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Pluralisms About Truth and Logic Nathan Kellen University of Connecticut - Storrs, [email protected]
University of Connecticut OpenCommons@UConn Doctoral Dissertations University of Connecticut Graduate School 8-9-2019 Pluralisms about Truth and Logic Nathan Kellen University of Connecticut - Storrs, [email protected] Follow this and additional works at: https://opencommons.uconn.edu/dissertations Recommended Citation Kellen, Nathan, "Pluralisms about Truth and Logic" (2019). Doctoral Dissertations. 2263. https://opencommons.uconn.edu/dissertations/2263 Pluralisms about Truth and Logic Nathan Kellen, PhD University of Connecticut, 2019 Abstract: In this dissertation I analyze two theories, truth pluralism and logical pluralism, as well as the theoretical connections between them, including whether they can be combined into a single, coherent framework. I begin by arguing that truth pluralism is a combination of realist and anti-realist intuitions, and that we should recognize these motivations when categorizing and formulating truth pluralist views. I then introduce logical functionalism, which analyzes logical consequence as a functional concept. I show how one can both build theories from the ground up and analyze existing views within the functionalist framework. One upshot of logical functionalism is a unified account of logical monism, pluralism and nihilism. I conclude with two negative arguments. First, I argue that the most prominent form of logical pluralism faces a serious dilemma: it either must give up on one of the core principles of logical consequence, and thus fail to be a theory of logic at all, or it must give up on pluralism itself. I call this \The Normative Problem for Logical Pluralism", and argue that it is unsolvable for the most prominent form of logical pluralism. Second, I examine an argument given by multiple truth pluralists that purports to show that truth pluralists must also be logical pluralists. -
Misconceived Relationships Between Logical Positivism and Quantitative Research: an Analysis in the Framework of Ian Hacking
DOCUMENT RESUME ED 452 266 TM 032 553 AUTHOR Yu, Chong Ho TITLE Misconceived Relationships between Logical Positivism and Quantitative Research: An Analysis in the Framework of Ian Hacking. PUB DATE 2001-04-07 NOTE 26p. PUB TYPE Opinion Papers (120) ED 2S PRICE MF01/PCO2 Plus Postage. 'DESCRIPTORS *Educational Research; *Research Methodology IDENTIFIERS *Logical Positivism ABSTRACT Although quantitative research methodology is widely applied by psychological researchers, there is a common misconception that quantitative research is based on logical positivism. This paper examines the relationship between quantitative research and eight major notions of logical positivism:(1) verification;(2) pro-observation;(3) anti-cause; (4) downplaying explanation;(5) anti-theoretical entities;(6) anti-metaphysics; (7) logical analysis; and (8) frequentist probability. It is argued that the underlying philosophy of modern quantitative research in psychology is in sharp contrast to logical positivism. Putting the labor of an out-dated philosophy into quantitative research may discourage psychological researchers from applying this research approach and may also lead to misguided dispute between quantitative and qualitative researchers. What is needed is to encourage researchers and students to keep an open mind to different methodologies and apply skepticism to examine the philosophical assumptions instead of accepting them unquestioningly. (Contains 1 figure and 75 references.)(Author/SLD) Reproductions supplied by EDRS are the best that can be made from the original document. Misconceived relationships between logical positivism and quantitative research: An analysis in the framework of Ian Hacking Chong Ho Yu, Ph.D. Arizona State University April 7, 2001 N N In 4-1 PERMISSION TO REPRODUCE AND DISSEMINATE THIS MATERIALHAS BEEN GRANTED BY Correspondence: TO THE EDUCATIONAL RESOURCES Chong Ho Yu, Ph.D. -
John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected]
John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected] LOGIC & PHILOSOPHICAL METHODOLOGY Introduction For present purposes “logic” will be understood to mean the subject whose development is described in Kneale & Kneale [1961] and of which a concise history is given in Scholz [1961]. As the terminological discussion at the beginning of the latter reference makes clear, this subject has at different times been known by different names, “analytics” and “organon” and “dialectic”, while inversely the name “logic” has at different times been applied much more broadly and loosely than it will be here. At certain times and in certain places — perhaps especially in Germany from the days of Kant through the days of Hegel — the label has come to be used so very broadly and loosely as to threaten to take in nearly the whole of metaphysics and epistemology. Logic in our sense has often been distinguished from “logic” in other, sometimes unmanageably broad and loose, senses by adding the adjectives “formal” or “deductive”. The scope of the art and science of logic, once one gets beyond elementary logic of the kind covered in introductory textbooks, is indicated by two other standard references, the Handbooks of mathematical and philosophical logic, Barwise [1977] and Gabbay & Guenthner [1983-89], though the latter includes also parts that are identified as applications of logic rather than logic proper. The term “philosophical logic” as currently used, for instance, in the Journal of Philosophical Logic, is a near-synonym for “nonclassical logic”. There is an older use of the term as a near-synonym for “philosophy of language”. -
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. -
Shorter Curriculum Vitae Akihiro Kanamori
SHORTER CURRICULUM VITAE AKIHIRO KANAMORI DESCRIPTION: Born 23 October 1948 in Tokyo, Japan; now a United States citizen. DEGREES: 1966-1970 California Institute of Technology, Bachelor of Science. 1970-1975 University of Cambridge (King's College), Doctor of Philosophy. Subject: Set Theory, Mathematics. Thesis: Ultrafilters over Uncountable Cardinals. Advisor: A.R.D. Mathias. This involved one year of research at: 1972-1973 University of Wisconsin, Madison. Advisor: K. Kunen. PROFESSIONAL EXPERIENCE: 1975-1977 Lectureship at the University of California, Berkeley. 1977-1981 Benjamin Pierce Assistant Professorship at Harvard University. 1981-1982 Assistant Professorship at Baruch College of the City University of New York. 1982-1983 Visiting Associate Professorship at Boston University. 1983-1992 Associate Professorship at Boston University. 1988-1989 Berman Visiting Professorship, Institute of Mathematics, Hebrew University of Jerusalem. 1992- Professorship at Boston University. 1995 Visiting Professorship, Institute of Mathematics, Hebrew Universiy of Jerusalem. 2002-2003 Senior Fellow of the Dibner Institute for the History of Science and Technology. Visiting Scholar at the Department of the History of Science at Harvard University. 1 2009-2010 Senior Fellow of the Lichtenberg-Kolleg, Institute for Advanced Study, G¨ottingen,Germany. Lecture Course on Set Theory, Mathematische Institut, G¨ottingen,Germany, June-July 2010. FELLOWSHIPS AND AWARDS: Marshall Scholarship (British Government), 1970-1972. Danforth Foundation Fellowship, 1970-1975. Woodrow Wilson Foundation Fellowship, 1970. 1984 New England Open Co-Champion of Chess. Equal First 1986 Greater Boston Chess Open. Equal Second, 1987 Massachusetts Chess Open Championship. Equal Sixth, 1989 Israel Open. Class Prize, 1992 New England Open Championship. 2002-2003 Senior Fellowship, Dibner Institute for the History of Science and Technology. -
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. -
Philosophy of Logic (Routledge Revivals)
Routledge Revivals Philosophy of Logic First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general. This page intentionally left blank Philosophy of Logic Hilary Putnam First published in the UK in 1972 by George Allen & Unwin Ltd This edition first published in 2010 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN Simultaneously published in the USA and Canada by Routledge 270 Madison Avenue, New York, NY 10016 Routledge is an imprint of the Taylor & Francis Group, an informa business © 1971 Hilary Putnam All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Publisher’s Note The publisher has gone to great lengths to ensure the quality of this reprint but points out that some imperfections in the original copies may be apparent. Disclaimer The publisher has made every effort to trace copyright holders and welcomes correspondence from those they have been unable to contact. A Library of Congress record exists under ISBN: 0041600096 ISBN 13: 978-0-415-58092-2 (hbk) ISBN 10: 0-415-58092-7 (hbk) Philosophy of Logic This page intentionally left blank This page intentionally left blank This page intentionally left blank Philosophy of Logic ••••• Hilary Putnam PHILOSOPHY OF LOGIC. -
Redalyc.Logical Consequence for Nominalists
THEORIA. Revista de Teoría, Historia y Fundamentos de la Ciencia ISSN: 0495-4548 [email protected] Universidad del País Vasco/Euskal Herriko Unibertsitatea España ROSSBERG, Marcus; COHNITZ, Daniel Logical Consequence for Nominalists THEORIA. Revista de Teoría, Historia y Fundamentos de la Ciencia, vol. 24, núm. 2, 2009, pp. 147- 168 Universidad del País Vasco/Euskal Herriko Unibertsitatea Donostia-San Sebastián, España Available in: http://www.redalyc.org/articulo.oa?id=339730809003 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Logical Consequence for Nominalists Marcus ROSSBERG and Daniel COHNITZ BIBLID [0495-4548 (2009) 24: 65; pp. 147-168] ABSTRACT: It has repeatedly been argued that nominalistic programmes in the philosophy of mathematics fail, since they will at some point or other involve the notion of logical consequence which is unavailable to the nominalist. In this paper we will argue that this is not the case. Using an idea of Nelson Goodman and W.V.Quine's which they developed in Goodman and Quine (1947) and supplementing it with means that should be nominalistically acceptable, we present a way to explicate logical consequence in a nominalistically acceptable way. Keywords: Philosophy of mathematics, nominalism, logical consequence, inferentialism, Nelson Goodman, W.V. Quine. 1. The Argument from Logical Consequence We do not have any strong convictions concerning the question of the existence or non- existence of abstract objects. We do, however, believe that ontological fastidiousness is prima facie a good attitude to adopt. -
Philosophy of Logic
Philosophy of Logic Fall 2005 - Winter 2006 Our goal over these two quarters is to think through a series of positions on the nature of logical truth. We’ll focus on the most fundamental questions: what is the ground of logical truth? (what makes logical truths true?), and how do we come to know these truths? I have in mind here the simplest of logical truths -- if it’s either red or green and it’s not red, then it must be green -- or the simplest of logical validities -- any situation in which all men are mortal and Socrates is a man is a situation in which Socrates is mortal. The default requirement for those taking the course for a grade (other than S/U) is three short papers (750-1250 words) due at the beginning of class in the 4th week, 7th week, and 10th week. Each paper should isolate one localized point in the readings and offer some analysis and/or critique. Other options are open to negotiation. I assume everyone has access to copies of: Carnap, Logical Syntax of Language. Frege, A Frege Reader (edited by Beany). Wittgenstein, Tractatus Logico-Philosophicus (preferably the Ogden translation). Philosophical Investigations. Course copies of Anscombe, Black, Fogelin, Hacker, Kenny, Kripke, McGinn, Mounce, Ostrow, Pears, Stenius, Stern (both books), and Wittgenstein’s Remarks on the Foundations of Mathematics will be kept in Brian Rogers’ office for borrowing. All other assigned reading (plus some extra material for the curious) will be available outside my office for photocopying. Please come to the first meeting prepared to discuss the Kant reading. -
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.