Terminology aside, the essential point is that propositions are not concrete (or material) objects. Nor, for that matter, are they mental entities; they are not “thoughts”as Frege had suggested in the nineteenth century. The theory that propositions are the bearers of truth-values also has been criticized. Nominalists object to the abstract character of propositions. Another complaint is that it’s not sufficiently clear when we have a case of the same propositions as opposed to similar propositions. This is much like the complaint that we can’t determine when two sentences have exactly the same meaning. The relationship between sentences and propositions is a serious philosophical problem.

131.5 Statements

Many authors consider statements as truth-bearers, though as with the term“proposition”there is divergence in def- inition and usage of that term. Sometimes 'statements' are taken to be meaningful-declarative-sentences; sometimes they are thought to be what is asserted by a meaningful-declarative-sentence. It is not always clear in which sense the word is used. This provides two possible definitions for the purposes of discussion as below. A particular concept of a statement was introduced by Strawson in the 1950s.,*[17]*[18]*[19] Consider the following:

• I: The author of Waverley is dead • J: The author of Ivanhoe is dead

• K: I am less than six feet tall • L: I am over six feet tall

• M: The conductor is a bachelor • N: The conductor is married

On the assumption that the same person wrote Waverley and Ivanhoe, the two distinct patterns of characters (meaningful- declarative-sentences) I and J make the same statement but express different propositions. The pairs of meaningful-declarative-sentences (K, L) & (M, N) have different meanings, but they are not necessarily contradictory, since K & L may have been asserted by different people, and M & N may have been asserted about different conductors. What these examples show is that we cannot identify that which is true or false (the statement) with the sentence used in making it; for the same sentence may be used to make different statements, some of them true and some of them false. (Strawson, P.F. (1952)*[19]) This suggests:

• Two meaningful-declarative-sentence-tokens which say the same thing of the same object(s) make the same statement.

Theory 3a

All and only statements are meaningful-declarative-sentences.

Theory 3b

All and only meaningful-declarative-sentences can be used to make statements

Statement is not always used in one or other of these ways. Arguments for Theory 3a 498 CHAPTER 131. TRUTH-BEARER

•“All and only statements are meaningful-declarative-sentences.”is either a stipulative definition or a descriptive definition. If the former, the stipulation is useful or it is not; if the latter, either the descriptive definition correctly describes English usage or it does not. In either case no arguments, as such, are applicable

Criticisms of Theory 3a

• If the term statement is synonymous with the term meaningful-declarative-sentence, then the applicable criti- cisms are the same as those outlined under sentence below

• If all and only meaningful-declarative-sentences are statements, as advanced by Theory 3a, then the terms are synonymous and we can just as well speak of the meaningful-declarative-sentences themselves as the truth- bearers – there is no distinct concept of statement to consider, and the term statement is literally redundant.

Arguments for Theory 3b

131.6 Thoughts

Frege (1919) argued that an indicative sentence in which we communicate or state something, contains both a thought and an assertion, it expresses the thought, and the thought is the sense of the sentence.*[20]

131.7 Notes

[1] Character A character is a typographic character (printed or written), a unit of speech, a phoneme, a series of dots and dashes (as sounds, magnetic pulses, printed or written), a flag or stick held at a certain angle, a gesture, a sign as use in sign language, a pattern or raised indentations (as in brail) etc. in other words the sort of things that are commonly described as the elements of an alphabet.

[2] Word-token A word-token is a pattern of characters. The pattern of characters A This toucan can catch a can contains six word-tokens The pattern of characters D He is grnd contains three word-tokens

[3] Word-type A word-type is an identical pattern of characters, . The pattern of characters A: This toucan can catch a can. contains five word-types (the word-token can occurring twice)

[4] Meaningful-word-token A meaningful-word-token is a meaningful word-token. grnd in D He is grnd. is not meaningful..

[5] Word-meaning Two word-tokens which mean the same are of the same word-meaning. Only those word-tokens which are meaningful-word-tokens can have the same meaning as another word-token. The pattern of characters A: This toucan can catch a can. contains six word-meanings. Although it contains only five word-types, the two occurrences of the word-token can have different meanings. On the assumption that bucket and pail mean the same, the pattern of characters B: If you have a bucket, then you have a pail contains ten word-tokens, seven word-types, and six word-meanings.

[6] Sentence-token A sentence-token is a pattern of word-tokens. The pattern of characters D: He is grnd is a sentence-token because grnd is a word-token (albeit not a meaningful word- token.)

[7] Meaningful-sentence-token A meaningful-sentence-token is a meaningful sentence-token or a meaningful pattern of meaningful-word-tokens. The pattern of characters D: He is grnd is not a sentence-token because grnd is not a meaningful word-token.

[8] Sentence-type Two sentence-tokens are of the same sentence-type if they are identical patterns of word-tokens characters, e.g. the sentence-tokens P: I'm Spartacus and Q: I'm Spartacus are of the same sentence-type.

[9] Meaningful-declarative-sentence-tokens A meaningful-declarative-sentence-token is a meaningful declarative-sentence- token. The pattern of characters F: Cats blows the wind is not a meaningful-declarative-sentence-token because it is grammatically ill-formed The pattern of characters G: This stone is thinking about Vienna is not a meaningful-declarative-sentence-token because thinking cannot be predicated of a stone 131.8. REFERENCES 499

The pattern of characters H: This circle is square is not a meaningful-declarative-sentence-token because it is internally inconsistent The pattern of characters D: He is grnd is not a meaningful-declarative-sentence-token because it contains a word-token (grnd) which is not a meaningful-word-token

[10] Meaningful-declarative-sentence-types Two meaningful-declarative-sentence-tokens are of the same meaningful-declarative- sentence-type if they are identical patterns of word-tokens characters, e.g. the sentence-tokens P: I'm Spartacus and Q: I'm Spartacus are of the same meaningful-declarative-sentence-type. In other words a sentence-type is a meaningful- declarative-sentence-type if all tokens of which are meaningful-declarative-sentence-tokens

[11] Nonsense-declarative-sentence-token A nonsense-declarative-sentence-token is a declarative-sentence-token which is not a meaningful-declarative-sentence-token. The patterns of characters F: Cats blows the wind, G: This stone is thinking about Vienna and H: This circle is square are nonsense-declarative-sentence-tokens because they are declarative-sentence-tokens but not meaningful-declarative- sentence-tokens. The pattern of characters D: He is grnd is not a nonsense-declarative-sentence-token because it is not a declarative-sentence-token because it contains a word-token (grnd) which is not a meaningful-word-token.

[12] Meaningful-declarative-sentence-token-use A meaningful-declarative-sentence-token-use occurs when and only when a meaningful-declarative-sentence-token is used declaratively, rather than, say, mentioned. The pattern of characters T: Spartacus did not eat all his spinach in London on Feb 11th 2009 is a meaningful-declarative- sentence-token but, in all probability, it has never be used declaratively and thus there have been no meaningful-declarative- sentence-token-uses of T. A meaningful-declarative-sentence-token can be used zero to many times. Two meaningful- declarative-sentence-tokens-uses of the same meaningful-declarative-sentence-type are identical if and only if they are identical events in time and space with identical users.

[13] Referring-expression An expression that can be used to pick out or refer to particular entity, such as definite descriptions and proper names

[14] Referential success a referring-expression’s success in identifying a particular entity OR a meaningful-declarative- sentence-token-use’s containing one or more referring-expression all of which succeed in identifying a particular entity

[15] Referential failure a referring-expression’s failure to identify a particular entity is referentially successful OR a meaningful- declarative-sentence-token-use’s containing one or more referring-expression that fail to identify a particular entity.

[16] Referentially-successful-meaningful-declarative-sentence-token-use A meaningful-declarative-sentence-token-use con- taining no referring-expression that fails to identify a particular entity. A use of a token of the meaningful-declarative- sentence-type U: The King of France is bald’' is a referentially-successful-meaningful-declarative-sentence-token-use if (and only if) the embedded referring-expression ‘The King of France’is referentially successful. No use of a token of the meaningful-declarative-sentence-type V: The highest prime has no factors other than itself and 1 is not a referentially- successful-meaningful-declarative-sentence-token-use since the embedded referring-expression The highest prime is always a referential failure.

[17] • Meaningful-declarative-sentence-types Two meaningful-declarative-sentence-tokens are of the same meaningful-declarative-sentence-type if they are identical pat- terns of word-tokens characters, e.g. the sentence-tokens P and Q above are of the same meaningful-declarative-sentence- type. In other words a sentence-type is a meaningful-declarative-sentence-type if its tokens of are meaningful-declarative- sentence-tokens

[18] Utterance: The term utterance is frequently used to mean meaningful-declarative-sentence-token. See e.g. Grice, Mean- ing, 1957 http://semantics.uchicago.edu/kennedy/classes/f09/semprag1/grice57.pdf

[19] Eternal Sentence: A sentence that stays forever true, or forever false, independently of any special circumstances under which they happen to be uttered or written. More exactly, a meaningful-declarative-sentence-type whose tokens have the same truth values. E.g. The whole is greater than the part is an eternal sentence, It is raining is not an eternal sentence but It rains in Boston, Mass., on July 15, 1968 is an eternal sentence

131.8 References

[1] e.g.

•“In symbolic logic, a statement (also called a proposition) is a complete declarative sentence, which is either true or false.”Vignette 17 Logic, Truth and Language 500 CHAPTER 131. TRUTH-BEARER

•“A statement is just that; it is a declaration about something—anything—a declaration which can be evaluated as either true or false. “I am reading this sentence”is a statement, and if indeed you have looked at it and compre- hended its meaning, then it is safe to say that that statement can be evaluated as true.”Fundamental Logic Concepts: Statement

[2] e.g. *“Some philosophers claim that declarative sentences of natural language have underlying logical forms and that these forms are displayed by formulas of a formal language. Other writers hold that (successful) declarative sentences express propositions; and formulas of formal languages somehow display the forms of these propositions.”Shapiro, Stewart (2008). Edward N. Zalta, ed. “Classical Logic”in The Stanford Encyclopedia of Philosophy (Fall 2008 ed.).

[3] Wolfram, Sybil (1989). Philosophical Logic. Routledge, London and New York. ISBN 0-415-02317-3.

[4] Occurrences of numerals

[5] name=declarative-sentence-token group="nb"> Declarative-sentence-token A declarative-sentence-token is a sentence- token which that can be used to communicate truth or convey information. The pattern of characters E: Are you happy? is not a declarative-sentence-token because it interrogative not declarative.

[6] Fisher (2008). Philosophy of Logic. ISBN 0-495-00888-5.

[7] Kneale, W&M (1962). The development of logic. Oxford. ISBN 0-19-824183-6. page 593

[8] see Wolfram, Sybil (1989) generally on the application of type–token distinction

[9] QUINE, W.V. (1970). Philosophy of Logic. Prentice Hall. ISBN 0-13-663625-X.

[10] See also 'First-order logic#Semantics

[11] e.g. Russell, Wittgenstein, and Stanford Encyclopedia of Philosophy URL = http://plato.stanford.edu/entries/facts/#FacPro: “By‘proposition’, we shall mean truth-bearer, and remain neutral as to whether truth-bearers are sentences, statements, beliefs or abstract objects expressed by sentences, for instance—except in section 2.4.1.”

[12] McGrath, Matthew, “Propositions”, The Stanford (Fall 2008 Edition), Edward N. Zalta (ed.), URL = ."The term ‘proposition’has a broad use in contemporary philos- ophy. It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other “propositional attitudes”(i.e., what is believed, doubted, etc.), the referents of that-clauses, and the meanings of sentences.”

[13] Mark, Richard (2006). “Propositions”. On one use of the term,“propositions”are objects of assertion, what successful uses of declarative sentences say. As such, they determine truth-values and truth conditions. On a second, they are the objects of certain psychological states (such as belief and wonder) ascribed with verbs that take sentential complements (such as believe and wonder ). On a third use, they are what are (or could be) named by the complements of such verbs. Many assume that propositions in one sense are propositions in the others.

[14]“Philosopher's tolerance towards propositions has been encouraged partly by ambiguity in the term 'proposition'. The term often is used simply for the sentences themselves, declarative sentences; and then some writers who do use the term for meanings of sentences are careless about the distinction between sentences and their meanings”Quine 1970, p. 2

[15] i.e. when expressed by a token-meaningful-declarative-sentence made by Spartacus, and when expressed by somebody other than Spartacus

[16]“Philosophers who favor propositions have said that propositions are needed because truth only of propositions, not of sentences [read meaningful-declarative-sentences Ed], is intelligible. An unsympathetic answer is that we can explain truth of sentences to be propositional in their own terms: sentences are true whose meanings are true propositions. Any failure of intelligibilty here is already his own fault.”Quine 1970 page 10

[17] Strawson, PF (1950). “On referring”. Mind 9. reprinted in Strawson 1971 and elsewhere

[18] Strawson, PF (1957).“Propositions, Concepts and Logical Truths”. The Philosophical Quarterly 7. reprinted in Strawson, P.F. (1971). Logico-Linguistic Papers. Methuen. ISBN 0-416-09010-9.

[19] Strawson, P.F. (1952). Introduction to Logical Theory. Methuen: London. p. 4. ISBN 0-416-68220-0.

[20] Frege (1919) Die Gedanke trans AM and Marcelle Quinton in Frege, G (1956). “The thought: A logical Enquiry”. Mind 65. reprinted in Strawson 1967. 131.9. EXTERNAL LINKS 501

131.9 External links

• Stanford Encyclopedia of Philosophy:

• Truth; 2.1 Sentences as truth-bearers; Glanzberg, Michael • The Correspondence Theory of Truth; 2. Truthbearers and Truthmakers; David, Marian Chapter 132

Tuple-generating dependency

In relational database theory, a tuple-generating dependency (TGD) is a certain kind of constraint on a relational database. It is a subclass of the class of embedded dependencies (EDs). A TGD is a sentence in first-order logic of the form: ∀x1 ... xn, P(x1, ..., xn) → ∃z1, ..., zk, Q(y1, ..., ym) where {z1, ..., zk} = {y1, ..., ym} \ {x1, ..., xn}, and P is a possibly empty and Q is a non-empty conjunction of relational atoms. A relational atom has the form R(w1, ..., wh) where each of the w, ..., wh, wi, wj, are variables or constants. An algorithm known as the chase takes as input an instance that may or may not satisfy a set of TGDs (or more generally EDs), and, if it terminates (which is a priori undecidable), outputs an instance that does satisfy the TGDs.

132.1 References

• Serge Abiteboul, Richard B. Hull, Victor Vianu: Foundations of Databases. Addison-Wesley, 1995.

• Alin Deutsch, FOL Modeling of Integrity Constraints, http://db.ucsd.edu/pubsFileFolder/305.pdf

502 Chapter 133

Universal logic

This article is about the abstract study of logics. For the digital circuits notion, see universal logic gate. Universal logic is the field of logic that studies the common features of all logical systems, aiming to be to logic what

universal algebra is to algebra. A number of approaches to universal logic have been proposed since the twentieth century, using model theoretic, and categorical approaches.

133.1 Development

The roots of universal logic may go as far back as some work of Alfred Tarski in the early twentieth century, but the modern notion was first presented in the 1990s by Swiss logician Jean-Yves Béziau.*[1]*[2] The term 'universal logic' has also been separately used by logicians such as Richard Sylvan and Ross Brady to refer to a new type of (weak) relevant logic.*[3]

503 504 CHAPTER 133. UNIVERSAL LOGIC

In the context defined by Béziau, three main approaches to universal logic have been explored in depth:*[4]

• An abstract model theory system axiomatized by Jon Barwise,*[5]

• a topological/categorical approach based on sketches (sometimes called categorical model theory),*[6]

• a categorical approach originating in Computer Science based on Goguen and Burstall's notion of institution.*[7]

While logic has been studied for centuries, Mossakowski et al commented in 2007 that“it is embarrassing that there is no widely acceptable formal definition of “a logic”.*[8] These approaches to universal logic thus aim to address and formalize the nature of what may be called 'logic' as a form of “sound reasoning”.*[8]

133.2 World Congresses and Schools on Universal Logic

Since 2007, Béziau has been organizing world congresses and schools on universal logic. These events bring together hundreds of researchers and students in the field and offer tutorials and research talks on a wide range of subjects. Traditionally, the congresses have a contest and a secret speaker whose identity is only revealed when his or her talk begins.

• First World Congress and School on Universal Logic, 26 March–3 April 2005, Montreux, Switzerland. Par- ticipants included Béziau, Dov Gabbay, and David Makinson. (Secret Speaker: Saul Kripke.)

• Second World Congress and School on Universal Logic, 16–22 August 2007, Xi'an, China.

• Third World Congress and School on Universal Logic, 18–25 April 2010, Lisbon, Portugal. (Secret Speaker: Jaakko Hintikka.)

• Fourth World Congress and School on Universal Logic, 29 March–7 April 2013, Rio de Janeiro, .

• Fifth World Congress and School on Universal Logic, 20–30 June 2015, Istanbul, .

133.3 Publications in the field

A journal dedicated to the field, Logica Universalis, with Béziau as editor-in-chief started to be published by Birkhäuser Basel (an imprint of Springer) in 2007.*[9] Springer also started to publish a book series on the topic, Studies in Uni- versal Logic, with Béziau as series editor.*[10] An anthology titled Universal Logic was published in 2012, giving a new light on the subject. *[11]

133.4 See also

• Abstract algebraic logic

• Definitions of logic

• Institution

133.5 References

[1] The Road to Universal Logic: Festschrift for 50th Birthday of Jean-Yves Béziau Volume I, edited by Arnold Koslow and Arthur Buchsbaum 2014 Birkhäuser ISBN 978-3319101927 pp 2-10

[2] Jean-Yves Béziau, ed. (2007). Logica universalis: towards a general theory of logic (2nd ed.). Springer. ISBN 978-3- 7643-8353-4. 133.6. EXTERNAL LINKS 505

[3] Brady, R. 2006. Universal Logic. Stanford: CSLI Publications. ISBN 1-57586-255-7.

[4] Răzvan Diaconescu (2008). Institution-independent model theory. Birkhäuser. pp. 2–3. ISBN 978-3-7643-8707-5.

[5] Jon Barwise. Axioms for abstract model theory. Annals of Mathematical Logic,7:221–265, 1974

[6] Steffen Lewitzka “A Topological Approach to Universal Logic”Logica Universalis 2007 Birkhauser pp 35-61

[7] Razvan Diaconescu,“Three decades of institution theory”in Universal Logic: An Anthology edited by Jean-Yves Béziau 2012 Springer ISBN 978-3-0346-0144-3 pp 309-322

[8] T. Mossakowski, J. A. Goguen, R. Diaconescu, A. Tarlecki,“What is a Logic?", Logica Universalis 2007 Birkhauser, pp. 113–133.

[9] http://www.springer.com/birkhauser/mathematics/journal/11787

[10] http://www.springer.com/series/7391

[11] Jean-Yves Béziau, ed. (2012). Universal Logic: an Anthology - From Paul Hertz to Dov Gabbay. Springer. ISBN 978-3- 0346-0144-3.

133.6 External links

• Logica Universalis Chapter 134

Unspoken rule

Unspoken rules are behavioral constraints imposed in organizations or societies that are not voiced or written down. They usually exist in unspoken and unwritten format because they form a part of the logical argument or course of ac- tion implied by tacit assumptions. Examples involving unspoken rules include unwritten and unofficial organizational hierarchies, organizational culture, and acceptable behavioral norms governing interactions between organizational members. For example, the captain of a ship is always expected to be the last to evacuate it in a disaster. Or, as Vince Waldron wrote, “A pet, once named, instantly becomes an inseparable member of the family.”*[1]

134.1 Employment and discrimination

In the workplace, unspoken rules can have a significant impact on one’s job satisfaction, advancement opportunities, and career trajectory. In sports, Scottish football club, Rangers until 1989 had an unwritten rule of not signing any player who was openly Catholic.*[2] Yorkshire County Cricket Club also historically had an unwritten rule that cricketers could only play for them if they were born within the historical county boundaries of Yorkshire.*[3]

134.2 Concept

Since the mid-1980s, a set of widely applied concepts used to reveal the hidden inner workings of organizations and society have commonly been referred to as 'unwritten rules'.*[4]*[5]*[6]*[7] Devised by Peter Scott-Morgan*[8]*[9]*[10] (and popularized by a best-selling business book in 1994 called The Unwritten Rules of the Game),*[11] these con- cepts have been used as the theoretical framework for a variety of academic research projects across different coun- tries,*[12]*[13]*[14] and are cited in numerous academic papers,*[15]*[16]*[17]*[18] scholarly books,*[19]*[20] as well as specialist postings on the internet.*[21]*[22]*[23] A professor at London Business School writes that in 1985 she became intrigued by these ideas when she first met Scott-Morgan “who at that time was beginning to develop a process which he called 'the unwritten rules of the game'.”*[24] This usage of the term 'unwritten rules' has been incorporated into a range of management thinking*[4]*[6]*[7] and is also highlighted in various business books*[24]*[25]*[26]*[27]*[28] as well as business-related posts on the inter- net.*[29]*[30]*[31] In addition, several management consultancies apply Unwritten Rules concepts.*[5]*[8]*[32]*[33]*[34]*[35] The international management consultancy Arthur D. Little has revealed that from the mid-1990s conducting an Un- written Rules assignment became something of a rite of passage amongst its 3000 consultants – on the theory that “once you’ve fed [the sensitive results] back to a CEO …and survived …then you can do anything.”*[36] There are numerous accounts of organizations that have applied Unwritten Rules , such as Citibank,*[37] Daimler-Benz,*[38] Hewlett-Packard,*[15] Lloyds TSB,*[18] the UK National Health Service,*[39] Philips Con- sumer Electronics,*[34] and the Argentinian national oil company YPF.*[40] The former-head of Process Review at British Petroleum has published that in 1992 his corporation's“search for best practice in the consulting world led to my meeting Peter Scott-Morgan and learning of his insights into understanding – and changing – the Unwritten Rules of the Game.”*[41] He then describes how BP tested, and became convinced of, the validity of Scott-Morgan's

506 134.3. BIBLIOGRAPHY 507 technique and went on to apply it in several major operating centers. There are specific accounts of its early use at BP's Wytch Farm oilfield.*[26]*[41]

134.3 Bibliography

• Level Playing Field Institute and Center for Survey Research and Analysis at the University of Connecticut (2003) The HOW-FAIR study 2003: How opportunities in the workplace and fairness affect intergroup rela- tions. Level Playing Field Institute, San Francisco. • Scott-Morgan, Peter (1994). The Unwritten Rules of the Game: Master Them, Shatter Them, and Break Through the Barriers to Organizational Change. McGraw Hill. ISBN 0-07-057075-2

134.4 See also

• Ius non scriptum • Lex non scripta • Unenumerated rights

134.5 References

[1] Vince Waldron, The Official Dick Van Dyke Show Book, Hal Leonard, 2001, p. 176.

[2] http://www.guardian.co.uk/football/blog/2009/jul/10/maurice-mo-johnston-rangers-celtic

[3] http://www.independent.co.uk/sport/cricket/new-yorkshire-ready-to-restore-forgotten-glories-676410.html

[4] “Strategic Denial: Unwritten Rules and Wishful Thinking”, Global Strategies Project Commentaries from Encyclopedia of World Problems and Human Potential, Union of International Associations.

[5] Haserot, Phyllis Weiss. “How to Change Unwritten Rules”, Practice Development Counsel website, from original article in The New York Law Journal, 25 May 1999.

[6] “Culture Matters”, Bridge the Gap Between “Knowing”and “Doing”, p. 3. Deloitte, 2011.

[7] “What Makes an Organization Effective?", p. 3, Points of View, Avocet Organizational Performance Inc.

[8] Capek, Frank “Why Customer Experience Initiatives Fail?, Customer Innovations website, 31 October 2007.

[9] Dr. Peter Scott-Morgan. Speaker Profiles. Celebrity Speakers website.

[10] The Hidden Logic of Business Performance. Boardroom Imperative, The Concours Group, 2004.

[11] Scott-Morgan, Peter (1994). The Unwritten Rules of the Game: Master Them, Shatter Them, and Break Through the Barriers to Organizational Change. McGraw Hill. ISBN 0-07-057075-2

[12] McGovern, Patrick (1995)“Learning from the Gurus: Managers' Responses to The Unwritten rules of the Game, Business Strategy Review, Volume 6, Issue 3, pp. 13-25. 23 September 1995.

[13] Background Boudewijns & Roemen Groep corporate website

[14] Sharpe, Jason et al. (2011)“Re-engineering Unwritten Rules: An Ethnographic Study of an Intra-Organizational Ecology” , Centre for Facilities Management Development, Sheffield Hallam University.

[15] McGovern, Patrick et al. (1997) “Human Resource Management On the Line? Human Resource Management Journal, Volume 7, Issue 4, pp. 12-29, July 1997.

[16] Springer, Jon (2003)“Shifting the Unwritten Rules of Organizational Behavior The Systems Thinker, Volume 14, Number 3, pp. 6-7. Pegasus Communications, April 2003.

[17] Price, Ilfryn (1995)“Organisational Memetics?: Organisational Learning as a Selection Process”Management Learning, Volume 26, Number 3, pp. 299-318. 508 CHAPTER 134. UNSPOKEN RULE

[18] Gratton, Lynda et al.“Linking Individual Performance to Business Strategy: The People Process Model”Human Resource Management, Volume 38, Number 1, pp. 17-31, Spring 1999.

[19] Cross, Robert et al. (2004). The Hidden Power of Social Networks: Understanding How Work Really Gets Done in Orga- nizations. Harvard Business Review Press. ISBN 978 1591392705

[20] Flood, Patrick et al. (Eds.) (2000). Managing Strategy Implementation. Wiley-Blackwell. ISBN 978-0-631-21766-4

[21] Bellinger, Gene. “The Unwritten Rules: The Way Things Really Work” Systems Thinking website, posted 2004.

[22] Managing People Exam, Section A, Question 2, Chartered Institute of Personnel and Development, 4 May 2005.

[23] Herold, Max. “Companies by Neurological Levels” Integrated SocioPsychology website, 28 September 2002.

[24] Gratton, Lynda (2000). Living Strategy: Putting People at the Heart of Corporate Purpose, p. 48. Prentice Hall. ISBN 0 273 65015 7

[25] De Flander, Jeroen (2010).“About the Experts”, Strategy Execution Heroes: Business Strategy Implementation and Strategic Management Demystified, p. viii. The Performance Factory. ISBN 978 908148731 3

[26] Price, If et al. (1998). Shifting the Patterns: Breaching the Memetic Codes of Corporate Performance, pp. 84-91. Manage- ment Books 2000. ISBN 1 85252 253 4

[27] Floyd, Chris (1997). Managing Technology for Corporate Success, pp. 197-8 and 223. Gower. ISBN 0 566 07991 7

[28] Erickson, Tamara (2008). Plugged In: The Generation Y Guide to Thriving at Work, pp. 240-3. Harvard Business School Press. ISBN 978 1422120606

[29] Boulton, Charles.“Knowing the Unwritten Rules Can Improve Service Delivery and Cut Costs”, PublicNet website about management and the public sector, 26 March 2010.

[30] “Cryptonomics” Experience Mind website, 31 October 2007.

[31] Christie, Lisa. “Unwritten Rules Determine Behavior”, Creative Leadership Coaching website, 27 April 2008.

[32]“Driving Safety Culture: Identification of Leadership Qualities for Effective Safety Management”, Final Report to Maritime and Coastguard Agency, Arthur D. Little, October 2004.

[33]“Understanding the unwritten rules of the game”The People Side of Risk Intelligence: Aligning Talent and Risk Management, p.17, Risk Intelligence Series, Issue 18. Deloitte, 2010.

[34] Camrass, Roger “Big Change – Beware the Unwritten Rules!", Business Leaders CIO Blog, 4 August 2011

[35] “About delta 5”, delta 5 Dutch consulting services website.

[36] Eagar, Rick (2006). “A brief history of Arthur D Little” Prism / 120 years of Arthur D. Little p. 34. Arthur D. Little.

[37] Gratton, Lynda et al. (1999). Strategic Human Resource Management: Corporate Rhetoric and Human Reality, Acknowl- edgements section. Oxford University Press. ISBN 978 0198782032

[38] Bergmann, Karin (1998). “Knowledge Management at Daimler-Benz's Passenger-Car Division, Prism, Issue 2, 1998. Arthur D. Little.

[39] Clark, Liz (1999). “Multi-Skilling for Success”, Facilities, Volume 17, Issue 7/8, pp. 272-9.

[40] Ross, Christopher (1994).“Recreating the Argentine National Oil Company: A Paradigm for Privatisation”Prism, Issue 2, 1994. Arthur D. Little

[41] Price, Ifryn (1993). “Aligning People and Processes During Business-Focused Change in BP Exploration”, Prism, Issue 4, 1993. Arthur D. Little Chapter 135

Vacuous truth

A vacuous truth is a statement that asserts that all members of the empty set have a certain property. For example, the statement “all cell phones in the room are turned off”may be true simply because there are no cell phones in the room. In this case, the statement “all cell phones in the room are turned on" would also be vacuously true, as would the conjunction of the two: “all cell phones in the room are turned on and turned off”. More formally, a relatively well-defined usage refers to a conditional statement with a false antecedent. One example of such a statement is “if Uluru is in France, then the Eiffel Tower is in Bolivia". Such statements are considered vacuous because the fact that the antecedent is false prevents using the statement to infer anything about the truth value of the consequent. They are true because a material conditional is defined to be true when the antecedent is false (regardless of whether the conclusion is true). In pure mathematics, vacuously true statements are not generally of interest by themselves, but they frequently arise as the base case of proofs by mathematical induction.*[1] This notion has relevance as well as in any other field which uses classical logic. Outside of mathematics, statements which can be characterized informally as vacuously true can be misleading. Such statements make reasonable assertions about qualified objects which do not actually exist. For example, a child might tell his or her parent“I ate every vegetable on my plate”, when there were no vegetables on the child's plate to begin with.

135.1 Scope of the concept

A statement S is “vacuously true”if it resembles the statement P ⇒ Q , where P is known to be false. Statements that can be reduced (with suitable transformations) to this basic form include the following universally quantified statements:

• ∀x : P (x) ⇒ Q(x) , where it is the case that ∀x : ¬P (x) .

• ∀x ∈ A : Q(x) , where the set A is empty.

• ∀ξ : Q(ξ) , where the symbol ξ is restricted to a type that has no representatives.

Vacuous truth most commonly appears in classical logic, which in particular is two-valued. However, vacuous truth also appears in, for example, intuitionistic logic in the same situations given above. Indeed, if P is false, P ⇒ Q will yield vacuous truth in any logic that uses the material conditional; if P is a necessary falsehood, then it will also yield vacuous truth under the strict conditional. Other non-classical logics (for example, relevance logic) may attempt to avoid vacuous truths by using alternative conditionals (for example, the counterfactual conditional).

509 510 CHAPTER 135. VACUOUS TRUTH

135.2 Examples

These examples, one from mathematics and one from natural language, might help illustrate the concept: “For any integer x, if x > 5 then x > 3.”*[2] – This statement is true non-vacuously (since some integers are greater than 5), but some of its implications are only vacuously true: for example, when x is the integer 2, the statement implies the vacuous truth that “if 2 > 5 then 2 > 3”. “You're my favorite nephew. In fact, you're my only nephew.”*[3] – Since the aunt does not have a nephew whom she likes more than the one she is addressing,“You're my favorite nephew”is true, but since she also does not have a nephew whom she likes less than the one she is addressing, it is a vacuous truth.

135.3 See also

• Triviality (mathematics) and Degeneracy (mathematics)

• Empty sum and Empty product • Tautology (logic) – another type of true statement that also fails to convey any substantive information

• Paradoxes of material implication, especially the Principle of explosion • De Morgan's Laws – specifically the law that a universal statement is true just in case no counterexample exists: ∀x P (x) ≡ ¬∃x ¬P (x)

135.4 References

[1] Baldwin, Douglas L.; Scragg, Greg W. (2011), Algorithms and Data Structures: The Science of Computing, Cengage Learn- ing, p. 261, ISBN 9781285225128.

[2] “What precisely is a vacuous truth?".

[3] “Conditional Assertions”.

135.5 Bibliography

• Blackburn, Simon (1994). “vacuous,”The Oxford Dictionary of Philosophy. Oxford: Oxford University Press, p. 388. • David H. Sanford (1999). “implication.”The Cambridge Dictionary of Philosophy, 2nd. ed., p. 420.

• Beer, Ilan; Ben-David, Shoham; Eisner, Cindy; Rodeh, Yoav (1997).“Efficient Detection of Vacuity in ACTL Formulas”. Lecture Notes in Computer Science 1254: 279–290. doi:10.1007/3-540-63166-6_28.

135.6 External links

• Conditional Assertions: Vacuous truth Chapter 136

Vagrant predicate

Vagrant predicates are logical constructions that exhibit an inherent limit to conceptual knowledge.*[1] Such pred- icates can be used in general descriptions but are self-contradictory when applied to particulars. For instance, there are numbers which have never been mentioned but no example can be given as this would contradict its definition. Vagrant predicates have been proposed and studied by Nicholas Rescher.

F is a vagrant predicate iff ( ∃ u)Fu is true while nevertheless Fu0 is false for each and every specifically identified * u0. [2] When infinity is thought as number greater than any given, a similar idea is conceived. However vagrancy needs not to be monotonous and occurs also within bounds. Rescher has used vagrant predicates to solve the vagueness problem.*[1]*[2]

136.1 References

[1] Rescher N., Unknowability, Lexington books, 2009

[2] Rescher N., Informal Logic, Vol. 28, No.4 (2008), pp. 282-294

511 Chapter 137

Vaisheshika

Vaisheshika or Vaiśeṣika (Sanskrit: वैशेिषक) is one of the six orthodox schools of Hinduism (Vedic systems) from ancient India. In its early stages, the Vaiśeṣika was an independent philosophy with its own metaphysics, epistemology, logic, ethics, and soteriology.*[1] Over time, the Vaiśeṣika system became similar in its philosophical procedures, ethical conclusions and soteriology to the Nyāya school of Hinduism, but retained its difference in epistemology and metaphysics. The epistemology of Vaiśeṣika school of Hinduism, like Buddhism, accepted only two reliable means to knowledge: perception and inference.*[2]*[3] Vaiśeṣika school and Buddhism both consider their respective scriptures as indis- putable and valid means to knowledge, the difference being that the scriptures held to be a valid and reliable source by Vaiśeṣikas were the . Vaisheshika school is known for its insights in naturalism,*[4] and it is a form of atomism in natural philosophy.*[5] It postulated that all objects in the physical universe are reducible to paramāṇu (atoms), and one's experiences are derived from the interplay of substance (a function of atoms, their number and their spatial arrangements), qual- ity, activity, commonness, particularity and inherence.*[6] Knowledge and liberation was achievable by complete understanding of the world of experience, according to Vaiśeṣika school of Hinduism.*[6] Vaiśeṣika darshana was founded by Kaṇāda Kashyapa around the 2nd century BC.*[6]

137.1 Overview

Although the Vaisheshika system developed independently from the Nyaya school of Hinduism, the two became similar and are often studied together. In its classical form, however, the Vaishesika school differed from the Nyaya in one crucial respect: where Nyaya accepted four sources of valid knowledge, the Vaishesika accepted only two.*[2]*[3] The epistemology of Vaiśeṣika school of Hinduism accepted only two reliable means to knowledge - perception and inference.*[2] Vaisheshika espouses a form of atomism, that the reality is composed of four substances (earth, water, air, fire). Each of these four are of two types, explains Ganeri,*[5] atomic (paramāṇu) and composite. An atom is that which is indestructible (anitya), indivisible, and has a special kind of dimension, called “small”(aṇu). A composite is that which is divisible into atoms. Whatever human beings perceive is composite, and even the smallest perceptible thing, namely, a fleck of dust, has parts, which are therefore invisible.*[5] The Vaiśeṣikas visualized the smallest composite thing as a “triad”(tryaṇuka) with three parts, each part with a “dyad”(dyaṇuka). Vaiśeṣikas believed that a dyad has two parts, each of which is an atom. Size, form, truths and everything that human beings experience as a whole is a function of atoms, their number and their spatial arrangements. Vaisheshika postulated that what one experiences is derived from (substance: a function of atoms, their number and their spatial arrangements), guna (quality), karma (activity), samanya (commonness), vishesha (particularity) and samavaya (inherence, inseparable connectedness of everything).*[6]*[7]

512 137.2. EPISTEMOLOGY 513

137.2 Epistemology

Hinduism identifies six Pramāṇas as epistemically reliable means to accurate knowledge and to truths:*[8] Pratyakṣa (perception), Anumāna (inference), Upamāna (comparison and analogy), Arthāpatti (postulation, derivation from circumstances), Anupalabdi (non-perception, negative/cognitive proof) and Śabda (word, testimony of past or present reliable experts).*[2]*[3]*[9] Of these Vaiśeṣika epistemology considered only pratyakṣa (perception) and anumāna (inference) as reliable means of valid knowledge.*[10] Nyaya school, related to Vaiśeṣika, accepts four out of these six.*[2]

• Pratyakṣa (प्रत्यक्ष) means perception. It is of two types: external and internal. External perception is described as that arising from the interaction of five senses and worldly objects, while internal perception is described by this school as that of inner sense, the mind.*[11]*[12] The ancient and medieval texts of Hinduism identify four requirements for correct perception:*[13] Indriyarthasannikarsa (direct experience by one's sen- sory organ(s) with the object, whatever is being studied), Avyapadesya (non-verbal; correct perception is not through hearsay, according to ancient Indian scholars, where one's sensory organ relies on accepting or reject- ing someone else's perception), Avyabhicara (does not wander; correct perception does not change, nor is it the result of deception because one's sensory organ or means of observation is drifting, defective, suspect) and Vyavasayatmaka (definite; correct perception excludes judgments of doubt, either because of one's failure to observe all the details, or because one is mixing inference with observation and observing what one wants to observe, or not observing what one does not want to observe).*[13] Some ancient scholars proposed “un- usual perception”as pramāṇa and called it internal perception, a proposal contested by other Indian scholars. The internal perception concepts included pratibha (intuition), samanyalaksanapratyaksa (a form of induction from perceived specifics to a universal), and jnanalaksanapratyaksa (a form of perception of prior processes and previous states of a 'topic of study' by observing its current state).*[14] Further, the texts considered and refined rules of accepting uncertain knowledge from Pratyakṣa-pranama, so as to contrast nirnaya (definite judgment, conclusion) from anadhyavasaya (indefinite judgment).*[15]

• Anumāna (अनुमान) means inference. It is described as reaching a new conclusion and truth from one or more observations and previous truths by applying reason.*[16] Observing smoke and inferring fire is an example of Anumana.*[11] In all except one Hindu philosophies,*[17] this is a valid and useful means to knowledge. The method of inference is explained by Indian texts as consisting of three parts: pratijna (hypothesis), hetu (a reason), and drshtanta (examples).*[18] The hypothesis must further be broken down into two parts, state the ancient Indian scholars: sadhya (that idea which needs to proven or disproven) and paksha (the object on which the sadhya is predicated). The inference is conditionally true if sapaksha (positive examples as evidence) are present, and if vipaksha (negative examples as counter-evidence) are absent. For rigor, the Indian philosophies also state further epistemic steps. For example, they demand Vyapti - the requirement that the hetu (reason) must necessarily and separately account for the inference in “all”cases, in both sapaksha and vipaksha.*[18]*[19] A conditionally proven hypothesis is called a nigamana (conclusion).*[20]

137.2.1 Syllogism

The syllogism of the Vaiśeṣika school was similar to that of the Nyāya school of Hinduism, but the names given by Praśastapāda to the 5 members of syllogism are different.*[21]

137.3 Literature of Vaisheshika

The earliest systematic exposition of the Vaisheshika is found in the Vaiśeṣika Sūtra of Kaṇāda (or Kaṇabhaksha). This treatise is divided into ten books. The two commentaries on the Vaiśeṣika Sūtra, Rāvaṇabhāṣya and Bhāradvā- javṛtti are no more extant. Praśastapāda’s Padārthadharmasaṁgraha (c. 4th century) is the next important work of this school. Though commonly known as bhāṣya of Vaiśeṣika Sūtra, this treatise is basically an independent work on the subject. The next Vaisheshika treatise, Candra’s Daśapadārthaśāstra (648) based on Praśastapāda’s treatise is available only in Chinese translation. The earliest commentary available on Praśastapāda’s treatise is Vyomaśiva’s Vyomavatī (8th century). The other three commentaries are Śridhara’s Nyāyakandalī (991), Udayana’s Kiranāvali (10th century) and Śrivatsa’s Līlāvatī (11th century). Śivāditya’s Saptapadārthī which also belongs to the same period, presents the Nyāya and the Vaiśeṣika principles as a part of one whole. Śaṁkara Miśra’s Upaskāra on Vaiśeṣika Sūtra is also an important work.*[22] 514 CHAPTER 137. VAISHESHIKA

137.4 The Categories or Padārtha

According to the Vaisheshika school, all things which exist, which can be cognised, and which can be named are padārthas (literal meaning: the meaning of a word), the objects of experience. All objects of experience can be classified into six categories, dravya (substance), guṇa (quality), karma (activity), sāmānya (generality), viśeṣa (par- ticularity) and samavāya (inherence). Later Vaiśeṣikas (Śrīdhara and Udayana and Śivāditya) added one more cate- gory (non-existence). The first three categories are defined as (which can perceived) and they have real objective existence. The last three categories are defined as budhyapekṣam (product of intellectual discrimination) and they are logical categories.*[23] 1.Dravya (substance): The substances are conceived as 9 in number. They are, pṛthvī (earth), ap (water), tejas (fire), vāyu (air), ākaśa (ether), kāla (time), dik (space), (self or soul) and (mind). The first five are called bhūtas, the substances having some specific qualities so that they could be perceived by one or the other external senses.*[24] 2.Guṇa (quality): The Vaiśeṣika Sūtra mentions 17 guṇas (qualities), to which Praśastapāda added another 7. While a substance is capable of existing independently by itself, a guṇa(quality) cannot exist so. The original 17 guṇas (quali- ties) are, rūpa (colour), rasa (taste), gandha (smell), sparśa (touch), saṁkhyā (number), parimāṇa (size/dimension/quantity), pṛthaktva (individuality), saṁyoga (conjunction/accompaniments), vibhāga (disjunction), paratva (priority), aparatva (posteriority), buddhi (knowledge), (pleasure), duḥkha (pain), icchā (desire), dveṣa (aversion) and (effort). To these Praśastapāda added gurutva (heaviness), dravatva (fluidity), sneha (), (merit), adharma (demerit), śabda (sound) and saṁskāra (faculty).*[25] 3.Karma (activity): The (activities) like guṇas (qualities) have no separate existence, they belong to the substances. But while a quality is a permanent feature of a substance, an activity is a transient one. Ākāśa (ether), kāla (time), dik (space) and ātman (self), though substances, are devoid of karma (activity).*[26] 4.Sāmānya (generality): Since there are plurality of substances, there will be relations among them. When a property is found common to many substances, it is called sāmānya.*[27] 5.Viśeṣa (particularity): By means of viśeṣa, we are able to perceive substances as different from one another. As the ultimate atoms are innumerable so are the viśeṣas.*[28] 6.Samavāya (inherence): Kaṇāda defined samavāya as the relation between the cause and the effect. Praśastapāda defined it as the relationship existing between the substances that are inseparable, standing to one another in the relation of the container and the contained. The relation of samavāya is not perceivable but only inferable from the inseparable connection of the substances.*[29]

137.5 The atomic theory

The early Vaiśeṣika texts presented the following syllogism to prove that all objects i.e. the four bhūtas, pṛthvī (earth), ap (water), tejas (fire) and vāyu (air) are made of indivisible paramāṇus (atoms): Assume that the matter is not made of indivisible atoms, and that it is continuous. Take a stone. One can divide this up into infinitely many pieces (since matter is continuous). Now, the Himalayan mountain range also has infinitely many pieces, so one may build another Himalayan mountain range with the infinite number of pieces that one has. One begins with a stone and ends up with the Himalayas, which is a paradox - so the original assumption that matter is continuous must be wrong, and so all objects must be made up of a finite number of paramāṇus (atoms). According to the Vaiśeṣika school, the trasareṇu (dust particles visible in the sunbeam coming through a small window hole) are the smallest mahat (perceivable) particles and defined as tryaṇukas (triads). These are made of three parts, each of which are defined as dvyaṇuka (dyad). The dvyaṇukas are conceived as made of two parts, each of which are defined as paramāṇu (atom). The paramāṇus (atoms) are indivisible and eternal, they can neither be created nor destroyed.*[30] Each paramāṇu (atom) possesses its own distinct viśeṣa (individuality).*[31] The measure of the partless atoms is known as parimaṇḍala parimāṇa. It is eternal and it cannot generate the measure of any other substance. Its measure is its own absolutely.*[32] 137.6. LATER DEVELOPMENTS 515

137.6 Later developments

Over the centuries, the school became closely identified with the Nyaya school of Indian philosophy, as nyāya- vaiśeṣika. The school suffered a natural decline in India after the 15th century.

137.7 Views by the School

The say that the visible universe is created from an original stock of atoms (janim asataḥ). As Kaṇāda's Vaiśeṣika Sūtra (7.1.26) states, nityaṃ parimaṇḍalam (that which is of the smallest size, the atom, is eternal), he and his followers also postulate eternality for other, nonatomic entities, including the souls who become embodied, and even a Supreme Soul. But in Vaiśeṣika cosmology the souls and the Supersoul play only token roles in the atomic production of the universe. The Brahma (2.2.12) says ubhayathāpi na karmatas tad-abhavaḥ. According to this sūtra, one cannot claim that, at the time of creation, atoms first combine together because they are impelled by some karmic impulse adhering in the atoms themselves, since atoms by themselves, in their primeval state before combining into complex objects, have no ethical responsibility that might lead them to acquire pious and sinful reactions. Nor can the initial combination of atoms be explained as a result of the residual karma of the living entities who lie dormant prior to creation, since these reactions are each jiva's own and cannot be transferred from them even to other jīvas, what to speak of inert atoms.

137.8 See also

• Darshanas

• Hindu philosophy

• Hinduism

• Nyaya (philosophy)

• Padārtha

• Vaiśeṣika Sūtra

• Atomism

• Naturalism (philosophy)

137.9 Notes

[1] Amita Chatterjee (2011), Nyāya-vaiśeṣika Philosophy, The Oxford Handbook of World Philosophy, doi:10.1093/oxfordhb/9780195328998.003.0012

[2] DPS Bhawuk (2011), Spirituality and Indian Psychology (Editor: Anthony Marsella), Springer, ISBN 978-1-4419-8109-7, page 172

[3] • Eliott Deutsche (2000), in : Indian Philosophy Vol 4 (Editor: Roy Perrett), Routledge, ISBN 978-0815336112, pages 245-248; • John A. Grimes, A Concise Dictionary of Indian Philosophy: Sanskrit Terms Defined in English, State University of New York Press, ISBN 978-0791430675, page 238

[4] Dale Riepe (1996), Naturalistic Tradition in Indian Thought, ISBN 978-8120812932, pages 227-246

[5] Analytical philosophy in early modern India J Ganeri, Stanford Encyclopedia of Philosophy

[6] Oliver Leaman, Key Concepts in Eastern Philosophy. Routledge, ISBN 978-0415173629, 1999, page 269.

[7] M Hiriyanna (1993), Outlines of Indian Philosophy, Motilal Banarsidass, ISBN 978-8120810860, pages 228-237 516 CHAPTER 137. VAISHESHIKA

[8] P Bilimoria (1993), Pramāṇa epistemology: Some recent developments, in Asian philosophy - Volume 7 (Editor: G Flois- tad), Springer, ISBN 978-94-010-5107-1, pages 137-154

[9] Gavin Flood, An Introduction to Hinduism, Cambridge University Press, ISBN 978-0521438780, page 225

[10] Chattopadhyaya 1986, p. 170

[11] MM Kamal (1998), The Epistemology of the Carvaka Philosophy, Journal of Indian and , 46(2): 13-16

[12] B Matilal (1992), Perception: An Essay in Indian Theories of Knowledge, Oxford University Press, ISBN 978-0198239765

[13] Karl Potter (1977), Meaning and Truth, in Encyclopedia of Indian Philosophies, Volume 2, Princeton University Press, Reprinted in 1995 by Motilal Banarsidass, ISBN 81-208-0309-4, pages 160-168

[14] Karl Potter (1977), Meaning and Truth, in Encyclopedia of Indian Philosophies, Volume 2, Princeton University Press, Reprinted in 1995 by Motilal Banarsidass, ISBN 81-208-0309-4, pages 168-169

[15] Karl Potter (1977), Meaning and Truth, in Encyclopedia of Indian Philosophies, Volume 2, Princeton University Press, Reprinted in 1995 by Motilal Banarsidass, ISBN 81-208-0309-4, pages 170-172

[16] W Halbfass (1991), Tradition and Reflection, State University of New York Press, ISBN 0-7914-0362-9, page 26-27

[17] Carvaka school is the exception

[18] James Lochtefeld, “Anumana”in The Illustrated Encyclopedia of Hinduism, Vol. 1: A-M, Rosen Publishing. ISBN 0-8239-2287-1, page 46-47

[19] Karl Potter (2002), Presuppositions of India's Philosophies, Motilal Banarsidass, ISBN 81-208-0779-0

[20] Monier Williams (1893), Indian Wisdom - Religious, Philosophical and Ethical Doctrines of the Hindus, Luzac & Co, London, page 61

[21] Radhakrishnan 2006, p. 75ff

[22] Radhakrishnan 2006, pp. 180–81

[23] Radhakrishnan 2006, pp. 183–86

[24] Chattopadhyaya 1986, p. 169

[25] Radhakrishnan 2006, p. 204

[26] Radhakrishnan 2006, pp. 208–09

[27] Radhakrishnan 2006, p. 209

[28] Radhakrishnan 2006, p. 215

[29] Radhakrishnan 2006, pp. 216–19

[30] Chattopadhyaya 1986, pp. 169–70

[31] Radhakrishnan 2006, p. 202

[32] Dasgupta 1975, p. 314

137.10 References

• Chattopadhyaya, D. (1986), Indian Philosophy: A Popular Introduction, People’s Publishing House, New Delhi, ISBN 81-7007-023-6.

• Dasgupta, Surendranath (1975), A History of Indian Philosophy, Vol. I, Motilal Banarsidass, Delhi, ISBN 978-81-208-0412-8.

• Radhakrishnan, S. (2006), Indian Philosophy, Vol. II, Oxford University Press, New Delhi, ISBN 0-19- 563820-4. 137.11. FURTHER READING 517

137.11 Further reading

• Bimal Matilal (1977), A History of Indian Literature - Nyāya-Vaiśeṣika, Otto Harrassowitz Verlag, ISBN 978-3447018074, OCLC 489575550 • Gopi Kaviraj (1961), Gleanings from the history and bibliography of the Nyaya-Vaisesika literature, Indian Studies: Past & Present, Volume 2, Number 4, OCLC 24469380

137.12 External links

• A summary of Vaisheshika physics • Shastra Nethralaya - Vaisheshika

• GRETIL e-text of the Vaiśeṣika Sūtras Chapter 138

Valuation-based system

Valuation-based system (VBS) is a framework for knowledge representation and inference. Real-world problems are modeled in this framework by a network of interrelated entities, called variables. The relationships between variables (possibly uncertain or imprecise) are represented by the functions called valuations. The two basic operations for performing inference in a VBS are combination and marginalization. Combination corresponds to the aggregation of knowledge, while marginalization refers to the focusing (coarsening) of it. VBSs were introduced by Prakash P. Shenoy in 1989 as general frameworks for managing uncertainty in expert systems.

138.1 Applications

VBS are used for knowledge representation in expert systems and data fusion.

138.2 Bibliography

• Shenoy, Prakash P. A valuation-based language for expert systems. Int. Journal of Approximate Reasoning, vol. 3, no. 2, pages 383-411, 1989.

• Shenoy, Prakash P. Valuation based systems: A framework for managing uncertainty in expert systems. In L. A. Zadeh and J. Kacprzyk, editors, Fuzzy Logic and the Management of Uncertainty, chapter 4, pages 83–104. Wiley, New York, 1992. • Shenoy, Prakash P. and Shafer, G. Axioms for probability and belief-function propagation. In J. Pearl G. Shafer, editor, Readings in uncertain reasoning, pages 575-610. San Mateo, CA: Morgan Kaufmann, 1990.

138.3 External links

• Application of VBS to a threat assessment problem

518 Chapter 139

Vector logic

Vector logic*[1]*[2] is an algebraic model of elementary logic based on matrix algebra. Vector logic assumes that the truth values map on vectors, and that the monadic and dyadic operations are executed by matrix operators.

139.1 Overview

Classic binary logic is represented by a small set of mathematical functions depending on one (monadic ) or two (dyadic) variables. In the binary set, the value 1 corresponds to true and the value 0 to false. A two-valued vector logic requires a correspondence between the truth-values true (t) and false (f), and two q-dimensional normalized column vectors composed by real numbers s and n, hence:

t 7→ s and f 7→ n

(where q ≥ 2 is an arbitrary natural number, and “normalized”means that the length of the vector is 1; usually s and n are orthogonal vectors). This correspondence generates a space of vector truth-values: V2 = {s,n}. The basic logical operations defined using this set of vectors lead to matrix operators. The operations of vector logic are based on the scalar product between q-dimensional column vectors: uT v = ⟨u, v⟩ : the orthonormality between vectors s and n implies that ⟨u, v⟩ = 1 if u = v , and ⟨u, v⟩ = 0 if u ≠ v .

139.1.1 Monadic operators

The monadic operators result from the application Mon : V2 → V2 , and the associated matrices have q rows and q columns. The two basic monadic operators for this two-valued vector logic are the identity and the negation:

• Identity: A logical identity ID(p)is represented by matrix I = ssT + nnT . This matrix operates as follows: T T Ip = p, p ∈ V2; due to the orthogonality of s respect to n, we have Is = ss s + nn s = s⟨s, s⟩ + n⟨n, s⟩ = s , and conversely In = n .

• Negation: A logical negation ¬p is represented by matrix N = nsT + snT Consequently, Ns = n and Nn = s. The involutory behavior of the logical negation, namely that ¬(¬p) equals p, corresponds with the fact that N2 = I. Is important to note that this vector logic identity matrix is not generally an identity matrix in the sense of matrix algebra.

139.1.2 Dyadic operators

The 16 two-valued dyadic operators correspond to functions of the type Dyad : V2 ⊗ V2 → V2 ; the dyadic matrices have q rows and q2 columns. The matrices that execute these dyadic operations are based on the properties of the Kronecker product. Two properties of this product are essential for the formalism of vector logic:

519 520 CHAPTER 139. VECTOR LOGIC

1. The mixed-product property If A, B, C and D are matrices of such size that one can form the matrix products AC and BD, then

(A ⊗ B)(C ⊗ D) = AC ⊗ BD

2. Distributive transpose The operation of transposition is distributive over the Kronecker product:

(A ⊗ B)T = AT ⊗ BT .

Using these properties, expressions for dyadic logic functions can be obtained:

• Conjunction. The conjunction (p∧q) is executed by a matrix that acts on two vector truth-values: C(u ⊗ v) .This matrix reproduces the features of the classical conjunction truth-table in its formulation:

C = s(s ⊗ s)T + n(s ⊗ n)T + n(n ⊗ s)T + n(n ⊗ n)T

and verifies

C(s ⊗ s) = s,

C(s ⊗ n) = C(n ⊗ s) = C(n ⊗ n) = n.

• Disjunction. The disjunction (p∨q) is executed by the matrix

D = s(s ⊗ s)T + s(s ⊗ n)T + s(n ⊗ s)T + n(n ⊗ n)T ,

D(s ⊗ s) = D(s ⊗ n) = D(n ⊗ s) = s D(n ⊗ n) = n.

• Implication. The implication corresponds in classical logic to the expression p → q ≡ ¬p ∨ q. The vector logic version of this equivalence leads to a matrix that represents this implication in vector logic: L = D(N ⊗ I) . The explicit expression for this implication is:

L = s(s ⊗ s)T + n(s ⊗ n)T + s(n ⊗ s)T + n(n ⊗ n)T ,

and the properties of classical implication are satisfied: L(s ⊗ s) = L(n ⊗ s) = L(n ⊗ n) = s and L(s ⊗ n) = n.

• Equivalence and Exclusive or. In vector logic the equivalence p≡q is represented by the following matrix:

E = s(s ⊗ s)T + n(s ⊗ n)T + n(n ⊗ s)T + s(n ⊗ n)T 139.1. OVERVIEW 521

E(s ⊗ s) = E(n ⊗ n) = s

E(s ⊗ n) = E(n ⊗ s) = n.

X = NE

X = n(s ⊗ s)T + s(s ⊗ n)T + s(n ⊗ s)T + n(n ⊗ n)T ,

X(s ⊗ s) = X(n ⊗ n) = n

X(s ⊗ n) = X(n ⊗ s) = s.

• NAND and NOR

The matrices S and P correspond to the Sheffer (NAND) and the Peirce (NOR) operations, respectively:

S = NC

P = ND

139.1.3 De Morgan's law

In the two-valued logic, the conjunction and the disjunction operations satisfy the De Morgan's law: p∧q≡¬(¬p∨¬q), and its dual: p∨q≡¬(¬p∧¬q)). For the two-valued vector logic this Law is also verified:

C(u ⊗ v) = ND(Nu ⊗ Nv) , where u and v are two logic vectors.

The Kronecker product implies the following :

C(u ⊗ v) = ND(N ⊗ N)(u ⊗ v).

Then it can be proved that in the two–dimensional vector logic the De Morgan's law is a law involving operators, and not only a law concerning operations:*[3]

C = ND(N ⊗ N) 522 CHAPTER 139. VECTOR LOGIC

139.1.4 Law of contraposition

In the classical propositional calculus, the Law of Contraposition p → q ≡ ¬q → ¬p is proved because the equivalence holds for all the possible combinations of truth-values of p and q.*[4] Instead, in vector logic, the law of contraposition emerges from a chain of equalities within the rules of matrix algebra and Kronecker products, as shown in what follows:

L(u ⊗ v) = D(N ⊗ I)(u ⊗ v) = D(Nu ⊗ v) = D(Nu ⊗ NNv) = D(NNv ⊗ Nu) = D(N ⊗ I)(Nv ⊗ Nu) = L(Nv ⊗ Nu)

This result is based in the fact that D, the disjunction matrix, represents a commutative operation.

139.2 Many-valued two-dimensional logic

Many-valued logic was developed by many researchers, particularly by Jan Łukasiewicz and allows extending logical operations to truth-values that include uncertainties.*[5] In the case of two-valued vector logic, uncertainties in the truth values can be introduced using vectors with s and n weighted by probabilities. Let f = ϵs + δn , with ϵ, δ ∈ [0, 1], ϵ + δ = 1 be this kind of “probabilistic”vectors. Here, the many-valued character of the logic is introduced a posteriori via the uncertainties introduced in the inputs.*[1]

139.2.1 Scalar projections of vector outputs

The outputs of this many-valued logic can be projected on scalar functions and generate a particular class of proba- bilistic logic with similarities with the many-valued logic of Reichenbach.*[6]*[7]*[8] Given two vectors u = αs+βn and v = α′s + β′n and a dyadic logical matrix G , a scalar probabilistic logic is provided by the projection over vector s:

V al(scalars) = sT G(vectors)

Here are the main results of these projections:

NOT (α) = sT Nu = 1 − α OR(α, α′) = sT D(u ⊗ v) = α + α′ − αα′ AND(α, α′) = sT C(u ⊗ v) = αα′ IMPL(α, α′) = sT L(u ⊗ v) = 1 − α(1 − α′) XOR(α, α′) = sT X(u ⊗ v) = α + α′ − 2αα′

The associated negations are:

NOR(α, α′) = 1 − OR(α, α′) NAND(α, α′) = 1 − AND(α, α′) EQUI(α, α′) = 1 − XOR(α, α′)

If the scalar values belong to the set {0, ½, 1}, this many-valued scalar logic is for many of the operators almost identical to the 3-valued logic of Łukasiewicz. Also, it has been proved that when the monadic or dyadic operators act over probabilistic vectors belonging to this set, the output is also an element of this set.*[3] 139.3. HISTORY 523

139.3 History

The approach has been inspired in neural network models based on the use of high-dimensional matrices and vec- tors.*[9]*[10] Vector logic is a direct translation into a matrix-vector formalism of the classical Boolean polyno- mials.*[11] This kind of formalism has been applied to develop a fuzzy logic in terms of complex numbers.*[12] Other matrix and vector approaches to logical calculus have been developed in the framework of quantum physics, computer science and optics.*[13]*[14]*[15] Early attempts to use linear algebra to represent logic operations can be referred to Peirce and Copilowish.*[16] The Indian biophysicist G.N. Ramachandran developed a formalism using algebraic matrices and vectors to represent many operations of classical Jain Logic known as Syad and Saptbhangi. Indian logic.*[17] It requires independent affirmative evidence for each assertion in a proposition, and does not make the assumption for binary complementation.

139.4 Boolean polynomials

George Boole established the development of logical operations as polynomials.*[11] For the case of monadic oper- ators (such as identity or negation), the Boolean polynomials look as follows:

f(x) = f(1)x + f(0)(1 − x)

The four different monadic operations result from the different binary values for the coefficients. Identity operation requires f(1) = 1 and f(0) = 0, and negation occurs if f(1) = 0 and f(0) = 1. For the 16 dyadic operators, the Boolean polynomials are of the form:

f(x, y) = f(1, 1)xy + f(1, 0)x(1 − y) + f(0, 1)(1 − x)y + f(0, 0)(1 − x)(1 − y)

The dyadic operations can be translated to this polynomial format when the coefficients f take the values indicated in the respective truth tables. For instance: the NAND operation requires that:

f(1, 1) = 0 and f(1, 0) = f(0, 1) = f(0, 0) = 1 .

These Boolean polynomials can be immediately extended to any number of variables, producing a large potential variety of logical operators. In vector logic, the matrix-vector structure of logical operators is an exact translation to the format of liner algebra of these Boolean polynomials, where the x and 1-x correspond to vectors s and n respectively (the same for y and 1-y). In the example of NAND, f(1,1)=n and f(1,0)=f(0,1)=f(0,0)=s and the matrix version becomes:

S = n(s ⊗ s)T + s[(s ⊗ n)T + (n ⊗ s)T + (n ⊗ n)T ]

139.5 Extensions

• Vector logic can be extended to include many truth values since large dimensional vector spaces allow to create many orthogonal truth values and the corresponding logical matrices.*[2] • Logical modalities can be fully represented in this context, with recursive process inspired in neural mod- els*[2]*[18] • Some cognitive problems about logical computations can be analyzed using this formalism, in particular re- cursive decisions. Any logical expression of classical propositional calculus can be naturally represented by a tree structure.*[4] This fact is retained by vector logic, and has been partially used in neural models focused in the investigation of the branched structure of natural languages.*[19]*[20]*[21]*[22]*[23]*[24] 524 CHAPTER 139. VECTOR LOGIC

• The computation via reversible operations as the Fredkin gate can be implemented in vector logic. This im- plementations provides explicit expressions for matrix operators that produce the input format and the output filtering necessary for obtaining computations*[2]*[3] • Elementary cellular automata can be analyzed using the operator structure of vector logic; this analysis leads to a spectral decomposition of the laws governing the its dynamics*[25]*[26] • In addition, based on this formalism, a discrete differential and integral calculus has been developed*[27]

139.6 See also

• Fuzzy logic • Quantum logic • Boolean algebra • Propositional calculus • George Boole • Jan Łukasiewicz

139.7 References

[1] Mizraji, E. (1992). Vector logics: the matrix-vector representation of logical calculus. Fuzzy Sets and Systems, 50, 179– 185, 1992

[2] Mizraji, E. (2008) Vector logic: a natural algebraic representation of the fundamental logical gates. Journal of Logic and Computation, 18, 97–121, 2008

[3] Mizraji, E. (1996) The operators of vector logic. Mathematical Logic Quarterly, 42, 27–39

[4] Suppes, P. (1957) Introduction to Logic, Van Nostrand Reinhold, New York.

[5] Łukasiewicz, J. (1980) Selected Works. L. Borkowski, ed., pp. 153–178. North-Holland, Amsterdam, 1980

[6] Rescher, N. (1969) Many-Valued Logic. McGraw–Hill, New York

[7] Blanché, R. (1968) Introduction à la Logique Contemporaine, Armand Colin, Paris

[8] Klir, G.J., Yuan, G. (1995) Fuzzy Sets and Fuzzy Logic. Prentice–Hall, New Jersey

[9] Kohonen, T. (1977) Associative Memory: A System-Theoretical Approach. Springer-Verlag, New York

[10] Mizraji, E. (1989) Context-dependent associations in linear distributed memories. Bulletin of Mathematical Biology, 50, 195–205

[11] Boole, G. (1854) An Investigation of the Laws of Thought, on which are Founded the Theories of Logic and Probabilities. Macmillan, London, 1854; Dover, New York Reedition, 1958

[12] Dick, S. (2005) Towards complex fuzzy logic. IEEE Transactions on Fuzzy Systems, 15,405–414, 2005

[13] Mittelstaedt, P. (1968) Philosophische Probleme der Modernen Physik, Bibliographisches Institut, Mannheim

[14] Stern, A. (1988) Matrix Logic: Theory and Applications. North-Holland, Amsterdam

[15] Westphal, J., Hardy, J. (2005) Logic as a vector system. Journal of Logic and Computation, 15, 751–765

[16] Copilowish, I.M. (1948) Matrix development of the calculus of relations. Journal of Symbolic Logic, 13, 193–203

[17] Jain, M.K. (2011) Logic of evidence-based inference propositions, Current Science, 1663–1672, 100

[18] Mizraji, E. (1994) Modalities in vector logic. Notre Dame Journal of Formal Logic, 35, 272–283

[19] Mizraji, E., Lin, J. (2002) The dynamics of logical decisions. Physica D, 168–169, 386–396 139.7. REFERENCES 525

[20] beim Graben, P., Potthast, R. (2009). Inverse problems in dynamic cognitive modeling. Chaos, 19, 015103

[21] beim Graben, P., Pinotsis, D., Saddy, D., Potthast, R. (2008). Language processing with dynamic fields. Cogn. Neurodyn., 2, 79–88

[22] beim Graben, P., Gerth, S., Vasishth, S.(2008) Towards dynamical system models of language-related brain potentials. Cogn. Neurodyn., 2, 229–255

[23] beim Graben, P., Gerth, S. (2012) Geometric representations for minimalist grammars. Journal of Logic, Language and Information, 21, 393-432 .

[24] Binazzi, A.(2012) Cognizione logica e modelli mentali. Studi sulla formazione, 1–2012, pag. 69–84

[25] Mizraji, E. (2006) The parts and the whole: inquiring how the interaction of simple subsystems generates complexity. International Journal of General Systems, 35, pp. 395–415.

[26] Arruti, C., Mizraji, E. (2006) Hidden potentialities. International Journal of General Systems, 35, 461–469.

[27] Mizraji, E. (2015) Differential and integral calculus for logical operations. A matrix–vector approach Journal of Logic and Computation 25, 613-638, 2015 Chapter 140

Warnier/Orr diagram

A Warnier/Orr diagram (also known as a logical construction of a program/system) is a kind of hierarchical flowchart that allows the description of the organization of data and procedures. They were initially developed in France by Jean-Dominique Warnier and in the United States by Kenneth Orr. This method aids the design of pro- gram structures by identifying the output and processing results and then working backwards to determine the steps and combinations of input needed to produce them. The simple graphic method used in Warnier/Orr diagrams makes the levels in the system evident and the movement of the data between them vivid.

140.1 Basic Elements

Warnier/Orr diagrams show the processes and sequences in which they are performed. Each process is defined in a hierarchical manner i.e. it consists of sets of subprocesses, that define it. At each level, the process is shown in bracket that groups its components. Since a process can have many different subprocesses, Warnier/Orr diagram uses a set of brackets to show each level of the system. Critical factors in s/w definition and development are iteration or repetition and alteration. Warnier/Orr diagrams show this very well.

140.2 Using Warnier/Orr diagrams

To develop a Warnier/Orr diagram, the analyst works backwards, starting with systems output and using output oriented analysis. On paper, the development moves from right to left . First, the intended output or results of the processing are defined. At the next level, shown by inclusion with a bracket, the steps needed to produce the output are defined. Each step in turn is further defined. Additional brackets group the processes required to produce the result on the next level. Warnier/Orr diagram offer some distinct advantages to systems experts. They are simple in appearance and easy to understand. Yet they are powerful design tools. They have advantage of showing groupings of processes and the data that must be passed from level to level. In addition, the sequence of working backwards ensures that the system will be result oriented. This method is useful for both data and process definition. It can be used for each independently, or both can be combined on the same diagram.

140.3 Constructs in Warnier/Orr diagrams

There are four basic constructs used on Warnier/Orr diagrams: hierarchy, sequence, repetition, and alternation. There are also two slightly more advanced concepts that are occasionally needed: concurrency and recursion.

526 140.3. CONSTRUCTS IN WARNIER/ORR DIAGRAMS 527

140.3.1 Hierarchy

Hierarchy is the most fundamental of all of the Warnier/Orr constructs. It is simply a nested group of sets and subsets shown as a set of nested brackets. Each bracket on the diagram (depending on how you represent it, the character is usually more like a brace "{" than a bracket "[", but we call them“brackets”) represents one level of hierarchy. The hierarchy or structure that is represented on the diagram can show the organization of data or processing. However, both data and processing are never shown on the same diagram.

140.3.2 Sequence

Sequence is the simplest structure to show on a Warnier/Orr diagram. Within one level of hierarchy, the features listed are shown in the order in which they occur. In other words, the step listed first is the first that will be executed (if the diagram reflects a process), while the step listed last is the last that will be executed. Similarly with data, the data field listed first is the first that is encountered when looking at the data, the data field listed last is the final one encountered.

140.3.3 Repetition

Repetition is the representation of a classic “loop”in programming terms. It occurs whenever the same set of data occurs over and over again (for a data structure) or whenever the same group of actions is to occur over and over again (for a processing structure). Repetition is indicated by placing a set of numbers inside parentheses beneath the repeating set. Typically there are two numbers listed in the parentheses, representing the fewest and the most number of times the set will repeat. By convention the first letter of the repeating set is the letter chosen to represent the maximum. While the minimum bound and maximum bound can technically be anything, they are most often either "(1,n)" as in the example, or "(0,n).”When used to depict processing, the "(1,n)" repetition is classically known as a “DoUntil” loop, while the "(0,n)" repetition is called a “DoWhile”loop. On the Warnier/Orr diagram, however, there is no distinction between the two different types of repetition, other than the minimum bound value. On occasion, the minimum and maximum bound are predefined and not likely to change: for instance the set“Day” occurs within the set “Month”from 28 to 31 times (since the smallest month has 28 days, the largest months, 31). This is not likely to change. And on occasion, the minimum and maximum are fixed at the same number. In general, though, it is a bad idea to "hard code" a constant other than “0”or “1”in a number of times clause —the design should be flexible enough to allow for changes in the number of times without changes to the design. For instance, if a company has 38 employees at the time a design is done, hard coding a “38”as the “number of employees”within company would certainly not be as flexible as designing "(1,n)". The number of times clause is always an operator attached to some set (i.e., the name of some bracket), and is never attached to an element (a diagram feature which does not decompose into smaller features). The reason for this will become more apparent as we continue to work with the diagrams. For now, you will have to accept this as a formation rule for a correct diagram.

140.3.4 Alternation

Alternation, or selection, is the traditional“decision”process whereby a determination is made to execute one process or another. The Exclusive OR symbol (the plus sign inside the circle) indicates that the sets immediately above and below it are mutually exclusive (if one is present the other is not). This diagram indicates that an Employee is either Management or Non-Management, one Employee cannot be both. It is also permissible to use a “negation bar” above an alternative in a manner similar to engineering notation. The bar is read by simply using the word “not”. Alternatives do not have to be binary as in the previous examples, but may be many-way alternatives.

140.3.5 Concurrency

Concurrency is one of the two advanced constructs used in the methodology. It is used whenever sequence is unimpor- tant. For instance, years and weeks operate concurrently (or at the same time) within our calendar. The concurrency 528 CHAPTER 140. WARNIER/ORR DIAGRAM operator is rarely used in program design (since most languages do not support true concurrent processing anyway), but does come into play when resolving logical and physical data structure clashes.

140.3.6 Recursion

Recursion is the least used of the constructs. It is used to indicate that a set contains an earlier or a less ordered version of itself. In the classic “bill of materials”problem components contain parts and other sub-components. Sub-components also contain sub-sub-components, and so on. The doubled bracket indicates that the set is recursive. Data structures that are truly recursive are rather rare.

140.4 See also

• Structure chart

140.5 References

140.6 External links

• Warnier

• Dave Higgins Consulting website and original source for Wikipedia entry. • James A. Senn, Analysis & Design of Information Systems, 2nd ed., McGraw-Hill Publishing Company

• Ken Orr Institute Chapter 141

Warrant (logic)

The Toulmin method is an informal method of reasoning. Created by the British philosopher Stephen Toulmin, it involves the data, claim, and warrant of an argument. These three parts of the argument are all necessary to support a good argument. The data is the evidence used to prove something. The claim is what you are proving with the data. The warrant is the assumption or principle that connects the data to the claim. All three parts are necessary.

141.1 Examples

For an example: “Harry was born in Bermuda, so Harry must be a British subject.” In the above sentence, the phrase “Harry was born in Bermuda”is the data. This is evidence to support the claim. The claim in the sentence above is “Harry must be a British subject.”The warrant is not explicitly stated in this sentence; it is implied. The warrant is something like this, “A man born in Bermuda will be a British subject.”It is not necessary to state the warrant in a sentence. Usually, one explains the warrant in following sentences. Other times, like in the sentence above, the speaker of the sentence assumes the listener already knows the fact that all people born in Bermuda are British subjects. Another example: “Steve bought apple juice for himself, so he must like apple juice.” This argument provides the data, claim, and warrant. The data would be the fact that Steve bought apple juice for himself. The claim is that Steve must like apple juice. The warrant is that people who buy apple juice, drink it, which means that they must like it, or else they wouldn't drink it. Again, the warrant is considered background knowledge and unnecessary to repeat in the argument. If one were to expound this argument, however, it would be helpful to explain the warrant.

141.2 Techniques

An author usually will not bother to explain the warrant because it is too obvious. It is usually an assumption or a generalization. However, the author must make sure the warrant is clear because the reader must understand the author's assumptions and why the author assumes these opinions. An example of an argument with an unclear warrant is like this: “Drug abuse is a serious problem in the United States. Therefore, the United States must help destroy drug production in Latin America.”This may leave the reader confused. By inserting the warrant in between the data and the claim, though, would make the argument clearer. Something like, “As long as drugs are manufactured in Latin America, they will be smuggled into the United States, and drug abuse will continue.”This phrase makes clear why the evidence relates to the claim. One must be cautious as to deciding whether or not to include the warrant in the argument because flaws in the argument could be obvious. Backing, rebuttals, and qualifiers are also typical additions to this argument. The backing is added logic or reasoning that may be needed to convince the audience and further support the warrant if it is not initially accepted. Rebuttals are used as a preemptive method against any counter-arguments. These acknowledge the limits of the claim, considering certain conditions where it would not hold true. Usually following is a counter-argument or presentation of new evidence to further support the original claim. Qualifiers are words that quantify the argument. They include words

529 530 CHAPTER 141. WARRANT (LOGIC) such as 'most', 'usually', 'always', 'never', 'absolutely' or 'sometimes'. These can either strongly assert arguments or make them vague and uncertain.

141.3 References

•“Reasoning.”The Bedford Reader. By X.J. Kennedy, Dorothy M. Kennedy, and Jane E. Aaron. Ed. Denise B. Wydra and Karen S. Henry. 9th ed. New York: Bedford/St. Martin's, 2006. p. 519-522. Chapter 142

What the Tortoise Said to Achilles

"What the Tortoise Said to Achilles", written by Lewis Carroll in 1895, for the philosophical journal Mind, is a brief allegorical dialogue on the foundations of logic. The title alludes to one of Zeno's paradoxes of motion, in which Achilles could never overtake the tortoise in a race. In Carroll's dialogue, the tortoise challenges Achilles to use the force of logic to make him accept the conclusion of a simple deductive argument. Ultimately, Achilles fails, because the clever tortoise leads him into an infinite regression.

142.1 Summary of the dialogue

The discussion begins by considering the following logical argument:

• A: “Things that are equal to the same are equal to each other”(Euclidean relation, a weakened form of the transitive property) • B: “The two sides of this triangle are things that are equal to the same” • Therefore Z: “The two sides of this triangle are equal to each other”

The Tortoise asks Achilles whether the conclusion logically follows from the premises, and Achilles grants that it obviously does. The Tortoise then asks Achilles whether there might be a reader of Euclid who grants that the argument is logically valid, as a sequence, while denying that A and B are true. Achilles accepts that such a reader might exist, and that he would hold that if A and B are true, then Z must be true, while not yet accepting that A and B are true. (A reader who denies the premises.) The Tortoise then asks Achilles whether a second kind of reader might exist, who accepts that A and B are true, but who does not yet accept the principle that if A and B are both true, then Z must be true. Achilles grants the Tortoise that this second kind of reader might also exist. The Tortoise, then, asks Achilles to treat the Tortoise as a reader of this second kind. Achilles must now logically compel the Tortoise to accept that Z must be true. (The tortoise is a reader who denies the argument form itself; the syllogism's conclusion, structure, or validity.) After writing down A, B, and Z in his notebook, Achilles asks the Tortoise to accept the hypothetical:

• C: “If A and B are true, Z must be true”

The Tortoise agrees to accept C, if Achilles will write down what it has to accept in his notebook, making the new argument:

• A: “Things that are equal to the same are equal to each other” • B: “The two sides of this triangle are things that are equal to the same” • C: “If A and B are true, Z must be true” • Therefore Z: “The two sides of this triangle are equal to each other”

531 532 CHAPTER 142. WHAT THE TORTOISE SAID TO ACHILLES

But now that the Tortoise accepts premise C, it still refuses to accept the expanded argument. When Achilles demands that “If you accept A and B and C, you must accept Z,”the Tortoise remarks that that's another hypothetical proposition, and suggests even if it accepts C, it could still fail to conclude Z if it did not see the truth of:

• D: “If A and B and C are true, Z must be true”

The Tortoise continues to accept each hypothetical premise once Achilles writes it down, but denies that the conclusion necessarily follows, since each time it denies the hypothetical that if all the premises written down so far are true, Z must be true:

“And at last we've got to the end of this ideal racecourse! Now that you accept A and B and C and D, of course you accept Z.” “Do I?" said the Tortoise innocently. “Let's make that quite clear. I accept A and B and C and D. Suppose I still refused to accept Z?" “Then Logic would take you by the throat, and force you to do it!" Achilles triumphantly replied.“Logic would tell you, 'You can't help yourself. Now that you've accepted A and B and C and D, you must accept Z!' So you've no choice, you see.” “Whatever Logic is good enough to tell me is worth writing down,”said the Tortoise. “So enter it in your notebook, please. We will call it (E) If A and B and C and D are true, Z must be true. Until I've granted that, of course I needn't grant Z. So it's quite a necessary step, you see?" “I see,”said Achilles; and there was a touch of sadness in his tone.

Thus, the list of premises continues to grow without end, leaving the argument always in the form:

• (1): “Things that are equal to the same are equal to each other” • (2): “The two sides of this triangle are things that are equal to the same” • (3): (1) and (2) ⇒ (Z) • (4): (1) and (2) and (3) ⇒ (Z) • ... • (n): (1) and (2) and (3) and (4) and ... and (n − 1) ⇒ (Z) • Therefore (Z): “The two sides of this triangle are equal to each other”

At each step, the Tortoise argues that even though he accepts all the premises that have been written down, there is some further premise (that if all of (1)–(n) are true, then (Z) must be true) that it still needs to accept before it is compelled to accept that (Z) is true.

142.2 Explanation

Lewis Carroll was showing that there is a regress problem that arises from modus ponens deductions.

P → Q, P ∴ Q The regress problem arises because a prior principle is required to explain logical principles, here modus ponens, and once that principle is explained, another principle is required to explain that principle. Thus, if the causal chain is to continue, the argument falls into infinite regress. However, if a formal system is introduced where modus ponens is simply a rule of inference defined by the system, then it can be abided by simply because it is so. For example, chess has particular rules that simply go without question and players must abide by them because they form the very 142.3. DISCUSSION 533

framework of the game. Likewise, a formal system of logic is defined by rules that are to be followed, by definition, without question. Having a defined formal system of logic stops the infinite regression—that is, the regression stops at the axioms or rules, per se, of the given game, system, etc. In propositional logic the logical implication is defined as follows: P implies Q if and only if the proposition not P or Q is a tautology. Hence de modus ponens, [P ∧ (P → Q)] ⇒ Q, is a valid logical conclusion according to the definition of logical implication just stated. Demonstrating the logical implication simply translates into verifying that the compound truth table produces a tautology. But the tortoise does not accept on faith the rules of propositional logic that this explanation is founded upon. He asks that these rules, too, be subject to logical proof. The Tortoise and Achilles don't agree on any definition of logical implication. In addition, the story hints at problems with the propositional solution. Within the system of propositional logic, no proposition or variable carries any semantic content. The moment any proposition or variable takes on semantic content, the problem arises again because semantic content runs outside the system. Thus, if the solution is to be said to work, then it is to be said to work solely within the given formal system, and not otherwise. Some logicians (Kenneth Ross, Charles Wright) draw a firm distinction between the conditional connective and the implication relation. These logicians use the phrase not p or q for the conditional connective and the term implies for an asserted implication relation.

142.3 Discussion

Several philosophers have tried to resolve Carroll's paradox. Bertrand Russell discussed the paradox briefly in § 38 of The Principles of Mathematics (1903), distinguishing between implication (associated with the form “if p, then q"), which he held to be a relation between unasserted propositions, and inference (associated with the form "p, therefore q"), which he held to be a relation between asserted propositions; having made this distinction, Russell could deny that the Tortoise's attempt to treat inferring Z from A and B is equivalent to, or dependent on, agreeing to the hypothetical “If A and B are true, then Z is true.” The Wittgensteinian philosopher discussed the paradox in The Idea of a Social Science and its Relation to Philosophy (1958), where he argued that the paradox showed that “the actual process of drawing an inference, which is after all at the heart of logic, is something which cannot be represented as a logical formula ... Learning to infer is not just a matter of being taught about explicit logical relations between propositions; it is learning to do something”(p. 57). Winch goes on to suggest that the moral of the dialogue is a particular case of a general lesson, to the effect that the proper application of rules governing a form of human activity cannot itself be summed up with a set of further rules, and so that “a form of human activity can never be summed up in a set of explicit precepts” (p. 53). Carroll's dialogue is apparently the first description of an obstacle to conventionalism about logical truth,*[1] later reworked in more sober philosophical terms by W.V.O. Quine.*[2]

142.4 See also

• Deduction theorem

• Homunculus argument

• Münchhausen trilemma

• Paradox

• Regress argument

• Rule of inference 534 CHAPTER 142. WHAT THE TORTOISE SAID TO ACHILLES

142.5 References

[1] Maddy, P. (December 2012).“The Philosophy of Logic”. Bulletin of Symbolic Logic 18 (4): 481–504. doi:10.2178/bsl.1804010. JSTOR 23316289.

[2] Quine, W.V.O. (1976). The Ways of Paradox, and Other Essays. Cambridge, MA: Havard University Press. ISBN 9780674948358. OCLC 185411480.

142.6 Sources

• Carroll, Lewis (1895).“What the Tortoise Said to Achilles”. Mind 104 (416): 691–693. doi:10.1093/mind/104.416.691. JSTOR 2254477. Reprinted in The Penguin Complete Lewis Carroll (Harmondsworth, Penguin, 1982), pp 1104-1108.

• Hofstadter, Douglas. Gödel, Escher, Bach: an Eternal Golden Braid. See the second dialogue, entitled“Two- Part Invention”. Hofstadter appropriated the characters of Achilles and the Tortoise for other, original, dialogues in the book which alternate contrapuntally with prose chapters. Hofstadter's Tortoise is of the male sex, though the Tortoise's sex is never specified by Carroll. The French translation of the book rendered the Tortoise's name as “Madame Tortue”. • A number of websites, including“What the Tortoise Said to Achilles”at the Lewis Carroll Society of North America, “What the Tortoise Said to Achilles” at Digital Text International, and “What the Tortoise Said to Achilles” at Fair Use Repository.

142.7 External links

• Works related to What the Tortoise Said to Achilles at Wikisource

• What the Tortoise Said to Achilles public domain audiobook at LibriVox Chapter 143

Window operator

In modal logic, the window operator △ is a with the following semantic definition: M, w |= △ϕ ⇐⇒ ∀u, M, u |= ϕ ⇒ Rwu for M = (W, R, f) a Kripke model and w, u ∈ W . Informally, it says that w “sees”every φ-world (or every φ-world is seen by w). This operator is not definable in the basic modal logic (i.e. some propositional non-modal language together with a single primitive “necessity”(universal) operator, often denoted by ' □ ', or its existential dual, often denoted by ' ♢ '). Notice that its truth condition is the converse of the truth condition for the standard “necessity”operator. For references to some of its applications, see the References section.

143.1 References

• Blackburn, P; de Rijke, M; Venema, Y (2002). Modal Logic. Cambridge University Press.

535 Chapter 144

Witness (mathematics)

In mathematical logic, a witness is a specific value t to be substituted for variable x of an existential statement of the form ∃x φ(x) such that φ(t) is true.

144.1 Examples

For example, a theory T of arithmetic is said to be inconsistent if there exists a proof in T of the formula “0=1”. The formula I(T), which says that T is inconsistent, is thus an existential formula. A witness for the inconsistency of T is a particular proof of “0 = 1”in T. Boolos, Burgess, and Jeffrey (2002:81) define the notion of a witness with the example, in which S is an n-place relation on natural numbers, R is an n-place recursive relation, and ↔ indicates logical equivalence (if and only if):

" S(x1, ..., xn) ↔ ∃y R(x1, . . ., xn, y)

"A y such that R holds of the xi may be called a 'witness' to the relation S holding of the xi (provided we understand that when the witness is a number rather than a person, a witness only testifies to what is true).”In this particular example, B-B-J have defined s to be (positively) recursively semidecidable, or simply semirecursive.

144.2 Henkin witnesses

In predicate calculus, a Henkin witness for a sentence ∃x ϕ(x) in a theory T is a term c such that T proves φ(c) (Hinman 2005:196). The use of such witnesses is a key technique in the proof of Gödel's completeness theorem presented by Leon Henkin in 1949.

144.3 Relation to game semantics

The notion of witness leads to the more general idea of game semantics. In the case of sentence ∃x ϕ(x) the winning strategy for the verifier is to pick a witness for ϕ . For more complex formulas involving universal quantifiers, the existence of a winning strategy for the verifier depends on the existence of appropriate Skolem functions. For example, if S denotes ∀x∃y ϕ(x, y) then an equisatisfiable statement for S is ∃f∀x ϕ(x, f(x)) . The Skolem function f (if it exists) actually codifies a winning strategy for the verifier of S by returning a witness for the existential sub-formula for every choice of x the falsifier might make.

144.4 See also

• Certificate (complexity), an analogous concept in computational complexity theory

536 144.5. REFERENCES 537

144.5 References

• George S. Boolos, John P. Burgess, and Richard C. Jeffrey, 2002, Computability and Logic: Fourth Edition, Cambridge University Press, ISBN 0-521-00758-5. • Leon Henkin, 1949, “The completeness of the first-order functional calculus”, Journal of Symbolic Logic v. 14 n. 3, pp. 159–166.

• Peter G. Hinman, 2005, Fundamentals of mathematical logic, A.K. Peters, ISBN 1-56881-262-0. • J. Hintikka and G. Sandu, 2009, “Game-Theoretical Semantics”in Keith Allan (ed.) Concise Encyclopedia of Semantics, Elsevier, ISBN 0-08095-968-7, pp. 341–343 Chapter 145

Zhegalkin polynomial

Zhegalkin (also Zegalkin or Gegalkine) polynomials form one of many possible representations of the operations of boolean algebra. Introduced by the Russian mathematician I. I. Zhegalkin in 1927, they are the polynomials of ordinary high school algebra interpreted over the integers mod 2. The resulting degeneracies of modular arithmetic result in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coefficients nor exponents. Coefficients are redundant because 1 is the only nonzero coefficient. Exponents are redundant because in arithmetic mod 2, x2 = x. Hence a polynomial such as 3x2y5z is congruent to, and can therefore be rewritten as, xyz.

145.1 Boolean equivalent

Prior to 1927 boolean algebra had been considered a calculus of logical values with logical operations of conjunction, disjunction, negation, etc. Zhegalkin showed that all boolean operations could be written as ordinary numeric poly- nomials, thinking of the logical constants 0 and 1 as integers mod 2. The logical operation of conjunction is realized as the arithmetic operation of multiplication xy, and logical exclusive-or as arithmetic addition mod 2, (written here as x⊕y to avoid confusion with the common use of + as a synonym for inclusive-or ∨). Logical complement ¬x is then derived from 1 and ⊕ as x⊕1. Since ∧ and ¬ form a sufficient basis for the whole of boolean algebra, meaning that all other logical operations are obtainable as composites of these basic operations, it follows that the polynomials of ordinary algebra can represent all boolean operations, allowing boolean reasoning to be performed reliably by ap- pealing to the familiar laws of high school algebra without the distraction of the differences from high school algebra that arise with disjunction in place of addition mod 2. An example application is the representation of the boolean 2-out-of-3 threshold or median operation as the Zhegalkin polynomial xy⊕yz⊕zx, which is 1 when at least two of the variables are 1 and 0 otherwise.

145.2 Formal properties

Formally a Zhegalkin monomial is the product of a finite set of distinct variables (hence square-free), including the empty set whose product is denoted 1. There are 2*n possible Zhegalkin monomials in n variables, since each monomial is fully specified by the presence or absence of each variable. A Zhegalkin polynomial is the sum (exclusive- or) of a set of Zhegalkin monomials, with the empty set denoted by 0. A given monomial's presence or absence in a polynomial corresponds to that monomial's coefficient being 1 or 0 respectively. The Zhegalkin monomials, being linearly independent, span a 2*n-dimensional vector space over the Galois field GF(2) (NB: not GF(2*n), whose multiplication is quite different). The 2*2*n vectors of this space, i.e. the linear combinations of those monomials as unit vectors, constitute the Zhegalkin polynomials. The exact agreement with the number of boolean operations on n variables, which exhaust the n-ary operations on {0,1}, furnishes a direct counting argument for completeness of the Zhegalkin polynomials as a boolean basis. This vector space is not equivalent to the free boolean algebra on n generators because it lacks complementation (bitwise logical negation) as an operation (equivalently, because it lacks the top element as a constant). This is not to say that the space is not closed under complementation or lacks top (the all-ones vector) as an element, but rather that the linear transformations of this and similarly constructed spaces need not preserve complement and top.

538 145.3. METHOD OF COMPUTATION 539

Those that do preserve them correspond to the boolean homomorphisms, e.g. there are four linear transformations from the vector space of Zhegalkin polynomials over one variable to that over none, only two of which are boolean homomorphisms.

145.3 Method of Computation

There are three known methods generally used for the computation of the Zhegalkin's polynomial.

• Using the Method of Indeterminate Coefficients

• By constructing the Perfect Disjunctive Canonical Form (PDCF)

• By using tables

145.3.1 The method of Indeterminate Coefficients

Using this method, a linear system consisting of all the tuples of the function and their values is generated. Solving the linear gives the coefficients of the Zhegalkin's polynomial.

145.3.2 Using PDCF

Using this method, the Perfect Disjunctive Canonical Form is computed first. Then the negations in the PDCF are replaced by an equivalent expression using the mod 2 sum of the variable and 1. The Disjunction signs are changed to addition mod 2, the brackets are opened, and the resulting Boolean expression is simplified. This simplification results in the Zhegalkin's polynomial.

145.4 Related work

In the same year as Zhegalkin's paper (1927) the American mathematician E.T. Bell published a sophisticated arithmetization of boolean algebra based on Dedekind's ideal theory and general modular arithmetic (as opposed to arithmetic mod 2). The much simpler arithmetic character of Zhegalkin polynomials was first noticed in the west (independently, communication between Soviet and western mathematicians being very limited in that era) by the American mathematician Marshall Stone in 1936 when he observed while writing up his celebrated Stone duality theorem that the supposedly loose analogy between boolean algebras and rings could in fact be formulated as an exact equivalence holding for both finite and infinite algebras, leading him to substantially reorganize his paper.

145.5 References

• Bell, Eric (1927). “Arithmetic of Logic”. Transactions of the American Mathematical Society (Transactions of the American Mathematical Society, Vol. 29, No. 3) 29 (3): 597–611. doi:10.2307/1989098. JSTOR 1989098.

• Gindikin, S.G. (1972). Algebraic Logic (Russian: алгебра логики в задачах). Moscow: Nauka (English translation Springer-Verlag 1985). ISBN 0-387-96179-8.

• Stone, Marshall (1936).“The Theory of Representations for Boolean Algebras”. Transactions of the American Mathematical Society (Transactions of the American Mathematical Society, Vol. 40, No. 1) 40 (1): 37–111. doi:10.2307/1989664. ISSN 0002-9947. JSTOR 1989664.

• Zhegalkin, Ivan Ivanovich (1927). “On the Technique of Calculating Propositions in Symbolic Logic”. Matematicheskii Sbornik 43: 9–28. 540 CHAPTER 145. ZHEGALKIN POLYNOMIAL

145.6 See also

• Ivan Ivanovich Zhegalkin

• Algebraic normal form • Boolean algebra (logic)

• Boolean domain

• Boolean function • Boolean-valued function 145.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 541

145.7 Text and image sources, contributors, and licenses

145.7.1 Text

• Antepredicament Source: https://en.wikipedia.org/wiki/Antepredicament?oldid=666504788 Contributors: Brian0918, Gregbard, Hugo999, Fadesga, Ottawa4ever, Omnipaedista and ChrisGualtieri • Apophasis Source: https://en.wikipedia.org/wiki/Apophasis?oldid=708418518 Contributors: Zundark, Shii, DocWatson42, Orangemike, Beland, Rich Farmbrough, Florian Blaschke, Jnestorius, Lysdexia, V2Blast, Arthena, Keenan Pepper, Ringbang, Garylhewitt, Wnjr, Ste- fanomione, BD2412, CheshireKatz, Koavf, Economy1, Mhbourne, Brec, Epte, Hakamadare, Korny O'Near, Cleared as filed, Jim62sch, Mscuthbert, Greatgavini, Bazonka, Can't sleep, clown will eat me, Tamfang, Jlarson, Chrylis, LoveMonkey, Jon Awbrey, ChrisCork, S ried, CmdrObot, ShelfSkewed, Unmitigated Success, King Hildebrand, Jonathan Tweet, WillowW, Aristophanes68, Pdxuser, Thijs!bot, LRG, Danny Reese, Chubbles, Julia Rossi, Ermeyers, Natty4bumpo, Murrax, Trumpet marietta 45750, Paularblaster, Wordsmith, Bear- ian, Synthebot, Joseph A. Spadaro, Cnilep, SieBot, ClueBot, Binksternet, Niceguyedc, No such user, DumZiBoT, MystBot, Addbot, Fyrael, Jarble, Goregore~enwiki, Luckas-bot, Yobot, AnomieBOT, Mauro Lanari, Rjanag, LlywelynII, Citation bot, LilHelpa, Makeswell, Omnipaedista, FrescoBot, Mlh loves avon, Pollinosisss, DASHBot, Jacobisq, Odysseus1479, Spannerjam, Helpful Pixie Bot, Tbanderson, BG19bot, 2WR1, Yggdrasilfire, An0dos325, Pcasilli, Oiyarbepsy, BrennaM1, SSTflyer, Josip888 and Anonymous: 49 • Argument map Source: https://en.wikipedia.org/wiki/Argument_map?oldid=706692880 Contributors: William Avery, Michael Hardy, Ctwardy, Crissov, Andrewman327, Tim Bell, El C, Mdd, Redvers, Rjwilmsi, Spencerk, Guslacerda, Fnorp, Welsh, John hartley, Smack- Bot, Elonka, McGeddon, Bluebot, RDBrown, EdgeOfEpsilon, Byelf2007, Grumpyyoungman01, Mets501, George100, CmdrObot, John Wilkins, Gregbard, Cimbalom, Al Lemos, Stannered, Widefox, Clan-destine, Argey, Mkorcy, Charlesbaldo, Lloyd borrett, Jodi.a.schneider, Hbent, Gwern, BigrTex, Martin451, Michael Allan, Asimong, EJF, OsamaBinLogin, WurmWoode, Jaded-view, XLinkBot, Philosophy- class HSOG, Addbot, Leiifsexxy, Diogenes5, Yobot, Jagungal, GEBStgo, Rhalah, Lotje, BertSeghers, T3dkjn89q00vl02Cxp1kqs3x7, Tfgordon, Cyberic71, Politechinst, Mcmatter, MarkWWW, Steveswei, Walk&check, Probitt, Thinkmelater, Biogeographist, Jakub24, Fatiherikli, Chesterlaww, Kahtar, Jack9832, Spirit Ethanol and Anonymous: 47 • Argumentation theory Source: https://en.wikipedia.org/wiki/Argumentation_theory?oldid=708224740 Contributors: Ed Poor, SimonP, Edward, Michael Hardy, Ronz, Hectorthebat, Reddi, Markhurd, Rbellin, Banno, Giftlite, David Johnson, Jason Quinn, Gracefool, Andy- cjp, Beland, CSTAR, Yayay, Paulscrawl, RayBirks, D6, Rich Farmbrough, Bitplane, Robertbowerman, Mr. Billion, El C, Jjron, Hu, RJFJR, Velho, Woohookitty, Sandius, Ruud Koot, Analogisub, Dedalus, Rjwilmsi, Koavf, Miskin, Naraht, Spencerk, Schweinsberg, Bgwhite, Wavelength, Anomalocaris, Rick Norwood, Dialectric, Tfine80, Pacogo7, Arthur Rubin, GraemeL, Trickstar, SmackBot, DCDuring, Josephprymak, Teemu Ruskeepää, Oli Filth, Nbarth, Ctbolt, Colonies Chris, Hallenrm, Fiziker, JonasRH, DMacks, Vic- tor Lopes, Byelf2007, Giovanni33, JzG, Antonielly, Ckatz, Grumpyyoungman01, ShakingSpirit, Fjbex, Eastlaw, CRGreathouse, N2e, ShelfSkewed, DragonRidr, Gregbard, Lindsay658, Clan-destine, Qwerty Binary, VoABot II, Gamkiller, Cic, Cathalwoods, DerHexer, BigrTex, RickardV, Moomot, Halber, TreasuryTag, Aerojmac, Rexroad, Tautologies, Cnilep, Farcaster, Jdcanfield, Wahrmund, Fiora- vante Patrone en, Trivialist, Erebus Morgaine, Henninb, EtudiantEco, Tired time, Arthur chos, Rexroad2, Addbot, MrOllie, KGBar- nett, Tassedethe, Yobot, BenDoGood, AnomieBOT, Materialscientist, Spidern, Joarsolo, Vanished user xlkvmskgm4k, JonDePlume, FrescoBot, Machine Elf 1735, Winterst, MeUser42, Jandalhandler, Cnwilliams, Lotje, Fox Wilson, EvanHarper, RjwilmsiBot, Ibbn, Tijfo098, ClueBot NG, Frietjes, Mpcoder, Helpful Pixie Bot, BG19bot, WikiHannibal, Brad7777, MHeder, K.salo.85, ChrisGualtieri, Akababa, Me, Myself, and I are Here, Biogeographist, Argthesis, Minhaqui, PhilipJudson, Beth.Alex123, Helpfuledits5678, Nøkkenbuer, Fabriziomacagno, Joseph H. Lewis and Anonymous: 106 • Ariadne's thread (logic) Source: https://en.wikipedia.org/wiki/Ariadne'{}s_thread_(logic)?oldid=603475455 Contributors: Ixfd64, Safor- rest, Nagelfar, Pmanderson, Duja, Oleg Alexandrov, Zotmeister, King of Hearts, Aeusoes1, WookieInHeat, Bruce Marlin, Glengordon01, Judgesurreal777, JRSpriggs, Gregbard, Adlewis, VolkovBot, Yungjui, ClueBot, Mild Bill Hiccup, Addbot, Ironholds, Tide rolls, Angel ivanov angelov, Xqbot, FrescoBot, Deltik, Spidey104, Zeal-power, Palosirkka and Anonymous: 19 • Austrian Ludwig Wittgenstein Society Source: https://en.wikipedia.org/wiki/Austrian_Ludwig_Wittgenstein_Society?oldid=666047239 Contributors: Markhurd, Wavelength, Valenciano, Magioladitis, I am One of Many and Anonymous: 1 • Belief revision Source: https://en.wikipedia.org/wiki/Belief_revision?oldid=684091316 Contributors: Michael Hardy, Charles Matthews, Hyacinth, Dvavasour, RossPatterson, Fvdham, BD2412, Tizio, FlaBot, Mathbot, Welsh, Closedmouth, Rouenpucelle, Mhss, Jason Koziol, Jon Awbrey, Kripkenstein, TheMightyOrb, Gregbard, Stephanhartmannde, David Eppstein, EdiTor, STBotD, LBehounek, Andrewaskew, CharlesGillingham, Kvihill, Dthomsen8, DarianSentient, Addbot, AnomieBOT, Cdrdata, Xqbot, Safinaskar, Woodlot, John of Reading, Sean r lynch, Jefferson2011 and Anonymous: 20 • Boolean network Source: https://en.wikipedia.org/wiki/Boolean_network?oldid=705077420 Contributors: Kku, Bloodshedder, Everyk- ing, Ehamberg, TedPavlic, Rjwilmsi, Salix alba, Nihiltres, Wavelength, EAderhold, Zargulon, Ojii-san, InverseHypercube, Betacommand, Ekr, Amosfolarin, Myasuda, Roy W. 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ZHEGALKIN POLYNOMIAL

• Circumscription (taxonomy) Source: https://en.wikipedia.org/wiki/Circumscription_(taxonomy)?oldid=675543074 Contributors: Nurg, Andycjp, Rich Farmbrough, Circeus, BD2412, Rosarinagazo, Nadiatalent, Geekdiva, TXiKiBoT, Una Smith, Andrewaskew, Addbot, FrescoBot, RedBot, JonRichfield, Sminthopsis84 and Anonymous: 1 • Classical logic Source: https://en.wikipedia.org/wiki/Classical_logic?oldid=697756237 Contributors: Hyacinth, Guppyfinsoup, Chalst, Qwertyus, Tizio, YurikBot, Pacogo7, Logician2, Jagged 85, Mhss, Muckapedia, Lambiam, Levineps, Simeon, Gregbard, Cydebot, STBot, VolkovBot, Anonymous , Jesin, BotMultichill, Soler97, Hazzzzzz12, Hugo Herbelin, Good Olfactory, Addbot, Fryed-peach, Legobot, Yobot, Pcap, AnakngAraw, GrouchoBot, MetaNest, RjwilmsiBot, Quondum, Tijfo098, GmeSalazar, Kasirbot, Helpful Pixie Bot, J R Gainey and Anonymous: 11 • Colorless green ideas sleep furiously Source: https://en.wikipedia.org/wiki/Colorless_green_ideas_sleep_furiously?oldid=703334191 Contributors: Damian Yerrick, Tbc~enwiki, Derek Ross, Taw, Tim Chambers, Shii, Stevertigo, Infrogmation, Kku, Chinju, Cyde, Delir- ium, Cadr, Mxn, Charles Matthews, Timwi, DJ Clayworth, HappyDog, Saltine, Mythrandia, AnonMoos, Wetman, Hajor, Sjorford, Brand- dobbe, Altenmann, Chancemill, Babbage, Meelar, Auric, Diderot, Bkell, JerryFriedman, Xanzzibar, Snobot, Niteowlneils, Maarten van Vliet, OldakQuill, JimWae, Lulu of the Lotus-Eaters, Kzzl, Dbachmann, Rjo, Zenohockey, Spearhead, Shoujun, Circeus, Army1987, Viriditas, ZayZayEM, Man vyi, Celada, Mark Dingemanse, Clydeiii, Jared81, Arthena, Caesura, SidP, Ish ishwar, Omphaloscope, IMe- owbot, Guthrie, Dtobias, LizardWizard, Mindmatrix, Drostie, Noetica, Toussaint, Gwil, Graham87, Qwertyus, Lxowle, Rjwilmsi, Koavf, FutureNJGov, Dougluce, Klosterdev, Celestianpower, Revolving Bugbear, Wasted Time R, Wavelength, Hairy Dude, Ste1n, Retodon8, Anetode, Ncsaint, G. Lakoff, Nikkimaria, Bondegezou, Finell, SmackBot, InverseHypercube, Bluebot, Jcc1, Thumperward, Imaginary- octopus, JesseRafe, Byelf2007, Lambiam, Bcasterline, JoshuaZ, Mgiganteus1, Peeky44, CmdrObot, TheHerbalGerbil, Gregbard, Warho- rus, Davius, Omicronpersei8, The Wednesday Island, G Purevdorj, Dylan Lake, Chester320, Darrenhusted, Puellanivis, AtticusX, Father Goose, Memotype, CapnPrep, Oren0, Adavidb, Senu, Ohms law, Heyitspeter, CatchNathan, Aaron Rotenberg, AllGloryToTheHyp- notoad, Lova Falk, Auspicion, Cnilep, Psbsub, Paradoctor, SDiZ, Ve4ernik, Drmies, SamuelTheGhost, ChrisHodgesUK, Wikiyuvraj, Lephantome, Addbot, C6541, LaaknorBot, Thrill going up, Lightbot, OlEnglish, Yobot, Carleas, Rjanag, Josepvalls, Citation bot, Ju- bileeclipman, Greatfermat, Shirik, Bxsullivan, FrescoBot, LucienBOT, Bukovets, Robo37, Citation bot 1, Crowe86, Andreldritch, Never give in, Pathock, Solomonfromfinland, Erpert, Akhil 0950, H3llBot, Anthiety, Flipgstring, Anoldtreeok, Fanyavizuri, AJGEC, Supe- rion maximus, Joefromrandb, Helpful Pixie Bot, Alastaira, Ollieinc, Writ Keeper, EricEnfermero, 331dot, 㓟, Epicgenius, BreakfastJr, Michipedian, Yomisma83, Monkbot, Amortias and Anonymous: 109 • Composition of Causes Source: https://en.wikipedia.org/wiki/Composition_of_Causes?oldid=606955316 Contributors: Michael Hardy, Ryelk, Gregbard, Pascal.Tesson, Alaibot, Proscript and JKeck • Concatenation theory Source: https://en.wikipedia.org/wiki/Concatenation_theory?oldid=693009672 Contributors: Michael Hardy, RHaworth, David Eppstein, BOOLE1847, Yobot, AnomieBOT, Omnipaedista, BattyBot, Polytope24, NABRASA, POLY1956 and Anonymous: 1 • Condensed detachment Source: https://en.wikipedia.org/wiki/Condensed_detachment?oldid=636800191 Contributors: Auric, Paul Au- gust, Pearle, Mr Adequate, RJFJR, Dan East, Dtrebbien, Nahaj, SmackBot, Unschool, HoodedMan, CredoFromStart, Ezrakilty, Greg- bard, Headbomb, Nick Number, Hsnieman, S (usurped also), Crisperdue, Barkeep, AnomieBOT, DrilBot, Francis Lima, RjwilmsiBot, Jochen Burghardt and Anonymous: 6 • Condition (philosophy) Source: https://en.wikipedia.org/wiki/Condition_(philosophy)?oldid=674962419 Contributors: Michael Hardy, JackofOz, Stemonitis, Gareth E. Kegg, BOOLE1847, Yobot, AnomieBOT, Omnipaedista, DivineAlpha, Adirlanz, POLY1956, Tyro13, Vieque and Anonymous: 1 • Conditional proof Source: https://en.wikipedia.org/wiki/Conditional_proof?oldid=697344562 Contributors: Tobias Hoevekamp, Si- monP, Ryguasu, Justin Johnson, AugPi, Sethmahoney, Fredrik, Sverdrup, Giftlite, Tadpole, Tagishsimon, AllyUnion, Bookandcoffee, Oleg Alexandrov, Graham87, Rjwilmsi, DVdm, Fram, Tropylium, SmackBot, McGeddon, Kurykh, Tisthammerw, Clconway, Can't sleep, clown will eat me, Mhym, Jon Awbrey, TenPoundHammer, Lambiam, JRSpriggs, CBM, Gregbard, Cydebot, Mona Lisa Smile, Heyitspeter, Odd nature, Yobot, Stmannew, Erik9bot, Planeswalkerdude and Anonymous: 6 • Conditional quantifier Source: https://en.wikipedia.org/wiki/Conditional_quantifier?oldid=577930382 Contributors: Jeffq, Nortexoid, CBM, CharlotteWebb, ChrisGualtieri and Jochen Burghardt • Conflation Source: https://en.wikipedia.org/wiki/Conflation?oldid=703109981 Contributors: Michael Hardy, Doradus, Banno, Alten- mann, Chris-gore, Isopropyl, TomViza, Eep², Lulu of the Lotus-Eaters, Antaeus Feldspar, Bender235, Linmhall, Pedro Aguiar, BRW, Woohookitty, Pol098, Mael Iosa, Imnotminkus, Jimp, Grafen, Deku-shrub, SmackBot, Nzd, Colonies Chris, Glover, Lisasmall, Neil- Fraser, Hvn0413, Balrog, Natezomby, Gogo Dodo, JamesAM, Once in a Blue Moon, Magioladitis, A.P.W., JaGa, R'n'B, Commons- Delinker, Penguinwithin, Osndok, Pmarshal, Enkyo2, Anchor Link Bot, Escape Orbit, Auntof6, Porcofederal, Lexi kate, Libcub, Thi- sIsMyWikipediaName, Leszek Jańczuk, Halloleo, Yobot, Jason Recliner, Esq., AnomieBOT, F W Nietzsche, Mapgenesis, FrescoBot, Voila-pourquoi, I dream of horses, RjwilmsiBot, Fianca, Bassam Jallad, Tango303, Nodove and Anonymous: 52 • Counterexample Source: https://en.wikipedia.org/wiki/Counterexample?oldid=681239775 Contributors: XJaM, Toby Bartels, SimonP, Michael Hardy, Darkwind, Александър, Lumos3, Henrygb, Borislav, CryptoDerk, Kbh3rd, Rgdboer, C S, Jumbuck, Gary, Ricky81682, Oleg Alexandrov, Jftsang, MFH, Marudubshinki, FreplySpang, Yurik, ElKevbo, Mathbot, Tardis, Salvatore Ingala, Jayme, YurikBot, Ian- Manka, ArséniureDeGallium, Jpbowen, Schmock, KnightRider~enwiki, SmackBot, Lambiam, Fvasconcellos, CBM, Gregbard, Ntsimp, Omicronpersei8, RichardVeryard, Widefox, JAnDbot, VoABot II, Albmont, Euku, Numbo3, Thegreenj, NiZhiDao, Thomas Larsen, ,Luckas-bot, Papppfaffe, ArthurBot ,حسام ,NHRHS2010, SieBot, Techman224, JackSchmidt, ClueBot, Alexbot, BOTarate, Addbot GrouchoBot, Frosted14, Winterwater, Robert hoffman, Setitup, Bk314159, ClueBot NG, JohnsonL623, Snotbot, Cntras, Kyorilys, BG19bot, GoShow, Crystallizedcarbon, Loraof and Anonymous: 55 • Counterintuitive Source: https://en.wikipedia.org/wiki/Counterintuitive?oldid=693196225 Contributors: The Anome, William M. Con- nolley, Pakaran, Phil Boswell, Deadbarnacle, DavidCary, Karol Langner, Moquel, WpZurp, Robin klein, MementoVivere, Violetriga, Johnkarp, Falcorian, Shreevatsa, Jeff3000, Tabletop, Marudubshinki, Graham87, Rjwilmsi, Paradoxs, CiaPan, DVdm, Bhny, Rodasmith, LesmanaZimmer, Długosz, Schlafly, Epipelagic, IstvanWolf, Gilliam, Robofish, LarrySDonald, The Man in Question, Iridescent, CR- Greathouse, Gregbard, Nabokov, Septagram, Mattisse, Edwardx, M0ffx, JimCubb, BryanC, Tedickey, DadaNeem, Squids and Chips, Robinson weijman, Viridiflavus~enwiki, Xkeops, Novas0x2a, Dthomsen8, Yobot, Julia W, AnomieBOT, Sda030, Dougofborg, Couter- bane, Nickyus, XXXpinoy777, Petrb, ClueBot NG, Wcherowi, Vatsal19, Helpful Pixie Bot, Zhnirlwaupp and Anonymous: 49 • Cratylism Source: https://en.wikipedia.org/wiki/Cratylism?oldid=700653926 Contributors: Furrykef, BD2412, SmackBot, Smoothtofu, Gregbard, Mirrormundo, Alaibot, Evan.knappenberger, VolkovBot, Philip Trueman, Addbot, Tbvdm, Omnipaedista, Diotemaheartsphi- losophy, Gdje je nestala duša svijeta, SkateTier, Ljsreader and Anonymous: 4 145.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 543

• Critical thinking Source: https://en.wikipedia.org/wiki/Critical_thinking?oldid=708232313 Contributors: Dreamyshade, Fubar Ob- fusco, William Avery, Hephaestos, D, Michael Hardy, Snoyes, Bueller 007, Markhurd, Fuzzywolfenburger, Pedant17, Hyacinth, Bevo, Raul654, Wetman, Merriam~enwiki, Tomchiukc, BitwiseMan, Altenmann, DHN, Sunray, Hadal, JesseW, Wereon, Alan Liefting, Matthew Stannard, Hargettp, Niteowlneils, Gilgamesh~enwiki, Siroxo, Rparle, Deus Ex, SoWhy, Andycjp, Xinit, Ian Yorston, Ablewisuk, Vor- pal suds, Djenner, Icairns, Beardless, WpZurp, Guppyfinsoup, Ayager, DanielCD, Discospinster, Xezbeth, Arthur Holland, Bender235, El C, Carlon, Lycurgus, Chalst, Jon the Geek, Bobo192, Meggar, Viriditas, Xevious, A.t.bruland, Valar, Mdd, Mark Dingemanse, Touqen, TracyRenee, SlimVirgin, Wtmitchell, Pioneer-12, Voxadam, Paulegg, Ogambear, Jensgb, Adamwiggins~enwiki, Kelly Mar- tin, TiesThatBind, Micaqueen, Phuzzy3d, Mindmatrix, VinM, Carlos Porto, Blair P. Houghton, Uncle G, Jeff3000, MONGO, CharlesC, Pictureuploader, Mfjps1, BD2412, Kbdank71, Chenxlee, Rjwilmsi, Koavf, JeffLong, Wiarthurhu, Bubba73, Jasoneth, Windchaser, Carina22, Chobot, Adoniscik, Wavelength, TexasAndroid, Phantomsteve, Nesbit, Manop, Gaius Cornelius, Wimt, Jimphilos, Wiki alf, Onias, Cleared as filed, Irishguy, Mikeblas, Misza13, Shotgunlee, DeadEyeArrow, ChrisGriswold, Closedmouth, Arthur Rubin, Xaxafrad, GraemeL, Chriswaterguy, Katieh5584, N Yo FACE, DVD R W, SmackBot, RedHouse18, Sticky Parkin, Brick Thrower, Bradtcordeiro, Cessator, Ema Zee, Gilliam, Wlmg, Betacommand, Skizzik, Frankahilario, Kmarinas86, David Ludwig, Jerry Ritcey, MartinPoulter, Miquonranger03, MalafayaBot, Silly rabbit, DHN-bot~enwiki, Hallenrm, Darth Panda, Argyriou, Stevenmitchell, Freek Verkerk, Zrulli, Flyguy649, Cybercobra, TedE, Drphilharmonic, Jon Awbrey, Fuzzypeg, Kukini, Ohconfucius, Dak, Kuru, Euchiasmus, Vgy7ujm, Feline Nursery, AstroChemist, Ckatz, Stjamie, 16@r, Noah Salzman, TastyPoutine, Tuspm, Dl2000, Iridescent, Vespine, R~enwiki, Eyere- sist, Phoenixrod, Courcelles, Dlohcierekim, George100, PatrickJCollins, JForget, Wolfdog, Vanished user sojweiorj34i4f, Lighthead, ShelfSkewed, Leujohn, Aubrey Jaffer, Penbat, Keithh, ProofReader, Gregbard, Abhishek26, Slazenger, Kpossin, Peterdjones, Gogo Dodo, Was a bee, B, Joeyadams, Biblbroks, LarryQ, Mitchoyoshitaka, Letranova, Thijs!bot, Epbr123, Second Quantization, Sigma- Algebra, Nick Number, Pfranson, LachlanA, Mentifisto, AntiVandalBot, RobotG, Milton Stanley, Sheilrod, Pseudo narcissus, Seaphoto, Brian0324, Someguyx, Stalik, JAnDbot, Barek, MER-C, Txomin, The Transhumanist, Michael Ly, Olaf, Lazera, Gavia immer, Xact, Bongwarrior, Marycontrary, Gamkiller, Kutu su~enwiki, Irapm, Cathalwoods, Hamiltonstone, Cpl Syx, Gomm, Naynerz, Lloyd borrett, XPensive, Oroso, Nescafe1001, MartinBot, Xumm1du, Arjun01, Ron2, Aksoldat, Terrek, MistyMorn, Maurice Carbonaro, Flatter- world, Parneix, Archie Thompson, Liamooo, Bonadea, S (usurped also), Thyer, Idioma-bot, Vranak, VolkovBot, Bricas, 8thstar, Philip Trueman, Rocmike3, Vipinhari, Jazzwick, Dcicchel, Oxfordwang, John Carter, Rsekanayake, Seachmall, Game31, Raymondwinn, An- drewaskew, Lova Falk, Dmcq, DanBlanchard, Steven Weston, Coffee, Xxavyer, Dawn Bard, Caltas, Steveisdumbgrrr, Smsarmad, Keilana, Flyer22 Reborn, Jojalozzo, Rcarlberg, Transhuman7, Martinlc, RichardWPaul, Plusaf, Svick, Cleverthought, Capitalismojo, SociableLib- eral, Nn123645, Dolphin51, Ossguy, ClueBot, Nzimmer911, Rjd0060, Mattgirling, Petorial, Unbuttered Parsnip, Niceguyedc, Jhkayejr, Alexbot, Jusdafax, Watchduck, Abrech, BirgerH, Rhododendrites, Arjayay, Landerclipse12, Hasteur, Paulbruton, Emilyiscute, Versus22, Humanengr, ClanCC, XLinkBot, Gerhardvalentin, Jonxwood, Tayste, Addbot, Royvdbb, Some jerk on the Internet, Unclebaldrick, Freak- mighty, Tcncv, CL, DougsTech, Drinkstootall, Jncraton, Vishnava, Socerizard, MrOllie, Download, CarsracBot, Chzz, Quercus solaris, Tassedethe, Lkpd, Tide rolls, Verbal, Jarble, Chickenman12345, Csdavis1, CommonSense2007, Luckas-bot, Yobot, Gory292, Ptbot- gourou, Fraggle81, Amirobot, LisaKachold, Azcolvin429, AnomieBOT, Aryeh M. Friedman, Sandris.Ā, Rick1223, Krelnik, Jim1138, Dwayne, Materialscientist, Ptiloser64, Citation bot, Felyza, Gondwanabanana, MarkAllenby, 4twenty42o, Nasnema, Srich32977, JanDe- Fietser, Omnipaedista, SassoBot, Aaron Kauppi, Jagungal, Javad61 12, Dan6hell66, FrescoBot, Diablotin, Liufeng, TurningWork, Dotty- dotdot, Flatronwide, Pasteurizer, Machine Elf 1735, Finalius, Kwiki, Winterst, Biker Biker, Pinethicket, Augustofretes, Leboite, Dazed- bythebell, Dmitry St, Jandalhandler, Marine79, VAP+VYK, Gamewizard71, Colchester121891, Villaonisland, Lotje, LilyKitty, Utegmy, ThinkEnemies, WillNess, Dj6ual, Onel5969, Mean as custard, Rcriticalthink, Bento00, Eekerz, NeonNights79, T3dkjn89q00vl02Cxp1kqs3x7, Immunize, Edlitz36, Zellskelington, Uradumbs, K6ka, Anirudh Emani, Woownick, Alexetc, Brookenoelmoore, Muckauppa, Rexodus, Alpha Quadrant (alt), Lapinkert, Tristan Davie, Wayne Slam, Architect21c, Peterh5322, Kbs 1990, L Kensington, Pfacione, Donner60, Phronetic, Kyew, FurryAminal, Tijfo098, NTox, Avid0g, Sven Manguard, ClueBot NG, Djklgk, MelbourneStar, This lousy T-shirt, Satellizer, Rathersilly, Vitalii-Fedorenko, Bped1985, Amr.rs, Trescious, Masssly, Widr, EHAF, Helpful Pixie Bot, Iste Praetor, HMSSo- lent, Juro2351, Rhennis, Bark3141, AlisonLeCornu, Allecher, Mark Arsten, Emcdona89, Aranea Mortem, Dentalplanlisa, OdikiaNeuro, Cgreen26, Willycactus, MrBill3, Smerch, FCT1980, G.E.Hoostal, Radsmachine, Sfarney, Mrt3366, Cyberbot II, EuroCarGT, Bina- ryPhoton, Tobaifo, Hhira13, CForcesMajor, Frosty, Nancyorschel, BurritoBazooka, Subitizer, Kwaddell28, SomeFreakOnTheInternet, Red-eyed demon, Tentinator, DavidLeighEllis, Keblibrarian, Ctackett3, TDJankins, Mandruss, Ginsuloft, Little Ninpire, Thennicke, Rozalfaro, Arson McFire, GrantWishes, LAciti, JaconaFrere, Opgrandmaster, Lisa Beck, KCToon, Mattybinks, Sexmonkeyreadyto- getfunky, P714-853-1212, AKS.9955, Prisencolin, Vanhung2210, RaxaWiki, Roborodger, AmiraNasr, Ardenbrat, Pff1971, Amortias, Dr.mbl, Holypod, KH-1, 12345bobby12345112233, Spss3000, KINGOFTO, Johnprince203, Nøkkenbuer, KasparBot, Next-geniusindia, Revant m, Datbubblegumdoe, Buzeaston, Hodges334, Spirit Ethanol, Activnadine and Anonymous: 594

• Deductive reasoning Source: https://en.wikipedia.org/wiki/Deductive_reasoning?oldid=707746576 Contributors: The Cunctator, Toby Bartels, Youandme, Mrwojo, DennisDaniels, Michael Hardy, TakuyaMurata, BenKovitz, EdH, DesertSteve, Charles Matthews, Dtgm, Hyacinth, Lumos3, Robbot, R3m0t, Romanm, AceMyth, Blainster, Tobias Bergemann, Ancheta Wis, Giftlite, Lethe, Guanaco, Bovlb, Archenzo, Jason Quinn, Piotrus, Karol Langner, Aequo, Stepp-Wulf, EricBright, TedPavlic, Kevinb, Stbalbach, PhilHibbs, Causa sui, Flammifer, Espoo, Jumbuck, Ryanmcdaniel, CyberSkull, Nasukaren, Garrisonroo, SidP, Kenyon, Tariqabjotu, Stephen, Velho, OwenX, Mindmatrix, TheNightFly, Ruud Koot, Jon Harald Søby, ZephyrAnycon, Teemu Leisti, BD2412, Nightscream, Koavf, Gmelli, Jweiss11, Tangotango, YAZASHI, Ggfevans, DirkvdM, FlaBot, Nihiltres, Fresheneesz, Skc0001, Alphachimp, Chobot, YurikBot, Borgx, Er- achima, DTRY, Rick Norwood, Holycharly, TriGen, EEMIV, Bota47, Shadro, Mjhy0926, SMcCandlish, Allens, Infinity0, GrinBot~enwiki, DVD R W, Sardanaphalus, SmackBot, Aim Here, KocjoBot~enwiki, Thunderboltz, Stephensuleeman, WookieInHeat, Ieopo, The great kawa, Gilliam, Q1w2e3, Mhss, Psiphiorg, Bluebot, ViolinGirl, MalafayaBot, George Rodney Maruri Game, Octahedron80, DHN- bot~enwiki, Javalenok, Chlewbot, Mr.Z-man, ConMan, Cybercobra, Jon Awbrey, RossF18, Byelf2007, The Ungovernable Force, Sasha- toBot, Nishkid64, Dbtfz, JoseREMY, IronGargoyle, Extremophile, Penagate, Comicist, Quaeler, Iridescent, K, Zarex, Van helsing, ChristineD, Neelix, Gregbard, Julian Mendez, Thijs!bot, LactoseTI, Marek69, Kborer, Noaa, AntiVandalBot, MaTT~enwiki, AaronY, IrishPete, Oliver Simon, BenMcLean, JAnDbot, Skomorokh, The Transhumanist, Agentnj, Hewinsj, GurchBot, Probios, Djradon, Kir- rages, Rupes, VoABot II, Arno Matthias, Snowded, Oxford Dictionary, Illspirit, Vanished user ty12kl89jq10, Cathalwoods, HemRaj Singh, Pbroks13, Pomte, Stjeanp, N4nojohn, J.delanoy, Trusilver, Shawn in Montreal, OAC, Tparameter, Jaxha, CompuChip, Hey- itspeter, DorganBot, Vinsfan368, Lallallal, Jonwilliamsl, Pasixxxx, MARVINWAGNER, Rucha58, Leoremy, TXiKiBoT, Technopat, A4bot, Msviolone, Philogo, Broadbot, Abdullais4u, Dprust, Andrewaskew, Graymornings, Lova Falk, Kusyadi, MCTales, Cnilep, Far- caster, Sfmammamia, SieBot, Paradoctor, Meldor, Mankar Camoran, Sunrise, Svick, DesolateReality, Mygerardromance, Escape Orbit, Troy 07, Kenji000, De728631, ClueBot, R000t, Philosophy.dude, Bfeylia, Neverquick, Excirial, Rohit nit, GoldenGoose100, PaulKin- caidSmith, SpikeToronto, Ember of Light, GlasGhost, Thingg, Vanished User 1004, Zenwhat, XLinkBot, BodhisattvaBot, Kwork2, Ger- hardvalentin, Saeed.Veradi, WikHead, Qwertytastic, Ewger, Addbot, CanadianLinuxUser, H92Bot, Glane23, GlobalSequence, Tide rolls, -Legobot, Luckas-bot, Yobot, 2D, Azcolvin429, AnomieBOT, Rubinbot, PresMan, Flewis, Prbclj25, Arthur ,سعی ,Lightbot, ScienceApe 544 CHAPTER 145. ZHEGALKIN POLYNOMIAL

Bot, Xqbot, Doezxcty, S h i v a (Visnu), Lord Archer, Capricorn42, Xephras, Hartkiller, Hjurgen, Ordning, Lord Bane, Ruby.red.roses, RowanEvans, RibotBOT, SEASONnmr, FrescoBot, Beclp, Pinethicket, I dream of horses, Rushbugled13, Mohehab, JimRech, Jandal- handler, NeuroBells123, Keri, Humble Rat, Difu Wu, Whisky drinker, Mean as custard, Badelmann, Tesseract2, DASHBot, EmausBot, Hedonistbot4000, Mo ainm, Tommy2010, Winner 42, TheGeomaster, JSquish, Fæ, Oncenawhile, MindShifts, Foreverlove642, Wayne Slam, Donner60, Tziemer991, Jimmynudes2, ClueBot NG, Drdoug5, Kimberleyporter, Fauzan, Jj1236, Tideflat, Amr.rs, Dictabeard, O.Koslowski, Masssly, Widr, Chillllls, Helpful Pixie Bot, BG19bot, Nichole773, Hallows AG, Wiki13, Luke13579, Richard84041, Nin- jagoat, Sopidex, Miszatomic, Dhruv-NJITWILL, Sgilmore10, Courtneysfoster, ChrisGualtieri, GoShow, Oligocene, ShangTsung87, Lu- gia2453, 93, MostlyListening, BreakfastJr, EMBViki, Strikingstar, VogelsangLorenzo, Ginsuloft, Mauriziogeri2013, Monkbot, Crocs.Sox, KasparBot, Sweepy, Sorte Slyngel, 黄佳玮, Trex363 and Anonymous: 373 • Defeasible reasoning Source: https://en.wikipedia.org/wiki/Defeasible_reasoning?oldid=690037128 Contributors: The Anome, Michael Hardy, Flammifer, Ramaz, Velho, Tizio, RussBot, Mavaddat, SmackBot, Slamb, Mhss, DMS, Cybercobra, Bejnar, TenPoundHammer, Ripounet, Gregbard, JaGa, R'n'B, Ontarioboy, Una Smith, The Tetrast, Thesatori, CharlesGillingham, Ronaldloui, Versus22, Dthomsen8, Addbot, LaaknorBot, Margin1522, D'Artagnol, DrilBot, RedBot, Hoobalkanoobal, WikiPeterD and Anonymous: 14 • Definable set Source: https://en.wikipedia.org/wiki/Definable_set?oldid=599620860 Contributors: Trovatore, CBM, Blargoner, Avaya1, Thehalfone, Hans Adler, HOTmag, Addbot, Jncraton, AndersBot, Yobot, JimVC3, FrescoBot, Carlonicolai, Plasticspork, Chimpionspeak, ChuispastonBot and Anonymous: 5 • Definitions of logic Source: https://en.wikipedia.org/wiki/Definitions_of_logic?oldid=688685740 Contributors: Peter Damian (original account), Charles Matthews, Chalst, Wasted Time R, Rmky87, Elonka, Melchoir, Gregbard, Jdclevenger, Hullaballoo Wolfowitz, Cic, Anarchia, Ontoraul, Jjzanath, Addbot, Yobot, AnomieBOT, Rhalah, Oncenawhile, NGPriest, ClueBot NG, Ezequiel454, PaulBustion87, SaundersLane and Anonymous: 8 • Degree of truth Source: https://en.wikipedia.org/wiki/Degree_of_truth?oldid=630169465 Contributors: Cherkash, Charles Matthews, Furrykef, Hyacinth, Auric, Paul Murray, Falcon Kirtaran, Stephan Leclercq, Paul August, Vanished user lp09qa86ft, Melaen, Simetrical, Rjwilmsi, Tadanisakari, Chris Capoccia, SmackBot, Hippo43, Dreadstar, Bjankuloski06en~enwiki, Simeon, Gregbard, Cydebot, Rgheck, Helgus, SE16, Addbot, ZéroBot, Nizamibilal1064, Nathanielfirst and Anonymous: 9 • Denying the antecedent Source: https://en.wikipedia.org/wiki/Denying_the_antecedent?oldid=707474534 Contributors: Bryan Derk- sen, Mrwojo, Zocky, Voidvector, HarmonicSphere, Iulianu, William M. Connolley, WhisperToMe, Rursus, Mattflaschen, Taak, Elembis, Silence, Kaveh, Sasquatch, Jeltz, Bookandcoffee, Angr, Waldir, KYPark, Gaga~enwiki, Supermorff, YurikBot, Pseudomonas, Jpeob, Shawnc, Ybbor, NickelShoe, SmackBot, Ck4829, Bluebot, Factorial, Furby100, Richard001, Andeggs, DavidHOzAu, Gihanuk, Greg- bard, Steel, 271828182, Isilanes, DavidSTaylor, Obscurans, Mark.camp, Hiddenhearts, Lyctc, Gen. Quon, Jamelan, Gerakibot, Danc- ingPhilosopher, CharlesGillingham, Jfromcanada, Addbot, Xaquseg, Luckas-bot, Gongshow, Gemtpm, Allformweek, ZéroBot, Akerans, ClueBot NG, Rkohar, ChrisGualtieri, SilverSylvester and Anonymous: 38 • Diagrammatic reasoning Source: https://en.wikipedia.org/wiki/Diagrammatic_reasoning?oldid=656440541 Contributors: Zundark, Edward, Ronz, Jaybee~enwiki, Mdd, GregorB, BD2412, Josh Parris, Allan McInnes, Gregbard, Goldenrowley, R'n'B, Dezignr, Gen- eral Reader, Lova Falk, Alethe, CharlesGillingham, Dthomsen8, Lightbot, AnomieBOT, J04n, Aaron Kauppi, Citation bot 1, Ebenezer- LePage, Fiftytwo thirty, Me6620, John of Reading, Xanchester, FSII and Mannschuh • Dialectica space Source: https://en.wikipedia.org/wiki/Dialectica_space?oldid=672954583 Contributors: Michael Hardy, Charles Matthews, Rich Farmbrough, Linas, Cronholm144, Gregbard, 28421u2232nfenfcenc, David Eppstein, Valeria.depaiva, Helper911 and Anonymous: 3 • Diamond 25 Source: https://en.wikipedia.org/wiki/Diamond_25?oldid=689684049 Contributors: Michael Hardy, David Eppstein, Largo- plazo and Mayorsp • Dichotomy Source: https://en.wikipedia.org/wiki/Dichotomy?oldid=698811215 Contributors: Fubar Obfusco, D, Michael Hardy, Mdupont, Pcb21, Hyacinth, Sabbut, Olathe, Jamesday, Fredrik, Altenmann, Stewartadcock, Wlievens, Dissident, Bradeos Graphon, Bensaccount, Maroux, Entropy (usurped), Gdr, Piotrus, Mukerjee, Rdsmith4, Asbestos, Trevor MacInnis, Kenb215, VBGFscJUn3, Alansohn, JY- olkowski, Snowolf, RJFJR, Uncle G, Bluemoose, Tsunade, Matilda, BD2412, Volland, Mayumashu, Drrngrvy, BMF81, YurikBot, Scot- tocracy, Ihope127, Crasshopper, Petri Krohn, RealityCheck, SmackBot, Bggoldie~enwiki, Cutter, Ultramandk, BiT, Ck4829, Zven, An- gel Olivera, Gohst, AndreRD, StevenGould, Charmedguy18, Jitterro, Ceoil, Notwist, Webucation, Iridescent, Balber1, Meng.benjamin, Ken Gallager, Gregbard, Cydebot, Ninguém, Twonex, W Hukriede, Nick Number, Ivan Vlasov, Colin MacLaurin, Ingolfson, Res2216firestar, JAnDbot, Barek, Purplezart, R'n'B, Trusilver, TyrS, Maurice Carbonaro, Mikael Häggström, Thradar, VolkovBot, Florrat~enwiki, Wiae, Mr. Absurd, Lova Falk, Albertus Aditya, Nbumbic, DancingPhilosopher, Considerable powers, Melcombe, Bee Cliff River Slob, XDanielx, Loren.wilton, ClueBot, SummerWithMorons, Mild Bill Hiccup, Polyamorph, Three-quarter-ten, Hasanadnantaha, Vegetator, NicoleT- edesco, Spitfire, Dthomsen8, Addbot, Murderface 623, CactusWriter, DrJos, Gizziiusa, Busterbarker2008, Luckas-bot, Yobot, Bratsche864, Obersachsebot, Carturo222, Xqbot, Omnipaedista, RibotBOT, 13alexander, Roseclearfield, LucienBOT, Notjustbnw, Machine Elf 1735, Winterst, I dream of horses, Hellknowz, Shanerobins, Standardfact, Lotje, Zujine, EmausBot, John of Reading, WikitanvirBot, Pile-Up, Edlitz36, HiW-Bot, ZéroBot, Grstein, Iketsi, Llightex, JonRichfield, ClueBot NG, Widr, Sean The Conspiracy, Calabe1992, ISTB351, Kyoakoa, XatosPwnsAlot, AvocatoBot, The Vintage Feminist, Lemnaminor, Chillinflute, Thelegendomer, Yija Honkgua, Loraof and Anonymous: 97 • Difference (philosophy) Source: https://en.wikipedia.org/wiki/Difference_(philosophy)?oldid=706747336 Contributors: Peter Damian (original account), Charles Matthews, Banno, Arthena, Stefanomione, Vegaswikian, Robertvan1, BirgitteSB, Mike Selinker, Sardanaphalus, SmackBot, Sct72, Lapaz, Gregbard, Cydebot, Magioladitis, VolkovBot, Muhandes, DerBorg, GKantaris, Addbot, Eumolpo, Armbrust, Omnipaedista, Erik9bot, RjwilmsiBot, Xanchester, Faus, PhnomPencil, Wikurtpedia, ArmbrustBot, Awwametal and Anonymous: 5 • Digital timing diagram Source: https://en.wikipedia.org/wiki/Digital_timing_diagram?oldid=666112325 Contributors: Heron, Michael Devore, Wtshymanski, Cburnett, David Haslam, BD2412, Rjwilmsi, Dirkbike, SmackBot, InverseHypercube, Radagast83, AntiVandal- Bot, Alphachimpbot, SieBot, Addbot, Jordsan, Herr Satz, Erik9bot, ZéroBot, ClueBot NG, WikiTryHardDieHard and Anonymous: 11 • Don't-care term Source: https://en.wikipedia.org/wiki/Don't-care_term?oldid=703143447 Contributors: Pnm, Charles Matthews, Bfinn, MattGiuca, HonoluluMan, Fresheneesz, Manop, SmackBot, Skapur, ShelfSkewed, Gregbard, Thijs!bot, NovaSTL, McSly, STBotD, VolkovBot, Macfanatic, Dindon~enwiki, Spinningspark, Ctxppc, DumZiBoT, Thesnark, Addbot, Download, Tassedethe, Qwertyytrewqqw- erty, Biezl, Yobot, Eric-Wester, AnomieBOT, Fast healthy fish, Djgdto, Deekyboy, ClueBot NG, Matthiaspaul, Widr, Helpful Pixie Bot, BG19bot, Jochen Burghardt, Whijus19, Some Gadget Geek and Anonymous: 15 145.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 545

• Embedded dependency Source: https://en.wikipedia.org/wiki/Embedded_dependency?oldid=631901767 Contributors: Wikidsp, Yobot, Wgolf, Clone200 and Anonymous: 1 • Empty name Source: https://en.wikipedia.org/wiki/Empty_name?oldid=585751815 Contributors: Fubar Obfusco, BoNoMoJo (old), EdH, Peter Damian (original account), Gzornenplatz, Karol Langner, Ryanmcdaniel, Shell Kinney, SmackBot, Oatmeal batman, Gregbard, Warhorus, Skomorokh, Anarchia, Yobot, ChrisGualtieri and Anonymous: 11 • Enumerative definition Source: https://en.wikipedia.org/wiki/Enumerative_definition?oldid=671192374 Contributors: Hyacinth, Rpyle731, Abdull, STGM, SmackBot, Bluebot, Jon Awbrey, PaulTanenbaum, Erik9bot, Ljsreader and Anonymous: 2 • Epicureanism Source: https://en.wikipedia.org/wiki/Epicureanism?oldid=702277951 Contributors: Brion VIBBER, Wesley, Andre En- gels, Youssefsan, Shii, Axon, Rcingham, Heron, Hephaestos, Fransvannes, Stevertigo, Tim Starling, TimShell, Kku, BoNoMoJo (old), Jakub~enwiki, Ellywa, ThirdParty, Angela, Peter Damian (original account), Rednblu, Markhurd, Wetman, Secretlondon, Jni, Halibutt, GreatWhiteNortherner, Tobias Bergemann, ManuelGR, Monedula, Risk one, Everyking, Brona, NeoJustin, Decoy, Irene1949, Euro- Tom, Sam Hocevar, Neutrality, Picapica, Lacrimosus, Xjy, SocratesJedi, Dbachmann, Paul August, Bender235, BenjBot, El C, PBTim, Tverbeek, Viriditas, Giraffedata, Jumbuck, Enirac Sum, Arthena, Logologist, Hu, Snowolf, Ross Burgess, Japanese Searobin, Velho, Simetrical, Mequa, Wikiklrsc, Palica, Jrtayloriv, President Rhapsody, Common Man, Bgwhite, Imaek, YurikBot, RussBot, Warshy, Chris Capoccia, Gaius Cornelius, Grafen, NickBush24, Ragesoss, Aldux, Emersoni, Tomisti, ClaesWallin, Zzuuzz, Andrew Lancaster, Jules.LT, SMcCandlish, Katieh5584, Sardanaphalus, SmackBot, Tomyumgoong, TobyK, InverseHypercube, Frasor, Kintetsubuffalo, Betacommand, Rmosler2100, Chris the speller, Christian Findlay, MalafayaBot, Scwlong, Richard001, -Ozone-, RossF18, Arglebar- gleIV, Breno, The Man in Question, Mallaccaos, Tokeefe, Tmcw, Neddyseagoon, Midnightblueowl, K, Courcelles, Eickenberg, Laplace's Demon, Danlev, Tawkerbot2, Chris55, CmdrObot, DominusEtDude, Neelix, Gregbard, Cydebot, Gogo Dodo, Mirrormundo, Dark- Link, DBaba, Nearfar, Thijs!bot, Epbr123, Itsmejudith, Escarbot, D. Webb, Bjenks, Myanw, JAnDbot, Foggytown, Skomorokh, The Transhumanist, Magioladitis, Doug Coldwell, Artegis, Steven Walling, Epicurusphilosophy, Aflyax, FisherQueen, Anarchia, Rettetast, Nigris, Nev1, Cdogg429, Thaurisil, SubwayEater, Andreabrugiony, Penguinwithin, Uberlizard, AntiSpamBot, Christopherjames911, RJASE1, Idioma-bot, Benio76, Philip Trueman, TXiKiBoT, Oshwah, Steveman1012, Ontoraul, Seraphim, Ru8ikscu8er, Cnilep, Pjoef, Epicurean87, SieBot, WereSpielChequers, Fabullus, ConfuciusOrnis, Wombatcat, Surferhere, Javierfv1212, Epikouros~enwiki, James Aaron, BenoniBot~enwiki, Johnanth, Dror dori, Tradereddy, Phil wink, Leranedo, ClueBot, BrocknessMonsta, Aaa3-other, Singingle- mon~enwiki, Mightymouse mm, Alexbot, Jusdafax, Maser Fletcher, Mindstalk, Coinmanj, Esimal, Catalographer, Anon6414, DumZ- iBoT, XLinkBot, ErkinBatu, Addbot, Lespaulocaster, The Sage of Stamford, Fluffernutter, Gzhanstong, LinkFA-Bot, Numbo3-bot, Lightbot, The Bushranger, Legobot, Luckas-bot, Yobot, LeonMT, Anradt, Eduen, Tempodivalse, AnomieBOT, Valueyou, Alphafish, DeadTotoro, ArthurBot, Xqbot, Srich32977, J04n, GrouchoBot, Omnipaedista, PM800, Jsp722, DasallmächtigeJ, FrescoBot, Lika2672, Lemon1001, Pinethicket, Dhalaughlin, SoccerMan2009, LiberatorG, Abnyc81, Tim1357, Pollinosisss, SmartyBoots, DarcWiki, MrAr- ifnajafov, EmausBot, Syncategoremata, RenamedUser01302013, Finn Bjørklid, Epichead, K6ka, ChuispastonBot, Mcc1789, Help- some, ClueBot NG, Nobody60, Duranvskp, Asukite, Jorge Gomes Sharky, Jacksonredden, Joshuajohnson555, Wayne aus, Davidiad, Brs92369, Fiddlersmouth, JCJC777, Reburn2, Vanished user sdij4rtltkjasdk3, BreakfastJr, JamboBeefyQueen, EMBViki, Cmckain14, SpotlightElucidation, Ostracod1, British-Hellenophile, Crow, LawrencePrincipe, VeNeMousKAT, KasparBot and Anonymous: 268 • Equality-generating dependency Source: https://en.wikipedia.org/wiki/Equality-generating_dependency?oldid=631901847 Contribu- tors: Fram, Wikidsp, Yobot, Wgolf, Clone200 and Anonymous: 1 • Erotetics Source: https://en.wikipedia.org/wiki/Erotetics?oldid=605471432 Contributors: Alaibot, Misarxist, Krzysztofgajewski, Dana boomer, Addbot, KamikazeBot, LilHelpa, ChrisGualtieri, Liz and Anonymous: 3 • Existential graph Source: https://en.wikipedia.org/wiki/Existential_graph?oldid=703275058 Contributors: Zundark, Toby Bartels, Michael Hardy, AugPi, TraxPlayer, Silverfish, Charles Matthews, Ancheta Wis, Giftlite, Gecko~enwiki, Poccil, Elwikipedista~enwiki, Mdd, Oleg Alexandrov, Tizio, Moorlock, IanManka, My Cat inn, Juliannechat, Mhss, Jon Awbrey, ArglebargleIV, Mets501, CBM, Sdorrance, An- drewHowse, Julian Mendez, Synergy, Ael 2, Eleuther, Avaya1, R'n'B, N4nojohn, The Tetrast, Sanfranman59, Kl4m, Djacnov~enwiki, Ktr101, Palnot, Addbot, Lightbot, Yobot, Jujutacular, Dexbot, Jochen Burghardt and Anonymous: 23 • Extensional context Source: https://en.wikipedia.org/wiki/Extensional_context?oldid=544118835 Contributors: Mrwojo, Karada, Hy- acinth, Grm wnr, Eliazar, Linas, Kim Stebel, SmackBot, Mets501, Gregbard, Settembrini~enwiki, EagleFan, InspectorTiger, Desolate- Reality, Addbot, Kyoakoa, YFdyh-bot and Anonymous: 10 • Extensional definition Source: https://en.wikipedia.org/wiki/Extensional_definition?oldid=623187284 Contributors: Charles Matthews, Hyacinth, Mindspillage, Sn0wflake, Pearle, Linas, Zunaid, Finell, SmackBot, Jon Awbrey, Biocosmologo, CBM, Hans Adler, Mpawel, Addbot, Xqbot, Erik9bot and Anonymous: 3 • Extensionalism Source: https://en.wikipedia.org/wiki/Extensionalism?oldid=687520038 Contributors: Jan Schreiber and I dream of horses • Fa (concept) Source: https://en.wikipedia.org/wiki/Fa_(concept)?oldid=701874434 Contributors: BD2412, Koavf, RekishiEJ, Gregbard, Magioladitis, Niceguyedc, Kanguole, AnomieBOT, FourLights, Van Gulik, Vanished user sdij4rtltkjasdk3 and Anonymous: 2 • Finite model property Source: https://en.wikipedia.org/wiki/Finite_model_property?oldid=640519278 Contributors: Nortexoid, Ott2, Salamurai, A3nm, Addbot, Yobot, Ansa211, Arthur MILCHIOR and Anonymous: 2 • Fluidics Source: https://en.wikipedia.org/wiki/Fluidics?oldid=706971925 Contributors: Heron, Michael Hardy, Booyabazooka, Charles Matthews, Steinsky, Altenmann, DavidCary, Stepp-Wulf, GraemeLeggett, BD2412, Jaraalbe, Bgwhite, DMahalko, Mskfisher, Deville, Grandmartin11, Jibjibjib, Bluebot, Radagast83, NeilFraser, Gobonobo, Nagle, Kvng, StuHarris, Gregbard, Cydebot, W.F.Galway, Al- phachimpbot, Magioladitis, Verkhovensky, Nikevich, D-rew, STBot, ColinClark, VolkovBot, SieBot, Jimmuhk, Jerryobject, Anchor Link Bot, XLinkBot, Addbot, Legobot, TaBOT-zerem, Almabot, RjwilmsiBot, BAICAN XXX, Microprocessor Man, ClueBot NG, Virtualerian, Wikimpan, Szescstopni, Fudgewunkles, Fsowkfufusua, Bowlesfluidics, Acayl, 高啼 and Anonymous: 28 • Formal ontology Source: https://en.wikipedia.org/wiki/Formal_ontology?oldid=642124638 Contributors: Deb, Ronz, Greenrd, Mbover- load, Agingjb, BD2412, DigitalThief, Mccready, SmackBot, Kostmo, Iridescent, Gregbard, Cydebot, JamesMcGuiggan, Daviddecraene, Luna Santin, Beanformer, Darklilac, Extendon, Gwern, STBot, R'n'B, FLoebe, Funandtrvl, Jimmaths, Ontoraul, Shadowlapis, Dan Polan- sky, Mkbergman, 1ForTheMoney, Libcub, Addbot, Yobot, Omnipaedista, FrescoBot, LittleWink, APGalton, Koi.lover, Foobarnix, Obankston, Technologist9, WebHubTel, Rds1970, Wiki2103, Vladimir Alexiev and Anonymous: 17 • Forward chaining Source: https://en.wikipedia.org/wiki/Forward_chaining?oldid=661541762 Contributors: Frecklefoot, Kku, Pakcw, Yayay, Chalst, Notreadbyhumans, Kotasik, Japanese Searobin, Jeff3000, Qwertyus, The Rambling Man, Tribaal, SmackBot, Mgreenbe, 546 CHAPTER 145. ZHEGALKIN POLYNOMIAL

Sawran~enwiki, Gschadow, Robofish, RichMorin, DBooth, CBM, Gregbard, Makaimc, WinBot, Styrofoam1994, Ravendarksky, Kp- miyapuram, McSly, VanishedUserABC, Tomaxer, Kgoarany, Ravanacker, Martarius, Mdebellis, Jusdafax, Addbot, Konieckropka, Louperi- bot, Stef joosten, BattyBot, The Illusive Man, Rupendra.chulyadyo, FuenAvalos, Monkbot, Supdiop and Anonymous: 20 • Freethought Source: https://en.wikipedia.org/wiki/Freethought?oldid=703565107 Contributors: Bryan Derksen, FvdP, Hephaestos, Ki- wimac, Rbrwr, Bdesham, Michael Hardy, Tim Starling, Bcrowell, JWSchmidt, LittleDan, BenKovitz, Netsnipe, Charles Matthews, Dino, Markhurd, JorgeGG, Twang, Deist, Sjorford, Robbot, Altenmann, Humanist, Tirmie, Gnomon Kelemen, Hadal, Emyth, Guy Peters, Alan Liefting, Gtrmp, Andries, Barbara Shack, Lupin, Zigger, Gracefool, AlistairMcMillan, Brockert, Chameleon, Gadfium, Andycjp, Bt- boy500, Icairns, Scott Burley, Burschik, Mike Rosoft, Sysy, Rich Farmbrough, Vsmith, Deh, Xezbeth, Dbachmann, ChadMiller, Arevich, Cladist, Bender235, RoyBoy, IFaqeer, Billymac00, Cmdrjameson, Armanvaziri, Srrostum, Thialfi, Plumbago, Calton, DreamGuy, Don- Quixote, SteinbDJ, Kenyon, Jeffrey O. Gustafson, Mindmatrix, MGTom, BlaiseFEgan, Toussaint, 790, BD2412, Rjwilmsi, Tizio, Univer- sist, Bubba73, Williamborg, Cassowary, Ian Pitchford, Pevernagie, Viznut, TruthInEvidence, DVdm, YurikBot, RobotE, NTBot~enwiki, RussBot, Pigman, Gardar Rurak, KSmrq, Gaius Cornelius, Nirvana2013, Joel7687, Tailpig, Abb3w, Wyldkat, LodeRunner, Dannyno, Cinik, Homagetocatalonia, Kelliotes, Laurence Boyce, SmackBot, Dsmccoy, NantucketNoon, Prototime, Jagged 85, ProveIt, Ck4829, Amitst, Gilliam, Portillo, Mainsail, Imaginaryoctopus, Tescher, Zachorious, Avb, JonHarder, Kennovak, Blueboar, Nakon, Ggpauly, DavidMann, RJBurkhart, Usernamefortonyd, Giovanni33, -ramz-, Slowmover, JorisvS, Musashiaharon, 041744, MTN~enwiki, Tuspm, Noleander, Colonel Warden, Lottamiata, Greenie2600, SciurusCarolinensis, Daniel5127, Nydas, George100, MicahDCochran, War- renallensmith, Dshin, WeggeBot, Ken Gallager, Gregbard, Matthew Treder, Fl295, Cydebot, Lupine Proletariat, Slp1, Pagana, Khatru2, ST47, Alsario, Jeffseaver, Kdfrawg, NaLalina, Energyfreezer, Khamar, Thijs!bot, Pstanton, Pinaki ghosh, KeithWeil, Edhubbard, Some- Human, Branchoff, Escarbot, AntiVandalBot, Brian0324, RDT2, Tigeroo, Bjenks, Ghmyrtle, Somerset219, Skomorokh, Snarkypuppet, Natehal, Neowulf, Bshushan, Obhidhan, Apostrophyx, Plantigrade, WhatamIdoing, Cgingold, Valerius Tygart, Skylights76, B9 hum- mingbird hovering, Mabu2, Victor Blacus, Numbo3, Maurice Carbonaro, Ian.thomson, AntiSpamBot, TragicHipster, Nebukadje, S. Craig Wilson, A nowhere guy, Idioma-bot, ABF, TravellerDMT-07, James Callahan, Tesscass, TXiKiBoT, Zamphuor, Jkeene, BertSen, Ay- matth2, Ctmt, Mannafredo, Jjmckool, Arch ghost of yule past, AlleborgoBot, Punkyfunky, SieBot, StMichael71, Chinesearabs, Mimihi- tam, SimonTrew, OKBot, Bee Cliff River Slob, WickerGuy, Twinsday, Hoejamma, Piriczki, SamuelTheGhost, DragonBot, Alexbot, Van- isheduser12345, Rhododendrites, PhiRho, -guardian, Johnuniq, Editor2020, DumZiBoT, WonderingAngel-aesc78, XLinkBot, Aunt Entropy, Addbot, Juangelos, Elmondo21st, Fluffernutter, ForgetfulDoryFish, Elm, Luckas-bot, Eduen, AnomieBOT, Britishisles, Infor- mationtheory, Dendlai, Bob Burkhardt, ArthurBot, Lennim, Arj1981, DSisyphBot, AV3000, Omnipaedista, RibotBOT, Sift&Winnow, Grentworthy, Nantucketnoon, FrescoBot, VS6507, AriTotle, Haeinous, Mizanthrop, Jonesey95, Yahia.barie, Softarget, Jujutacular, Nedclark, Derrick.cooks, LilyKitty, WillNess, Nederlandse Leeuw, RjwilmsiBot, Noommos, EmausBot, Lebrouillard, Werieth, PBS- AWB, RationalKat22, Wikignome0530, Tobeprecise, RayneVanDunem, Mcc1789, EdoBot, Therewillbehotcake, ClueBot NG, BownCh- ingy, Catlemur, Humanist1859, Libertythought, Lynnettian, Dream of Nyx, Helpful Pixie Bot, BG19bot, Peteranderson6663, Ranikh, JohnChrysostom, Altaïr, Zock70, MisterMorton, BattyBot, Latheist, ChrisGualtieri, Mutoso, YFdyh-bot, Khazar2, Nathanielfirst, In- diandrum, Hmainsbot1, Quid-oportet-dixit, Blackman1000, Srdela, NewYorkerDean, Roboti Tung, SchizophrenicDingo, ColdNorth- Wind2, Schrauwers, Kanadkanhere, DangerouslyPersuasiveWriter, FreeThinkingBeing, EvergreenFir, Swagyolo420blzit, Monkbot, Zu- moarirodoka, Geoff cooper00, Cr1ms0nKitty, King of all fruit, Elmidae, Atho Phink, Acayl, Lux-hibou and Anonymous: 280 • Graphoid Source: https://en.wikipedia.org/wiki/Graphoid?oldid=677781423 Contributors: The Anome, Michael Hardy, Rjwilmsi, RJE42, Magioladitis, Kvihill, Yobot, John of Reading, BG19bot, MaryGaulke and Anonymous: 1 • Herbrandization Source: https://en.wikipedia.org/wiki/Herbrandization?oldid=439798612 Contributors: AshtonBenson, Bjones, Linas, SmackBot, MarshBot, Mere Interlocutor, MorganGreen and Anonymous: 2 • History of logic Source: https://en.wikipedia.org/wiki/History_of_logic?oldid=705754470 Contributors: LC~enwiki, Michael Hardy, Gabbe, Stephen C. Carlson, William M. Connolley, Pratyeka, Denny, Peter Damian (original account), Reddi, Stone, Markhurd, Tp- bradbury, Goethean, Tobias Bergemann, Giftlite, Everyking, 20040302, Siroxo, Junuxx, Kusunose, CSTAR, Alsocal, Hbmartin, D6, Freakofnurture, JTN, Rich Farmbrough, Wclark, Paul August, Bender235, Chalst, Liberatus, Jguk 2, PWilkinson, Snowolf, Velho, Mel Etitis, Linas, Mindmatrix, BD2412, Rjwilmsi, Ucucha, Margosbot~enwiki, Tedder, Bgwhite, YurikBot, Deeptrivia, Gaius Cornelius, Marklara, Aftermath, Grafen, MX44, Zagalejo, Tony1, MrSativa, Tomisti, Josh3580, Fram, Amalthea, SmackBot, Fanblade, Jagged 85, Mhss, Clconway, Henning Makholm, Clicketyclack, Ceoil, Ocanter, Bilby, Beetstra, SandyGeorgia, Iridescent, Mrdthree, CBM, TheTito, Myasuda, Gregbard, Cydebot, Doug Weller, DumbBOT, Malleus Fatuorum, Itsmejudith, Nick Number, AntiVandalBot, Luna Santin, Heysan, Doc Tropics, Itistoday, Hermel, Matthew Fennell, Appraiser, Ling.Nut, Baccyak4H, Seberle, CCS81, R'n'B, CommonsDelinker, Nev1, Cpiral, DarwinPeacock, DadaNeem, KD Tries Again, DorganBot, Steel1943, Tesscass, Ontoraul, Clarince63, The Tetrast, Philogo, Logan, Newbyguesses, SieBot, Kumioko (renamed), Svick, Dabomb87, 3rdAlcove, Francvs, Emptymountains, Classicalecon, Athenean, ClueBot, Nsk92, Singinglemon~enwiki, JustinClarkCasey, Hans Adler, BOTarate, Pichpich, Algebran, Addbot, Logicist, Ronhjones, Pe- ter Damian (old), Luckas-bot, Yobot, Fraggle81, AnomieBOT, Eumolpo, ChildofMidnight, Why isn't anyone watching the history of the logic?, Anna Frodesiak, Peter Damian, FrescoBot, Mfwitten, Citation bot 1, Tkuvho, Pinethicket, U8701, RedBot, Hriber, Generalboss3, Omkargokhale, Dewritech, Syncategoremata, RA0808, Here today, gone tomorrow, Outriggr, HistorianofLogic, Soni Ruchi, Logic His- torian, From the other side, Werieth, ZéroBot, H3llBot, Here for a bit, ClueBot NG, Wcherowi, Masssly, Widr, Australopithecus2, Helpful Pixie Bot, Calabe1992, DBigXray, BigEars42, BG19bot, CitationCleanerBot, Harizotoh9, Flosfa, MisterCake, Theconsequen- tialist, Qetuth, Ighso, Jochen Burghardt, Epicgenius, CsDix, Melody Lavender, Majo statt Senf, NABRASA, Oliveristhingone, Dick P Nuss, Monkbot, KH-1, Jebulon and Anonymous: 112 • HPO formalism Source: https://en.wikipedia.org/wiki/HPO_formalism?oldid=542220137 Contributors: Dspradau, Gregbard, FogDevil, StevenJohnston, II MusLiM HyBRiD II and Anonymous: 3 • Imperative logic Source: https://en.wikipedia.org/wiki/Imperative_logic?oldid=682365447 Contributors: CyborgTosser, BD2412, Gaius Cornelius, Sdorrance, Gregbard, Paradoctor, Fratrep, Thedarkfourth, Yobot, AnomieBOT, Lunchbox11, SporkBot, Tijfo098, ClueBot NG, Helpful Pixie Bot, Marfuas, WovenRoses22 and Anonymous: 7 • Inclusion (logic) Source: https://en.wikipedia.org/wiki/Inclusion_(logic)?oldid=609028697 Contributors: Michael Hardy, Jpbowen, Smack- Bot, Mathman1550, SMasters, Gregbard, KConWiki, Interchange88, Ironholds, Yobot and Here today, gone tomorrow • Index of logic articles Source: https://en.wikipedia.org/wiki/Index_of_logic_articles?oldid=693700744 Contributors: Michael Hardy, Docu, Nanobug, Andrewa, Александър, Whkoh, Charles Matthews, Dcoetzee, Greenrd, WhisperToMe, Ldo, Paul Klenk, Psychonaut, Wile E. Heresiarch, Filemon, Alan Liefting, Zigger, Barnaby dawson, Guppyfinsoup, Jim Henry, Silence, ZeroOne, Aranel, Chalst, Ricky81682, BD2412, Quiddity, Mathbot, N8cantor, Trovatore, Fram, SmackBot, Bluebot, Fplay, Syrcatbot, Sdorrance, Gregbard, Cy- debot, Gogo Dodo, Danger, The Transhumanist, N4nojohn, IdLoveOne, Squids and Chips, Funandtrvl, Ontoraul, Wiae, Maxim, Neparis, 145.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 547

Kumioko (renamed), NuclearWarfare, SchreiberBike, Verbal, AnomieBOT, Kibi78704, Gamewizard71, Hpvpp, LordJeff, UnUsedCon- doms, Tijfo098, Logicalgregory, Wbm1058, BG19bot, Mark Arsten and Anonymous: 10 • Informal logic Source: https://en.wikipedia.org/wiki/Informal_logic?oldid=674012098 Contributors: Chris Q, DennisDaniels, Kwertii, Pcb21, Docu, Julesd, Dysprosia, Markhurd, Furrykef, Hyacinth, Tomchiukc, Leonard G., CSTAR, Paulscrawl, Guppyfinsoup, Chalst, Femto, A.t.bruland, Velho, Simetrical, Mindmatrix, BD2412, Rjwilmsi, Gurch, Mattpeck, Tsch81, NawlinWiki, NostinAdrek, Closed- mouth, GraemeL, SmackBot, Stev0, Bluebot, Colonies Chris, Jon Awbrey, Byelf2007, Grumpyyoungman01, Meco, JohnsonRalph, Moreschi, Gregbard, Kpossin, Steel, Al Lemos, Escarbot, Doremítzwr, WinBot, Deeplogic, Clan-destine, Cathalwoods, Gomm, Gwern, R'n'B, RickardV, Adedayoojo, Homo logos, Nburden, Igglebop, Guillaume2303, Ontoraul, GlassFET, Kumioko (renamed), ClueBot, Napzilla, Staticshakedown, Addbot, SpBot, Lightbot, Luckas-bot, Luce nordica, AnomieBOT, LilHelpa, Omnipaedista, Gerald Roark, FrescoBot, LucienBOT, Machine Elf 1735, Abductive, Wotnow, John of Reading, ZéroBot, Tijfo098, ClueBot NG, Helpful Pixie Bot, GabeIglesia and Anonymous: 38 • Inquiry Source: https://en.wikipedia.org/wiki/Inquiry?oldid=707602087 Contributors: Michael Hardy, Paul A, Glenn, Hyacinth, Andy- cjp, Kusunose, Reinthal, Rich Farmbrough, El C, Jhertel, Arthena, Versageek, Oleg Alexandrov, Velho, Jeffrey O. Gustafson, DePiep, DoubleBlue, FlaBot, Gurch, DragonHawk, Trovatore, DeadEyeArrow, Closedmouth, Reyk, NielsenGW, John Broughton, SmackBot, Michael%Sappir, C.Fred, Wittylama, HalfShadow, Mhss, Mitsuhirato, Jon Awbrey, JzG, 16@r, Slakr, RichardF, Gregbard, Gogo Dodo, Julian Mendez, Letranova, Escarbot, JAnDbot, Robina Fox, Hut 8.5, VoABot II, Brigit Zilwaukee, Yolanda Zilwaukee, J.delanoy, Ig- natzmice, Izno, Pasixxxx, Bovineboy2008, The Tetrast, GlobeGores, LeaveSleaves, Maxim, Jesin, Graymornings, DancesWithGrues, Newbyguesses, France3470, Oda Mari, Aboluay, ClueBot, Jbening, The Thing That Should Not Be, Rjd0060, Victor Dahlmann, Hans Adler, Buchanan's Navy Sec, Overstay, Marsboat, Calor, Versus22, Unco Guid, Viva La Information Revolution!, Autocratic Uzbek, Poke Salat Annie, Flower Mound Belle, Ps07swt, Navy Pierre, Mrs. Lovett's Meat Puppets, Chester County Dude, Southeast Penna ,Washburnmav, KamikazeBot, Queen of the Dishpan, McKaot ,דוד שי ,Poppa, Delaware Valley Girl, Libcub, Addbot, Mabdul, Mootros AnomieBOT, Rubinbot, Materialscientist, TheAMmollusc, Urbansuperstar~enwiki, Stefanson, Aaron Kauppi, FrescoBot, Cass Opolis, Dinamik-bot, WTM, AnselmiJuan, Cit helper, ChuispastonBot, Ratpow, Petrb, ClueBot NG, Movses-bot, Cntras, Masssly, MerlIwBot, PhnomPencil, Gorthian, Cyberbot II, RU123, Mr. Guye, SteenthIWbot, Biogeographist, Shauryagandhi19, Emedoh, Spirit Ethanol and Anonymous: 51 • Intension Source: https://en.wikipedia.org/wiki/Intension?oldid=704946232 Contributors: Mav, Andre Engels, Hari, Toby Bartels, Ryguasu, Michael Hardy, Ducker, Murijja, Kimiko, Fuzheado, Gandalf61, Bfinn, Lucidish, STGM, ·~enwiki, Solemnavalanche, Caesura, Oleg Alexandrov, Kelly Martin, Fefifofum~enwiki, Salix alba, RussBot, Welsh, Tomisti, Eaefremov, SmackBot, InverseHypercube, Peter Iso- talo, Mhss, Hraefen, Gregwmay, Jon Awbrey, Metamagician3000, Byelf2007, Physis, David Legrand, Lenoxus, Jordanotto, Gregbard, Thijs!bot, HappyInGeneral, DanielLevitin, Mr pand, Eleuther, Jirka6, Nopira, Way of Inquiry, R'n'B, Edvard Munchkin, Tdbreaux, Insan- ity Incarnate, Moonriddengirl, Adam Francis Carr, Mild Bill Hiccup, CohesionBot, Jaime Saldarriaga, SchreiberBike, Addbot, Lightbot, Denispir, AnomieBOT, Omnipaedista, Louperibot, Winterst, CircularReason, PPdd, Neil P. Quinn, Ferndias, Helpful Pixie Bot, Pacerier, Mark viking, Vieque and Anonymous: 22 • International Wittgenstein Symposium Source: https://en.wikipedia.org/wiki/International_Wittgenstein_Symposium?oldid=666691026 Contributors: Markhurd, Gregbard, John of Reading, ArticlesForCreationBot, ChrisGualtieri, FoCuSandLeArN, I am One of Many and Anonymous: 3 • Inverse consequences Source: https://en.wikipedia.org/wiki/Inverse_consequences?oldid=681561091 Contributors: Charles Matthews, Epipelagic, Cydebot, Wikid77, Tomsega, Andrewaskew, Jncraton, Yobot and Meclee • Language, Proof and Logic Source: https://en.wikipedia.org/wiki/Language%2C_Proof_and_Logic?oldid=679312048 Contributors: Giraffedata, PWilkinson, BeeJay~enwiki, RHaworth, Stefanomione, RadioFan, Tom Morris, Gregbard, Katharineamy, Squids and Chips, Phil Bridger, Yobot, Rhalah, GoingBatty, AvicBot, Yodamgod, ClueBot NG, Dexbot and Anonymous: 2 • Lexical definition Source: https://en.wikipedia.org/wiki/Lexical_definition?oldid=660024352 Contributors: Hyacinth, Wolfkeeper, Mind- spillage, Szyslak, Jag123, Scope creep, SmackBot, Byelf2007, Igoldste, CmdrObot, Gogo Dodo, Klausness, Dafyddg, Adriansrfr, Addbot, Rey Productions, Jim1138, Erik9bot, TobeBot, Emembergin, Wikipelli, ChrisGualtieri, Isarra (HG), Vrdennisj and Anonymous: 11 • List of paradoxes Source: https://en.wikipedia.org/wiki/List_of_paradoxes?oldid=708239276 Contributors: The Anome, Ed Poor, Michael Hardy, Chinju, Karada, Cyan, Dcoetzee, Markhurd, Hyacinth, Shantavira, Henrygb, Blainster, Mattflaschen, Giftlite, Dben- benn, Gene Ward Smith, Subsolar, ALE!, Quarl, DragonflySixtyseven, Indolering, Robin klein, Salem, Eliazar, Mike Rosoft, RossPatter- son, Discospinster, Florian Blaschke, Cagliost, Bender235, Andrejj, Rgdboer, Root4(one), Alderbourne, Cretog8, Reinyday, Teorth, Jemfinch, Free Bear, Rgclegg, Batmanand, Mysdaao, Joris Gillis, Kgashok, Sciurinæ, Spambit, Tiger Khan, Kay Dekker, Firsfron, Jeffrey O. Gustafson, Justavo, Igny, Jeff3000, Mangojuice, Btyner, Joerg Kurt Wegner, Mandarax, IIBewegung, BD2412, Melesse, Sjakkalle, Koavf, Urbane Legend, Trlovejoy, Salix alba, The wub, FayssalF, Spencerk, DVdm, Whosasking, Wavelength, Jimp, Shawn81, KSchutte, Anomalocaris, Rohitbd, Trovatore, Schlafly, JocK, Retired username, Davilla~enwiki, Frogular, Syrthiss, Nlu, SMcCandlish, Reyk, Caballero1967, The 13th 4postle~enwiki, Serendipodous, AndrewWTaylor, Snalwibma, Brizimm, SmackBot, WilliamThweatt, Pokipsy76, Eternal th33v, Kopaka649, Skizzik, Afa86, Amatulic, Jprg1966, Liamdaly620, Ted87, Colonies Chris, Emurphy42, Scw- long, Can't sleep, clown will eat me, GeorgeMoney, Korako, Thewebb, Charles Merriam, Cubbi, MathStatWoman, DMacks, Byelf2007, Dollyknot, Lambiam, Attys, Sunik.lee, Rwald, STyx, Teutanic, Loadmaster, Makyen, Cstella23, Ambuj.Saxena, Novangelis, Cmbalin, Darry2385, White Ash, Team.chaotix, Skorpion87, Lage~enwiki, Scarlet Lioness, Tauʻolunga, Winston Spencer, Leon01323, INkubusse, CRGreathouse, Aaronak, Bogdan Preunca, BeenAroundAWhile, Gregbard, CumbiaDude, Grahamrichter, Reywas92, ST47, Michael C Price, DumbBOT, Chrislk02, Eubulide, Epbr123, Qwyrxian, LeeG, Ucanlookitup, N5iln, Why1991, Leon7, D.H, Michas pi, Daimanta, Widefox, Mashiah Davidson, Billyoneal, Storkk, Somerset219, Acroterion, VoABot II, Kuyabribri, Yakushima, Arno Matthias, Albmont, Eroshiyda, Swpb, Baccyak4H, Animum, Chris G, Agamemnon117, DerHexer, QuasiChameleon, Sarathklal, Lunakeet, Otvaltak, Stephen- chou0722, MartinBot, AntiochCollege, Jigsy, D2B, EagleEyes, R'n'B, Hilltoppers, HEL, Brix., J.delanoy, AstroHurricane001, Bogey97, MistyMorn, Peytonbland, Necropedal, Fartknocker, Gmazeroff, Brassmouth, Tokyogirl79, Sao123, Katharineamy, Trumpet marietta 45750, BryanDavis, Gemena, Jake Rilko, Mach Boy, Robertgreer, Action Jackson IV, LakewoodBrian, Ja 62, Soccerman90, Bennerhingl, Hammersoft, VolkovBot, DrDentz, Geoffw1948, Philip Trueman, Blahber, Jhon montes24, Oshwah, Cosmic Latte, Robinson weijman, Hqb, Carlangas, Anonymous Dissident, Karmos, Feldgrau~enwiki, Konamiuss, Feong changer, Martin451, Don4of4, UncleZeiv, Aaron Rotenberg, Jackfork, Jmc6171, Wiae, Falcon8765, @pple, Sylent, Seresin, Asdfazerty, Bensondaled, Jr.lizardking, Why Not A Duck, Dmcq, AlleborgoBot, Neparis, Littleherby2412, Pdfpdf, Eminence Grise, Defender 911, Ishvara7, PlanetStar, Paradoctor, TriBulated, Dawn Bard, Dryfee, Yintan, Soler97, Flyer22 Reborn, Thehotelambush, AlexHOUSE, ClassA42, TwoTeasChris, Eouw0o83hf, Kortag- gio, RQJ, ClueBot, Loginks, Dominiquedoandd, Sammmttt, Julianhall, Wysprgr2005, Drmies, Der Golem, Mild Bill Hiccup, HovisM, 548 CHAPTER 145. ZHEGALKIN POLYNOMIAL

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• Logic Source: https://en.wikipedia.org/wiki/Logic?oldid=708210270 Contributors: AxelBoldt, Vicki Rosenzweig, The Anome, Toby Bartels, Ryguasu, Hirzel, Dwheeler, Stevertigo, Edward, Patrick, Chas zzz brown, Michael Hardy, Lexor, TakuyaMurata, Bagpuss, Looxix~enwiki, Ahoerstemeier, Notheruser, BigFatBuddha, Александър, Glenn, Marco Krohn, Rossami, Tim Retout, Rotem Dan, Evercat, EdH, DesertSteve, Caffelice~enwiki, Mxn, Michael Voytinsky, Peter Damian (original account), Rzach, Charles Matthews, Dcoetzee, Paul Stansifer, Dysprosia, Jitse Niesen, Xiaodai~enwiki, Markhurd, MikeS, Carol Fenijn, SEWilco, Samsara, J D, Shizhao, Olathe, Jusjih, Ldo, Banno, Chuunen Baka, Robbot, Iwpg, Fredrik, R3m0t, Altenmann, MathMartin, Rorro, Rholton, Saforrest, Borislav, Robertoalencar, Michael Snow, Raeky, Guy Peters, Jooler, Filemon, Ancheta Wis, Exploding Boy, Giftlite, Recentchanges, Inter, Wolf- keeper, Lee J Haywood, COMPATT, Everyking, Rookkey, Malyctenar, Andris, Bovlb, Jason Quinn, Sundar, 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alba, Crazynas, Ligulem, Bary- onic Being, Titoxd, FlaBot, Kwhittingham, Latka, Mathbot, Twipley, SportsMaster, RexNL, AndriuZ, Quuxplusone, Celendin, Influ- ence, R Lee E, JegaPRIME, Malhonen, Spencerk, Chobot, DVdm, EamonnPKeane, Roboto de Ajvol, Wavelength, Deeptrivia, KSmrq, Endgame~enwiki, Polyvios, CambridgeBayWeather, KSchutte, NawlinWiki, Rick Norwood, SEWilcoBot, Mipadi, Brimstone~enwiki, LaszloWalrus, AJHalliwell, Trovatore, Pontifexmaximus, Chunky Rice, Cleared as filed, Nick, Darkfred, Wjwma, Googl, Mendicott, StuRat, Open2universe, ChrisGriswold, Nikkimaria, OEMCUST, Nahaj, Extreme Unction, Allens, Sardanaphalus, Johndc, Smack- Bot, Lestrade, InverseHypercube, Pschelden, Jim62sch, Jagged 85, WookieInHeat, Josephprymak, Timotheus Canens, Srnec, Lone- someDrifter, Collingsworth, Gilliam, Skizzik, RichardClarke, Heliostellar, Chris the speller, Jaymay, Da nuke, Unbreakable MJ, MK8, Andrew Parodi, Kevin Hanse, MalafayaBot, Clconway, Sciyoshi~enwiki, Go for it!, Mikker, Zsinj, Can't sleep, clown will eat me, Misg- nomer, Grover cleveland, Fuhghettaboutit, Cybercobra, Nakon, Jiddisch~enwiki, Richard001, MEJ119, Kabain52, Lacatosias, Jon Aw- brey, DMacks, Henning Makholm, Ged UK, Ceoil, Byelf2007, SashatoBot, Lambiam, Dbtfz, Deaconse, UberCryxic, FrozenMan, Heim- stern, Shlomke, Shadowlynk, F15 sanitizing eagle, Prince153, WithstyleCMC, Hvn0413, Meco, RichardF, Novangelis, Vagary, Pam- plmoose, KJS77, Hu12, Levineps, BananaFiend, K, Lottamiata, Catherineyronwode, Mrdthree, Igoldste, Themanofnines, Adambiswanger1, Satarnion, Tawkerbot2, Galex, SkyWalker, CRGreathouse, CBM, Editorius, Rubberchix, Gregbard, Kpossin, Cydebot, [email protected], Jasperdoomen, Samuell, Quinnculver, Peterdjones, Travelbird, Pv2b, Drksl, JamesLucas, Julian Mendez, Dancter, Tawkerbot4, Shiru- lashem, Doug Weller, DumbBOT, Garik, Progicnet, Mattisse, Letranova, Thijs!bot, Epbr123, Kredal, Smee, Marek69, AgentPepper- mint, OrenBochman, Dawnseeker2000, Escarbot, Eleuther, Mentifisto, Vafthrudnir, AntiVandalBot, Peoppenheimer, Majorly, Gioto, Hidayat ullah, GeePriest, Dougher, Sluzzelin, JAnDbot, Narssarssuaq, MER-C, The Transhumanist, Avaya1, Zizon, Frankie816, Sa- vant13, Dr mindbender, LittleOldMe, Bongwarrior, VoABot II, SDas, JNW, Arno Matthias, Appraiser, Gamkiller, Smihael, Caesarjb- squitti, Midgrid, Bubba hotep, Moopiefoof, GeorgeFThomson, Virtlink, David Eppstein, Epsilon0, DerHexer, Waninge, Exbuzz, Mart- inBot, Wylve, CommonsDelinker, EdBever, C.R.Selvakumar, J.delanoy, Trusilver, Jbessie, Fictionpuss, Cpiral, RJMalko, McSly, Light- est~enwiki, Classicalsubjects, Mrg3105, Daniel5Ko, The Transhumanist (AWB), Policron, MetsFan76, Kenneth M Burke, Steel1943, Idioma-bot, Spellcast, WraithM, VolkovBot, Cireshoe, Rucha58, Macedonian, Hotfeba, Indubitably, Fundamental metric tensor, Jim- maths, Djhmoore, Aesopos, Rei-bot, Llamabr, Ontoraul, Philogo, Leafyplant, Sanfranman59, Abdullais4u, Cullowheean, Wiae, Maxim, Myscience, LIBLAHLIBLAHTIMMAH, Synthebot, Rurik3, Koolo, Nagy, Symane, PGWG, W4chris, Prom2008, NHRHS2010, Rada- gast3, Demmy, JonnyJD, Newbyguesses, Linguist1, SieBot, StAnselm, Maurauth, Gerakibot, RJaguar3, Yintan, Bjrslogii, Soler97, Til Eulenspiegel, Flyer22 Reborn, DanEdmonds, Undead Herle King, Crowstar, Redmarkviolinist, Spinethetic, Thelogicthinker, Dancing- Philosopher, Svick, Valeria.depaiva, Adhawk, Sginc, Tognopop, CBM2, 3rdAlcove, PsyberS, Francvs, Classicalecon, Athenean, Mr. Granger, Atif.t2, Martarius, ClueBot, Andrew Nutter, Snigbrook, The Thing That Should Not Be, Taroaldo, Ukabia, TheOldJacobite, Boing! said Zebedee, Niceguyedc, Blanchardb, DragonBot, Jessieslame, Excirial, Alexbot, Jusdafax, Watchduck, AENAON, Nucle- arWarfare, Arjayay, SchreiberBike, Thingg, JDPhD, Scalhotrod, Budelberger, Skunkboy74, Gerhardvalentin, Duncan, Saeed.Veradi, Mcgauley08, NellieBly, Noctibus, Aunt Entropy, Jjfuller123, Spidz, Addbot, Rdanneskjold, Proofreader77, Atethnekos, Sully111, Logi- cist, Vitruvius3, Rchard2scout, Glane23, Uber WoMensch!, Chzz, Favonian, LinkFA-Bot, AgadaUrbanit, Numbo3-bot, Ehrenkater, Tide rolls, Lightbot, Macro Shell, Zorrobot, Jarble, JEN9841, Aarsalankhalid, GorgeUbuasha, Yobot, Arcvirgos 08, Jammie101, Francos22, 145.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 549

Azcolvin429, MassimoAr, AnomieBOT, Jim1138, IRP, AdjustShift, Melune, NickK, Neurolysis, ArthurBot, Gemtpm, Blueeyedbomb- shell, Junho7391, Xqbot, Gilo1969, The Land Surveyor, Tyrol5, A157247, Petropoxy (Lithoderm Proxy), Uarrin, GrouchoBot, Hifcelik, ,Tales23, GhalyBot, Aaron Kauppi, GliderMaven, FrescoBot, Liridon, D'Artagnol, Tobby72 ,, مهدی جمشیدی78.26פריימן, ,Omnipaedista D'ohBot, Mewulwe, Itisnotme, Cannolis, Rhalah, Citation bot 1, Chenopodiaceous, AstaBOTh15, Gus the mouse, Pinethicket, Vicenar- ian, A8UDI, Ninjasaves, Seryred123, Wikiain, Maokart444, Gamewizard71, FoxBot, TobeBot, Burritoburritoburrito, Mysticcooperfox, ,Merlinsorca, Literateur, Jarpup, Whisky drinker, Mean as custard, Rlnewma ,בן גרשון ,Lotje, GregKaye, Vistascan, Vrenator, Duoduoduo TjBot, Walkinxyz, EmausBot, Orphan Wiki, The Kytan Apprentice, Pologic, Faolin42, Jedstamas, Wham Bam Rock II, Solarra, ZéroBot, Leminh91, Josve05a, Shuipzv3, Mar4d, Wayne Slam, Frigotoni, Rcsprinter123, FrankFlanagan, L Kensington, Danmuz, Eranderson, Donner60, Chewings72, Puffin, GKaczinsky, ChuispastonBot, NTox, Poopnubblet, Xanchester, Rememberway, ClueBot NG, W.Kaleem, Jack Greenmaven, SusikMkr, Quantamflux, Validlessness, Bastianperrot, Wdchk, Snotbot, Masssly, Widr, Lawsonstu, ESL75, Helpful Pixie Bot, Anav2221, BG19bot, Daniel Zsenits, Norma Romm, PTJoshua, Northamerica1000, Graham11, Geegeeg, JohnChrysostom, Frze, Chjohnson39, Alex.Ramek, CJMacalister, CitationCleanerBot, Jilliandivine, Flosfa, Chrisct1993, Brad7777, Lrq3000, Mewhho18, A.coolmcfly, Compulogger, Cyberbot II, Roger Smalling, The Illusive Man, NanishaOpaenyak, Rhlozier, EagerToddler39, Dexbot, Mar- ius siuram, Табалдыев Ысламбек, Omanchandy007, RideLightning, Jochen Burghardt, Wieldthespade, Hippocamp, Wickid123, Mat- ,Tentinator, EvergreenFir, Babitaarora, Ugog Nizdast ,גלעד ניר ,ticusmadness, JMCF125, NIXONDIXON, CsDix, I am One of Many Melody Lavender, JustBerry, Skansi.sandro, Ginsuloft, Robf00f1235, Calvinator8, The Annoyed Logician, Liz, GreyWinterOwl, By- Dash, Jbob13, Henniepenny, Matthew Derick B Cruz, Filedelinkerbot, Sherlock502, Fvdedphill, Norwo037, Karnaoui, Pat132, The Expedia, Sbcdave, Muneeb Masoud, Jacksplay, Asdklf;, Esicam, ChamithN, Ntuser123, Cthulhu is love cthulhu is life, Loraof, Juliet- deltalima, Adamrobson28, Josmust222, Rubbish computer, Layfi, KcBessy, SamiLayfi, Lanzdsey, SoSivr, Human3015, ZanderEdmunds, KasparBot, Bestusername-ign, Sparky Macgillicuddy, Mithisharma, MindForgedManacle, Citation requested but not required, Nyetoson, SaundersLane, TBNRGiazo, Spirit Ethanol, Harmon758, Isuredid, Rachel Benedict and Anonymous: 776 • Logic of Argumentation Source: https://en.wikipedia.org/wiki/Logic_of_Argumentation?oldid=680600421 Contributors: Stuartyeates, MenoBot, Rankersbo, Anupmehra, BattyBot, ChrisGualtieri, Jamesmcmahon0, PhilipJudson and Usa60527 • Logic of class Source: https://en.wikipedia.org/wiki/Logic_of_class?oldid=641407033 Contributors: Brianhe, TheTito, Thijs!bot, Ale- jandrocaro35, Addbot, Luckas-bot, Yobot, Materialscientist, Xqbot, EatsShootsAndLeaves, LucienBOT, EdoBot, Masssly, As for the bucket, Blwething, Nerdfighter1302, Faehjfle and Anonymous: 2 • Logic of information Source: https://en.wikipedia.org/wiki/Logic_of_information?oldid=700874354 Contributors: Goethean, Paul Au- gust, El C, Versageek, Jeffrey O. Gustafson, SmackBot, Incnis Mrsi, C.Fred, Jon Awbrey, JzG, Coredesat, Slakr, Xinyu, Floridi~enwiki, KnightLago, Gregbard, Maria Vargas, Gogo Dodo, Hut 8.5, Yolanda Zilwaukee, The Tetrast, Seb26, Coffee, Flyer22 Reborn, Maelgwn- bot, Fadesga, Rjd0060, Overstay, Marsboat, 1ForTheMoney, Unco Guid, Viva La Information Revolution!, Autocratic Uzbek, Poke Salat Annie, Flower Mound Belle, Navy Pierre, Mrs. Lovett's Meat Puppets, Chester County Dude, Southeast Penna Poppa, Delaware Valley Girl, Iceblock, Petrb, ClueBot NG, Masssly, Me, Myself, and I are Here, 314Username and Anonymous: 8 • Logical abacus Source: https://en.wikipedia.org/wiki/Logical_abacus?oldid=646766518 Contributors: Tobias Hoevekamp, Bryan Derk- sen, Charles Matthews, Dysprosia, Rich Farmbrough, PWilkinson, BD2412, Tktktk, Gregbard, MER-C, David Eppstein, Commons- Delinker, Robertgreer, Lights, Coffee, Vrsane, Hans Adler, Thehelpfulbot, PBS-AWB, Clcindia, Jbustamante-NJITWill, Joefromrandb, Masssly, Solomon7968, Jochen Burghardt, DavidLeighEllis and Anonymous: 9 • Logical determinism Source: https://en.wikipedia.org/wiki/Logical_determinism?oldid=670542878 Contributors: Yobot and Peter Damian • Logical extreme Source: https://en.wikipedia.org/wiki/Logical_extreme?oldid=633551383 Contributors: AlexR, Skysmith, Altenmann, Diderot, MSGJ, Florian Blaschke, Tene, DanMS, Synaptidude, Nate1481, SmackBot, Mdd4696, Yaf, Ultraviolent, Alaibot, Ryanl91, JL- Bot, Addbot, Yobot, LilHelpa, Erik9bot, Machine Elf 1735, GoingBatty, Masssly, Meatsgains, Don of Cherry, DoctorKubla, Michipedian, TheTallSomething and Anonymous: 5 • Logical harmony Source: https://en.wikipedia.org/wiki/Logical_harmony?oldid=708401153 Contributors: AugPi, Jason Quinn, Lu- cidish, Chalst, Luke stebbing, Bookandcoffee, Qwertyus, Kjlewis, Neil Leslie, Mhss, Bluebot, Lambiam, Disambiguator, Qwfp, Addbot, HRoestBot, ChuispastonBot, Frietjes, Masssly and Anonymous: 3 • Logical pluralism Source: https://en.wikipedia.org/wiki/Logical_pluralism?oldid=700972092 Contributors: Kzollman • Loosely associated statements Source: https://en.wikipedia.org/wiki/Loosely_associated_statements?oldid=592910308 Contributors: Gregbard, Miracle Pen, Saedon and Anonymous: 1 • Markov's principle Source: https://en.wikipedia.org/wiki/Markov'{}s_principle?oldid=690804104 Contributors: Michael Hardy, EmilJ, AshtonBenson, Btyner, Hairy Dude, SmackBot, Lambiam, Gregbard, Cydebot, Hqb, Classicalecon, Hugo Herbelin, Addbot, Yobot, ZéroBot and Anonymous: 11 • Metamathematics Source: https://en.wikipedia.org/wiki/Metamathematics?oldid=707308391 Contributors: Toby Bartels, Ams80, Charles Matthews, Dysprosia, Jitse Niesen, Hyacinth, Fredrik, MathMartin, Almit39, Leibniz, Bender235, Ardric47, CuriousOne, Eric Herboso, Oleg Alexandrov, Justavo, Guardian of Light, Marudubshinki, Dpv, Somesh, FlaBot, Mathbot, Margosbot~enwiki, DVdm, Ksyrie, Arthur Rubin, Sardanaphalus, SmackBot, Adam M. Gadomski, Xyzzyplugh, Byelf2007, Lambiam, Metric, Mets501, Vanisaac, CBM, Gregbard, MDE, M a s, Thijs!bot, Husond, Maurice Carbonaro, Hair Commodore, Anonymous Dissident, JhsBot, Sapphic, ArdClose, Editor2020, Addbot, Dyaa, Lightbot, Kiril Simeonovski, Luckas-bot, AnomieBOT, Bci2, Crzer07, The Wiki ghost, Nikiriy, TobeBot, Wikitanvir- Bot, 4meter4, ZéroBot, ChuispastonBot, Kasirbot, Paolo Lipparini, Brirush, Hamerbro, Echoet, Suelru, Cosmia Nebula, WillemienH, KasparBot and Anonymous: 29 • Multiple-conclusion logic Source: https://en.wikipedia.org/wiki/Multiple-conclusion_logic?oldid=405064210 Contributors: BD2412, Salix alba, Rick Norwood, SmackBot, Dbtfz, CBM and Westerdundrun • Mutual exclusivity Source: https://en.wikipedia.org/wiki/Mutual_exclusivity?oldid=686535727 Contributors: Vicki Rosenzweig, Zun- dark, Michael Hardy, Kku, Gaurav, Docu, Quickbeam, Charles Matthews, Dysprosia, Tobias Bergemann, Giftlite, Neilc, Bender235, An- thony Appleyard, Ricky81682, Angelic Wraith, Oleg Alexandrov, Bluemoose, Yuriybrisk, Qwertyus, Grammarbot, Concordia, ACrush, Godlord2, Roboto de Ajvol, Capitalist, Sardanaphalus, SmackBot, NYKevin, Cybercobra, Rory096, Wvbailey, SpyMagician, Levineps, Courcelles, Gregbard, Yaris678, Mazzid, Mattisse, Neil916, Daniel.kho, Richard n, Martinkunev, Magioladitis, Jotate, Anarchia, Ret- tetast, The Anonymous One, AstroHurricane001, Maurice Carbonaro, Neon white, AntiSpamBot, Alnokta, Lights, Epson291, Technopat, Salvar, Wiae, Burntsauce, SieBot, Puffin1234567, Happysailor, Leranedo, The Thing That Should Not Be, HexaChord, Addbot, Aykantspel, Ccacsmss, Darth Cracker, SpBot, Legobot, Fraggle81, AnomieBOT, Iamatom, Xqbot, Koolkyle123, Bsquare4ac, Duoduoduo, Side- ways713, Keegscee, Czhanacek, Bento00, Tesseract2, Furtheraptitudes, ZéroBot, Chharvey, Δ, ClueBot NG, Primergrey, Congresser, Karl 334, Vvlaura, Kmraghu61, John Aiello, Alarty, SDG23uas, Vorapre, Amitakiwate and Anonymous: 122 550 CHAPTER 145. ZHEGALKIN POLYNOMIAL

• Mutual knowledge (logic) Source: https://en.wikipedia.org/wiki/Mutual_knowledge_(logic)?oldid=692158293 Contributors: Waldir, Bgwhite, RobertHannah89, Yobot, VoiceOfTheCommons and Anonymous: 3 • Münchhausen trilemma Source: https://en.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma?oldid=703880545 Contributors: The Anome, Kwertii, Llywrch, Ec5618, Rursus, Luis Dantas, Bfinn, Dmmaus, Rich Farmbrough, Dbachmann, Diego Moya, Keenan Pep- per, Snowolf, Kazvorpal, Lawrence King, BD2412, Qwertyus, Koavf, TimSC, Cholmes75, Bob Hu, Tom Morris, SmackBot, Rtc, Scot- tForschler, H-J-Niemann, Cybercobra, Leon..., Byelf2007, AnonEMouse, Loadmaster, Kujeger, Connection, CRGreathouse, ShelfSkewed, Gregbard, Jasperdoomen, Miguel de Servet, Concept14, Artawiki, TomS TDotO, Signalhead, Broadbot, Weetjesman, Alcmaeonid, Dan Polansky, LeadSongDog, Vanished user oij8h435jweih3, Martarius, Singinglemon~enwiki, Jusdafax, Spirals31, Hans Adler, Yurizuki, XLinkBot, Addbot, Luckas-bot, KamikazeBot, AnomieBOT, Rubinbot, E235, Machine Elf 1735, DrilBot, Skyerise, Tesseract2, Emaus- Bot, WikitanvirBot, ZéroBot, The Nut, Ὁ οἶστρος, LezheLady, ClueBot NG, Lemuellio, Curb Chain, Wbm1058, Kanghuitari, Me, Myself, and I are Here, Rfassbind, Alexwho314, MSZ372, Fpetillo and Anonymous: 44 • Natural kind Source: https://en.wikipedia.org/wiki/Natural_kind?oldid=707932782 Contributors: Zundark, Charles Matthews, Hy- acinth, Sunray, Karol Langner, Martpol, Longhair, Smalljim, Cyberchimp, CHE~enwiki, Tomisti, Closedmouth, SmackBot, Betacom- mand, Bluebot, Richard001, ArglebargleIV, Gnome (Bot), Murzim, Gregbard, Nick.wiebe, Jdvelasc, M Payne, Danny lost, Crunk- car, Mkbergman, Twinsday, Chabalala, Hans Adler, Aquillyne, Bus Bax, SchreiberBike, Addbot, Redheylin, Fraggle81, AnomieBOT, FrescoBot, Machine Elf 1735, Lotje, EmausBot, Budpowellduet, Brian Tomasik, DavidHunterWalsh, Cyberbot II, Mdpacer, Jochen Burghardt, Monkbot and Anonymous: 24 • Neutrality (philosophy) Source: https://en.wikipedia.org/wiki/Neutrality_(philosophy)?oldid=697517894 Contributors: Jayjg, Woohookitty, Pol098, Kbdank71, Wavelength, Emmanuelm, NawlinWiki, SmackBot, Aeternus, Penbat, Gregbard, Doug Weller, Magioladitis, Ap- praiser, Grantsky, Shawn in Montreal, UnitedStatesian, Haseo9999, Metadat, SieBot, StAnselm, Rhododendrites, NJGW, Jytdog, Dawynn, AnomieBOT, MercuryApex, Omnipaedista, EmausBot, KernSibbald, Diamondland, ClueBot NG, Hillbillyholiday, BreakfastJr, Trysty- nAlxander and Anonymous: 16 • Non-monotonic logic Source: https://en.wikipedia.org/wiki/Non-monotonic_logic?oldid=665286331 Contributors: Seth Ilys, Ancheta Wis, Giftlite, Karol Langner, Shib71, Aranel, Vanished user lp09qa86ft, Flammifer, Arthena, Mindmatrix, Wdyoung, Kzollman, Ruud Koot, Tizio, Banazir, ColdFeet, YurikBot, TexasAndroid, Nfm, NotInventedHere, TechPurism, Dbtfz, Futurerustic, CRGreathouse, Simeon, Gregbard, Woland37, Thenub314, LittleOldMe old, Una Smith, Jamelan, Andrewaskew, JonnyJD, Kumioko (renamed), Alge- bran, Addbot, Luckas-bot, Yobot, 4th-otaku, I dream of horses, Tijfo098, ClueBot NG, Jspar2234 and Anonymous: 28 • Nonfirstorderizability Source: https://en.wikipedia.org/wiki/Nonfirstorderizability?oldid=525081734 Contributors: The Anome, Michael Hardy, P0lyglut, Ben Standeven, Nortexoid, SmackBot, MrDrBob, Dbtfz, CBM, Asenine, DOI bot, Citation bot, Omnipaedista, Bride- OfKripkenstein, Citation bot 1, Tkuvho, Tijfo098 and Anonymous: 6 • Normal form (natural deduction) Source: https://en.wikipedia.org/wiki/Normal_form_(natural_deduction)?oldid=532304588 Con- tributors: Paul August, Chalst, RxS, Hairy Dude, SmackBot, Addbot, Piano non troppo, Erik9bot, ClueBot NG and Anonymous: 2 • Object of the mind Source: https://en.wikipedia.org/wiki/Object_of_the_mind?oldid=599331603 Contributors: William M. Connol- ley, Greenrd, Banno, Mdd, Wtshymanski, Oleg Alexandrov, Woohookitty, SDC, Marudubshinki, Dpv, Planetneutral, RussBot, Welsh, Tomisti, Pegship, Fang Aili, SmackBot, Melchoir, Bradtcordeiro, Oneismany, Bluebot, Doczilla, ChrisCork, Gregbard, Cydebot, Peter- djones, Mattisse, Barticus88, Magioladitis, DGG, Shay Guy, Calen11, Dlabtot, CohesionBot, ZuluPapa5, Addbot, LilHelpa, Gondwana- banana, Aaron Kauppi, Wawawemn, Adi4094, FiachraByrne, Gibbja, Hmainsbot1 and Anonymous: 2 • Ontological commitment Source: https://en.wikipedia.org/wiki/Ontological_commitment?oldid=678991514 Contributors: The Anome, Michael Hardy, EvanProdromou, Peter Damian (original account), Charles Matthews, Goethean, Daniel Brockman, Siroxo, Creidieki, Kenb215, PWilkinson, BD2412, Rjwilmsi, Tangotango, MZMcBride, Pacaro, That Guy, From That Show!, Bluebot, Gregbard, An- drewHowse, Pascal.Tesson, Danny lost, Snowded, Reedy Bot, Ontoraul, Brews ohare, Hans Adler, Yobot, AnomieBOT, J04n, FrescoBot, Howdood, Tyrannus Mundi, Mogism, Jochen Burghardt, Geigr, Monkbot and Anonymous: 9 • Original proof of Gödel's completeness theorem Source: https://en.wikipedia.org/wiki/Original_proof_of_G%C3%B6del'{}s_completeness_ theorem?oldid=679095433 Contributors: AxelBoldt, General Wesc, Mav, Ktsquare, Mjb, Michael Hardy, Eric119, Dysprosia, Giftlite, Mboverload, Rich Farmbrough, Guanabot, AlanBarrett, Ben Standeven, Pt, Nickj, Grue, Randall Holmes, 3mta3, Oleg Alexandrov, Woohookitty, Rjwilmsi, Tim!, Salix alba, Sodin, Trovatore, Zvika, Charles Moss, Dan Gluck, Zero sharp, CBM, Gregbard, Cydebot, Headbomb, Jjaazz, R'n'B, Mad7777, SieBot, Hans Adler, Lightbot, Legobot, Yobot, Citation bot 1, Bbbbbbbbba, Bahersabry and Anony- mous: 25 • Ostensive definition Source: https://en.wikipedia.org/wiki/Ostensive_definition?oldid=698250986 Contributors: Andres, Charles Matthews, Hyacinth, Banno, Pengo, Ancheta Wis, Mporter, Gamaliel, RayBirks, Mindspillage, Rich Farmbrough, Oolong, Zerofoks, Velho, Bgwhite, Shinmawa, SmackBot, Nbarth, Sokolesq, Jon Awbrey, DavidOaks, Sdorrance, Sense1, Jsteph, Pustelnik, Thijs!bot, Dogaroon, NLOleson, It Is Me Here, Natg 19, Mild Bill Hiccup, Alexbot, Addbot, DOI bot, CarsracBot, Luckas-bot, AnomieBOT, Citation bot, DSisyphBot, Citation bot 1, Ciarandean, HairySwede, Helpful Pixie Bot, Op47, BattyBot, Monkbot and Anonymous: 15 • Outline of logic Source: https://en.wikipedia.org/wiki/Outline_of_logic?oldid=701334901 Contributors: Michael Hardy, Greenrd, Markhurd, Tobias Bergemann, Robin Hood~enwiki, Chalst, EmilJ, PWilkinson, Woohookitty, BD2412, Qwertyus, Jake Wartenberg, Tedder, Grafen, Trovatore, Googl, Nikkimaria, Arthur Rubin, Auroranorth, SmackBot, David Kernow, Nexus Seven, RichardF, Courcelles, CRGreathouse, CBM, Sdorrance, Myasuda, Gregbard, Alaibot, The Transhumanist, True Genius, JaGa, Anarchia, N4nojohn, The Transhumanist (AWB), Ontoraul, The Tetrast, Maxim, Capitalismojo, Drmies, Niceguyedc, Auntof6, Robert Skyhawk, Watchduck, Libcub, Verbal, Minnecolo- gies, RJGray, Aaron Kauppi, Thehelpfulbot, Wikiain, Gamewizard71, Morton Shumway, Hpvpp, Bomazi, Wcherowi, Masssly, Han- lon1755, Letsbefiends, Jochen Burghardt, SoSivr and Anonymous: 4 • Per fas et nefas Source: https://en.wikipedia.org/wiki/Per_fas_et_nefas?oldid=705347084 Contributors: Jmchambers90, Antiquary, Addbot, AnomieBOT, Omnipaedista, EmausBot, Snotbot, Hfilben, Itc editor2 and Finnusertop • Persuasive definition Source: https://en.wikipedia.org/wiki/Persuasive_definition?oldid=698393458 Contributors: AxelBoldt, Mrwojo, Michael Hardy, Hyacinth, Xanzzibar, JimWae, Mindspillage, Circeus, Rjwilmsi, Allens, Mountain Goat, Chris the speller, Cybercobra, Richard001, Loodog, Jafet, AntiSpamBot, SieBot, Sanya3, Dlabtot, Addbot, Download, Yobot, Amirobot, AnomieBOT, Berkeley626, Jafet.vixle, Omnipaedista, DrilBot, RjwilmsiBot, Tesseract2, ClueBot NG, Helpful Pixie Bot, BG19bot, Mark Arsten, Cwobeel, Monkbot, Ihaveacatonmydesk, Fabriziomacagno and Anonymous: 22 145.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 551

• Philosophic burden of proof Source: https://en.wikipedia.org/wiki/Philosophic_burden_of_proof?oldid=708083811 Contributors: Chealer, Aetheling, Discospinster, Bender235, Phiwum, Diego Moya, Kazvorpal, Rjwilmsi, Brandmeister (old), Malcolma, Trickstar, Smack- Bot, Rtc, The one092001, Huon, Fuhghettaboutit, Richard001, Ariel Pontes, Banedon, Stalik, Wiploc, Jablomih, Zach99998, John- Manuel, Sanya3, Sunrise, Denisarona, Rhododendrites, Simon Villeneuve, Dthomsen8, Piratejosh85, Mitch Ames, Addbot, Dawynn, AgadaUrbanit, AnomieBOT, Mark Renier, Jonesey95, Calmer Waters, GrapedApe, Philocentric, Nederlandse Leeuw, Becritical, Tesser- act2, Tinss, GoingBatty, Ursus Lapideus, Solomonfromfinland, AgentTheGreat, ChuispastonBot, ClueBot NG, Rtucker913, Spannerjam, Jeraphine Gryphon, Pacerier, MisterCake, Pirhayati, Dawkinsgirl, Epicgenius, Digimaster002, Rekowo, Balljust, AnuragVohra27, Lee- grc, Millardjmelnyk, Yibby is so cool, KasparBot, Ephemerance, Yellowishmagenta, Spirit Ethanol, Benblur4 and Anonymous: 41 • Philosophy of logic Source: https://en.wikipedia.org/wiki/Philosophy_of_logic?oldid=683216785 Contributors: Michael Hardy, An- drewman327, Rich Farmbrough, Bender235, Before My Ken, BD2412, Welsh, SmackBot, Sct72, BWCNY, Gregbard, JamesAM, VolkovBot, Ontoraul, The Tetrast, Philogo, Tomaxer, Paradoctor, Володимир Ф, Francvs, Hans Adler, SchreiberBike, Dthomsen8, ,Paraconsistent, ArthurBot, Xqbot, Omnipaedista, Aaron Kauppi, FrescoBot, Gamewizard71 ,ماني ,Addbot, Redheylin, BOOLE1847 GregKaye, Zhongguoren2, Hriber, Satellizer, WikiPuppies, MerlIwBot, Helpful Pixie Bot, PhnomPencil, John Aiello, Jochen Burghardt, Liz, Vieque, GeneralizationsAreBad and Anonymous: 13 • Polarity item Source: https://en.wikipedia.org/wiki/Polarity_item?oldid=676244958 Contributors: Michael Hardy, Kku, Furrykef, Angr, BillC, BD2412, Rjwilmsi, Neither, Jprg1966, Cybercobra, Physis, Iridescent, Alexey Feldgendler, JustAGal, Albany NY, CapnPrep, Legendsword, Cnilep, Addbot, DOI bot, Citation bot 1, Mundart, RjwilmsiBot, Kazoos, Victor Yus and Anonymous: 12 • Portal:Logic Source: https://en.wikipedia.org/wiki/Portal%3ALogic?oldid=644099213 Contributors: Banno, Alan Liefting, Mtmelen- dez, Bjankuloski06en~enwiki, RekishiEJ, Gregbard, Cydebot, GeneChase, Luna Santin, Anarchia, Brest, NewEnglandYankee, Fu- nandtrvl, VolkovBot, Iohannes Animosus, Addbot, CompulsiveLairLindsay, Ciphers, Moxy, Mean as custard, John of Reading, Afterwald, Northamerica1000, Firozshaikklubs, YiFeiBot, John of Reading Bot, Aappt and Anonymous: 5 • Post's lattice Source: https://en.wikipedia.org/wiki/Post'{}s_lattice?oldid=648793836 Contributors: EmilJ, LOL, Rjwilmsi, David Epp- stein, Addbot, SpBot, Luckas-bot, Gamewizard71, Helpful Pixie Bot, Janniksiebert and Anonymous: 2 • Pragmatic mapping Source: https://en.wikipedia.org/wiki/Pragmatic_mapping?oldid=627025792 Contributors: RJFJR, Wavelength, SmackBot, Chris the speller, Gregbard, After Midnight, Mike V, Calliopejen1, John W. Oller, Jr., Addbot, Ben Ben, Yobot, FrescoBot, Gamewizard71, Helpful Pixie Bot, Kyoakoa, ChrisGualtieri, DoctorKubla and Jihadcola • Pragmatic maxim Source: https://en.wikipedia.org/wiki/Pragmatic_maxim?oldid=648705035 Contributors: William Avery, Nickg, Banno, Rursus, Kaldari, Versageek, Oleg Alexandrov, Woohookitty, Uncle G, RxS, Gwernol, JuJube, Carlosguitar, Jon Awbrey, JzG, Coredesat, Slakr, Hetar, CBM, Gregbard, Thijs!bot, Hut 8.5, Ars Tottle, Mistercupcake, Steven J. Anderson, The Tetrast, Seb26, En- viroboy, Newbyguesses, Maelgwnbot, Mr. Stradivarius, ClueBot, Trainshift, Pluto Car, Unco Guid, Viva La Information Revolution!, Autocratic Uzbek, Navy Pierre, Mrs. Lovett's Meat Puppets, Unknown Justin, West Goshen Guy, Southeast Penna Poppa, Delaware Valley Girl, Addbot, Ariel Black, The Wiki ghost, RedBot, Gamewizard71, Helpful Pixie Bot, UAwiki and Anonymous: 12 • Pragmatic theory of truth Source: https://en.wikipedia.org/wiki/Pragmatic_theory_of_truth?oldid=664572951 Contributors: Charles Matthews, Banno, Rursus, Everyking, Nathan Ladd, Runnerupnj, Samulili, Staeiou, Jeff3000, BD2412, Rjwilmsi, Ben moss, Bhny, Gaius Cornelius, Atfyfe, Cleared as filed, Igiffin, SmackBot, Kmarinas86, Jon Awbrey, FlyHigh, Byelf2007, Lambiam, K, CmdrObot, Gregbard, Nick Number, Edokter, Alastair Haines, JamesBWatson, Way of Inquiry, Anarchia, R'n'B, Ianmathwiz7, Belovedfreak, Mistercupcake, Begewe, The Tetrast, Philogo, Larklight, Andrewaskew, Cnilep, Newbyguesses, JL-Bot, ClueBot, Arunsingh16, Yobot, AnomieBOT, LilHelpa, J04n, Omnipaedista, FrescoBot, 10Kthings, ClueBot NG, Icantfindanyusername, Brian Tomasik, Weird.Tesseract, Khazar2, Squiver and Anonymous: 20 • Principle of bivalence Source: https://en.wikipedia.org/wiki/Principle_of_bivalence?oldid=705780608 Contributors: LC~enwiki, Bryan Derksen, Zundark, Ixfd64, Justin Johnson, Evercat, Charles Matthews, Hyacinth, NSash, Decoy, Guanabot, Paul August, Tsujigiri~enwiki, Chalst, Wareh, Beige Tangerine, Nortexoid, Lysdexia, Anthony Appleyard, Snowolf, RJFJR, Oleg Alexandrov, Velho, Linas, Pruss, Apokrif, Btyner, Graham87, Brighterorange, YurikBot, Hairy Dude, Cedar101, SmackBot, Rtc, Mhss, Nbarth, Frap, Jon Awbrey, Byelf2007, Wvbailey, Mets501, CBM, Gregbard, Letranova, R'n'B, WOSlinker, Don4of4, Modocc, Hugo Herbelin, RPHv, Matěj Grabovský, Legobot, Luckas-bot, Yobot, AnomieBOT, Hriber, Chharvey, Tijfo098, ClueBot NG, Helpful Pixie Bot, Blue Mist 1, Onge- potchket, Harizotoh9, Dexbot, SoledadKabocha, Camila Cavalcanti Nery, Mathematical Truth and Anonymous: 41 • Principle of nonvacuous contrast Source: https://en.wikipedia.org/wiki/Principle_of_nonvacuous_contrast?oldid=666502212 Contrib- utors: Charles Matthews, Stemonitis, Filipem, Gregbard, Ludvikus, Fadesga, Addbot, Yobot, AnomieBOT, ClueBot NG and Anonymous: 1 • Propositional representation Source: https://en.wikipedia.org/wiki/Propositional_representation?oldid=675892395 Contributors: Banno, Enochlau, Pearle, InShaneee, RJFJR, GregorB, Sasuke Sarutobi, Gardar Rurak, ZabMilenko, Spliffy, SmackBot, Kostmo, Scoty6776, Neelix, Gregbard, Guy Macon, Fabrictramp, Roastytoast, Katharineamy, Phil Bridger, Mynameisnotpj, FrescoBot, AvicAWB, BattyBot, DoctorKubla and Anonymous: 6 • Quantifier (logic) Source: https://en.wikipedia.org/wiki/Quantifier_(logic)?oldid=706967817 Contributors: Hyacinth, Trylks, R.e.b., Arthur Rubin, Chrisahn, AnomieBOT, John of Reading, Quondum, Pacerier, David.moreno72, Jochen Burghardt and Anonymous: 4 • Quantization (linguistics) Source: https://en.wikipedia.org/wiki/Quantization_(linguistics)?oldid=706115211 Contributors: Altenmann, Ish ishwar, Wavelength, Neither, Lambiam, AndrewHowse, Hans Adler, Addbot, The Wiki ghost, Crispulop and Anonymous: 6 • Regular modal logic Source: https://en.wikipedia.org/wiki/Regular_modal_logic?oldid=450911891 Contributors: Nortexoid, Spug, Simeon, Addbot and AvicAWB • Relevance Source: https://en.wikipedia.org/wiki/Relevance?oldid=706804333 Contributors: Edward, Ihcoyc, Ahoerstemeier, Scott, Charles Matthews, Hyacinth, Metasquares, Pingveno, Micru, Macrakis, Lucidish, Rich Farmbrough, Pmsyyz, Aecis, EmilJ, Stesmo, Small- jim, Foobaz, Adrian~enwiki, SpeedyGonsales, PWilkinson, Runner1928, John Quiggin, RainbowOfLight, Brookie, Tabletop, Magis- ter Mathematicae, BD2412, Tommy Kronkvist, FlaBot, YurikBot, Hairy Dude, RL0919, Roger Lindsay, Paul Erik, GrinBot~enwiki, SmackBot, McGeddon, WillAndrews, Gilliam, Silly rabbit, Rklawton, JorisvS, Physis, Dreftymac, Mpoulshock, Megatronium, Greg- bard, Themightyquill, Thijs!bot, AntiVandalBot, JAnDbot, Ecurrey, Arno Matthias, Father Goose, Cpl Syx, Mcfar54, Dan Pelleg, Mar- tinBot, Arjun01, AstroHurricane001, Yonidebot, SimDarthMaul, Vranak, Zmnsr1, Fences and windows, Nedelisky, Michaeldsuarez, Maranlar, Neparis, Flyer22 Reborn, JSpung, Mr. Stradivarius, ClueBot, Mike Klaassen, Blanchardb, RenamedUser jaskldjslak903, Aw- ickert, Excirial, Jusdafax, PixelBot, BirgerH, Rebele, BarretB, Noctibus, Gunnex, Addbot, Ezekiel 7:19, Ccacsmss, West.andrew.g, Tide 552 CHAPTER 145. ZHEGALKIN POLYNOMIAL

rolls, Lightbot, OlEnglish, Zorrobot, Jarble, Luckas-bot, Yobot, THEN WHO WAS PHONE?, AnomieBOT, Materialscientist, Arthur- Bot, Shadowjams, Wissling, Pinethicket, Lotje, Reach Out to the Truth, John of Reading, Tommy2010, Stefania75~enwiki, Bertman3, L Kensington, ClueBot NG, Fauzan, MerlIwBot, Helpful Pixie Bot, HMSSolent, Leonxlin, Briancondron, Sriharsh1234, New worl, WikiEnthusiastNumberTwenty-Two, Grey.dreyk, Qwertyxp2000, QKsu, XLSXANDER24, Lilybaizer and Anonymous: 100 • Segment addition postulate Source: https://en.wikipedia.org/wiki/Segment_addition_postulate?oldid=686968981 Contributors: Michael Hardy, Oliver Pereira, Causa sui, Siddhant, Wavelength, Zagalejo, Dspradau, Mathman1550, Matthew Yeager, ClueBot, Jusdafax, Nu- clearWarfare, MatthewVanitas, Addbot, Yobot, Radix38, Sdfl;jsakdf;asdjk, Alpha Quadrant, Donner60, ClueBot NG, Mysterytrey, Ugog Nizdast, HMSLavender, Loraof, BaxterC6 and Anonymous: 24 • Self-reference Source: https://en.wikipedia.org/wiki/Self-reference?oldid=701643140 Contributors: WojPob, Mav, The Anome, Man- ning Bartlett, Ed Poor, Enchanter, Rootbeer, Anthere, Mintguy, Hephaestos, Patrick, Michael Hardy, Tim Starling, Oliver Pereira, Jah- sonic, MartinHarper, Matthewmayer, GTBacchus, Karada, Paddu, Tregoweth, Ahoerstemeier, EntmootsOfTrolls, Slovakia, Error, Tim ,Khym Chanur, Jerzy, Skrim, Ldo, Jeffq ,דוד ,Retout, IMSoP, Palfrey, Harris7, Dysprosia, Furrykef, David Shay, Omegatron, Bevo Sjorford, Fredrik, PBS, Chocolateboy, Goethean, Rfc1394, Geogre, Saforrest, Wereon, Anthony, Diberri, Matthew Stannard, Alerante, Giftlite, Dbenbenn, DocWatson42, DavidCary, Jao, Haeleth, Orangemike, Theon~enwiki, Wikiwikifast, Matt Crypto, HorsePunchKid, Joeblakesley, Zondor, Grunt, RevRagnarok, AlexChurchill, Freakofnurture, DanielCristofani, Jim Henry, Discospinster, Max Terry, Xezbeth, TheJames, Paul August, MattTM, Danakil, Brian0918, PhilHibbs, JRM, Jonathan Drain, Grick, Excalibre, Richi, Stjef, Ni- hil~enwiki, Sp00n17, Massar, Naif, Msauve, ShawnVW, H2g2bob, DV8 2XL, LukeSurl, Ceyockey, Dtobias, Zntrip, Mjpotter, Angr, Mindmatrix, BillC, Jacobolus, Kosher Fan, Pufferfish101, Dionyziz, Macaddct1984, MushroomCloud, Marudubshinki, Mandarax, Gra- ham87, Keeves, Qwertyus, Rjwilmsi, KYPark, Edbrims, Dpark, Cakedamber, Baryonic Being, Titoxd, SchuminWeb, Jax-wp, Gurch, Mstroeck, King of Hearts, Ahpook, Wavelength, Ilanpi, Hyad, Sasuke Sarutobi, Kyorosuke, Thane, Deskana, TheLH, JDoorjam, Yano, William R. Buckley, Alex43223, Noam~enwiki, Wknight94, Alecmconroy, 21655, Johndburger, Closedmouth, Warreed, Garion96, Profero, Jinzo7272, NetRolller 3D, SmackBot, Moeron, McGeddon, Piroteknix, Nerd42, Jwestbrook, Dyslexic agnostic, Oneismany, Chris the speller, MartinPoulter, Nbarth, Kindall, JonHarder, Matchups, AnthonyMartin, Gaddy1975, Wes!, Khoikhoi, Igor the Lion, Chris3145, Luigi III, TenPoundHammer, BrownHairedGirl, Tktktk, Steipe, Physis, 041744, 16@r, Loadmaster, Shimmera, Jeiki , Dicklyon, Mikem1234, EdC~enwiki, Cbuckley, Mikekearn, Kencf0618, Impy4ever, Lesion, Shultz III, Buddy13, Yashgaroth, Peter1c, Mapsax, CRGreathouse, CmdrObot, Outriggr (2006-2009), Seven of Nine, Bmk, Chantessy, Gregbard, Cricketseven, Dragon's Blood, Cambrant, Danman3459, Peterdjones, Gogo Dodo, Inkington, Eubulide, Thijs!bot, Oryanw~enwiki, Timo3, Al Lemos, TheTruthiness, KamStak23, Nycdi, Rps, AnAj, Oddity-, JAnDbot, Smiddle, Tiberone, Trey314159, Chevellefan11, Ophion, Hullaballoo Wolfowitz, Tkang, Chemical Engineer, Esanchez7587, Logicbox, Gwern, B9 hummingbird hovering, Anarchia, Smokizzy, Anandcv, AstroHur- ricane001, Numbo3, Maurice Carbonaro, Chiswick Chap, Ritarius, Vanished user 47736712, Sbaxt641, Atheuz, Botx, Zephyrus11, Dorftrottel, VolkovBot, MasterPeanut, Maghnus, Uagehry456, Jamelan, Jesin, VanishedUserABC, Cjc13, Mblub, CortexSurfer, SieBot, YonaBot, Paradoctor, Ernie shoemaker, Toddst1, JLKrause, Svick, Anchor Link Bot, JL-Bot, JosefAssad, Iceberg1031, MarkSMann, De728631, DionysosProteus, Arakunem, Wildroot, P.T.isfirst, Trivialist, John J. Bulten, Rabbitslayer21, Jumbolino, Three-quarter-ten, PixelBot, Sun Creator, JasonAQuest, Kakofonous, Tezero, Alevy1234, XLinkBot, Koreindian, Rror, Slogan120, Tayste, Addbot, Ot- terathome, MrOllie, AgadaUrbanit, Tassedethe, Emdrgreg, Lightbot, Jarble, Luckas-bot, Yobot, II MusLiM HyBRiD II, AnomieBOT, 1exec1, Trevithj, Galoubet, Eumolpo, ArthurBot, GrouchoBot, Titi2~enwiki, G3548dm, Berylcloud, Worldrimroamer, Ace9999, Asafox, Boxplot, Pinethicket, RedBot, Foobarnix, Fosforos, Dungeonscaper, Hobbes Goodyear, Beyond My Ken, WikitanvirBot, Singularity2, ClueBot NG, MRFazry, JohnsonL623, MaximalIdeal, Helpful Pixie Bot, Curb Chain, Bouket, Lifeformnoho, HelloAndroid, Mrryk- ler, Lyrewyn, Maximuspryme, Meteor sandwich yum, VictorLucas, Parabolooidal, Ice ax1940ice pick, Chocolatechip65, 74-tungsten, KasparBot, Toh59, Fibonacci2358, Ironak314 and Anonymous: 252 • Self-refuting idea Source: https://en.wikipedia.org/wiki/Self-refuting_idea?oldid=706591710 Contributors: Michael Hardy, Greenrd, Furrykef, Goethean, Beefman, Kevin Dorner, Bender235, Longhair, Giraffedata, Mailer diablo, DreamGuy, Stemonitis, Ruud Koot, Miss Madeline, Rjwilmsi, NeonMerlin, Reinis, Hairy Dude, Joel7687, Roy Brumback, Amberrock, Snalwibma, SmackBot, Colonies Chris, Lambiam, Zelaron, Neelix, Kyrisch, Gregbard, Peterdjones, Vertium, WikiSlasher, Gioto, SteveWolfer, NBeale, Dsp13, Anarchia, Vir- tualDelight, Merzul, Jmrowland, Harfarhs, Geoffw1948, Nozzer42, Sunrise, Bowei Huang 2, ClueBot, Ideal gas equation, EdibleKarma, ManicBrit, Singinglemon~enwiki, Sun Creator, Adamfinmo, Editor2020, Helixweb, Favonian, Sprachpfleger, PMLawrence, Wargo, AnomieBOT, LilHelpa, Sketchmoose, Omnipaedista, Fortdj33, Machine Elf 1735, LittleWink, John of Reading, Solomonfromfinland, Lambda.calc, Libertaar, Mcc1789, ClueBot NG, Godsoflogic, Primergrey, Bloodwashed, Helpful Pixie Bot, JohnChrysostom, Brad- lake10, XMadej, Harizotoh9, 23haveblue, Zaxnaaog, PhaseVelocity, John Doddridge, Critical-interval, Faizan and Anonymous: 60 • Ship of Theseus Source: https://en.wikipedia.org/wiki/Ship_of_Theseus?oldid=708299149 Contributors: General Wesc, The Anome, Deb, Hephaestos, Michael Hardy, Kalki, Karada, Stan Shebs, Julesd, Error, Scott, Samw, Etaoin, Vargenau, Nikola Smolenski, Ehn, Timwi, Nickg, Furrykef, Hyacinth, AnonMoos, Chrism, Pjedicke, SchmuckyTheCat, Auric, Paul G, Wereon, Ruakh, Xanzzibar, Pengo, David Gerard, Bogdanb, Anville, 20040302, Chowbok, Noe, Roisterer, Kuralyov, Sfoskett, Karl-Henner, Hilarleo, DMG413, Porges, DanielCristofani, Night Gyr, Edward Z. Yang, Tverbeek, Root4(one), Simfish, John Vandenberg, Viriditas, Scu98rkr, Oop, Fatphil, Lys- dexia, Anthony Appleyard, CyberSkull, Keenan Pepper, PatrickFisher, Ashley Pomeroy, Macl, Dhartung, Melaen, Omphaloscope, P Ingerson, Ringbang, Richwales, Dismas, Oleg Alexandrov, Jävligsvengelska, Vorn, Miss Madeline, Smmurphy, TotoBaggins, GregorB, Gerbrant, BD2412, Kotukunui, Salix alba, Heah, Miserlou, Cassowary, DirkvdM, MWAK, Naraht, SchuminWeb, Djrobgordon, Nihiltres, Jameshfisher, Nimur, Ali Davoodifar, SteveBaker, Esslk, Chobot, YurikBot, Rob T Firefly, Hydrargyrum, Gaius Cornelius, Anomalo- caris, Mixvio, Dysmorodrepanis~enwiki, Bluebird47, Dogcow, Isolani, Abb3w, Mobilesworking, T-rex, Salmanazar, Pawyilee, Nikki- maria, Sturmovik, John Broughton, A bit iffy, SmackBot, Dweller, Marcusscotus1, Zazaban, InverseHypercube, McGeddon, WikiuserNI, Unyoyega, Eskimbot, Kintetsubuffalo, Septegram, Chris the speller, Endroit, Thumperward, Freshmutt, Matt9090, Audigex, Nbarth, Namangwari, KingAlanI, Hallenrm, Oatmeal batman, Emurphy42, Mooncow, Konczewski, GRuban, Metageek, Jpaulm, Cybercobra, TGC55, Minipie8, Kuronue, Wybot, Terrasidius, Andeggs, Lambiam, Axem Titanium, MYT, Ckatz, Stratadrake, Loadmaster, Bless sins, Filanca, Spiff666, Ryulong, Dr.K., Peter Horn, Cls14, Bottesini, Switchercat, CRGreathouse, Eponymous-Archon, JETM, Byziden, Beatriceblue, SeanMon, Teratornis, Editor at Large, Jon C., JAF1970, Catsmoke, Davidhorman, Hcobb, SummerPhD, Rps, D. Webb, Tempest115, Robot Revery, Matthew Fennell, Midnightdreary, The elephant, Wasell, Arno Matthias, Albmont, Oskay, Cuebon, Nyt- tend, Ka-ru, Dave Muscato, Martynas Patasius, R'n'B, KTo288, Jaggerblade, Arriva436, Arrow740, RickardV, Mighty Antar, Heshy613, VolkovBot, TXiKiBoT, Oshwah, Sluttymike, Technopat, Shureg, Nda98, Rubseb, Jamelan, Bporopat, Clevemire321, WereSpielChe- quers, Rohanabm, Flyer22 Reborn, Dans, Otherfates, SaniOKh, Magic9mushroom, Beefeather~enwiki, Berjangles, Tesi1700, Wee Curry Monster, Randy Kryn, Martarius, MBK004, ClueBot, EelkeSpaak, Xszcw, Piledhigheranddeeper, Singinglemon~enwiki, Ozy42, Trivi- alist, Mr. Laser Beam, Twinsemi, Catalographer, Antediluvian67, Psymier, DumZiBoT, Borock, Wyatt915, Addbot, Patd987, Anders Sandberg, Bte99, Tehslug, Tide rolls, Yobot, Jabberwockgee, John renard, AnomieBOT, Captain Quirk, Galoubet, Materialscientist, Im- 145.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 553

peratorExercitus, Citation bot, Bradshaws1, Ubcule, Misha Vargas, Twirligig, FrescoBot, Wearmysocks1611, Citation bot 1, Insoc, An- gelicaAgurbash, Lotje, 4, Supervidin, Redfish18, WildBot, Ericwag, EmausBot, John of Reading, WikitanvirBot, Lucien504, Mordgier, GoingBatty, Exok, Wikipelli, ZéroBot, Life in General, Crowblurt, Spiffulent, Anglais1, Donner60, Dwstultz, El Alternativo, ClueBot NG, TheConduqtor, Delusion23, Morgan Riley, Helpful Pixie Bot, BG19bot, Ashleyfrieze, W.andrea, TangoTizerWolfstone, Michael Barera, Duxwing, Dienekles, Steve M Kane, BattyBot, Simeon Dahl, Phronesis8, Saifaa, PAB1990, Extrasensory90, CsDix, Debouch, Sajin Jiju, Stevenson7869, Barukh Habba, Marc André Miron, Sotrecycle, Jackmcbarn, Ssmani2000, Explorerofftheworld, Noyster, De- vanita, Monkbot, RJANKA, Signedzzz, Máel Milscothach, CeraWithaC, Ted Peeples, BrainyJon and Anonymous: 327 • Simple non-inferential passage Source: https://en.wikipedia.org/wiki/Simple_non-inferential_passage?oldid=578957871 Contributors: Salimfadhley, BD2412, Gregbard, LilHelpa and Saedon • Situational analysis Source: https://en.wikipedia.org/wiki/Situational_analysis?oldid=562584988 Contributors: Gregbard, Msrasnw, Addbot, EmausBot and Anonymous: 1 • Social software (social procedure) Source: https://en.wikipedia.org/wiki/Social_software_(social_procedure)?oldid=698862824 Con- tributors: MarXidad, Edward, Kku, Drstuey, Netoholic, RTucker, Mdd, Landroni, MIT Trekkie, OwenX, Rjwilmsi, SmackBot, Vaughan Pratt, Alaibot, Natalie Erin, Mack2, MartinDK, RockMFR, Fsatari, Farishta Satari, Yash M, Margin1522, Cdrdata, Omnipaedista, DPar- doeWilson, Jonkerz and Anonymous: 8 • Soku hi Source: https://en.wikipedia.org/wiki/Soku_hi?oldid=667341447 Contributors: Andycjp, Ouro, SmackBot, Sohale, Betacom- mandBot, Materialscientist, Omnipaedista, ClueBot NG, Yomanking, Markp1053 and Anonymous: 1 • Specialization (logic) Source: https://en.wikipedia.org/wiki/Specialization_(logic)?oldid=418789112 Contributors: The Anome, Patrick, MartinHarper, Snoyes, Saforrest, Sj, Cambyses, Christopherlin, Utcursch, Jeshii, Twinxor, RoyBoy, Bobo192, Rd232, Andrew Gray, ZeiP, Suruena, RainbowOfLight, Defixio, FlaBot, Fram, DVD R W, SmackBot, KnowledgeOfSelf, Cazort, Can't sleep, clown will eat me, Celarnor, Richard001, 16@r, Adambiswanger1, Chrislk02, Clamster5, Catgut, Mmustafa~enwiki, Infrangible, J.delanoy, Rehrer, So- liloquial, Jamelan, Frank Romein~enwiki, Seanust, Latics, JL-Bot, ClueBot, Excirial, Hegsbfgszd, Zwasqx, Awehrfjkashkfjd, La Pianista, Jadtnr1, Addbot, Luckas-bot, Erik9bot, EdoBot and Anonymous: 40 • Tacit assumption Source: https://en.wikipedia.org/wiki/Tacit_assumption?oldid=558358688 Contributors: Rednblu, Adam78, Mecan- ismo, Cask05, SmackBot, DMS, JHunterJ, CmdrObot, Egriffin, X96lee15, Lenticel, Goblinman, MironGainz and Anonymous: 1 • Term logic Source: https://en.wikipedia.org/wiki/Term_logic?oldid=690969140 Contributors: Ed Poor, Enchanter, Michael Hardy, AugPi, EdH, Peter Damian (original account), Charles Matthews, Timwi, Dysprosia, Wik, Markhurd, Maximus Rex, Hyacinth, Robbot, Fredrik, Stewartadcock, Ruakh, Filemon, Giftlite, Siroxo, Gubbubu, Beland, Pmanderson, Deelkar, Paul August, Elwikipedista~enwiki, Chalst, Wood Thrush, BrokenSegue, Nortexoid, PWilkinson, Amerindianarts, Mark Dingemanse, Ricky81682, George Hernandez, Linas, Oriondown, BD2412, Grammarbot, Haya shiloh, Wavelength, Leuliett, SEWilcoBot, Cleared as filed, Doncram, Reyk, Tevildo, Joan- neB, Bernd in Japan, GrinBot~enwiki, Sardanaphalus, SmackBot, Jagged 85, The great kawa, Yamaguchi 先生, Mhss, Oatmeal batman, Byelf2007, Anapraxic, CmdrObot, Gregbard, Cydebot, Gimmetrow, Barticus88, Bmorton3, DuncanHill, Gwern, Philcha, Jevansen, Djh- moore, Ontoraul, The Tetrast, Kumioko (renamed), Le vin blanc, JustinBlank, Andrewmlang, The Thing That Should Not Be, Imperfectly- Informed, Excirial, CohesionBot, PixelBot, Wordwright, MilesAgain, JDPhD, Palnot, Good Olfactory, Addbot, Markenrode, LightSpec- tra, Tassedethe, BOOLE1847, Lightbot, Vasiľ, Yobot, Ordre Nativel, AnomieBOT, LilHelpa, GrouchoBot, Peter Damian, Omnipaedista, Rb1205, Machine Elf 1735, Winterst, Dhanyavaada, Dude1818, Pollinosisss, Wikielwikingo, EmausBot, Acather96, Moswento, Buf- faboy, Rememberway, ClueBot NG, Jeraphine Gryphon, Regulov, BG19bot, JohnChrysostom, Hansen Sebastian, Hmainsbot1, Hariket, Jochen Burghardt, Tyro13, TE5ITA and Anonymous: 53 • Testability Source: https://en.wikipedia.org/wiki/Testability?oldid=693955496 Contributors: NeonMerlin, Gurch, The Rambling Man, Pinecar, Arado, Shanel, Retired username, SmackBot, Javalenok, Richard001, Jon Awbrey, Triper~enwiki, Levineps, George100, Neelix, Gregbard, Magioladitis, WLU, N4nojohn, VolkovBot, MusicScience, Sunrise, Gorillasapiens, Blue bear sd, Pmronchi, Xme, Addbot, Deamon138, Luckas-bot, Contributor124, DexDor, EmausBot, RockMagnetist, Zyxwv99, Lugia2453, Dodi 8238 and Anonymous: 17 • Tetralemma Source: https://en.wikipedia.org/wiki/Tetralemma?oldid=541276286 Contributors: Deeptrivia, Sardanaphalus, Jagged 85, Octahedron80, Zero sharp, Harej bot, Gregbard, Kilva, Itistoday, Gwern, B9 hummingbird hovering, TXiKiBoT, Owlmonkey, AJ3D, DumZiBoT, Addbot, Ego White Tray and Anonymous: 7 • The Game of Logic Source: https://en.wikipedia.org/wiki/The_Game_of_Logic?oldid=676660257 Contributors: Fadesga, Attila.lendvai, Suslindisambiguator, ChrisGualtieri, MrNiceGuy1113, Cyrej and Anonymous: 1 • Theoretical definition Source: https://en.wikipedia.org/wiki/Theoretical_definition?oldid=703897577 Contributors: Hyacinth, Hen- rygb, Mindspillage, Allens, Junglecat, SmackBot, Byelf2007, Gregbard, Maurice Carbonaro, Hodja Nasreddin, CharlesGillingham, PCHS-NJROTC, Addbot, Yobot, Pdragy, AnomieBOT, Erik9bot, Xttina.Garnet, Tesseract2, Fæ, Widr, BG19bot, Qetuth, SteenthIWbot and Anonymous: 9 • Train of thought Source: https://en.wikipedia.org/wiki/Train_of_thought?oldid=696556621 Contributors: Altenmann, Woohookitty, The wub, Ewlyahoocom, Gurch, SmackBot, DCDuring, Cybercobra, Eastlaw, ShelfSkewed, JFreeman, Greensburger, Jonsmallwood2004, Bushcarrot, Jamal.abdulhaq~enwiki, Pundit8086, Invertzoo, ClueBot, Julianhall, Eliminati, Hickypedia, SoxBot III, Antti29, Addbot, Luckas-bot, Pcap, Capricorn42, Aaron Kauppi, FrescoBot, Dennis714, Anluoridge, Makecat-bot, Tentinator and Anonymous: 40 • Transferable belief model Source: https://en.wikipedia.org/wiki/Transferable_belief_model?oldid=694678627 Contributors: Michael Hardy, GregorB, Rjwilmsi, SmackBot, Bluebot, Belush, Curdeius, David Eppstein, Melcombe, Trivialist, Addbot, Jarble, Ptbotgourou, Hxd1011 and Anonymous: 5 • Trikonic Source: https://en.wikipedia.org/wiki/Trikonic?oldid=705774538 Contributors: Ioapetraka, Mdd, SmackBot, Bluebot, Stefan2, Trikonic, WVhybrid, Jvhertum, LookingGlass, The Tetrast, Sfan00 IMG, Cavarrone, J04n and Anonymous: 4 • Truth-bearer Source: https://en.wikipedia.org/wiki/Truth-bearer?oldid=683373726 Contributors: Banno, Barbara Shack, Andycjp, DanielDe- maret, Bender235, Pearle, BD2412, Nihiltres, Gurch, Bgwhite, Rick Norwood, Tomisti, Iph, Sct72, Jon Awbrey, Dicklyon, RedRoller- skate, Gregbard, Cydebot, Skomorokh, Salad Days, Magioladitis, Heyitspeter, Philogo, Andrewaskew, StAnselm, Iamthedeus, Qwfp, Dthomsen8, Addbot, Queenmomcat, Ettrig, Yobot, AnomieBOT, Eumolpo, LilHelpa, Locobot, FrescoBot, BrideOfKripkenstein, Ci- tation bot 1, SixPurpleFish, John of Reading, Beta M, Helpful Pixie Bot, Dr Lindsay B Yeates, POLY1956, Wofalamatalahoop and Anonymous: 10 • Tuple-generating dependency Source: https://en.wikipedia.org/wiki/Tuple-generating_dependency?oldid=629175028 Contributors: Drag- onflySixtyseven, Wikidsp, Angela.bonifati, Krainah.moufida and Anonymous: 1 554 CHAPTER 145. ZHEGALKIN POLYNOMIAL

• Universal logic Source: https://en.wikipedia.org/wiki/Universal_logic?oldid=706698635 Contributors: Tabletop, SmackBot, Dbtfz, WereSpielChe- quers, Hans Adler, Lightbot, Omnipaedista, The Interior, Oursipan, Ludovic stern, RjwilmsiBot, GKaczinsky, Tijfo098, Helpful Pixie Bot, BG19bot, SaundersLane and Anonymous: 10 • Unspoken rule Source: https://en.wikipedia.org/wiki/Unspoken_rule?oldid=698767380 Contributors: Ed Poor, Nagelfar, Ukexpat, Mdd, Knuckles, Wavelength, Mongol, Malcolma, Cask05, Ghumpa, Bluebot, Kstinch, Vina-iwbot~enwiki, Khazar, Doczilla, TheTito, Greg- bard, Mattisse, 17Drew, NatureBoyMD, Idioma-bot, SieBot, Trivialist, Xme, EgraS, Addbot, CanadianLinuxUser, The C of E, Yobot, Omnipaedista, Mondotta, Tiganusi, Donner60, ClueBot NG, LisaNotsimpson, Amintly, Knife-in-the-drawer, PurpleLego and Anony- mous: 11 • Vacuous truth Source: https://en.wikipedia.org/wiki/Vacuous_truth?oldid=689791865 Contributors: AxelBoldt, Lee Daniel Crocker, Eloquence, Wesley, The Anome, Tarquin, Toby Bartels, Ryguasu, Montrealais, Michael Hardy, Dominus, Chinju, Karada, Ihcoyc, Jpa- tokal, Jiang, Sethmahoney, Timwi, JeffTL, Dysprosia, Jitse Niesen, Spikey, SirPeebles, BenRG, Catskul, ChrisO~enwiki, Chancemill, Henrygb, JerryFriedman, David Gerard, Matt Gies, Giftlite, Ian Maxwell, Jao, Tubular, Dratman, Jason Quinn, Sundar, Eequor, Bob- blewik, Achituv~enwiki, Rdsmith4, Sam Hocevar, Vivacissamamente, Kate, Rich Farmbrough, Paul August, JRM, Army1987, Foobaz, Mpeisenbr, Batmanand, Cromwellt, Kznf, Mindmatrix, Marudubshinki, Rjwilmsi, Salix alba, RobertG, DVdm, Hairy Dude, Trovatore, Julienlecomte, Ytram99, A Doon, Garion96, AndrewWTaylor, SmackBot, Elonka, Melchoir, McGeddon, Chris the speller, Trebor, Javalenok, Salmar, BrownHairedGirl, Agradman, Antonielly, Loadmaster, Mets501, Saxbryn, Etafly, JRSpriggs, CRGreathouse, CBM, Thomasmeeks, Gregbard, Cydebot, Haifadude, Julian Mendez, Pcu123456789, Michael Falkov, Widefox, Hamaryns, David Eppstein, Boston, N4nojohn, Trumpet marietta 45750, LokiClock, SQL, Xeno8, ClueBot, Snigbrook, Vivio Testarossa, Alexey Muranov, Schreiber- Bike, Ra2007, EdgeNavidad, Legobot, Yobot, Stmannew, Johnrenehbbs, Cobaltcigs, JimsMaher, Aschroet, Frietjes, Solomon7968, DPL bot, Kephir, Loraof, Majorminors and Anonymous: 63 • Vagrant predicate Source: https://en.wikipedia.org/wiki/Vagrant_predicate?oldid=442106129 Contributors: Rpyle731, Drbreznjev, Ael 2 and Yobot • Vaisheshika Source: https://en.wikipedia.org/wiki/Vaisheshika?oldid=707814496 Contributors: Ambarish, Everyking, LordSimonof- Shropshire, Aponar Kestrel, CALR, S.K., John Vandenberg, Mjaganna, Nk, Ogress, Dangerous-Boy, Prater~enwiki, Koavf, FlaBot, Chobot, DaGizza, RobotE, Deeptrivia, Sophroniscus, Trovatore, Joel7687, Tomisti, Jagged 85, Srkris, Ohnoitsjamie, Chris the speller, Madmedea~enwiki, Leaflord, Hitneosh, Deepak D'Souza, Hu12, Gregbard, Cydebot, Babub, Thijs!bot, Joy1963, WinBot, .anacondabot, Maurice Carbonaro, VolkovBot, TXiKiBoT, AlleborgoBot, Arjun024, SieBot, TJRC, Arbor to SJ, Jotterbot, DumZiBoT, Wakari07, Mitsube, Addbot, Favonian, Luckas-bot, Yobot, AnomieBOT, Kingpin13, Xqbot, Omnipaedista, Jsp722, Lecheminlu, Aditya soni, Un- jpmaiya, Prophetvcn, RjwilmsiBot, Shashikgp, EmausBot, WikitanvirBot, TeleComNasSprVen, Kkm010, Rahulsummon, Vassiliades, ClueBot NG, Dream of Nyx, Helpful Pixie Bot, Titodutta, BG19bot, Amitrochates, ChrisGualtieri, Arcandam, Makecat-bot, Suradut- tashandilya, Ms Sarah Welch, KH-1, Knife-in-the-drawer, Doctor Sonya and Anonymous: 41 • Valuation-based system Source: https://en.wikipedia.org/wiki/Valuation-based_system?oldid=533060408 Contributors: Jason Quinn, Lockley, SmackBot, Frap, Alaibot, Fabrictramp, Brewcrewer, Benavoli, Addbot and Anonymous: 2 • Vector logic Source: https://en.wikipedia.org/wiki/Vector_logic?oldid=688735011 Contributors: Michael Hardy, Chris the speller, Mya- suda, Almadana, Yobot, FrescoBot, Josve05a, Frietjes, DPL bot and Anonymous: 8 • Warnier/Orr diagram Source: https://en.wikipedia.org/wiki/Warnier/Orr_diagram?oldid=632403855 Contributors: Crissov, Markhurd, Klemen Kocjancic, Mdd, Ksnow, Davehiggins, GregorB, SmackBot, Bluebot, Madmedea~enwiki, OrphanBot, Naveen.maurya, Richard- Veryard, Tony Myers, Destynova, Windymilla, MartinBot, Kimleonard, JL-Bot, Thingg, WikHead, Addbot, Yobot, AnomieBOT, Ot- tomachin, WikitanvirBot and Anonymous: 10 • Warrant (logic) Source: https://en.wikipedia.org/wiki/Toulmin_method?oldid=703602450 Contributors: RJFJR, SMcCandlish, Greg- bard, Arno Matthias, Bradv, MatthewVanitas, Addbot, LilHelpa, Ncarr20, Omnipaedista, Trth1097, Widr, BattyBot, DoctorKubla, Em- Phillimore, Sacrko and Anonymous: 1 • What the Tortoise Said to Achilles Source: https://en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles?oldid=703548180 Con- tributors: AxelBoldt, Magnus Manske, Toby Bartels, Enchanter, Ryguasu, Ahoerstemeier, ThirdParty, AugPi, Charles Matthews, Paul Stansifer, Radgeek, David Shay, AnonMoos, BenRG, Aleph4, Mattflaschen, Marc Venot, Everyking, Savant1984, Czrisher, Floorsheim, Cretog8, Pyrosim, Algorithm, Alan Canon, Porcher, Rjwilmsi, Tim!, Koavf, Cfortunato, KSchutte, Saric, Mais oui!, SmackBot, Bluebot, MeekSaffron, TKD, Hgilbert, Andeggs, Risssa, Ohconfucius, Byelf2007, Ace Frahm, Gregbard, Cydebot, JPG-GR, Rxtreme, Cognita, Llygadebrill~enwiki, DrDentz, Bawm79, Mikemoral, Tranquilled, Sun Creator, Hans Adler, Addbot, Hisarmwasherleg, Favonian, Trin- iMuñoz, Narayan, Ptbotgourou, AnomieBOT, Dromioofephesus, Omnipaedista, Aaron Kauppi, Wikiain, AATroop, TjBot, Zafar142003, ZéroBot, Jenks24, Стюарт Радзинский, Accelerometer, Wbm1058, BG19bot, Paolo Lipparini, MisterCake, Kephir, Necocco123, BreakfastJr, Monkbot, Trackteur, Marfuas, Kadath9969 and Anonymous: 40 • Window operator Source: https://en.wikipedia.org/wiki/Window_operator?oldid=388597503 Contributors: Nortexoid, Jay1279, Ad- dbot and Yobot • Witness (mathematics) Source: https://en.wikipedia.org/wiki/Witness_(mathematics)?oldid=641963431 Contributors: Michael Hardy, RussBot, Wvbailey, CBM, David Eppstein, WereSpielChequers, BlazerKnight, Tijfo098, Helpful Pixie Bot, Qetuth and ChrisGualtieri • Zhegalkin polynomial Source: https://en.wikipedia.org/wiki/Zhegalkin_polynomial?oldid=693765060 Contributors: Michael Hardy, Macrakis, Bkkbrad, Rjwilmsi, GBL, Vaughan Pratt, CRGreathouse, Myasuda, Gregbard, Alaibot, Dougher, Jeepday, Hans Adler, Addbot, DOI bot, Legobot, Luckas-bot, Yobot, Citation bot, Citation bot 1, Nwezeakunelson and Anonymous: 1

145.7.2 Images

• File:34th_G8_summit_member_20080707.jpg Source: https://upload.wikimedia.org/wikipedia/commons/f/f1/34th_G8_summit_member_ 20080707.jpg License: Public domain Contributors: whitehouse.gov Original artist: photo by Eric Draper • File:Absolute_complement.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/8c/Absolute_complement.svg License: CC- BY-SA-3.0 Contributors: Based on GFDL Image:Absolute complement (set teory, Venn diagram).PNG. Recreated in SVG by User: Brighterorange. Original artist: User:Brighterorange 145.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 555

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Original artist: ? • File:Deliberations_of_Congress.jpg Source: https://upload.wikimedia.org/wikipedia/commons/2/22/Deliberations_of_Congress.jpg License: Public domain Contributors: New York Herald (Credit: The Granger Collection, NY) Original artist: W. A. 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Desktop Project. Original artist: The people from the Tango! project. And according to the meta-data in the file, specifically: “Andreas Nilsson, and Jakub Steiner (although minimally).” • File:Emoji_u1f4bb.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d7/Emoji_u1f4bb.svg License: Apache License 2.0 Contributors: https://code.google.com/p/noto/ Original artist: Google • File:Epicurus_bust2.jpg Source: https://upload.wikimedia.org/wikipedia/commons/1/14/Epicurus_bust2.jpg License: Public domain Contributors: Transferred from en.wikipedia; transferred to Commons by User:Jayantanth using CommonsHelper. Original artist: Original uploader was Interstate295revisited at en.wikipedia • File:Ernst_Zermelo_1900s.jpg Source: https://upload.wikimedia.org/wikipedia/commons/e/ed/Ernst_Zermelo_1900s.jpg License: Pub- lic domain Contributors: http://www.gettyimages.co.uk/detail/news-photo/portrait-of-the-german-mathematician-ernst-zermelo-1900s-news-photo/ 141551246 Original artist: Unknown (Mondadori Publishers) • File:Fennerartoflogic-small.jpg Source: https://upload.wikimedia.org/wikipedia/commons/6/6a/Fennerartoflogic-small.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: Peter Damian • File:Flag_of_Canada.svg Source: https://upload.wikimedia.org/wikipedia/en/c/cf/Flag_of_Canada.svg License: PD Contributors: ? Original artist: ? • File:Flag_of_Europe.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b7/Flag_of_Europe.svg License: Public domain Contributors: • File based on the specification given at [1]. Original artist: User:Verdy p, User:-xfi-, User:Paddu, User:Nightstallion, User:Funakoshi, User:Jeltz, User:Dbenbenn, User:Zscout370 145.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 557

• File:Flag_of_France.svg Source: https://upload.wikimedia.org/wikipedia/en/c/c3/Flag_of_France.svg License: PD Contributors: ? Orig- inal artist: ? • File:Flag_of_Germany.svg Source: https://upload.wikimedia.org/wikipedia/en/b/ba/Flag_of_Germany.svg License: PD Contributors: ? Original artist: ? • File:Flag_of_Italy.svg Source: https://upload.wikimedia.org/wikipedia/en/0/03/Flag_of_Italy.svg License: PD Contributors: ? Original artist: ? • File:Flag_of_Japan.svg Source: https://upload.wikimedia.org/wikipedia/en/9/9e/Flag_of_Japan.svg License: PD Contributors: ? Orig- inal artist: ? • File:Flag_of_Russia.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f3/Flag_of_Russia.svg License: PD Contributors: ? Orig- inal artist: ? • File:Flag_of_the_United_Kingdom.svg Source: https://upload.wikimedia.org/wikipedia/en/a/ae/Flag_of_the_United_Kingdom.svg Li- cense: PD Contributors: ? Original artist: ? • File:Flag_of_the_United_States.svg Source: https://upload.wikimedia.org/wikipedia/en/a/a4/Flag_of_the_United_States.svg License: PD Contributors: ? Original artist: ? • File:Fluidic_AND_XOR.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/02/Fluidic_AND_XOR.svg License: Public domain Contributors: Own work Original artist: Joseph Fyfield • File:Fluidicamplifier.gif Source: https://upload.wikimedia.org/wikipedia/commons/2/2a/Fluidicamplifier.gif License: Public domain Contributors: ? Original artist: ? • File:Folder_Hexagonal_Icon.svg Source: https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg License: Cc- by-sa-3.0 Contributors: ? Original artist: ? • File:G.W.F._Hegel_(by_Sichling,_after_Sebbers).jpg Source: https://upload.wikimedia.org/wikipedia/commons/b/b6/G.W.F._Hegel_ %28by_Sichling%2C_after_Sebbers%29.jpg License: Public domain Contributors: http://www.hegel.net/en/gwh3.htm Original artist: Julius Ludwig Sebbers • File:Gandhi_thinking_mood_1931.jpg Source: https://upload.wikimedia.org/wikipedia/commons/4/4f/Gandhi_thinking_mood_1931. jpg License: Public domain Contributors: This image comes from the Google-hosted LIFE Photo Archive where it is available under the filename d6117dfe96af68e8. Original artist: Unknownwikidata:Q4233718 • File:George_Boole_color.jpg Source: https://upload.wikimedia.org/wikipedia/commons/c/ce/George_Boole_color.jpg License: Pub- lic domain Contributors: http://schools.keldysh.ru/sch444/museum/1_17-19.htm Original artist: Unknownwikidata:Q4233718 • File:Giuseppe_Peano.jpg Source: https://upload.wikimedia.org/wikipedia/commons/3/3a/Giuseppe_Peano.jpg License: Public domain Contributors: School of Mathematics and Statistics, University of St Andrews, Scotland [1] Original artist: Unknownwikidata:Q4233718 • File:Gottfried_Wilhelm_von_Leibniz.jpg Source: https://upload.wikimedia.org/wikipedia/commons/6/6a/Gottfried_Wilhelm_von_ Leibniz.jpg License: Public domain Contributors: /gbrown/philosophers/leibniz/BritannicaPages/Leibniz/LeibnizGif.html Original artist: Christoph Bernhard Francke • File:Greek_letter_uppercase_Phi.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/96/Greek_letter_uppercase_Phi.svg License: GPLv3 Contributors: A character from the font Linux Libertine. Original artist: • SVG by Tryphon • File:Helmetarg.png Source: https://upload.wikimedia.org/wikipedia/commons/6/6b/Helmetarg.png License: CC BY-SA 3.0 Contribu- tors: Own work Original artist: John Wilkins • File:Hou710_BooleanNetwork.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9c/Hou710_BooleanNetwork.svg Li- cense: CC BY-SA 4.0 Contributors: Own work Original artist: Hou710 • File:IMG_Tree.gif Source: https://upload.wikimedia.org/wikipedia/commons/6/6e/IMG_Tree.gif License: CC-BY-SA-3.0 Contribu- tors: Transferred from zh.wikipedia to Commons by Shizhao using CommonsHelper. Original artist: The original uploader was Mhss at Chinese Wikipedia • File:Illustration_to_Euclid'{}s_proof_of_the_Pythagorean_theorem.svg Source: https://upload.wikimedia.org/wikipedia/commons/ 2/26/Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg License: WTFPL Contributors: ? Original artist: ? • File:Independent_premises.jpg Source: https://upload.wikimedia.org/wikipedia/commons/a/a6/Independent_premises.jpg License: CC BY-SA 4.0 Contributors: Own work Original artist: Philosophyclass HSOG • File:Intermediate_conclusion.jpg Source: https://upload.wikimedia.org/wikipedia/commons/6/6b/Intermediate_conclusion.jpg License: CC BY-SA 4.0 Contributors: Own work Original artist: Philosophyclass HSOG • File:Internet_map_1024.jpg Source: https://upload.wikimedia.org/wikipedia/commons/d/d2/Internet_map_1024.jpg License: CC BY 2.5 Contributors: Originally from the English Wikipedia; description page is/was here. Original artist: The Opte Project 558 CHAPTER 145. ZHEGALKIN POLYNOMIAL

• File:It-.png Source: https://upload.wikimedia.org/wikipedia/commons/2/2b/It-%E6%B3%95.png License: Public domain Contribu- tors: Own work Original artist: Arlas! it • File:Kripke.JPG Source: https://upload.wikimedia.org/wikipedia/commons/d/d4/Kripke.JPG License: Public domain Contributors: Own work Original artist: Oursipan • File:Kurt_gödel.jpg Source: https://upload.wikimedia.org/wikipedia/en/4/42/Kurt_g%C3%B6del.jpg License: ? Contributors: http://www.arithmeum.uni-bonn.de/en/events/285 Original artist: unknown • File:LampFlowchart.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/91/LampFlowchart.svg License: CC-BY-SA- 3.0 Contributors: vector version of Image:LampFlowchart.png Original artist: svg by Booyabazooka

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Original artist: ? • File:Logical_connectives_Hasse_diagram.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3e/Logical_connectives_ Hasse_diagram.svg License: Public domain Contributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk) • File:Logo_for_the_Austrian_Ludwig_Wittgenstein_Society.jpg Source: https://upload.wikimedia.org/wikipedia/en/a/ac/Logo_for_ the_Austrian_Ludwig_Wittgenstein_Society.jpg License: Fair use Contributors: http://www.alws.at/index.php/press/ Original artist: ? • File:Lord_Warden_Cinque_Ports_(Lord_Boyce).svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4b/Lord_Warden_ Cinque_Ports_%28Lord_Boyce%29.svg License: CC BY 3.0 Contributors: Own work by en:User:GrahamPadruig Original artist: Origi- nally uploaded to the English language Wikipedia by GrahamPadruig on 16 December 2007 (log). • File:Lucretius,_De_rerum_natura.jpg Source: https://upload.wikimedia.org/wikipedia/commons/3/3e/Lucretius%2C_De_rerum_natura. jpg License: Public domain Contributors: http://www.ibiblio.org/expo/vatican.exhibit/exhibit/f-medicine_bio/Medicine_bio.html Origi- nal artist: In Latin, Copied by Girolamo di Matteo de Tauris for Sixtus IV, Italy, 1483 • File:Ludwig_Wittgenstein_and_pupils_in_Puchberg.jpg Source: https://upload.wikimedia.org/wikipedia/en/e/e6/Ludwig_Wittgenstein_ and_pupils_in_Puchberg.jpg License: Fair use Contributors: ? Original artist: ? • File:MANNapoli_124545_plato'{}s_academy_mosaic.jpg Source: https://upload.wikimedia.org/wikipedia/commons/5/50/MANNapoli_ 124545_plato%27s_academy_mosaic.jpg License: Public domain Contributors: Own work Jebulon Original artist: ? • File:Maquina.png Source: https://upload.wikimedia.org/wikipedia/commons/3/3d/Maquina.png License: Public domain Contributors: en.wikipedia Original artist: Schadel (http://turing.izt.uam.mx) • File:Merge-arrow.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/aa/Merge-arrow.svg License: Public domain Con- tributors: ? Original artist: ? • File:Merge-arrows.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/52/Merge-arrows.svg License: Public domain Con- tributors: ? Original artist: ? • File:Monty_open_door.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3f/Monty_open_door.svg License: Public do- main Contributors: Own work Original artist: Cepheus • File:Muenchhausen_Herrfurth_7_500x789.jpg Source: https://upload.wikimedia.org/wikipedia/commons/3/3b/Muenchhausen_Herrfurth_ 7_500x789.jpg License: Public domain Contributors: http://www.goethezeitportal.de/wissen/illustrationen/gottfried-august-buerger/die-abenteuer-des-freiherrn-von-muenchhausen/ muenchhausens-abenteuer-in-bildern-von-oskar-herrfurth-folge-2.html Original artist: Oskar Herrfurth • File:Native_gold_nuggets.jpg Source: https://upload.wikimedia.org/wikipedia/commons/6/69/Native_gold_nuggets.jpg License: Pub- lic domain Contributors: ? Original artist: ? • File:Nuvola_apps_kalzium.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/8b/Nuvola_apps_kalzium.svg License: LGPL Contributors: Own work Original artist: David Vignoni, SVG version by Bobarino • File:Nuvola_apps_korganizer.svg Source: https://upload.wikimedia.org/wikipedia/en/e/e6/Nuvola_apps_korganizer.svg License: ? Con- tributors: ? Original artist: ? • File:Objections.png Source: https://upload.wikimedia.org/wikipedia/commons/2/28/Objections.png License: CC BY-SA 3.0 Contrib- utors: Own work Original artist: John Wilkins • File:Office-book.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a8/Office-book.svg License: Public domain Contrib- utors: This and myself. Original artist: Chris Down/Tango project • File:Om_symbol.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b7/Om_symbol.svg License: Public domain Contrib- utors: No machine-readable source provided. Own work assumed (based on copyright claims). Original artist: No machine-readable author provided. Rugby471 assumed (based on copyright claims). • File:Ouroboros.png Source: https://upload.wikimedia.org/wikipedia/commons/f/fa/Ouroboros.png License: Public domain Contribu- tors: ? Original artist: ? • File:PD-icon.svg Source: https://upload.wikimedia.org/wikipedia/en/6/62/PD-icon.svg License: PD Contributors: ? Original artist: ? 145.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 559

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Original artist: ? • File:PeirceAlphaGraphs.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/29/PeirceAlphaGraphs.svg License: CC-BY- SA-3.0 Contributors: from PNG file of the same name by GottschallCH Original artist: Poccil • File:People_icon.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg License: CC0 Contributors: Open- Clipart Original artist: OpenClipart • File:Pinocchio_paradox.png Source: https://upload.wikimedia.org/wikipedia/commons/2/2b/Pinocchio_paradox.png License: Public domain Contributors: File:Pinocchio paradox.jpg Original artist: Carlo Chiostri, User:Mbz1 • File:Pixelbay_Question_mark_Word_Art.png Source: https://upload.wikimedia.org/wikipedia/en/2/23/Pixelbay_Question_mark_Word_ Art.png License: PD Contributors: http://pixabay.com/en/question-inquiry-unknown-mystery-511568/ Original artist: johnhain • File:Pmdsgdbhxdfgb2.jpg Source: https://upload.wikimedia.org/wikipedia/commons/4/44/Pmdsgdbhxdfgb2.jpg License: Public do- main Contributors: http://en.wikipedia.org/wiki/File:Pmdsgdbhxdfgb2.jpg Original artist: Nick Dillinger • File:Portal-puzzle.svg Source: https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg License: Public domain Contributors: ? Original artist: ? • File:Post-lattice-centre.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/26/Post-lattice-centre.svg License: GFDL Con- tributors: Own work Original artist: User:EmilJ • File:Post-lattice.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/19/Post-lattice.svg License: CC BY-SA 3.0 Contrib- utors: Own work Original artist: EmilJ • File:Proprep1.png Source: https://upload.wikimedia.org/wikipedia/commons/2/20/Proprep1.png License: CC-BY-SA-3.0 Contribu- tors: Originally from en.wikibooks; description page is/was here. Original artist: Original uploader was Mrausch at en.wikibooks • File:Proprep2.png Source: https://upload.wikimedia.org/wikipedia/commons/1/15/Proprep2.png License: CC-BY-SA-3.0 Contribu- tors: Originally from en.wikibooks; description page is/was here. 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Original artist: ? • File:Syntax_tree.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/aa/Syntax_tree.svg License: Public domain Contrib- utors: Own work Original artist: Aaron Rotenberg • File:Tea_Leaf_Paradox_Stirring.ogv Source: https://upload.wikimedia.org/wikipedia/commons/0/0f/Tea_Leaf_Paradox_Stirring.ogv License: CC BY-SA 3.0 Contributors: Own work Original artist: Apoc2400 • File:Tetrapharmakos_PHerc_1005_col_5.png Source: https://upload.wikimedia.org/wikipedia/commons/8/8f/Tetrapharmakos_PHerc_ 1005_col_5.png License: Public domain Contributors: http://163.1.169.40/cgi-bin/library?e=q-000-00--$-$0PHerc-$-$00-0-0-$-$0prompt-10--$-$4----de2-$-$0-1l-$-$1-en-50--$-$20-about-1005-$-$00031-001-1-0utfZz-8-00& a=d&c=PHerc&cl=search&d=HASHa4f001cac3c2d3ebb39748 Original artist: Giuseppe (?) 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