Logical Consequence: a Defense of Tarski*
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Logical Truth, Contradictions, Inconsistency, and Logical Equivalence
Logical Truth, Contradictions, Inconsistency, and Logical Equivalence Logical Truth • The semantical concepts of logical truth, contradiction, inconsistency, and logi- cal equivalence in Predicate Logic are straightforward adaptations of the corre- sponding concepts in Sentence Logic. • A closed sentence X of Predicate Logic is logically true, X, if and only if X is true in all interpretations. • The logical truth of a sentence is proved directly using general reasoning in se- mantics. • Given soundness, one can also prove the logical truth of a sentence X by provid- ing a derivation with no premises. • The result of the derivation is that X is theorem, ` X. An Example • (∀x)Fx ⊃ (∃x)Fx. – Suppose d satisfies ‘(∀x)Fx’. – Then all x-variants of d satisfy ‘Fx’. – Since the domain D is non-empty, some x-variant of d satisfies ‘Fx’. – So d satisfies ‘(∃x)Fx’ – Therefore d satisfies ‘(∀x)Fx ⊃ (∃x)Fx’, QED. • ` (∀x)Fx ⊃ (∃x)Fx. 1 (∀x)Fx P 2 Fa 1 ∀ E 3 (∃x)Fx 1 ∃ I Contradictions • A closed sentence X of Predicate Logic is a contradiction if and only if X is false in all interpretations. • A sentence X is false in all interpretations if and only if its negation ∼X is true on all interpretations. • Therefore, one may directly demonstrate that a sentence is a contradiction by proving that its negation is a logical truth. • If the ∼X of a sentence is a logical truth, then given completeness, it is a theorem, and hence ∼X can be derived from no premises. 1 • If a sentence X is such that if it is true in any interpretation, both Y and ∼Y are true in that interpretation, then X cannot be true on any interpretation. -
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. -
Relative Interpretations and Substitutional Definitions of Logical
Relative Interpretations and Substitutional Definitions of Logical Truth and Consequence MIRKO ENGLER Abstract: This paper proposes substitutional definitions of logical truth and consequence in terms of relative interpretations that are extensionally equivalent to the model-theoretic definitions for any relational first-order language. Our philosophical motivation to consider substitutional definitions is based on the hope to simplify the meta-theory of logical consequence. We discuss to what extent our definitions can contribute to that. Keywords: substitutional definitions of logical consequence, Quine, meta- theory of logical consequence 1 Introduction This paper investigates the applicability of relative interpretations in a substi- tutional account of logical truth and consequence. We introduce definitions of logical truth and consequence in terms of relative interpretations and demonstrate that they are extensionally equivalent to the model-theoretic definitions as developed in (Tarski & Vaught, 1956) for any first-order lan- guage (including equality). The benefit of such a definition is that it could be given in a meta-theoretic framework that only requires arithmetic as ax- iomatized by PA. Furthermore, we investigate how intensional constraints on logical truth and consequence force us to extend our framework to an arithmetical meta-theory that itself interprets set-theory. We will argue that such an arithmetical framework still might be in favor over a set-theoretical one. The basic idea behind our definition is both to generate and evaluate substitution instances of a sentence ' by relative interpretations. A relative interpretation rests on a function f that translates all formulae ' of a language L into formulae f(') of a language L0 by mapping the primitive predicates 0 P of ' to formulae P of L while preserving the logical structure of ' 1 Mirko Engler and relativizing its quantifiers by an L0-definable formula. -
The Analytic-Synthetic Distinction and the Classical Model of Science: Kant, Bolzano and Frege
Synthese (2010) 174:237–261 DOI 10.1007/s11229-008-9420-9 The analytic-synthetic distinction and the classical model of science: Kant, Bolzano and Frege Willem R. de Jong Received: 10 April 2007 / Revised: 24 July 2007 / Accepted: 1 April 2008 / Published online: 8 November 2008 © The Author(s) 2008. This article is published with open access at Springerlink.com Abstract This paper concentrates on some aspects of the history of the analytic- synthetic distinction from Kant to Bolzano and Frege. This history evinces con- siderable continuity but also some important discontinuities. The analytic-synthetic distinction has to be seen in the first place in relation to a science, i.e. an ordered system of cognition. Looking especially to the place and role of logic it will be argued that Kant, Bolzano and Frege each developed the analytic-synthetic distinction within the same conception of scientific rationality, that is, within the Classical Model of Science: scientific knowledge as cognitio ex principiis. But as we will see, the way the distinction between analytic and synthetic judgments or propositions functions within this model turns out to differ considerably between them. Keywords Analytic-synthetic · Science · Logic · Kant · Bolzano · Frege 1 Introduction As is well known, the critical Kant is the first to apply the analytic-synthetic distinction to such things as judgments, sentences or propositions. For Kant this distinction is not only important in his repudiation of traditional, so-called dogmatic, metaphysics, but it is also crucial in his inquiry into (the possibility of) metaphysics as a rational science. Namely, this distinction should be “indispensable with regard to the critique of human understanding, and therefore deserves to be classical in it” (Kant 1783, p. -
The Substitutional Analysis of Logical Consequence
THE SUBSTITUTIONAL ANALYSIS OF LOGICAL CONSEQUENCE Volker Halbach∗ dra version ⋅ please don’t quote ónd June óþÕä Consequentia ‘formalis’ vocatur quae in omnibus terminis valet retenta forma consimili. Vel si vis expresse loqui de vi sermonis, consequentia formalis est cui omnis propositio similis in forma quae formaretur esset bona consequentia [...] Iohannes Buridanus, Tractatus de Consequentiis (Hubien ÕÉßä, .ì, p.óóf) Zf«±§Zh± A substitutional account of logical truth and consequence is developed and defended. Roughly, a substitution instance of a sentence is dened to be the result of uniformly substituting nonlogical expressions in the sentence with expressions of the same grammatical category. In particular atomic formulae can be replaced with any formulae containing. e denition of logical truth is then as follows: A sentence is logically true i all its substitution instances are always satised. Logical consequence is dened analogously. e substitutional denition of validity is put forward as a conceptual analysis of logical validity at least for suciently rich rst-order settings. In Kreisel’s squeezing argument the formal notion of substitutional validity naturally slots in to the place of informal intuitive validity. ∗I am grateful to Beau Mount, Albert Visser, and Timothy Williamson for discussions about the themes of this paper. Õ §Z êZo±í At the origin of logic is the observation that arguments sharing certain forms never have true premisses and a false conclusion. Similarly, all sentences of certain forms are always true. Arguments and sentences of this kind are for- mally valid. From the outset logicians have been concerned with the study and systematization of these arguments, sentences and their forms. -
Yuji NISHIYAMA in This Article We Shall Focus Our Attention Upon
The Notion of Presupposition* Yuji NISHIYAMA Keio University In this article we shall focus our attention upon the clarificationof the notion of presupposition;First, we shall be concerned with what kind of notion is best understood as the proper notion of presupposition that grammars of natural languageshould deal with. In this connection,we shall defend "logicalpresupposi tion". Second, we shall consider how to specify the range of X and Y in the presuppositionformula "X presupposes Y". Third, we shall discuss several difficultieswith the standard definition of logical presupposition. In connection with this, we shall propose a clear distinction between "the definition of presupposi tion" and "the rule of presupposition". 1. The logical notion of presupposition What is a presupposition? Or, more particularly, what is the alleged result of a presupposition failure? In spite of the attention given to it, this question can hardly be said to have been settled. Various answers have been suggested by the various authors: inappropriate use of a sentence, failure to perform an illocutionary act, anomaly, ungrammaticality, unintelligibility, infelicity, lack of truth value-each has its advocates. Some of the apparent disagreementmay be only a notational and terminological, but other disagreement raises real, empirically significant, theoretical questions regarding the relationship between logic and language. However, it is not the aim of this article to straighten out the various proposals about the concept of presupposition. Rather, we will be concerned with one notion of presupposition,which stems from Frege (1892)and Strawson (1952),that is: (1) presupposition is a condition under which a sentence expressing an assertive proposition can be used to make a statement (or can bear a truth value)1 We shall call this notion of presupposition "the statementhood condition" or "the conditionfor bivalence". -
The Development of Mathematical Logic from Russell to Tarski: 1900–1935
The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives . -
12 Propositional Logic
CHAPTER 12 ✦ ✦ ✦ ✦ Propositional Logic In this chapter, we introduce propositional logic, an algebra whose original purpose, dating back to Aristotle, was to model reasoning. In more recent times, this algebra, like many algebras, has proved useful as a design tool. For example, Chapter 13 shows how propositional logic can be used in computer circuit design. A third use of logic is as a data model for programming languages and systems, such as the language Prolog. Many systems for reasoning by computer, including theorem provers, program verifiers, and applications in the field of artificial intelligence, have been implemented in logic-based programming languages. These languages generally use “predicate logic,” a more powerful form of logic that extends the capabilities of propositional logic. We shall meet predicate logic in Chapter 14. ✦ ✦ ✦ ✦ 12.1 What This Chapter Is About Section 12.2 gives an intuitive explanation of what propositional logic is, and why it is useful. The next section, 12,3, introduces an algebra for logical expressions with Boolean-valued operands and with logical operators such as AND, OR, and NOT that Boolean algebra operate on Boolean (true/false) values. This algebra is often called Boolean algebra after George Boole, the logician who first framed logic as an algebra. We then learn the following ideas. ✦ Truth tables are a useful way to represent the meaning of an expression in logic (Section 12.4). ✦ We can convert a truth table to a logical expression for the same logical function (Section 12.5). ✦ The Karnaugh map is a useful tabular technique for simplifying logical expres- sions (Section 12.6). -
Presupposition: a Semantic Or Pragmatic Phenomenon?
Arab World English Journal (AWEJ) Volume. 8 Number. 3 September, 2017 Pp. 46 -59 DOI: https://dx.doi.org/10.24093/awej/vol8no3.4 Presupposition: A Semantic or Pragmatic Phenomenon? Mostafa OUALIF Department of English Studies Faculty of Letters and Humanities Ben M’sik, Casablanca Hassan II University, Casablanca, Morocco Abstract There has been debate among linguists with regards to the semantic view and the pragmatic view of presupposition. Some scholars believe that presupposition is purely semantic and others believe that it is purely pragmatic. The present paper contributes to the ongoing debate and exposes the different ways presupposition was approached by linguists. The paper also tries to attend to (i) what semantics is and what pragmatics is in a unified theory of meaning and (ii) the possibility to outline a semantic account of presupposition without having recourse to pragmatics and vice versa. The paper advocates Gazdar’s analysis, a pragmatic analysis, as the safest grounds on which a working grammar of presupposition could be outlined. It shows how semantic accounts are inadequate to deal with the projection problem. Finally, the paper states explicitly that the increasingly puzzling theoretical status of presupposition seems to confirm the philosophical contention that not any fact can be translated into words. Key words: entailment, pragmatics, presupposition, projection problem, semantic theory Cite as: OUALIF, M. (2017). Presupposition: A Semantic or Pragmatic Phenomenon? Arab World English Journal, 8 (3). DOI: https://dx.doi.org/10.24093/awej/vol8no3.4 46 Arab World English Journal (AWEJ) Volume 8. Number. 3 September 2017 Presupposition: A Semantic or Pragmatic Phenomenon? OUALIF I. -
Logical Empiricism and the Verification Theory
Philosophy 308: The Language Revolution Hamilton College Fall 2015 Russell Marcus Classes #14–15: Logical Empiricism and the Verification Theory I. Wittgenstein and the Logical Empiricists We started our exploration into the philosophy of language with Frege’s “On Sense and Reference.” We noted several questions about the nature of senses, but pursued reference, first. Now, we turn to senses, or meanings. In the next few weeks, we will look more closely at some attempts to define, refine, or eliminate meanings in semantic theory. As we will start with the logical empiricist’s verifiability theory of meaning, some background to logical empiricism might be useful. Frege’s work was partly a response to the metaphysical speculations widespread in nineteenth-century idealism. The logical empiricists are also responding to idealism, and speculative metaphysics generally. They were intent on ridding philosophy of what they deemed to be pseudo-problems, pseudo-questions, and meaningless language. Focused on science, they derided such concerns as: A. The meaning of life B. The existence (or non-existence) of God C. Whether the world was created, with all its historical remnants and memories, say, five minutes ago D. Why there is something rather than nothing E. Emergent evolutionary theory, and the elan vital F. Freudian psychology G. Marxist theories of history We can see the logical empiricist’s metaphysical quarry clearly, in Ayer’s derision (pp 11-12) of sentences like AL. AL The absolute is lazy. Logical empiricism is more-often called logical positivism, or just positivism. Those are misleading terms. Logical empiricism is often characterized as British empiricism (e.g. -
• Speech Errors (Review) • Issues in Lexicalization • LRP in Language Production (Review) • N200 (N2b) • N200 in Language Production • Semantic Vs
Plan • Speech Errors (Review) • Issues in Lexicalization • LRP in Language Production (Review) • N200 (N2b) • N200 in Language Production • Semantic vs. Phonological Information • Semantic vs. Syntactic Information • Syntactic vs. Phonological Information • Time Course of Word Production in Picture Naming Eech Sperrors • What can we learn from these things? • Anticipation Errors – a reading list Æ a leading list • Exchange Errors – fill the pool Æ fool the pill • Phonological, lexical, syntactic • Speech is planned in advance – Distance of exchange, anticipation errors suggestive of how far in advance we “plan” Word Substitutions & Word Blends • Semantic Substitutions • Lexicon is organized – That’s a horse of another color semantically AND Æ …a horse of another race phonologically • Phonological Substitutions • Word selection must happen – White Anglo-Saxon Protestant after the grammatical class of Æ …prostitute the target has been • Semantic Blends determined – Edited/annotated Æ editated – Nouns substitute for nouns; verbs for verbs • Phonological Blends – Substitutions don’t result in – Gin and tonic Æ gin and topic ungrammatical sentences • Double Blends – Arrested and prosecuted Æ arrested and persecuted Word Stem & Affix Morphemes •A New Yorker Æ A New Yorkan (American) • Seem to occur prior to lexical insertion • Morphological rules of word formation engaged during speech production Stranding Errors • Nouns & Verbs exchange, but inflectional and derivational morphemes rarely do – Rather, they are stranded • I don’t know that I’d -
Substitution and Propositional Proof Complexity
Substitution and Propositional Proof Complexity Sam Buss Abstract We discuss substitution rules that allow the substitution of formulas for formula variables. A substitution rule was first introduced by Frege. More recently, substitution is studied in the setting of propositional logic. We state theorems of Urquhart’s giving lower bounds on the number of steps in the substitution Frege system for propositional logic. We give the first superlinear lower bounds on the number of symbols in substitution Frege and multi-substitution Frege proofs. “The length of a proof ought not to be measured by the yard. It is easy to make a proof look short on paper by skipping over many links in the chain of inference and merely indicating large parts of it. Generally people are satisfied if every step in the proof is evidently correct, and this is permissable if one merely wishes to be persuaded that the proposition to be proved is true. But if it is a matter of gaining an insight into the nature of this ‘being evident’, this procedure does not suffice; we must put down all the intermediate steps, that the full light of consciousness may fall upon them.” [G. Frege, Grundgesetze, 1893; translation by M. Furth] 1 Introduction The present article concentrates on the substitution rule allowing the substitution of formulas for formula variables.1 The substitution rule has long pedigree as it was a rule of inference in Frege’s Begriffsschrift [9] and Grundgesetze [10], which contained the first in-depth formalization of logical foundations for mathematics. Since then, substitution has become much less important for the foundations of Sam Buss Department of Mathematics, University of California, San Diego, e-mail: [email protected], Supported in part by Simons Foundation grant 578919.