Tarski, Frege and the Liar Paradox

Philosophy ◽  
1971 ◽  
Vol 46 (176) ◽  
pp. 133-147 ◽  
Author(s):  
Sloman Aaron
Keyword(s):  
Liar Paradox ◽  
Higher Order ◽  
Colloquial Language ◽  
Law Of Excluded Middle ◽  
The Law ◽  
The Liar ◽  
The Liar Paradox ◽  
Excluded Middle

A.1. Some philosophers, including Tarski and Russell, have concluded from a study of various versions of the Liar Paradox ‘that there must be a hierarchy of languages, and that the words “true” and “false”, as applied to statements in any given language, are themselves words belonging to a language of higher order’. In his famous essay on truth Tarski claimed that ‘colloquial’ language is inconsistent as a result of its property of ‘universality’: that is, whatever can be said at all can in principle be said in it, with an extended vocabularly if necessary. Thus, in English we can talk about English expressions, what they denote, what they say, whether what they say is true or false, and so on: English contains its own metalanguage. This universality enables us to construct sentences which say of themselves that they are false, and by applying the law of excluded middle to them we easily derive a contradiction. Tarski concludes that ‘these antinomies seem to provide a proof that every language which is universal in the above sense, and for which the normal laws of logic hold, must be inconsistent’ (op. cit., pp. 164—5). He then proposes to avoid such contradictions by the use of a hierarchy of languages such that statements about any one language can be made only in a different language at a higher level.

Revenge, I

2018 ◽  
Author(s):  
Keith Simmons
Keyword(s):  
Natural Language ◽  
Liar Paradox ◽  
Critical Discussion ◽  
Kripke’S Theory Of Truth ◽  
Law Of Excluded Middle ◽  
Truth Value ◽  
The Law ◽  
The Liar ◽  
The Liar Paradox ◽  
Excluded Middle

Chapter 8 is the first of two chapters on the phenomenon of revenge paradoxes, paradoxes which, roughly speaking, are constructed out of the very terms of a purported solution. The chapter begins by exploring the difficulties that revenge presents for Kripke’s theory of truth, in either of two versions: a version that admits truth value gaps, and a paracomplete version which rejects the law of excluded middle. The chapter goes on to critically examine Field’s paracomplete theory of truth and its treatment of revenge, arguing that Field’s theory is couched in terms that are artificial and too far removed from natural language. The chapter concludes with a critical discussion of Priest’s dialetheist approach to the Liar paradox, according to which there are true contradictions. It is argued that Priest’s theory is itself subject to revenge paradoxes.

Core Logic and the Paradoxes

2017 ◽  
Author(s):  
Neil Tennant
Keyword(s):  
Normal Form ◽  
Set Theory ◽  
Free Logic ◽  
Semantic Paradoxes ◽  
Law Of Excluded Middle ◽  
The Law ◽  
The Liar ◽  
Excluded Middle

The Law of Excluded Middle is not to be blamed for any of the logico-semantic paradoxes. We explain and defend our proof-theoretic criterion of paradoxicality, according to which the ‘proofs’ of inconsistency associated with the paradoxes are in principle distinct from those that establish genuine inconsistencies, in that they cannot be brought into normal form. Instead, the reduction sequences initiated by paradox-posing proofs ‘of ⊥’ do not terminate. This criterion is defended against some recent would-be counterexamples by stressing the need to use Core Logic’s parallelized forms of the elimination rules. We show how Russell’s famous paradox in set theory is not a genuine paradox; for it can be construed as a disproof, in the free logic of sets, of the assumption that the set of all non-self-membered sets exists. The Liar (by contrast) is still paradoxical, according to the proof-theoretic criterion of paradoxicality.

scholarly journals A Tale of Excluding the Middle

2021 ◽  
Vol 27 (1) ◽  
pp. 20-30
Author(s):  
Jc Beall ◽  
Graham Priest
Keyword(s):  
Liar Paradox ◽  
Liar Sentence ◽  
The Liar ◽  
The Liar Paradox ◽  
Excluded Middle

he paper discusses a number of interconnected points concerning negation, truth, validity and the liar paradox. In particular, it discusses an argument for the dialetheic nature of the liar sentence which draws on Dummett’s teleological account of truth. Though one way of formulating this fails, a different way succeeds. The paper then discusses the role of the Principle of Excluded Middle in the argument, and of the thought that truth in a model should be a model of truth.

Pluralism and the Liar

2017 ◽  
Author(s):  
Cory Wright ◽  
Bradley Armour-Garb
Keyword(s):  
Alternative Treatment ◽  
Liar Paradox ◽  
Recent Attempt ◽  
In Virtue Of ◽  
The Liar ◽  
The Liar Paradox

Pluralists maintain that there is more than one truth property in virtue of which bearers are true. Unfortunately, it is not yet clear how they diagnose the liar paradox or what resources they have available to treat it. This chapter considers one recent attempt by Cotnoir (2013b) to treat the Liar. It argues that pluralists should reject the version of pluralism that Cotnoir assumes, discourse pluralism, in favor of a more naturalized approach to truth predication in real languages, which should be a desideratum on any successful pluralist conception. Appealing to determination pluralism instead, which focuses on truth properties, it then proposes an alternative treatment to the Liar that shows liar sentences to be undecidable.

Microlanguages, Vagueness, and Paradox

2017 ◽  
Author(s):  
Peter Ludlow ◽  
Bradley Armour-Garb
Keyword(s):  
Recent Work ◽  
Dynamic Theory ◽  
Liar Paradox ◽  
Context Sensitivity ◽  
Static Picture ◽  
The Liar ◽  
The Liar Paradox

This chapter follows recent work in philosophy, linguistics, and psychology, which rejects the standard, static picture of languages and highlights its context sensitivity—a dynamic theory of the nature of language. On the view advocated, human languages are things that we build on a conversation-by-conversation basis. The author calls such languages microlanguages. The chapter argues that thinking of languages in terms of microlanguages yields interesting consequences for how we should think about the liar paradox. In particular, we will see that microlanguages have admissible conditions that preclude liar-like sentences. On the view presented in the chapter, liar sentences are not even sentences of any microlanguage that we might construct (or assertorically utter). Accordingly, the proper approach to such a paradoxical sentence is to withhold the sentence—not permitting it to be admitted into our microlanguage unless, or until, certain sharpening occurs.

Introduction

2017 ◽  
Author(s):  
Bradley Armour-Garb ◽  
Bradley Armour-Garb
Keyword(s):  
Liar Paradox ◽  
The Liar ◽  
The Liar Paradox

In this chapter, after introducing a few versions of the liar paradox and identifying the pathology that the versions of the paradox appear to present, the author considers some proposals for how to understand ‘paradox’ and goes on to offer a particular reading of that notion. He then identifies a number of projects the completion of which would contribute to our understanding—or, in some cases, our resolution—of the liar paradox and, after considering certain “treatments” of the paradox, highlights certain “revenge” problems that arise for such treatments. In the concluding section, the author summarizes each of the chapters that are contained in the volume.

scholarly journals The Liar Without Relativism

Erkenntnis ◽  
2021 ◽  
Author(s):  
Poppy Mankowitz
Keyword(s):  
Natural Language ◽  
Recent Literature ◽  
Liar Paradox ◽  
Full Understanding ◽  
Truth Values ◽  
The Liar ◽  
The Liar Paradox ◽  
Semantic Relativism

AbstractSome in the recent literature have claimed that a connection exists between the Liar paradox and semantic relativism: the view that the truth values of certain occurrences of sentences depend on the contexts at which they are assessed. Sagi (Erkenntnis 82(4):913–928, 2017) argues that contextualist accounts of the Liar paradox are committed to relativism, and Rudnicki and Łukowski (Synthese 1–20, 2019) propose a new account that they classify as relativist. I argue that a full understanding of how relativism is conceived within theories of natural language shows that neither of the purported connections can be maintained. There is no reason why a solution to the Liar paradox needs to accept relativism.

Reconsidering Kant’s Rejection of Indirect Arguments in Transcendental Philosophy

2021 ◽  
pp. 1-19
Author(s):  
Marcel Buß
Keyword(s):  
Immanuel Kant ◽  
Explanatory Power ◽  
Transcendental Philosophy ◽  
Point Of View ◽  
Transcendental Arguments ◽  
Mathematical Proofs ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Excluded Middle ◽  
Insight Into

Abstract Immanuel Kant states that indirect arguments are not suitable for the purposes of transcendental philosophy. If he is correct, this affects contemporary versions of transcendental arguments which are often used as an indirect refutation of scepticism. I discuss two reasons for Kant’s rejection of indirect arguments. Firstly, Kant argues that we are prone to misapply the law of excluded middle in philosophical contexts. Secondly, Kant points out that indirect arguments lack some explanatory power. They can show that something is true but they do not provide insight into why something is true. Using mathematical proofs as examples, I show that this is because indirect arguments are non-constructive. From a Kantian point of view, transcendental arguments need to be constructive in some way. In the last part of the paper, I briefly examine a comment made by P. F. Strawson. In my view, this comment also points toward a connection between transcendental and constructive reasoning.

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