Propositional and Predicate Calculus
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Chapter 2 Introduction to Classical Propositional
CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC 1 Motivation and History The origins of the classical propositional logic, classical propositional calculus, as it was, and still often is called, go back to antiquity and are due to Stoic school of philosophy (3rd century B.C.), whose most eminent representative was Chryssipus. But the real development of this calculus began only in the mid-19th century and was initiated by the research done by the English math- ematician G. Boole, who is sometimes regarded as the founder of mathematical logic. The classical propositional calculus was ¯rst formulated as a formal ax- iomatic system by the eminent German logician G. Frege in 1879. The assumption underlying the formalization of classical propositional calculus are the following. Logical sentences We deal only with sentences that can always be evaluated as true or false. Such sentences are called logical sentences or proposi- tions. Hence the name propositional logic. A statement: 2 + 2 = 4 is a proposition as we assume that it is a well known and agreed upon truth. A statement: 2 + 2 = 5 is also a proposition (false. A statement:] I am pretty is modeled, if needed as a logical sentence (proposi- tion). We assume that it is false, or true. A statement: 2 + n = 5 is not a proposition; it might be true for some n, for example n=3, false for other n, for example n= 2, and moreover, we don't know what n is. Sentences of this kind are called propositional functions. We model propositional functions within propositional logic by treating propositional functions as propositions. -
Logophoricity in Finnish
Open Linguistics 2018; 4: 630–656 Research Article Elsi Kaiser* Effects of perspective-taking on pronominal reference to humans and animals: Logophoricity in Finnish https://doi.org/10.1515/opli-2018-0031 Received December 19, 2017; accepted August 28, 2018 Abstract: This paper investigates the logophoric pronoun system of Finnish, with a focus on reference to animals, to further our understanding of the linguistic representation of non-human animals, how perspective-taking is signaled linguistically, and how this relates to features such as [+/-HUMAN]. In contexts where animals are grammatically [-HUMAN] but conceptualized as the perspectival center (whose thoughts, speech or mental state is being reported), can they be referred to with logophoric pronouns? Colloquial Finnish is claimed to have a logophoric pronoun which has the same form as the human-referring pronoun of standard Finnish, hän (she/he). This allows us to test whether a pronoun that may at first blush seem featurally specified to seek [+HUMAN] referents can be used for [-HUMAN] referents when they are logophoric. I used corpus data to compare the claim that hän is logophoric in both standard and colloquial Finnish vs. the claim that the two registers have different logophoric systems. I argue for a unified system where hän is logophoric in both registers, and moreover can be used for logophoric [-HUMAN] referents in both colloquial and standard Finnish. Thus, on its logophoric use, hän does not require its referent to be [+HUMAN]. Keywords: Finnish, logophoric pronouns, logophoricity, anti-logophoricity, animacy, non-human animals, perspective-taking, corpus 1 Introduction A key aspect of being human is our ability to think and reason about our own mental states as well as those of others, and to recognize that others’ perspectives, knowledge or mental states are distinct from our own, an ability known as Theory of Mind (term due to Premack & Woodruff 1978). -
Predicate Logic
Logical Form & Predicate Logic Jean Mark Gawron Linguistics San Diego State University [email protected] http://www.rohan.sdsu.edu/∼gawron Logical Form & Predicate Logic – p. 1/30 Predicates and Arguments Logical Form & Predicate Logic – p. 2/30 Logical Form The Logical Form of English sentences can be represented by formulae of Predicate Logic, What do we mean by the logical form of a sentence? We mean: we can capture the truth conditions of complex English expressions in predicate logic, given some account of the denotations (extensions) of the simple expressions (words). Logical Form & Predicate Logic – p. 3/30 The Leading Idea: Extensions Nouns, verbs, and adjectives have predicates as their translations. What does ’predicate’ mean? ’Predicate’ means what it means in predicate logic. Logical Form & Predicate Logic – p. 4/30 Intransitive verbs Translating into predicate logic (1) a. Johnwalks. b. walk(j) (2) a. [[j]] = John b. “The extension of ’j’ is the individual John.” c. [[walk]] = {x | x walks } d. “The extension of ’walk’ is the set of walkers” (3) a. [[John walks]] = [[walk(j)]] b. [[walk(j)]] = true iff [[j]] ∈ [[walk]] Logical Form & Predicate Logic – p. 5/30 Extensional denotations • Remember two kinds of denotation. For work with logic, denotations are always extensions. • We call a denotation such as [[walk]] (a set) an extension, because it is defined by the set of things the word walk describes or “extends over” • The denotation [[j]] is also extensional because it is defined by the individual the word John describes. • Next we extend extensional denotation to transitive verbs, nouns, and sentences. -
Aristotle's Illicit Quantifier Shift: Is He Guilty Or Innocent
Aristos Volume 1 Issue 2 Article 2 9-2015 Aristotle's Illicit Quantifier Shift: Is He Guilty or Innocent Jack Green The University of Notre Dame Australia Follow this and additional works at: https://researchonline.nd.edu.au/aristos Part of the Philosophy Commons, and the Religious Thought, Theology and Philosophy of Religion Commons Recommended Citation Green, J. (2015). "Aristotle's Illicit Quantifier Shift: Is He Guilty or Innocent," Aristos 1(2),, 1-18. https://doi.org/10.32613/aristos/ 2015.1.2.2 Retrieved from https://researchonline.nd.edu.au/aristos/vol1/iss2/2 This Article is brought to you by ResearchOnline@ND. It has been accepted for inclusion in Aristos by an authorized administrator of ResearchOnline@ND. For more information, please contact [email protected]. Green: Aristotle's Illicit Quantifier Shift: Is He Guilty or Innocent ARISTOTLE’S ILLICIT QUANTIFIER SHIFT: IS HE GUILTY OR INNOCENT? Jack Green 1. Introduction Aristotle’s Nicomachean Ethics (from hereon NE) falters at its very beginning. That is the claim of logicians and philosophers who believe that in the first book of the NE Aristotle mistakenly moves from ‘every action and pursuit aims at some good’ to ‘there is some one good at which all actions and pursuits aim.’1 Yet not everyone is convinced of Aristotle’s seeming blunder.2 In lieu of that, this paper has two purposes. Firstly, it is an attempt to bring some clarity to that debate in the face of divergent opinions of the location of the fallacy; some proposing it lies at I.i.1094a1-3, others at I.ii.1094a18-22, making it difficult to wade through the literature. -
Pronouns, Logical Variables, and Logophoricity in Abe Author(S): Hilda Koopman and Dominique Sportiche Source: Linguistic Inquiry, Vol
MIT Press Pronouns, Logical Variables, and Logophoricity in Abe Author(s): Hilda Koopman and Dominique Sportiche Source: Linguistic Inquiry, Vol. 20, No. 4 (Autumn, 1989), pp. 555-588 Published by: MIT Press Stable URL: http://www.jstor.org/stable/4178645 Accessed: 22-10-2015 18:32 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. MIT Press is collaborating with JSTOR to digitize, preserve and extend access to Linguistic Inquiry. http://www.jstor.org This content downloaded from 128.97.27.20 on Thu, 22 Oct 2015 18:32:27 UTC All use subject to JSTOR Terms and Conditions Hilda Koopman Pronouns, Logical Variables, Dominique Sportiche and Logophoricity in Abe 1. Introduction 1.1. Preliminaries In this article we describe and analyze the propertiesof the pronominalsystem of Abe, a Kwa language spoken in the Ivory Coast, which we view as part of the study of pronominalentities (that is, of possible pronominaltypes) and of pronominalsystems (that is, of the cooccurrence restrictionson pronominaltypes in a particulargrammar). Abe has two series of thirdperson pronouns. One type of pronoun(0-pronoun) has basically the same propertiesas pronouns in languageslike English. The other type of pronoun(n-pronoun) very roughly corresponds to what has been called the referential use of pronounsin English(see Evans (1980)).It is also used as what is called a logophoric pronoun-that is, a particularpronoun that occurs in special embedded contexts (the logophoric contexts) to indicate reference to "the person whose speech, thought or perceptions are reported" (Clements (1975)). -
Chapter 6 Mirativity and the Bulgarian Evidential System Elena Karagjosova Freie Universität Berlin
Chapter 6 Mirativity and the Bulgarian evidential system Elena Karagjosova Freie Universität Berlin This paper provides an account of the Bulgarian admirative construction andits place within the Bulgarian evidential system based on (i) new observations on the morphological, temporal, and evidential properties of the admirative, (ii) a criti- cal reexamination of existing approaches to the Bulgarian evidential system, and (iii) insights from a similar mirative construction in Spanish. I argue in particular that admirative sentences are assertions based on evidence of some sort (reporta- tive, inferential, or direct) which are contrasted against the set of beliefs held by the speaker up to the point of receiving the evidence; the speaker’s past beliefs entail a proposition that clashes with the assertion, triggering belief revision and resulting in a sense of surprise. I suggest an analysis of the admirative in terms of a mirative operator that captures the evidential, temporal, aspectual, and modal properties of the construction in a compositional fashion. The analysis suggests that although mirativity and evidentiality can be seen as separate semantic cate- gories, the Bulgarian admirative represents a cross-linguistically relevant case of a mirative extension of evidential verbal forms. Keywords: mirativity, evidentiality, fake past 1 Introduction The Bulgarian evidential system is an ongoing topic of discussion both withre- spect to its interpretation and its morphological buildup. In this paper, I focus on the currently poorly understood admirative construction. The analysis I present is based on largely unacknowledged observations and data involving the mor- phological structure, the syntactic environment, and the evidential meaning of the admirative. Elena Karagjosova. -
Toward a Shared Syntax for Shifted Indexicals and Logophoric Pronouns
Toward a Shared Syntax for Shifted Indexicals and Logophoric Pronouns Mark Baker Rutgers University April 2018 Abstract: I argue that indexical shift is more like logophoricity and complementizer agreement than most previous semantic accounts would have it. In particular, there is evidence of a syntactic requirement at work, such that the antecedent of a shifted “I” must be a superordinate subject, just as the antecedent of a logophoric pronoun or the goal of complementizer agreement must be. I take this to be evidence that the antecedent enters into a syntactic control relationship with a null operator in all three constructions. Comparative data comes from Magahi and Sakha (for indexical shift), Yoruba (for logophoric pronouns), and Lubukusu (for complementizer agreement). 1. Introduction Having had an office next to Lisa Travis’s for 12 formative years, I learned many things from her that still influence my thinking. One is her example of taking semantic notions, such as aspect and event roles, and finding ways to implement them in syntactic structure, so as to advance the study of less familiar languages and topics.1 In that spirit, I offer here some thoughts about how logophoricity and indexical shift, topics often discussed from a more or less semantic point of view, might have syntactic underpinnings—and indeed, the same syntactic underpinnings. On an impressionistic level, it would not seem too surprising for logophoricity and indexical shift to have a common syntactic infrastructure. Canonical logophoricity as it is found in various West African languages involves using a special pronoun inside the finite CP complement of a verb to refer to the subject of that verb. -
The Predicate View of Names
The Predicate View of Names Aidan Gray March 28, 2008 There is, and has for some time been, a consensus among philosophers interested in lan- guage that proper names in natural language are properly thought of on the model of individual constants in a formal language. This is to say, at least, that their semantic function is to pick out individuals. This as opposed to, e.g., intransitive verbs, whose semantic function is to pick out properties of individuals (let’s say) or quantifiers phrases whose function is to pick out prop- erties of properties (let’s say). There has been disagreement as to the details of this approach — for instance, in grasping the meaning of the name must I simply know which object it refers to or must I also know the manner in which the name presents the object? — but the basic premise has rarely been questioned. What reasons have philosophers had to assimilate names to individual constants? The ques- tion immediately strikes one as odd. The oddness is, in part, an expression of fact the picture is so deeply held as to constitute part of our framework for asking questions about language. More interestingly, perhaps, the question strikes one as odd because it seems like, as philosophers, we understand individual constants on the model of names just as much as we understand names on the model of individual constants. Individual constants were introduced into formal languages to do the job that names do in natural languages. Though this may well be true, it has little bearing on the question at hand. -
Scope Ambiguity in Syntax and Semantics
Scope Ambiguity in Syntax and Semantics Ling324 Reading: Meaning and Grammar, pg. 142-157 Is Scope Ambiguity Semantically Real? (1) Everyone loves someone. a. Wide scope reading of universal quantifier: ∀x[person(x) →∃y[person(y) ∧ love(x,y)]] b. Wide scope reading of existential quantifier: ∃y[person(y) ∧∀x[person(x) → love(x,y)]] 1 Could one semantic representation handle both the readings? • ∃y∀x reading entails ∀x∃y reading. ∀x∃y describes a more general situation where everyone has someone who s/he loves, and ∃y∀x describes a more specific situation where everyone loves the same person. • Then, couldn’t we say that Everyone loves someone is associated with the semantic representation that describes the more general reading, and the more specific reading obtains under an appropriate context? That is, couldn’t we say that Everyone loves someone is not semantically ambiguous, and its only semantic representation is the following? ∀x[person(x) →∃y[person(y) ∧ love(x,y)]] • After all, this semantic representation reflects the syntax: In syntax, everyone c-commands someone. In semantics, everyone scopes over someone. 2 Arguments for Real Scope Ambiguity • The semantic representation with the scope of quantifiers reflecting the order in which quantifiers occur in a sentence does not always represent the most general reading. (2) a. There was a name tag near every plate. b. A guard is standing in front of every gate. c. A student guide took every visitor to two museums. • Could we stipulate that when interpreting a sentence, no matter which order the quantifiers occur, always assign wide scope to every and narrow scope to some, two, etc.? 3 Arguments for Real Scope Ambiguity (cont.) • But in a negative sentence, ¬∀x∃y reading entails ¬∃y∀x reading. -
30. Tense Aspect Mood 615
30. Tense Aspect Mood 615 Richards, Ivor Armstrong 1936 The Philosophy of Rhetoric. Oxford: Oxford University Press. Rockwell, Patricia 2007 Vocal features of conversational sarcasm: A comparison of methods. Journal of Psycho- linguistic Research 36: 361−369. Rosenblum, Doron 5. March 2004 Smart he is not. http://www.haaretz.com/print-edition/opinion/smart-he-is-not- 1.115908. Searle, John 1979 Expression and Meaning. Cambridge: Cambridge University Press. Seddiq, Mirriam N. A. Why I don’t want to talk to you. http://notguiltynoway.com/2004/09/why-i-dont-want- to-talk-to-you.html. Singh, Onkar 17. December 2002 Parliament attack convicts fight in court. http://www.rediff.com/news/ 2002/dec/17parl2.htm [Accessed 24 July 2013]. Sperber, Dan and Deirdre Wilson 1986/1995 Relevance: Communication and Cognition. Oxford: Blackwell. Voegele, Jason N. A. http://www.jvoegele.com/literarysf/cyberpunk.html Voyer, Daniel and Cheryl Techentin 2010 Subjective acoustic features of sarcasm: Lower, slower, and more. Metaphor and Symbol 25: 1−16. Ward, Gregory 1983 A pragmatic analysis of epitomization. Papers in Linguistics 17: 145−161. Ward, Gregory and Betty J. Birner 2006 Information structure. In: B. Aarts and A. McMahon (eds.), Handbook of English Lin- guistics, 291−317. Oxford: Basil Blackwell. Rachel Giora, Tel Aviv, (Israel) 30. Tense Aspect Mood 1. Introduction 2. Metaphor: EVENTS ARE (PHYSICAL) OBJECTS 3. Polysemy, construal, profiling, and coercion 4. Interactions of tense, aspect, and mood 5. Conclusion 6. References 1. Introduction In the framework of cognitive linguistics we approach the grammatical categories of tense, aspect, and mood from the perspective of general cognitive strategies. -
12 Propositional Logic
CHAPTER 12 ✦ ✦ ✦ ✦ Propositional Logic In this chapter, we introduce propositional logic, an algebra whose original purpose, dating back to Aristotle, was to model reasoning. In more recent times, this algebra, like many algebras, has proved useful as a design tool. For example, Chapter 13 shows how propositional logic can be used in computer circuit design. A third use of logic is as a data model for programming languages and systems, such as the language Prolog. Many systems for reasoning by computer, including theorem provers, program verifiers, and applications in the field of artificial intelligence, have been implemented in logic-based programming languages. These languages generally use “predicate logic,” a more powerful form of logic that extends the capabilities of propositional logic. We shall meet predicate logic in Chapter 14. ✦ ✦ ✦ ✦ 12.1 What This Chapter Is About Section 12.2 gives an intuitive explanation of what propositional logic is, and why it is useful. The next section, 12,3, introduces an algebra for logical expressions with Boolean-valued operands and with logical operators such as AND, OR, and NOT that Boolean algebra operate on Boolean (true/false) values. This algebra is often called Boolean algebra after George Boole, the logician who first framed logic as an algebra. We then learn the following ideas. ✦ Truth tables are a useful way to represent the meaning of an expression in logic (Section 12.4). ✦ We can convert a truth table to a logical expression for the same logical function (Section 12.5). ✦ The Karnaugh map is a useful tabular technique for simplifying logical expres- sions (Section 12.6). -
Discrete Mathematics, Chapter 1.4-1.5: Predicate Logic
Discrete Mathematics, Chapter 1.4-1.5: Predicate Logic Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 1 / 23 Outline 1 Predicates 2 Quantifiers 3 Equivalences 4 Nested Quantifiers Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 2 / 23 Propositional Logic is not enough Suppose we have: “All men are mortal.” “Socrates is a man”. Does it follow that “Socrates is mortal” ? This cannot be expressed in propositional logic. We need a language to talk about objects, their properties and their relations. Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 3 / 23 Predicate Logic Extend propositional logic by the following new features. Variables: x; y; z;::: Predicates (i.e., propositional functions): P(x); Q(x); R(y); M(x; y);::: . Quantifiers: 8; 9. Propositional functions are a generalization of propositions. Can contain variables and predicates, e.g., P(x). Variables stand for (and can be replaced by) elements from their domain. Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 4 / 23 Propositional Functions Propositional functions become propositions (and thus have truth values) when all their variables are either I replaced by a value from their domain, or I bound by a quantifier P(x) denotes the value of propositional function P at x. The domain is often denoted by U (the universe). Example: Let P(x) denote “x > 5” and U be the integers. Then I P(8) is true. I P(5) is false.