1 Chapter 1 Two Versions of the Liar Paradox
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Logic, Reasoning, and Revision
Logic, Reasoning, and Revision Abstract The traditional connection between logic and reasoning has been under pressure ever since Gilbert Harman attacked the received view that logic yields norms for what we should believe. In this paper I first place Harman’s challenge in the broader context of the dialectic between logical revisionists like Bob Meyer and sceptics about the role of logic in reasoning like Harman. I then develop a formal model based on contemporary epistemic and doxastic logic in which the relation between logic and norms for belief can be captured. introduction The canons of classical deductive logic provide, at least according to a widespread but presumably naive view, general as well as infallible norms for reasoning. Obviously, few instances of actual reasoning have such prop- erties, and it is therefore not surprising that the naive view has been chal- lenged in many ways. Four kinds of challenges are relevant to the question of where the naive view goes wrong, but each of these challenges is also in- teresting because it allows us to focus on a particular aspect of the relation between deductive logic, reasoning, and logical revision. Challenges to the naive view that classical deductive logic directly yields norms for reason- ing come in two sorts. Two of them are straightforwardly revisionist; they claim that the consequence relation of classical logic is the culprit. The remaining two directly question the naive view about the normative role of logic for reasoning; they do not think that the philosophical notion of en- tailment (however conceived) is as relevant to reasoning as has traditionally been assumed. -
Dialetheists' Lies About the Liar
PRINCIPIA 22(1): 59–85 (2018) doi: 10.5007/1808-1711.2018v22n1p59 Published by NEL — Epistemology and Logic Research Group, Federal University of Santa Catarina (UFSC), Brazil. DIALETHEISTS’LIES ABOUT THE LIAR JONAS R. BECKER ARENHART Departamento de Filosofia, Universidade Federal de Santa Catarina, BRAZIL [email protected] EDERSON SAFRA MELO Departamento de Filosofia, Universidade Federal do Maranhão, BRAZIL [email protected] Abstract. Liar-like paradoxes are typically arguments that, by using very intuitive resources of natural language, end up in contradiction. Consistent solutions to those paradoxes usually have difficulties either because they restrict the expressive power of the language, orelse because they fall prey to extended versions of the paradox. Dialetheists, like Graham Priest, propose that we should take the Liar at face value and accept the contradictory conclusion as true. A logical treatment of such contradictions is also put forward, with the Logic of Para- dox (LP), which should account for the manifestations of the Liar. In this paper we shall argue that such a formal approach, as advanced by Priest, is unsatisfactory. In order to make contradictions acceptable, Priest has to distinguish between two kinds of contradictions, in- ternal and external, corresponding, respectively, to the conclusions of the simple and of the extended Liar. Given that, we argue that while the natural interpretation of LP was intended to account for true and false sentences, dealing with internal contradictions, it lacks the re- sources to tame external contradictions. Also, the negation sign of LP is unable to represent internal contradictions adequately, precisely because of its allowance of sentences that may be true and false. -
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. -
Mathematical Logic
Proof Theory. Mathematical Logic. From the XIXth century to the 1960s, logic was essentially mathematical. Development of first-order logic (1879-1928): Frege, Hilbert, Bernays, Ackermann. Development of the fundamental axiom systems for mathematics (1880s-1920s): Cantor, Peano, Zermelo, Fraenkel, Skolem, von Neumann. Giuseppe Peano (1858-1932) Core Logic – 2005/06-1ab – p. 2/43 Mathematical Logic. From the XIXth century to the 1960s, logic was essentially mathematical. Development of first-order logic (1879-1928): Frege, Hilbert, Bernays, Ackermann. Development of the fundamental axiom systems for mathematics (1880s-1920s): Cantor, Peano, Zermelo, Fraenkel, Skolem, von Neumann. Traditional four areas of mathematical logic: Proof Theory. Recursion Theory. Model Theory. Set Theory. Core Logic – 2005/06-1ab – p. 2/43 Gödel (1). Kurt Gödel (1906-1978) Studied at the University of Vienna; PhD supervisor Hans Hahn (1879-1934). Thesis (1929): Gödel Completeness Theorem. 1931: “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”. Gödel’s First Incompleteness Theorem and a proof sketch of the Second Incompleteness Theorem. Core Logic – 2005/06-1ab – p. 3/43 Gödel (2). 1935-1940: Gödel proves the consistency of the Axiom of Choice and the Generalized Continuum Hypothesis with the axioms of set theory (solving one half of Hilbert’s 1st Problem). 1940: Emigration to the USA: Princeton. Close friendship to Einstein, Morgenstern and von Neumann. Suffered from severe hypochondria and paranoia. Strong views on the philosophy of mathematics. Core Logic – 2005/06-1ab – p. 4/43 Gödel’s Incompleteness Theorem (1). 1928: At the ICM in Bologna, Hilbert claims that the work of Ackermann and von Neumann constitutes a proof of the consistency of arithmetic. -
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. -
In Defence of Constructive Empiricism: Metaphysics Versus Science
To appear in Journal for General Philosophy of Science (2004) In Defence of Constructive Empiricism: Metaphysics versus Science F.A. Muller Institute for the History and Philosophy of Science and Mathematics Utrecht University, P.O. Box 80.000 3508 TA Utrecht, The Netherlands E-mail: [email protected] August 2003 Summary Over the past years, in books and journals (this journal included), N. Maxwell launched a ferocious attack on B.C. van Fraassen's view of science called Con- structive Empiricism (CE). This attack has been totally ignored. Must we con- clude from this silence that no defence is possible against the attack and that a fortiori Maxwell has buried CE once and for all, or is the attack too obviously flawed as not to merit exposure? We believe that neither is the case and hope that a careful dissection of Maxwell's reasoning will make this clear. This dis- section includes an analysis of Maxwell's `aberrance-argument' (omnipresent in his many writings) for the contentious claim that science implicitly and per- manently accepts a substantial, metaphysical thesis about the universe. This claim generally has been ignored too, for more than a quarter of a century. Our con- clusions will be that, first, Maxwell's attacks on CE can be beaten off; secondly, his `aberrance-arguments' do not establish what Maxwell believes they estab- lish; but, thirdly, we can draw a number of valuable lessons from these attacks about the nature of science and of the libertarian nature of CE. Table of Contents on other side −! Contents 1 Exordium: What is Maxwell's Argument? 1 2 Does Science Implicitly Accept Metaphysics? 3 2.1 Aberrant Theories . -
Semantical Paradox* Tyler Burge
4 Semantical Paradox* Tyler Burge Frege remarked that the goal of all sciences is truth, but that it falls to logic to discern the laws of truth. Perceiving that the task of determining these laws went beyond Frege’s conception of it, Tarski enlarged the jurisdiction of logic, establishing semantics as truth’s lawyer.1 At the core of Tarski’s theory of truth and validity was a diagnosis of the Liar paradox according to which natural language was hopelessly infected with contradiction. Tarski construed himself as treating the disease by replacing ordinary discourse with a sanitized, artificial construction. But those interested in natural language have been dissatisfied with this medication. The best ground for dis satisfaction is that the notion of a natural language’s harboring contradictions is based on an illegitimate assimilation of natural language to a semantical system. According to that assimilation, part of the nature of a “language” is a set of postulates that purport to be true by virtue of their meaning or are at least partially constitutive of that “language”. Tarski thought that he had identified just such postulates in natural language as spawning inconsistency. But postulates are contained in theories that are promoted by people. Natural languages per se do not postulate or Tyler Burge, “Semantical Paradox", reprinted from The Journal of Philosophy 76 (1979), 169-98. Copyright © 1979 The Journal of Philosophy. Reprinted by permission of the Editor of The Journal of Philosophy and the author. * I am grateful to Robert L. Martin for several helpful discussions; to Herbert Enderton for proving the consistency (relative to that of arithmetic) of an extension of Construction C3; to Charles Parsons for stimulating exchanges back in 1973 and 1974; and to the John Simon Guggenheim Foundation for its support. -
The Liar Paradox As a Reductio Ad Absurdum Argument
University of Windsor Scholarship at UWindsor OSSA Conference Archive OSSA 3 May 15th, 9:00 AM - May 17th, 5:00 PM The Liar Paradox as a reductio ad absurdum argument Menashe Schwed Ashkelon Academic College Follow this and additional works at: https://scholar.uwindsor.ca/ossaarchive Part of the Philosophy Commons Schwed, Menashe, "The Liar Paradox as a reductio ad absurdum argument" (1999). OSSA Conference Archive. 48. https://scholar.uwindsor.ca/ossaarchive/OSSA3/papersandcommentaries/48 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: The Liar Paradox as a Reductio ad Absurdum Author: Menashe Schwed Response to this paper by: Lawrence Powers (c)2000 Menashe Schwed 1. Introduction The paper discusses two seemingly separated topics: the origin and function of the Liar Paradox in ancient Greek philosophy and the Reduction ad absurdum mode of argumentation. Its goal is to show how the two topics fit together and why they are closely connected. The accepted tradition is that Eubulides of Miletos was the first to formulate the Liar Paradox correctly and that the paradox was part of the philosophical discussion of the Megarian School. Which version of the paradox was formulated by Eubulides is unknown, but according to some hints given by Aristotle and an incorrect version given by Cicero1, the version was probably as follows: The paradox is created from the Liar sentence ‘I am lying’. -
Intransitivity and Future Generations: Debunking Parfit's Mere Addition Paradox
Journal of Applied Philosophy, Vol. 20, No. 2,Intransitivity 2003 and Future Generations 187 Intransitivity and Future Generations: Debunking Parfit’s Mere Addition Paradox KAI M. A. CHAN Duties to future persons contribute critically to many important contemporaneous ethical dilemmas, such as environmental protection, contraception, abortion, and population policy. Yet this area of ethics is mired in paradoxes. It appeared that any principle for dealing with future persons encountered Kavka’s paradox of future individuals, Parfit’s repugnant conclusion, or an indefensible asymmetry. In 1976, Singer proposed a utilitarian solution that seemed to avoid the above trio of obstacles, but Parfit so successfully demonstrated the unacceptability of this position that Singer abandoned it. Indeed, the apparently intransitivity of Singer’s solution contributed to Temkin’s argument that the notion of “all things considered better than” may be intransitive. In this paper, I demonstrate that a time-extended view of Singer’s solution avoids the intransitivity that allows Parfit’s mere addition paradox to lead to the repugnant conclusion. However, the heart of the mere addition paradox remains: the time-extended view still yields intransitive judgments. I discuss a general theory for dealing with intransitivity (Tran- sitivity by Transformation) that was inspired by Temkin’s sports analogy, and demonstrate that such a solution is more palatable than Temkin suspected. When a pair-wise comparison also requires consideration of a small number of relevant external alternatives, we can avoid intransitivity and the mere addition paradox without infinite comparisons or rejecting the Independence of Irrelevant Alternatives Principle. Disagreement regarding the nature and substance of our duties to future persons is widespread. -
Expressive Completeness
Truth-Functional Completeness 1. A set of truth-functional operators is said to be truth-functionally complete (or expressively adequate) just in case one can take any truth-function whatsoever, and construct a formula using only operators from that set, which represents that truth-function. In what follows, we will discuss how to establish the truth-functional completeness of various sets of truth-functional operators. 2. Let us suppose that we have an arbitrary n-place truth-function. Its truth table representation will have 2n rows, some true and some false. Here, for example, is the truth table representation of some 3- place truth function, which we shall call $: Φ ψ χ $ T T T T T T F T T F T F T F F T F T T F F T F T F F T F F F F T This truth-function $ is true on 5 rows (the first, second, fourth, sixth, and eighth), and false on the remaining 3 (the third, fifth, and seventh). 3. Now consider the following procedure: For every row on which this function is true, construct the conjunctive representation of that row – the extended conjunction consisting of all the atomic sentences that are true on that row and the negations of all the atomic sentences that are false on that row. In the example above, the conjunctive representations of the true rows are as follows (ignoring some extraneous parentheses): Row 1: (P&Q&R) Row 2: (P&Q&~R) Row 4: (P&~Q&~R) Row 6: (~P&Q&~R) Row 8: (~P&~Q&~R) And now think about the formula that is disjunction of all these extended conjunctions, a formula that basically is a disjunction of all the rows that are true, which in this case would be [(Row 1) v (Row 2) v (Row 4) v (Row6) v (Row 8)] Or, [(P&Q&R) v (P&Q&~R) v (P&~Q&~R) v (P&~Q&~R) v (~P&Q&~R) v (~P&~Q&~R)] 4. -
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. -
A Set of Solutions to Parfit's Problems
NOÛS 35:2 ~2001! 214–238 A Set of Solutions to Parfit’s Problems Stuart Rachels University of Alabama In Part Four of Reasons and Persons Derek Parfit searches for “Theory X,” a satisfactory account of well-being.1 Theories of well-being cover the utilitarian part of ethics but don’t claim to cover everything. They say nothing, for exam- ple, about rights or justice. Most of the theories Parfit considers remain neutral on what well-being consists in; his ingenious problems concern form rather than content. In the end, Parfit cannot find a theory that solves each of his problems. In this essay I propose a theory of well-being that may provide viable solu- tions. This theory is hedonic—couched in terms of pleasure—but solutions of the same form are available even if hedonic welfare is but one aspect of well-being. In Section 1, I lay out the “Quasi-Maximizing Theory” of hedonic well-being. I motivate the least intuitive part of the theory in Section 2. Then I consider Jes- per Ryberg’s objection to a similar theory. In Sections 4–6 I show how the theory purports to solve Parfit’s problems. If my arguments succeed, then the Quasi- Maximizing Theory is one of the few viable candidates for Theory X. 1. The Quasi-Maximizing Theory The Quasi-Maximizing Theory incorporates four principles. 1. The Conflation Principle: One state of affairs is hedonically better than an- other if and only if one person’s having all the experiences in the first would be hedonically better than one person’s having all the experiences in the second.2 The Conflation Principle sanctions translating multiperson comparisons into single person comparisons.