Defationary Truth and Truth-Aptness Illuminated by Language and Norms: Paul Horwich, Huw Price, and Michael Lynch Jeff Ouny G
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CV, Paul Horwich, March 2017
Curriculum Vitae Paul Horwich Department of Philosophy 212 998 8320 (tel) New York University 212 995 4178 (fax) 5 Washington Place [email protected] New York, NY 10003 EDUCATION Cornell University (Philosophy) Ph.D. 1975 Cornell University (Philosophy) M.A. 1973 Yale University (Physics and Philosophy) M.A. 1969 Oxford University (Physics) B.A. 1968 TITLE OF DOCTORAL THESIS: The Metric and Topology of Time. EMPLOYMENT Spring 2007 Visiting Professor, Department of Philosophy, University of Tokyo Fall 2006 Visiting Professor of Philosophy, Ecole Normale Superieure, Paris 2005–present Professor, Department of Philosophy, New York University 2000–2005 Kornblith Distinguished Professor, Philosophy Program, Graduate Center of the City University of New York Spring 1998 Visiting Professor of Philosophy, University of Sydney 1994–2000 Professor, Department of Philosophy, University College London Fall 1994 Associate Research Director, Institute d'Histoire et Philosophie des Sciences et Technique, CNRS, Paris 1987–1994 Professor, Department of Linguistics And Philosophy, Massachusetts Institute of Technology 1980–1987 Associate Professor of Philosophy, MIT Fall 1978 Visiting Assistant Professor of Philosophy, University of California at Los Angeles 1973–1980 Assistant Professor of Philosophy, MIT CV, Paul Horwich, March 2017 GRANTS AND FELLOWSHIPS 2008–9 Guggenheim Fellowship Spring 2007 Fellowship from the Japan Society for the Promotion of Science 2007 U.S. National Endowment for the Humanities Fellowship Fall 1988 U.S. National Science Foundation -
Notes on Mathematical Logic David W. Kueker
Notes on Mathematical Logic David W. Kueker University of Maryland, College Park E-mail address: [email protected] URL: http://www-users.math.umd.edu/~dwk/ Contents Chapter 0. Introduction: What Is Logic? 1 Part 1. Elementary Logic 5 Chapter 1. Sentential Logic 7 0. Introduction 7 1. Sentences of Sentential Logic 8 2. Truth Assignments 11 3. Logical Consequence 13 4. Compactness 17 5. Formal Deductions 19 6. Exercises 20 20 Chapter 2. First-Order Logic 23 0. Introduction 23 1. Formulas of First Order Logic 24 2. Structures for First Order Logic 28 3. Logical Consequence and Validity 33 4. Formal Deductions 37 5. Theories and Their Models 42 6. Exercises 46 46 Chapter 3. The Completeness Theorem 49 0. Introduction 49 1. Henkin Sets and Their Models 49 2. Constructing Henkin Sets 52 3. Consequences of the Completeness Theorem 54 4. Completeness Categoricity, Quantifier Elimination 57 5. Exercises 58 58 Part 2. Model Theory 59 Chapter 4. Some Methods in Model Theory 61 0. Introduction 61 1. Realizing and Omitting Types 61 2. Elementary Extensions and Chains 66 3. The Back-and-Forth Method 69 i ii CONTENTS 4. Exercises 71 71 Chapter 5. Countable Models of Complete Theories 73 0. Introduction 73 1. Prime Models 73 2. Universal and Saturated Models 75 3. Theories with Just Finitely Many Countable Models 77 4. Exercises 79 79 Chapter 6. Further Topics in Model Theory 81 0. Introduction 81 1. Interpolation and Definability 81 2. Saturated Models 84 3. Skolem Functions and Indescernables 87 4. Some Applications 91 5. -
An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics
STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters 10.1090/stml/087 An Introduction to Ramsey Theory STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters Editorial Board Satyan L. Devadoss John Stillwell (Chair) Rosa Orellana Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 05D10, 03-01, 03E10, 03B10, 03B25, 03D20, 03H15. Jan Reimann was partially supported by NSF Grant DMS-1201263. For additional information and updates on this book, visit www.ams.org/bookpages/stml-87 Library of Congress Cataloging-in-Publication Data Names: Katz, Matthew, 1986– author. | Reimann, Jan, 1971– author. | Pennsylvania State University. Mathematics Advanced Study Semesters. Title: An introduction to Ramsey theory: Fast functions, infinity, and metamathemat- ics / Matthew Katz, Jan Reimann. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Student mathematical library; 87 | “Mathematics Advanced Study Semesters.” | Includes bibliographical references and index. Identifiers: LCCN 2018024651 | ISBN 9781470442903 (alk. paper) Subjects: LCSH: Ramsey theory. | Combinatorial analysis. | AMS: Combinatorics – Extremal combinatorics – Ramsey theory. msc | Mathematical logic and foundations – Instructional exposition (textbooks, tutorial papers, etc.). msc | Mathematical -
Model Completeness and Relative Decidability
Model Completeness and Relative Decidability Jennifer Chubb, Russell Miller,∗ & Reed Solomon March 5, 2019 Abstract We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is im- mediate that for a computable model A of a computably enumerable, model complete theory, the entire elementary diagram E(A) must be decidable. We prove that indeed a c.e. theory T is model complete if and only if there is a uniform procedure that succeeds in deciding E(A) from the atomic diagram ∆(A) for all countable models A of T . Moreover, if every presentation of a single isomorphism type A has this property of relative decidability, then there must be a procedure with succeeds uniformly for all presentations of an expansion (A,~a) by finitely many new constants. We end with a conjecture about the situation when all models of a theory are relatively decidable. 1 Introduction The broad goal of computable model theory is to investigate the effective arXiv:1903.00734v1 [math.LO] 2 Mar 2019 aspects of model theory. Here we will carry out exactly this process with the model-theoretic concept of model completeness. This notion is well-known and has been widely studied in model theory, but to our knowledge there has never been any thorough examination of its implications for computability in structures with the domain ω. We now rectify this omission, and find natural and satisfactory equivalents for the basic notion of model completeness of a first-order theory. Our two principal results are each readily stated: that a ∗The second author was supported by NSF grant # DMS-1362206, Simons Foundation grant # 581896, and several PSC-CUNY research awards. -
More Model Theory Notes Miscellaneous Information, Loosely Organized
More Model Theory Notes Miscellaneous information, loosely organized. 1. Kinds of Models A countable homogeneous model M is one such that, for any partial elementary map f : A ! M with A ⊆ M finite, and any a 2 M, f extends to a partial elementary map A [ fag ! M. As a consequence, any partial elementary map to M is extendible to an automorphism of M. Atomic models (see below) are homogeneous. A prime model of T is one that elementarily embeds into every other model of T of the same cardinality. Any theory with fewer than continuum-many types has a prime model, and if a theory has a prime model, it is unique up to isomorphism. Prime models are homogeneous. On the other end, a model is universal if every other model of its size elementarily embeds into it. Recall a type is a set of formulas with the same tuple of free variables; generally to be called a type we require consistency. The type of an element or tuple from a model is all the formulas it satisfies. One might think of the type of an element as a sort of identity card for automorphisms: automorphisms of a model preserve types. A complete type is the analogue of a complete theory, one where every formula of the appropriate free variables or its negation appears. Types of elements and tuples are always complete. A type is principal if there is one formula in the type that implies all the rest; principal complete types are often called isolated. A trivial example of an isolated type is that generated by the formula x = c where c is any constant in the language, or x = t(¯c) where t is any composition of appropriate-arity functions andc ¯ is a tuple of constants. -
LOGIC I 1. the Completeness Theorem 1.1. on Consequences
LOGIC I VICTORIA GITMAN 1. The Completeness Theorem The Completeness Theorem was proved by Kurt G¨odelin 1929. To state the theorem we must formally define the notion of proof. This is not because it is good to give formal proofs, but rather so that we can prove mathematical theorems about the concept of proof. {Arnold Miller 1.1. On consequences and proofs. Suppose that T is some first-order theory. What are the consequences of T ? The obvious answer is that they are statements provable from T (supposing for a second that we know what that means). But there is another possibility. The consequences of T could mean statements that hold true in every model of T . Do the proof theoretic and the model theoretic notions of consequence coincide? Once, we formally define proofs, it will be obvious, by the definition of truth, that a statement that is provable from T must hold in every model of T . Does the converse hold? The question was posed in the 1920's by David Hilbert (of the 23 problems fame). The answer is that remarkably, yes, it does! This result, known as the Completeness Theorem for first-order logic, was proved by Kurt G¨odel in 1929. According to the Completeness Theorem provability and semantic truth are indeed two very different aspects of the same phenomena. In order to prove the Completeness Theorem, we first need a formal notion of proof. As mathematicians, we all know that a proof is a series of deductions, where each statement proceeds by logical reasoning from the previous ones. -
The Structure of a Semantic Theory Author(S): Jerrold J
Linguistic Society of America The Structure of a Semantic Theory Author(s): Jerrold J. Katz and Jerry A. Fodor Source: Language, Vol. 39, No. 2 (Apr. - Jun., 1963), pp. 170-210 Published by: Linguistic Society of America Stable URL: http://www.jstor.org/stable/411200 . Accessed: 22/01/2014 11:13 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]. Linguistic Society of America is collaborating with JSTOR to digitize, preserve and extend access to Language. http://www.jstor.org This content downloaded from 193.219.45.7 on Wed, 22 Jan 2014 11:13:27 AM All use subject to JSTOR Terms and Conditions THE STRUCTURE OF A SEMANTIC THEORY JERROLDJ. KATZ JERRY A. FODOR MassachusettsInstitute of Technology 1. Introduction.This paperl does not attempt to present a semantic theory of a natural language,but rather to characterizethe form of such a theory. A semantic theory of a natural languageis part of a linguisticdescription of that language. Our problem, on the other hand, is part of the general theory of language, fully on a par with the problem of characterizingthe structure of grammarsof natural languages. A characterizationof the abstract form of a semantic theory is given by a metatheory which answers such questions as these: What is the domain of a semantictheory? What are the descriptiveand explanatory goals of a semantic theory? What mechanismsare employed in pursuit of these goals? What are the empiricaland methodologicalconstraints upon a semantic theory? The present paper approachesthe problem of characterizingthe form of semantic theories by describingthe structureof a semantic theory of English. -
The Explanatory Power of a New Proof: Henkin's Completeness Proof
The explanatory power of a new proof: Henkin’s completeness proof John Baldwin February 25, 2017 Mancosu writes But explanations in mathematics do not only come in the form of proofs. In some cases expla- nations are sought in a major recasting of an entire discipline. ([Mancosu, 2008], 142) This paper takes up both halves of that statement. On the one hand we provide a case study of the explanatory value of a particular milestone proof. In the process we examine how it began the recasting of a discipline. Hafner and Mancosu take a broad view towards the nature of mathematical explanation. They ar- gue that before attempting to establish a model of explanation, one should develop a ‘taxonomy of re- current types of mathematical explanation’ ([Hafner and Mancosu, 2005], 221) and preparatory to such a taxonomy propose to examine in depth various examples of proofs. In [Hafner and Mancosu, 2005] and [Hafner and Mancosu, 2008], they study deep arguments in real algebraic geometry and analysis to test the models of explanation of Kitcher and Steiner. In their discussion of Steiner’s model they challenge1 the as- sertion [Resnik and Kushner, 1987] that Henkin’s proof of the completeness theorem is explanatory, asking ‘what the explanatory features of this proof are supposed to consist of?’ As a model theorist the challenge to ‘explain’ the explanatory value of this fundamental argument is irresistible. In contrasting the proofs of Henkin and Godel,¨ we seek for the elements of Henkin’s proofs that permit its numerous generalizations. In Section 2 we try to make this analysis more precise through Steiner’s notion of characterizing property. -
Adopted from Pdflib Image Sample
1 WITH REFERENCE TO TRUTH: STI~IES IN REFERENTIAL SEMANTICS by DOUGLAS FILLMORE CANNON A.B., Harvard University 1973 SUBMITTED TO THE DEPARTMENT OF LINGUISTICS AND PHILOSOPHY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1982 <e> Douglas Fillmore Cannon 1982 The author hereby grants to M.I.T. permission to reproduce and to dis tribute publicly copies of this thesis document in ~hole or in part. Signature of Author__________~-.;~~~---•. ~....;;;""",,---.::l----- Certified by------------------__._1r--------- I < George Boolas Thesis Supervisor Accepted by aL~ «) ~"0 ~ilijarvis Thomson MA5~HUS~~fl~I~~~artment Graduate Committee OF TEtHNOlOSY Archives JUL 8 1982 2 WITH REFERENCE TO TRUTH: STUDIES IN REFERENTIAL SEMANTICS by DOUGLAS FILLMORE CANNON Submitted to the Department of Linguistics and Philosophy on April 8, 1982, in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Philosophy ABSTRACT In the first parts of my thesis I explore two philosophical programs in the area of referential semantics, namely, rigid designation accounts of proper names and naturalistic theories of truth. I conclude with an inquiry into the theory of truth for mathematics and its relationship to mathematical Platonism. In Part One, I confront Kripke's well-known views with Quine's pro posal that proper names correspond to a kind of predicate. I argue that the belief that proper names are rigid designators is unjustified and that many questions about the reference of terms in various possible worlds have no determinate answer. I take issue with Kripke's emphasis on the question, "How is the reference of names determined?", and suggest that it reflects dubious philosophical presuppositions. -
Horwich-Truth.Pdf
Horwich, Paul , Professor of Philosophy , University College London Truth Print ISBN 0198752237, 1998 Contents 1The Minimal Theory 1 A Sketch of the Minimalist Conception 1 The Space of Alternative Theories 8 Summary of Alleged Difficulties 12 The following is a list of the questions and problems regarding minimalism to which replies and solutions will be proposed in the course of this essay 2The Proper Formulation 15 1. 'Of what kinds are the entities to which truth may be attributed?' 16 2. 'What are the fundamental principles of the minimal theory of truth?' 17 3. 'It seems unlikely that instances of the equivalence schema could possibly suffice to explain all of the great variety of facts about truth.' 20 4. 'The minimal theory must be incomplete, for it says nothing about the relationships between truth and affiliated phenomena such as verification, practical success, reference, logical validity, and assertion.' 23 5. 'Even if the minimal theory is, in some sense, "adequate'' and "pure'', it is nevertheless unsatisfactory, being so cumbersome that it cannot even be explicitly formulated.' 25 end p.xiii 6. 'If there were really no more to a complete theory of truth than a list of biconditionals like "The proposition that snow is white is true if and only if snow is white'', then, since one could always say "p'' rather than "The proposition that p is true'', it would be inexplicable that our language should contain the word "true'': there would be no point in having such a notion.' 31 7. 'The minimal theory fails to specify what are meant by attributions of truth. -
Incompatible Ω-Complete Theories∗
Incompatible Ω-Complete Theories∗ Peter Koellner and W. Hugh Woodin July 25, 2009 Abstract In 1985 the second author showed that if there is a proper class of mea- surable Woodin cardinals and V B1 and V B2 are generic extensions of V B1 B2 2 satisfying CH then V and V agree on all Σ1-statements. In terms of the strong logic Ω-logic this can be reformulated by saying that un- der the above large cardinal assumption ZFC + CH is Ω-complete for 2 2 Σ1. Moreover, CH is the unique Σ1-statement with this feature in the 2 sense that any other Σ1-statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for all of third-order arithmetic. Going further, for each specifiable segment Vλ of the uni- verse of sets (for example, one might take Vλ to be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for the theory of Vλ. If such theories exist, extend one another, and are unique in the sense that any other such theory B with the same level of Ω-completeness as A is actually Ω-equivalent to A over ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms comple- menting large cardinal axioms. -
Model Theory, Volume 47, Number 11
fea-pillay.qxp 10/16/00 9:03 AM Page 1373 Model Theory Anand Pillay odel theory is a branch of mathemat- pathology. This is of course true in many ways. In ical logic. It has been considered as a fact, possibly the deepest results of twentieth- subject in its own right since the 1950s. century logic, such as the Gödel incompleteness I will try to convey something of the theorems, the independence of the continuum model-theoretic view of the world of hypothesis from the standard axioms of set theory, Mmathematical objects. The use of the word “model” and the negative solution to Hilbert’s Tenth in model theory is somewhat different from (and Problem, have this character. However, from al- even opposed to) usage in ordinary language. In most the beginning of modern logic there have the latter, a model is a copy or ideal representation been related, but in a sense opposite, trends, often of some real object or phenomena. A model in the within the developing area of model theory. In the sense of model theory, on the other hand, is 1920s Tarski gave his decision procedure for supposed to be the real thing. It is a mathematical elementary Euclidean geometry. A fundamental structure, and that which it is a model of (a set of result, the “compactness theorem” for first-order axioms, say) plays the role of the idealization. logic, was noticed by Gödel, Mal’cev, and Tarski in Abraham Robinson was fond of the expression the 1930s and was, for example, used by Mal’cev “metamathematics” as a description of mathemat- to obtain local theorems in group theory.