DOCSLIB.ORG

Degrees and Deflationism Jared Henderson University of Connecticut - Storrs, [email protected]

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Degrees and Deflationism Jared Henderson University of Connecticut - Storrs, Jared.Henderson@Uconn.Edu University of Connecticut Masthead Logo OpenCommons@UConn Doctoral Dissertations University of Connecticut Graduate School 4-12-2019 Degrees and Deflationism Jared Henderson University of Connecticut - Storrs, [email protected] Follow this and additional works at: https://opencommons.uconn.edu/dissertations Recommended Citation Henderson, Jared, "Degrees and Deflationism" (2019). Doctoral Dissertations. 2099. https://opencommons.uconn.edu/dissertations/2099 Degrees and Deflationism Jared Henderson , PhD University of Connecticut, 2019 Abstract: This dissertation investigates the limits of deflationary theories of truth and offers in their place a degree-theoretic conception of truth. The primary focus is on the intersections of theories of truth and theories of linguistic meaning, particularly as found in contemporary linguistics. I offer two degree theories: one substantive and one deflationary. It is arguedthat the substantive theory is the best descriptive theory of truth, but that the deflationary theory is an attractive revisionary theory. Degrees and Deflationism Jared Henderson B.A., Ohio University , 2013 M.A., Boston University, 2015 M.A., University of Connecticut, 2018 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy at the University of Connecticut 2019 i Copyright by Jared Henderson 2019 ii APPROVAL PAGE Doctor of Philosophy Dissertation Degrees and Deflationism Presented by Jared Henderson, B.A., M.A. Major Advisor Keith Simmons Associate Advisor Stewart Shapiro Associate Advisor Michael P. Lynch Associate Advisor Magdalena Kaufmann University of Connecticut 2019 iii Acknowledgements This is a dissertation about degrees about truth, and in particular about how to make sense of the fact that some truthbearers are more true than others. If it weren’t for any of the people I am about to thank, the contents of this dissertation would be significantly less true. I need to thank the philosophical community at the University of Connecticut, and espe- cially my fellow graduate students. I have grown so much as a philospoher since I joined this department four years ago, and I have routinely been challenged to be more clear, more precise, and more interesting in my arguments by sparring with the philosophers here. In particular, I thank Robin Jenkins, Nathan Kellen, Teresa Allen, Junyeol Kim, Rasa Davidaviciute, Colin McCullough-Benner, Hanna Gun, Ken Ito, Toby Napoletano, Dorit Bar-On, Donald Baxter, Jc Beall, Paul Bloomfield, Bill Lycan, Dave Ripley, Marcus Rossberg, and Lionel Shapiro. I also need to thank some folks who commented on the work of this dissertation or on work that didn’t quite make it in. In particular, I thank Jamin Asay for the meal in Hong Kong and discussions of truthmaking, Juliet Floyd for advising my master’s thesis on Frege on truth at Boston University, Dave Liebesman for comments on almost every paper I have written since 2013, and Chris Kennedy for correspondence on gradable adjectives. I also wish to thank some people in the broader Storrs community, especially the good peo- ple of St. Mark’s Episcopal Chapel, for personal support throughout my graduate career. In particular I thank Bennett Brockman and Brian Blayer. I now turn to my committee members: Stewart Shapiro, Michael Lynch, and Magda Kauf- mann. Stewart read papers of mine before we had ever met in person, and when he visited Storrs he was as helpful and active of a committee member as one could ask for. I benefited greatly from thinking about philosophy, semantics, and the relationship between the two with Stewart. iv Thank you, Stewart, and I hope you don’t think there is too much metaphysical speculation in this dissertation. Michael’s work, especially his book Truth as One and Many, is one of the primary reasons I am interested in the metaphysics of truth. Michael pushed me early on to go beyond debates about deflationism and to put forward a novel theory of truth. That’s whatI’ve tried to do, Michael. Magda has been my primary connection to linguistics since I arrived at UConn. I greatly benefited from her semantics seminar, and it was a comment Magda made about gradable adjectives and the truth predicate that lead me to write the chapter that is the heart of this dissertation. I cannot thank you enough for that, Magda! My primary philosophical influence is Keith Simmons. I approached Keith about my dis- sertation before I knew what I wanted to write on. (I believe I proposed three different topics in our first meeting: deflationism, ontological commitment, and plurals!) Under his guidance, I wrote the present work. He provided extensive comments on more drafts than I can remem- ber, and often I would realize days or weeks later just how helpful his questions in our meetings were. Without Keith, this would be a very different (and worse) dissertation. I am incredibly fortunate to have had you as my advisor, Keith. Thank you for nurturing this project for the past three years. Finally, I thank my family for their support. I especially thank Mengyu Hu, my fiancée, for her everlasting care and for all the laughs. v Contents 1 Scope and Methodology 1 1.1 Introduction ................................. 2 1.2 Theories of truth ............................... 2 1.2.1 Truth ................................. 3 1.2.2 Theories ............................... 4 1.3 Deflationism ................................. 5 1.4 Outline of the dissertation ........................... 7 1.5 Notes on notation ............................... 9 2 Deflationary Semantics 10 2.1 Introduction ................................. 11 2.2 The lightweight reading ............................ 13 2.3 The heavyweight reading ........................... 16 2.4 Objections: Utility and Mystery ........................ 19 2.5 Response to the Theoretical Utility Objection ................ 21 2.5.1 Generalized quantifier theory ..................... 22 vi 2.5.2 Sentential relations .......................... 26 2.5.3 What paraphrases show ........................ 28 2.6 Response to the Mystery Objection ...................... 29 2.7 Conclusion .................................. 32 3 Deflationism and Dependence 34 3.1 Introduction ................................. 35 3.2 The equivalence strategy ........................... 37 3.3 The implicature strategy ............................ 40 3.3.1 Implicature? ............................. 41 3.3.2 A dilemma .............................. 42 3.4 The logicality strategy ............................. 44 3.4.1 true and and ............................. 46 3.4.2 An error theory ............................ 48 3.5 Conclusion .................................. 49 4 True as a Gradable Adjective 51 4.1 Introduction ................................. 52 4.2 True as a gradable adjective .......................... 54 4.2.1 Markers of gradability ........................ 55 4.2.2 Degree semantics ........................... 57 4.2.3 Scales ................................. 59 4.2.4 Semantics for tall and true ...................... 60 4.3 Expressivism about true ............................ 65 vii 4.3.1 Objections to expressivism about true ................ 67 4.3.2 Absolute standards .......................... 69 4.3.3 Probabilities and degrees of truth ................... 70 4.4 Degrees of non-alethic properties? ...................... 71 4.5 Truth as a gradable property .......................... 74 4.5.1 Correspondence ........................... 75 4.5.2 Anti-realism and pluralism ...................... 76 4.6 Truth-related notions ............................. 78 4.6.1 T-schema ............................... 79 4.6.2 Assertion ............................... 80 4.6.3 Logical consequence ......................... 81 4.7 Objections and replies ............................. 83 4.7.1 Frege’s objection ........................... 84 4.7.2 Inappropriate precision ........................ 85 4.7.3 Real metaphysics? .......................... 86 4.8 Conclusion .................................. 88 5 Degrees of Correspondence 89 5.1 Introduction ................................. 90 5.2 Three desiderata ................................ 91 5.2.1 Dependence ............................. 91 5.2.2 More than a platitude ......................... 94 5.2.3 Truthbearers ............................. 95 5.3 Structural Correspondence .......................... 97 viii 5.3.1 Truthbearer structure ......................... 97 5.3.2 Commitment to facts ......................... 99 5.3.3 Structure does not admit of degrees ................. 100 5.3.4 The upshot .............................. 100 5.4 Objectual Correspondence .......................... 101 5.4.1 Aristotle’s theory of truth ...................... 101 5.4.2 Aquinas and Frege .......................... 101 5.5 Correspondence via Aboutness ........................ 103 5.5.1 The basic account .......................... 104 5.5.2 Degrees of objectual correspondence ................. 109 5.5.3 Similarity of objects .......................... 110 5.5.4 Degrees of truth via similarity .................... 113 5.5.5 Maximality, Minimality, and Degrees ................. 116 5.6 Objections and Replies ............................ 118 5.6.1 Aboutness .............................. 118 5.6.2 Context ................................ 120 5.6.3 Incommensurability ......................... 122 5.7 Alternative metaphysical pictures ....................... 123 5.7.1 Coherence .............................. 124 5.7.2 Verification .............................
Load more

Recommended publications
  • A Simple Logic for Comparisons and Vagueness
    THEODORE J. EVERETT A SIMPLE LOGIC FOR COMPARISONS AND VAGUENESS ABSTRACT. I provide an intuitive, semantic account of a new logic for comparisons (CL), in which atomic statements are assigned both a classical truth-value and a “how much” value or extension in the range [0, 1]. The truth-value of each comparison is determined by the extensions of its component sentences; the truth-value of each atomic depends on whether its extension matches a separate standard for its predicate; everything else is computed classically. CL is less radical than Casari’s comparative logics, in that it does not allow for the formation of comparative statements out of truth-functional molecules. I argue that CL provides a better analysis of comparisons and predicate vagueness than classical logic, fuzzy logic or supervaluation theory. CL provides a model for descriptions of the world in terms of comparisons only. The sorites paradox can be solved by the elimination of atomic sentences. In his recent book on vagueness, Timothy Williamson (1994) attacks accounts of vagueness arising from either fuzzy logic or supervaluation theory, both of which have well-known problems, stemming in part from their denial of classical bivalence. He then gives his own, epistemic ac- count of vagueness, according to which the vagueness of a predicate is fundamentally a matter of our not knowing whether or not it applies in every case. My main purpose in this paper is not to criticize William- son’s positive view, but to provide an alternative, non-epistemic account of predicate vagueness, based on a very simple logic for comparisons (CL), which preserves bivalence.
    [Show full text]
  • Does Informal Logic Have Anything to Learn from Fuzzy Logic?
    University of Windsor Scholarship at UWindsor OSSA Conference Archive OSSA 3 May 15th, 9:00 AM - May 17th, 5:00 PM Does informal logic have anything to learn from fuzzy logic? John Woods University of British Columbia Follow this and additional works at: https://scholar.uwindsor.ca/ossaarchive Part of the Philosophy Commons Woods, John, "Does informal logic have anything to learn from fuzzy logic?" (1999). OSSA Conference Archive. 62. https://scholar.uwindsor.ca/ossaarchive/OSSA3/papersandcommentaries/62 This Paper is brought to you for free and open access by the Conferences and Conference Proceedings at Scholarship at UWindsor. It has been accepted for inclusion in OSSA Conference Archive by an authorized conference organizer of Scholarship at UWindsor. For more information, please contact [email protected]. Title: Does informal logic have anything to learn from fuzzy logic? Author: John Woods Response to this paper by: Rolf George (c)2000 John Woods 1. Motivation Since antiquity, philosophers have been challenged by the apparent vagueness of everyday thought and experience. How, these philosophers have wanted to know, are the things we say and think compatible with logical laws such as Excluded Middle?1 A kindred attraction has been the question of how truth presents itself — relatively and in degrees? in approximations? in resemblances, or in bits and pieces? F.H. Bradley is a celebrated (or as the case may be, excoriated) champion of the degrees view of truth. There are, one may say, two main views of error, the absolute and relative. According to the former view there are perfect truths, and on the other side there are sheer errors..
    [Show full text]
  • Intransitivity and Vagueness
    Intransitivity and Vagueness Joseph Y. Halpern∗ Cornell University Computer Science Department Ithaca, NY 14853 [email protected] http://www.cs.cornell.edu/home/halpern Abstract preferences. Moreover, it is this intuition that forms the ba- sis of the partitional model for knowledge used in game the- There are many examples in the literature that suggest that indistinguishability is intransitive, despite the fact that the in- ory (see, e.g., [Aumann 1976]) and in the distributed sys- distinguishability relation is typically taken to be an equiv- tems community [Fagin, Halpern, Moses, and Vardi 1995]. alence relation (and thus transitive). It is shown that if the On the other hand, besides the obvious experimental obser- uncertainty perception and the question of when an agent vations, there have been arguments going back to at least reports that two things are indistinguishable are both care- Poincare´ [1902] that the physical world is not transitive in fully modeled, the problems disappear, and indistinguishabil- this sense. In this paper, I try to reconcile our intuitions ity can indeed be taken to be an equivalence relation. More- about indistinguishability with the experimental observa- over, this model also suggests a logic of vagueness that seems tions, in a way that seems (at least to me) both intuitively to solve many of the problems related to vagueness discussed appealing and psychologically plausible. I then go on to ap- in the philosophical literature. In particular, it is shown here ply the ideas developed to the problem of vagueness. how the logic can handle the Sorites Paradox. To understand the vagueness problem, consider the well- n+1 1 Introduction known Sorites Paradox: If grains of sand make a heap, then so do n.
    [Show full text]
  • On the Relationship Between Fuzzy Autoepistemic Logic and Fuzzy Modal Logics of Belief
    JID:FSS AID:6759 /FLA [m3SC+; v1.204; Prn:31/03/2015; 9:42] P.1(1-26) Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss On the relationship between fuzzy autoepistemic logic and fuzzy modal logics of belief ∗ Marjon Blondeel d, ,1, Tommaso Flaminio a,2, Steven Schockaert b, Lluís Godo c,3, Martine De Cock d,e a Università dell’Insubria, Department of Theorical and Applied Science, Via Mazzini 5, 21100 Varese, Italy b Cardiff University, School of Computer Science and Informatics, 5The Parade, Cardiff, CF24 3AA, UK c Artficial Intelligence Research Institute (IIIA) – CSIC, Campus UAB, 08193 Bellaterra, Spain d Ghent University, Department of Applied Mathematics, Computer Science and Statistics, Krijgslaan 281 (S9), 9000 Gent, Belgium e University of Washington Tacoma – Center for Web and Data Science, 1900 Commerce Street, Tacoma, WA 98402-3100, USA Received 12 February 2014; received in revised form 30 September 2014; accepted 26 February 2015 Abstract Autoepistemic logic is an important formalism for nonmonotonic reasoning originally intended to model an ideal rational agent reflecting upon his own beliefs. Fuzzy autoepistemic logic is a generalization of autoepistemic logic that allows to represent an agent’s rational beliefs on gradable propositions. It has recently been shown that, in the same way as autoepistemic logic generalizes answer set programming, fuzzy autoepistemic logic generalizes fuzzy answer set programming as well. Besides being related to answer set programming, autoepistemic logic is also closely related to several modal logics. To investigate whether a similar relationship holds in a fuzzy logical setting, we firstly generalize the main modal logics for belief to the setting of finitely-valued c Łukasiewicz logic with truth constants Łk, and secondly we relate them with fuzzy autoepistemic logics.
    [Show full text]
  • Tarski's Threat to the T-Schema
    Tarski’s Threat to the T-Schema MSc Thesis (Afstudeerscriptie) written by Cian Chartier (born November 10th, 1986 in Montr´eal, Canada) under the supervision of Prof. dr. Frank Veltman, and submitted to the Board of Examiners in partial fulfillment of the requirements for the degree of MSc in Logic at the Universiteit van Amsterdam. Date of the public defense: Members of the Thesis Committee: August 30, 2011 Prof. Dr. Frank Veltman Prof. Dr. Michiel van Lambalgen Prof. Dr. Benedikt L¨owe Dr. Robert van Rooij Abstract This thesis examines truth theories. First, four relevant programs in philos- ophy are considered. Second, four truth theories are compared according to a range of criteria. The truth theories are categorised according to Leitgeb’s eight criteria for a truth theory. The four truth theories are then compared with each-other based on three new criteria. In this way their relative use- fulness in pursuing some of the aims of the four programs is evaluated. This presents a springboard for future work on truth: proposing ideas for differ- ent truth theories that advance unambiguously different programs on the basis of the four truth theories proposed. 1 Acknowledgements I first thank my supervisor Prof. Dr. Frank Veltman for regularly read- ing and providing constructive feedback, recommending some papers along the way. This thesis has been much improved by his corrections and input, including extensive discussions and suggestions of the ideas proposed here. His patience in working with me on the thesis has been much appreciated. Among the other professors in the ILLC, Dr. Robert van Rooij also provided helpful discussion.
    [Show full text]
  • Consequence and Degrees of Truth in Many-Valued Logic
    Consequence and degrees of truth in many-valued logic Josep Maria Font Published as Chapter 6 (pp. 117–142) of: Peter Hájek on Mathematical Fuzzy Logic (F. Montagna, ed.) vol. 6 of Outstanding Contributions to Logic, Springer-Verlag, 2015 Abstract I argue that the definition of a logic by preservation of all degrees of truth is a better rendering of Bolzano’s idea of consequence as truth- preserving when “truth comes in degrees”, as is often said in many-valued contexts, than the usual scheme that preserves only one truth value. I review some results recently obtained in the investigation of this proposal by applying techniques of abstract algebraic logic in the framework of Łukasiewicz logics and in the broader framework of substructural logics, that is, logics defined by varieties of (commutative and integral) residuated lattices. I also review some scattered, early results, which have appeared since the 1970’s, and make some proposals for further research. Keywords: many-valued logic, mathematical fuzzy logic, truth val- ues, truth degrees, logics preserving degrees of truth, residuated lattices, substructural logics, abstract algebraic logic, Leibniz hierarchy, Frege hierarchy. Contents 6.1 Introduction .............................2 6.2 Some motivation and some history ...............3 6.3 The Łukasiewicz case ........................8 6.4 Widening the scope: fuzzy and substructural logics ....9 6.5 Abstract algebraic logic classification ............. 12 6.6 The Deduction Theorem ..................... 16 6.7 Axiomatizations ........................... 18 6.7.1 In the Gentzen style . 18 6.7.2 In the Hilbert style . 19 6.8 Conclusions .............................. 21 References .................................. 23 1 6.1 Introduction Let me begin by calling your attention to one of the main points made by Petr Hájek in the introductory, vindicating section of his influential book [34] (the italics are his): «Logic studies the notion(s) of consequence.
    [Show full text]
  • A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics
    FLORENTIN SMARANDACHE A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS (fourth edition) NL(A1 A2) = ( T1 ({1}T2) T2 ({1}T1) T1T2 ({1}T1) ({1}T2), I1 ({1}I2) I2 ({1}I1) I1 I2 ({1}I1) ({1} I2), F1 ({1}F2) F2 ({1} F1) F1 F2 ({1}F1) ({1}F2) ). NL(A1 A2) = ( {1}T1T1T2, {1}I1I1I2, {1}F1F1F2 ). NL(A1 A2) = ( ({1}T1T1T2) ({1}T2T1T2), ({1} I1 I1 I2) ({1}I2 I1 I2), ({1}F1F1 F2) ({1}F2F1 F2) ). ISBN 978-1-59973-080-6 American Research Press Rehoboth 1998, 2000, 2003, 2005 FLORENTIN SMARANDACHE A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS (fourth edition) NL(A1 A2) = ( T1 ({1}T2) T2 ({1}T1) T1T2 ({1}T1) ({1}T2), I1 ({1}I2) I2 ({1}I1) I1 I2 ({1}I1) ({1} I2), F1 ({1}F2) F2 ({1} F1) F1 F2 ({1}F1) ({1}F2) ). NL(A1 A2) = ( {1}T1T1T2, {1}I1I1I2, {1}F1F1F2 ). NL(A1 A2) = ( ({1}T1T1T2) ({1}T2T1T2), ({1} I1 I1 I2) ({1}I2 I1 I2), ({1}F1F1 F2) ({1}F2F1 F2) ). ISBN 978-1-59973-080-6 American Research Press Rehoboth 1998, 2000, 2003, 2005 1 Contents: Preface by C. Le: 3 0. Introduction: 9 1. Neutrosophy - a new branch of philosophy: 15 2. Neutrosophic Logic - a unifying field in logics: 90 3. Neutrosophic Set - a unifying field in sets: 125 4. Neutrosophic Probability - a generalization of classical and imprecise probabilities - and Neutrosophic Statistics: 129 5. Addenda: Definitions derived from Neutrosophics: 133 2 Preface to Neutrosophy and Neutrosophic Logic by C.
    [Show full text]
  • Reading T. S. Eliot Reading Spinoza
    READING T. S. ELIOT READING SPINOZA A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Yoshiaki Mihara January 2013 © 2013 Yoshiaki Mihara READING T. S. ELIOT READING SPINOZA Yoshiaki Mihara, Ph. D. Cornell University 2013 “[The ordinary man] falls in love, or reads Spinoza, and these two experiences have nothing to do with each other” – a good reader of T. S. Eliot’s criticism probably knows this famous passage; a good researcher on Eliot’s apprenticeship in philosophy perhaps knows that he did actually read Spinoza; and yet, in the current Eliot studies, reading Eliot and reading Spinoza seem to have nothing to do with each other. In this dissertation, I attempt to reconstruct Eliot’s reading of Spinoza as faithfully and comprehensively as possible, by closely analyzing the marginalia in Eliot’s copy of Spinoza’s Opera, housed in the Archive Centre of King’s College, Cambridge. At the same time, the Spinozist context for Eliot’s apprenticeship at Harvard and Oxford (with the intermission of “a romantic year” in Paris) is also to be presented, which is, in fact, a glaring absence in the philosophical branch of Eliot studies. In addition to these positivistic contributions, I also take a theoretical approach so as to demonstrate how illuminative Eliot’s reading of Spinoza can be for understanding the characteristic style (or “ethology”) of Eliot’s reading in general (i.e., Theory), by way of extensively analyzing the unpublished as well as published materials of Eliot’s “academic philosophizing” that culminated in his doctoral dissertation on F.
    [Show full text]
  • Modality and Factivity: One Perspective on the Meaning of the English Modal Auxiliaries
    MODALITY AND FACTIVITY: ONE PERSPECTIVE ON THE MEANING OF THE ENGLISH MODAL AUXILIARIES by NICOLA M,BREWER Submitted in accordance with the requirements for the degree of PhD The University of Leeds Department of Linguistics and Phonetics December 1987 ii ABSTRACT This study concentrates on modality as expressed by the set of modal auxiliaries and seeks to establish that these verbs share semantic as well as syntactic properties by identifying a single core meaning which they share. The relationship between modality and factivity is examined with the aim of gaining an insight into the former, more complex concept. When viewed from this perspective, the defining characteristic of all the modal auxiliary verbs in almost all of their uses is found to be nonfactivity. The meanings expressed by this set of verbs are classified according to a framework derived from modal logic consisting of three basic types of modality each of which relates to a different set of laws or principles; the relative factivity associated with the modal auxiliaries is seen to vary with the nature of modality as defined and classified by this framework. Within each of the three types of modality, a semantic scale is identified and modality is described as a gradable concept for which scalar analysis is appropriate, both within and beyond these three scales. Relative factivity is also shown to vary according to the degree of modality expressed by each of the modal verbs. The nature and degree of modality expressed interact with features of the linguistic (and pragmatic) context to determine the particular factive or a contrafactive interpretation conveyed by a given modal auxiliary token.
    [Show full text]
  • Irrationality and Human Reasoning Michael Aristidou Louisiana State University and Agricultural and Mechanical College
    Louisiana State University LSU Digital Commons LSU Master's Theses Graduate School 2004 Irrationality and human reasoning Michael Aristidou Louisiana State University and Agricultural and Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_theses Part of the Arts and Humanities Commons Recommended Citation Aristidou, Michael, "Irrationality and human reasoning" (2004). LSU Master's Theses. 2918. https://digitalcommons.lsu.edu/gradschool_theses/2918 This Thesis is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Master's Theses by an authorized graduate school editor of LSU Digital Commons. For more information, please contact [email protected]. IRRATIONALITY AND HUMAN REASONING A Thesis Submitted to the Graduate Faculty of the Louisiana State University in partial fulfillment of the requirements of the degree of Master of Arts in The Department of Philosophy by Michael Aristidou BSc, Aristotle University of Thessaloniki, 1999 MSc, Louisiana State University, 2001 May, 2004 To my parents Theodotos and Kristia, to my sister Eliza, and to my brother Konstantinos for all their love and support through all these years. ii ACKNOWLEDGMENTS I would like to thank my advisor Dr. Jon Cogburn for his great patience and valuable instruction on my thesis. I would also like to thank, Dr. Mary Sirridge for her advising and help through my graduate years in Philosophy, the members of my thesis committee Dr. Husain Sarkar and Dr. Ian Crystal for their cooperation, and Taneka Welch for her constant willingness to discuss issues related to my thesis, and her technical support.
    [Show full text]
  • First-Order Logic
    Chapter 10 First-Order Logic 10.1 Overview First-Order Logic is the calculus one usually has in mind when using the word ‘‘logic’’. It is expressive enough for all of mathematics, except for those concepts that rely on a notion of construction or computation. However, dealing with more advanced concepts is often somewhat awkward and researchers often design specialized logics for that reason. Our account of first-order logic will be similar to the one of propositional logic. We will present ² The syntax, or the formal language of first-order logic, that is symbols, formulas, sub-formulas, formation trees, substitution, etc. ² The semantics of first-order logic ² Proof systems for first-order logic, such as the axioms, rules, and proof strategies of the first-order tableau method and refinement logic ² The meta-mathematics of first-order logic, which established the relation between the semantics and a proof system In many ways, the account of first-order logic is a straightforward extension of propositional logic. One must, however, be aware that there are subtle differences. 10.2 Syntax The syntax of first-order logic is essentially an extension of propositional logic by quantification 8 and 9. Propositional variables are replaced by n-ary predicate symbols (P , Q, R) which may be instantiated with either variables (x, y, z, ...) or parameters (a, b, ...). Here is a summary of the most important concepts. 101 102 CHAPTER 10. FIRST-ORDER LOGIC (1) Atomic formulas are expressions of the form P c1::cn where P is an n-ary predicate symbol and the ci are variables or parameters.
    [Show full text]
  • A Subsystem of Classical Propositional Logic
    VALUATIONS FOR DIRECT PROPOSITIONAL LOGIC In (1969) a subsystem of classical propositional logic - called direct logic since all indirect inferences are excluded from it - was formulated as a generalized sequential calculus. With its rules the truth-value (true (t) or false (0) of the consequent can be determined from the truth-values of the premisses. In this way the semantical stipulations are formulated construc• tively as in a calculus of natural deduction, i.e. semantics itself is a formal system of rules. The rules for propositional composita define an extension of basic calculi containing only truth rules for atomic sentences. The sen• tences provable in the extensions of all basic calculi, i.e. in the proposi• tional calculus itself, are the logically true sentences. The principle of bi- valence is not presupposed for the basic calculi. That it holds for a specific calculus - and then also in its propositional extension - is rather a theorem to be proved by metatheoretic means. In this approach a semantics in the usual sense, i.e. a semantics based on a concept of valuation, has no place since the calculus of direct logic itself is already conceived of as a system of semantical rules. Nevertheless it is of some interest to see that there is an adequate and intuitively plau• sible valuation-semantics for this calculus. The language L used in what follows is the usual language of proposi• tional logic with —i, A and n> as basic operators.1 A classical or total valuation of L is a function V mapping the set of all sentences of L into {t, f} so that K(-i A) = t iff V(A) = f, V(A A B) = t iff V(A) = V(B) = t.
    [Show full text]