scholarly journals Gödel, Tarski, Church, and The Liar

2003 ◽  
Vol 9 (1) ◽  
pp. 3-25 ◽  
Author(s):  
György Serény
Keyword(s):  
Common Knowledge ◽  
Liar Paradox ◽  
Ordinary Language ◽  
Incompleteness Theorem ◽  
Exact Nature ◽  
Language Version ◽  
Close Relationship ◽  
Logical Paradoxes ◽  
The Liar ◽  
The Liar Paradox

The fact that Gödel's famous incompleteness theorem and the archetype of all logical paradoxes, that of the Liar, are related closely is, of course, not only well known, but is a part of the common knowledge of the community of logicians. Indeed, almost every more or less formal treatment of the theorem makes a reference to this connection. Gödel himself remarked in the paper announcing his celebrated result (cf. [7]):The analogy between this result and Richard's antinomy leaps to the eye;there is also a close relationship with the ‘liar’ antinomy, since … we are… confronted with a proposition which asserts its own unprovability.In the light of the fact that the existence of this connection is commonplace it is all the more surprising that very little can be learnt about its exact nature except perhaps that it is some kind of similarity or analogy. There is, however, a lot more to it than that. Indeed, as we shall try to show below, the general ideas underlying the three central theorems concerning internal limitations of formal deductive systems can be taken as different ways to resolve the Liar paradox. More precisely, it will turn out that an abstract formal variant of the Liar paradox, which can almost straightforwardly inferred from its original ordinary language version, is a possible common generalization of (both the syntactic and semantic versions of) Gödel's incompleteness theorem, the theorem of Tarski on the undefinability of truth, and that of Church concerning the undecidability of provability.

Four Alleged Paradoxes in Legal Reasoning

1979 ◽  
Vol 38 (2) ◽  
pp. 373-391 ◽  
Author(s):  
Laurence Goldstein
Keyword(s):  
Set Theory ◽  
Legal Theory ◽  
Legal Reasoning ◽  
Liar Paradox ◽  
Productive Activity ◽  
Fields Of Study ◽  
Logical Paradoxes ◽  
Axiomatic Set Theory ◽  
The Liar ◽  
The Liar Paradox

There can be no doubt that the existence of paradoxes has stimulated vigorous and highly productive activity in philosophy and in logic. Take two famous examples: The Liar Paradox, which arises from a sentence such asL This statement is false.and the Russell Paradox which arises from the sentenceR The class of all classes which are not members of themselvesis a member of itself.The first of these has been a source of anguish for over 2,000 years. The second has engaged the serious attention of logicians for over three quarters of a century. Investigation of paradoxes of this sort has spawned whole new fields of study, such as technical semantics and axiomatic set theory.It has been claimed that legal reasoning is infected with paradoxes and that these paradoxes are similar in structure to those, like the two we have cited, which are of interest to the logician. If this claim were true one of two consequences would follow. Either the jurisprudent would face what would in all likelihood be a protracted struggle with these legal paradoxes resulting, perhaps, in significant additions to legal theory, or else, if these paradoxes were sufficiently similar to those of the logician, he might try to utilise the logician's results to solve his own legal puzzles.The first alternative, though attractive to a theoretician, may appear rather dismal to those engaged in the business of law. Whereas reflection on the logical paradoxes can lead to only more refined abstractions—the philosopher's meat and drink—legal theory is rather intimately connected with practical affairs.

scholarly journals Is the Liar Paradox a semantic paradox?

2019 ◽  
Vol 13 (1) ◽  
pp. 285-293
Author(s):  
Vsevolod Ladov
Keyword(s):  
20Th Century ◽  
Liar Paradox ◽  
Point Of View ◽  
Philosophy Of Logic ◽  
Semantic Paradox ◽  
Ancient Times ◽  
And Mathematics ◽  
Logical Paradoxes ◽  
The Liar ◽  
The Liar Paradox

The Liar Paradox has been widely discussed from the ancient times and preserved its importance in contemporary philosophy of logic and mathematics. At the beginning of the 20th century, F.P. Ramsey asserted that the Liar Paradox is different from pure logical paradoxes such as Russell’s paradox. The Liar Paradox is connected with language and can be considered a semantic paradox. Ramsey's point of view has become widespread in the logic of the 20th century. The author of the article questions this view. It is argued that the Liar Paradox cannot be unequivocally attributed to the semantic paradoxes and therefore Ramsey's point of view should be revised.

scholarly journals PROVING UNPROVABILITY

2016 ◽  
Vol 10 (1) ◽  
pp. 92-115
Author(s):  
BRUNO WHITTLE
Keyword(s):  
Liar Paradox ◽  
Positive Answer ◽  
Incompleteness Theorem ◽  
The Liar ◽  
The Liar Paradox

AbstractThis paper addresses the question: given some theory T that we accept, is there some natural, generally applicable way of extending T to a theory S that can prove a range of things about what it itself (i.e., S) can prove, including a range of things about what it cannot prove, such as claims to the effect that it cannot prove certain particular sentences (e.g., 0 = 1), or the claim that it is consistent? Typical characterizations of Gödel’s second incompleteness theorem, and its significance, would lead us to believe that the answer is ‘no’. But the present paper explores a positive answer. The general approach is to follow the lead of recent (and not so recent) approaches to truth and the Liar paradox.

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.

A NOTE ON THE LIAR-PARADOX

Mind ◽  
1955 ◽  
Vol LXIV (256) ◽  
pp. 543-543 ◽  
Author(s):  
A. P. USHENKO
Keyword(s):  
Liar Paradox ◽  
The Liar ◽  
The Liar Paradox
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