Notes on the Founding of Logics and Metalogic: Aristotle, Boole, and Tarski
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Poznań Studies in the Philosophy of the Sciences and the Humanities), 21Pp
Forthcoming in: Uncovering Facts and Values, ed. A. Kuzniar and J. Odrowąż-Sypniewska (Poznań Studies in the Philosophy of the Sciences and the Humanities), 21pp. Abstract Alfred Tarski seems to endorse a partial conception of truth, the T-schema, which he believes might be clarified by the application of empirical methods, specifically citing the experimental results of Arne Næss (1938a). The aim of this paper is to argue that Næss’ empirical work confirmed Tarski’s semantic conception of truth, among others. In the first part, I lay out the case for believing that Tarski’s T-schema, while not the formal and generalizable Convention-T, provides a partial account of truth that may be buttressed by an examination of the ordinary person’s views of truth. Then, I address a concern raised by Tarski’s contemporaries who saw Næss’ results as refuting Tarski’s semantic conception. Following that, I summarize Næss’ results. Finally, I will contend with a few objections that suggest a strict interpretation of Næss’ results might recommend an overturning of Tarski’s theory. Keywords: truth, Alfred Tarski, Arne Næss, Vienna Circle, experimental philosophy Joseph Ulatowski ORDINARY TRUTH IN TARSKI AND NÆSS 1. Introduction Many of Alfred Tarski's better known papers on truth (e.g. 1944; 1983b), logical consequence (1983c), semantic concepts in general (1983a), or definability (1948) identify two conditions that successful definitions of “truth,” “logical consequence,” or “definition” require: formal correctness and material (or intuitive) adequacy.1 The first condition Tarski calls “formal correctness” because a definition of truth (for a given formal language) is formally correct when it is constructed in a manner that allows us to avoid both circular definition and semantic paradoxes. -
Logic and Its Metatheory
Logic and its Metatheory Philosophy 432 Spring 2017 T & TH: 300-450pm Olds Hall 109 Instructor Information Matthew McKeon Office: 503 South Kedzie Hall Office hours: 1-2:30pm on T & TH and by appointment. Telephone: Dept. Office—355-4490 E-mail: [email protected] Required Text Courseware Package (Text plus CD software): Jon Barwise and John Etchemendy. Language, Proof and Logic. 2nd edition. Stanford, CA: CLSI Publications, 2011. The courseware package includes several software tools that students will use to complete homework assignments and the self-diagnostic exercises from the text. DO NOT BUY USED BOOKS, BECAUSE THEY LACK THE CD, WHICH CONTAINS THE SOFTWARE. The software, easy to learn and use, makes some of the more technical aspects of symbolic logic accessible to introductory students and it is fun to use (or so I think). It is OK to have an earlier edition of the text as long as you have the CD with the password key that will allow you to access the text’s online site (https://ggweb.gradegrinder.net/lpl). Here you can update the software suite and download the 2nd edition of the text. At the site you can also purchase the software and an electronic version of the text. Primary Course Objectives • Deepen your understanding of the concept of logical consequence and sharpen your ability to see that one sentence is a logical consequence of others. • Introduce you to the basic methods of proof (both formal and informal), and teach you how to use them effectively to prove that one sentence is a logical consequence of others. -
The Metamathematics of Putnam's Model-Theoretic Arguments
The Metamathematics of Putnam's Model-Theoretic Arguments Tim Button Abstract. Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted mod- els. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I exam- ine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to meta- mathematical challenges. Copyright notice. This paper is due to appear in Erkenntnis. This is a pre-print, and may be subject to minor changes. The authoritative version should be obtained from Erkenntnis, once it has been published. Hilary Putnam famously attempted to use model theory to draw metaphys- ical conclusions. Specifically, he attacked metaphysical realism, a position characterised by the following credo: [T]he world consists of a fixed totality of mind-independent objects. (Putnam 1981, p. 49; cf. 1978, p. 125). Truth involves some sort of correspondence relation between words or thought-signs and external things and sets of things. (1981, p. 49; cf. 1989, p. 214) [W]hat is epistemically most justifiable to believe may nonetheless be false. (1980, p. 473; cf. 1978, p. 125) To sum up these claims, Putnam characterised metaphysical realism as an \externalist perspective" whose \favorite point of view is a God's Eye point of view" (1981, p. 49). Putnam sought to show that this externalist perspective is deeply untenable. To this end, he treated correspondence in terms of model-theoretic satisfaction. -
Set-Theoretic Geology, the Ultimate Inner Model, and New Axioms
Set-theoretic Geology, the Ultimate Inner Model, and New Axioms Justin William Henry Cavitt (860) 949-5686 [email protected] Advisor: W. Hugh Woodin Harvard University March 20, 2017 Submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Mathematics and Philosophy Contents 1 Introduction 2 1.1 Author’s Note . .4 1.2 Acknowledgements . .4 2 The Independence Problem 5 2.1 Gödelian Independence and Consistency Strength . .5 2.2 Forcing and Natural Independence . .7 2.2.1 Basics of Forcing . .8 2.2.2 Forcing Facts . 11 2.2.3 The Space of All Forcing Extensions: The Generic Multiverse 15 2.3 Recap . 16 3 Approaches to New Axioms 17 3.1 Large Cardinals . 17 3.2 Inner Model Theory . 25 3.2.1 Basic Facts . 26 3.2.2 The Constructible Universe . 30 3.2.3 Other Inner Models . 35 3.2.4 Relative Constructibility . 38 3.3 Recap . 39 4 Ultimate L 40 4.1 The Axiom V = Ultimate L ..................... 41 4.2 Central Features of Ultimate L .................... 42 4.3 Further Philosophical Considerations . 47 4.4 Recap . 51 1 5 Set-theoretic Geology 52 5.1 Preliminaries . 52 5.2 The Downward Directed Grounds Hypothesis . 54 5.2.1 Bukovský’s Theorem . 54 5.2.2 The Main Argument . 61 5.3 Main Results . 65 5.4 Recap . 74 6 Conclusion 74 7 Appendix 75 7.1 Notation . 75 7.2 The ZFC Axioms . 76 7.3 The Ordinals . 77 7.4 The Universe of Sets . 77 7.5 Transitive Models and Absoluteness . -
Against Logical Form
Against logical form Zolta´n Gendler Szabo´ Conceptions of logical form are stranded between extremes. On one side are those who think the logical form of a sentence has little to do with logic; on the other, those who think it has little to do with the sentence. Most of us would prefer a conception that strikes a balance: logical form that is an objective feature of a sentence and captures its logical character. I will argue that we cannot get what we want. What are these extreme conceptions? In linguistics, logical form is typically con- ceived of as a level of representation where ambiguities have been resolved. According to one highly developed view—Chomsky’s minimalism—logical form is one of the outputs of the derivation of a sentence. The derivation begins with a set of lexical items and after initial mergers it splits into two: on one branch phonological operations are applied without semantic effect; on the other are semantic operations without phono- logical realization. At the end of the first branch is phonological form, the input to the articulatory–perceptual system; and at the end of the second is logical form, the input to the conceptual–intentional system.1 Thus conceived, logical form encompasses all and only information required for interpretation. But semantic and logical information do not fully overlap. The connectives “and” and “but” are surely not synonyms, but the difference in meaning probably does not concern logic. On the other hand, it is of utmost logical importance whether “finitely many” or “equinumerous” are logical constants even though it is hard to see how this information could be essential for their interpretation. -
Alfred Tarski His:Bio:Tar: Sec Alfred Tarski Was Born on January 14, 1901 in Warsaw, Poland (Then Part of the Russian Empire)
bio.1 Alfred Tarski his:bio:tar: sec Alfred Tarski was born on January 14, 1901 in Warsaw, Poland (then part of the Russian Empire). Described as \Napoleonic," Tarski was boisterous, talkative, and intense. His energy was often reflected in his lectures|he once set fire to a wastebasket while disposing of a cigarette during a lecture, and was forbidden from lecturing in that building again. Tarski had a thirst for knowledge from a young age. Although later in life he would tell students that he stud- ied logic because it was the only class in which he got a B, his high school records show that he got A's across the board| even in logic. He studied at the Univer- sity of Warsaw from 1918 to 1924. Tarski Figure 1: Alfred Tarski first intended to study biology, but be- came interested in mathematics, philosophy, and logic, as the university was the center of the Warsaw School of Logic and Philosophy. Tarski earned his doctorate in 1924 under the supervision of Stanislaw Le´sniewski. Before emigrating to the United States in 1939, Tarski completed some of his most important work while working as a secondary school teacher in Warsaw. His work on logical consequence and logical truth were written during this time. In 1939, Tarski was visiting the United States for a lecture tour. During his visit, Germany invaded Poland, and because of his Jewish heritage, Tarski could not return. His wife and children remained in Poland until the end of the war, but were then able to emigrate to the United States as well. -
Truth-Bearers and Truth Value*
Truth-Bearers and Truth Value* I. Introduction The purpose of this document is to explain the following concepts and the relationships between them: statements, propositions, and truth value. In what follows each of these will be discussed in turn. II. Language and Truth-Bearers A. Statements 1. Introduction For present purposes, we will define the term “statement” as follows. Statement: A meaningful declarative sentence.1 It is useful to make sure that the definition of “statement” is clearly understood. 2. Sentences in General To begin with, a statement is a kind of sentence. Obviously, not every string of words is a sentence. Consider: “John store.” Here we have two nouns with a period after them—there is no verb. Grammatically, this is not a sentence—it is just a collection of words with a dot after them. Consider: “If I went to the store.” This isn’t a sentence either. “I went to the store.” is a sentence. However, using the word “if” transforms this string of words into a mere clause that requires another clause to complete it. For example, the following is a sentence: “If I went to the store, I would buy milk.” This issue is not merely one of conforming to arbitrary rules. Remember, a grammatically correct sentence expresses a complete thought.2 The construction “If I went to the store.” does not do this. One wants to By Dr. Robert Tierney. This document is being used by Dr. Tierney for teaching purposes and is not intended for use or publication in any other manner. 1 More precisely, a statement is a meaningful declarative sentence-type. -
The Mechanical Problems in the Corpus of Aristotle
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications, Classics and Religious Studies Department Classics and Religious Studies July 2007 The Mechanical Problems in the Corpus of Aristotle Thomas Nelson Winter University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/classicsfacpub Part of the Classics Commons Winter, Thomas Nelson, "The Mechanical Problems in the Corpus of Aristotle" (2007). Faculty Publications, Classics and Religious Studies Department. 68. https://digitalcommons.unl.edu/classicsfacpub/68 This Article is brought to you for free and open access by the Classics and Religious Studies at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications, Classics and Religious Studies Department by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Th e Mechanical Problems in the Corpus of Aristotle Th omas N. Winter Lincoln, Nebraska • 2007 Translator’s Preface. Who Wrote the Mechanical Problems in the Aristotelian Corpus? When I was translating the Mechanical Problems, which I did from the TLG Greek text, I was still in the fundamentalist authorship mode: that it survives in the corpus of Aristotle was then for me prima facie Th is paper will: evidence that Aristotle was the author. And at many places I found in- 1) off er the plainest evidence yet that it is not Aristotle, and — 1 dications that the date of the work was apt for Aristotle. But eventually, 2) name an author. I saw a join in Vitruvius, as in the brief summary below, “Who Wrote Th at it is not Aristotle does not, so far, rest on evidence. -
George Boole?
iCompute For more fun computing lessons and resources visit: Who was George Boole? 8 He was an English mathematician 8 He believed that human thought could be George Boole written down as ‘rules’ 8 His ideas led to boolean logic which is Biography for children used by computers today The story of important figures in the history of computing George Boole (1815 – 1864) © iCompute 2015 www.icompute -uk.com iCompute Why is George Boole important? 8 He invented a set of rules for thinking that are used by computers today 8 The rules were that some statements can only ever be ‘true’ or ‘false’ 8 His idea was first used in computers as switches either being ‘on’ or ‘off’ 8 Today this logic is used in almost every device and almost every line of computer code His early years nd 8 Born 2 November 1815 8 His father was a struggling shoemaker 8 George had had very little education and taught himself maths, French, German and Latin © iCompute 2015 www.icompute -uk.com iCompute 8 He also taught himself Greek and published a translation of a Greek poem in 1828 at the age of 14! 8 Aged 16, the family business collapsed and George began working as a teacher to support the family 8 At 19 he started his own school 8 In 1840 he began having his books about mathematics published 8 In 1844, he was given the first gold medal for Mathematics by the Royal Society 8 Despite never having been to University himself, in 1849 he became professor of Mathematics at Queens College Cork in Ireland 8 He married Mary Everett in 1855 8 They had four daughters between 1956-1864 -
1 Peter Mclaughlin the Question of the Authenticity of the Mechanical
Draft: occasionally updated Peter McLaughlin The Question of the Authenticity of the Mechanical Problems Sept. 30, 2013 Until the nineteenth century there was little doubt among scholars as to the authenticity of the Aristotelian Mechanical Problems. There were some doubters among the Renaissance humanists, but theirs were general doubts about the authenticity of a large class of writings, 1 not doubts based on the individual characteristics of this particular work. Some Humanists distrusted any text that hadn’t been passed by the Arabs to the Latin West in the High Middle Ages. However, by the end of the 18th century after Euler and Lagrange, the Mechanical Problems had ceased to be read as part of science and had become the object of history of science; and there the reading of the text becomes quite different from the Renaissance 2 readings. In his Histoire des mathématiques J.E. Montucla (1797) dismisses the Mechanical Problems with such epithets as “entirely false,” “completely ridiculous,” and “puerile.” William Whewell remarks in the History of the Inductive Sciences3 that “in scarcely any one instance are the answers, which Aristotle gives to his questions, of any value.” Neither of them, however, cast doubt on the authenticity of the work. Abraham Kaestner’s Geschichte der Mathematik (1796–1800) mentions doubts – but does not share them.4 Serious doubts about the authenticity of the Mechanical Problems as an individual work seem to be more a consequence of the disciplinary constitution of classical philology, particularly in nineteenth-century Germany. Some time between about 1830 and 1870, the opinion of most philologists shifted from acceptance to denial of the authenticity of the 5 Mechanical Problems. -
The Etienne Gilson Series 21
The Etienne Gilson Series 21 Remapping Scholasticism by MARCIA L. COLISH 3 March 2000 Pontifical Institute of Mediaeval Studies This lecture and its publication was made possible through the generous bequest of the late Charles J. Sullivan (1914-1999) Note: the author may be contacted at: Department of History Oberlin College Oberlin OH USA 44074 ISSN 0-708-319X ISBN 0-88844-721-3 © 2000 by Pontifical Institute of Mediaeval Studies 59 Queen’s Park Crescent East Toronto, Ontario, Canada M5S 2C4 Printed in Canada nce upon a time there were two competing story-lines for medieval intellectual history, each writing a major role for scholasticism into its script. Although these story-lines were O created independently and reflected different concerns, they sometimes overlapped and gave each other aid and comfort. Both exerted considerable influence on the way historians of medieval speculative thought conceptualized their subject in the first half of the twentieth cen- tury. Both versions of the map drawn by these two sets of cartographers illustrated what Wallace K. Ferguson later described as “the revolt of the medievalists.”1 One was confined largely to the academy and appealed to a wide variety of medievalists, while the other had a somewhat narrower draw and reflected political and confessional, as well as academic, concerns. The first was the anti-Burckhardtian effort to push Renaissance humanism, understood as combining a knowledge and love of the classics with “the discovery of the world and of man,” back into the Middle Ages. The second was inspired by the neo-Thomist revival launched by Pope Leo XIII, and was inhabited almost exclusively by Roman Catholic scholars. -
Truth-Conditions
PLIN0009 Semantic Theory Spring 2020 Lecture Notes 1 1 What this module is about This module is an introduction to truth-conditional semantics with a focus on two impor- tant topics in this area: compositionality and quantiication. The framework adopted here is often called formal semantics and/or model-theoretical semantics, and it is characterized by its essential use of tools and concepts developed in mathematics and logic in order to study semantic properties of natural languages. Although no textbook is required, I list some introductory textbooks below for your refer- ence. • L. T. F. Gamut (1991) Logic, Language, and Meaning. The University of Chicago Press. • Irene Heim & Angelika Kratzer (1998) Semantics in Generative Grammar. Blackwell. • Thomas Ede Zimmermann & Wolfgang Sternefeld (2013) Introduction to Semantics: An Essential Guide to the Composition of Meaning. De Gruyter Mouton. • Pauline Jacobson (2014) Compositional Semantics: An Introduction to the Syntax/Semantics. Oxford University Press. • Daniel Altshuler, Terence Parsons & Roger Schwarzschild (2019) A Course in Semantics. MIT Press. There are also several overview articles of the ield by Barbara H. Partee, which I think are enjoyable. • Barbara H. Partee (2011) Formal semantics: origins, issues, early impact. In Barbara H. Partee, Michael Glanzberg & Jurģis Šķilters (eds.), Formal Semantics and Pragmatics: Discourse, Context, and Models. The Baltic Yearbook of Cognition, Logic, and Communica- tion, vol. 6. Manhattan, KS: New Prairie Press. • Barbara H. Partee (2014) A brief history of the syntax-semantics interface in Western Formal Linguistics. Semantics-Syntax Interface, 1(1): 1–21. • Barbara H. Partee (2016) Formal semantics. In Maria Aloni & Paul Dekker (eds.), The Cambridge Handbook of Formal Semantics, Chapter 1, pp.