Yablo’s Paradox: Is the Infinite Liar Lying to Us?

2019 ◽  
Vol 56 (3) ◽  
pp. 88-102
Author(s):  
Andrei V. Nekhaev ◽  
Keyword(s):  
Liar Paradox ◽  
Semantic Content ◽  
Basic Structure ◽  
Truth Predicate ◽  
Semantic Paradoxes ◽  
Yablo’S Paradox ◽  
Truth Values ◽  
Traditional Notion ◽  
Self Reference

In 1993, the American logic S. Yablo was proposed an original infinitive formulation of the classical ≪Liar≫ paradox. It questioned the traditional notion of self-reference as the basic structure for semantic paradoxes. The article considers the arguments underlying two different approaches to analysis of proposals of the ≪Infinite Liar≫ and understanding of the genuine sources for semantic paradoxes. The first approach (V. Valpola, G.-H. von Wright, T. Bolander, etc.) imposes responsibility for the emergence of semantic paradoxes on the negation of the truth predicate. It deprives the ≪Infinite Liar≫ sentences of consistent truth values. The second approach is based on a modified version of anaphoric prosententialism (D. Grover, R. Brandom, etc.). The concepts of truth and falsehood are treated as special anaphoric operators. Logical constructs similar to the ≪Infinite Liar≫ do not attribute any definite truth values to sentences from which they are composed, but only state certain types of relations between the semantic content of such sentences.

SUPER LIARS

2010 ◽  
Vol 3 (3) ◽  
pp. 374-414 ◽  
Author(s):  
PHILIPPE SCHLENKER
Keyword(s):  
Fixed Points ◽  
Detailed Analysis ◽  
Expressive Power ◽  
Truth Predicate ◽  
Truth Values ◽  
Kripke’S Theory Of Truth ◽  
Domain Restrictions ◽  
The Liar ◽  
Self Reference ◽  
Linguistic Means

Kripke’s theory of truth offered a trivalent semantics for a language which, like English, contains a truth predicate and means of self-reference; but it did so by severely restricting the expressive power of the logic. In Kripke’s analysis, the Liar (e.g., This very sentence is not true) receives the indeterminate truth value, but this fact cannot be expressed in the language; by contrast, it is straightforward to say in English that the Liar is something other than true. Kripke’s theory also fails to handle the Strengthened Liar, which can be expressed in English as: This very sentence is something other than true. We develop a theory which seeks to overcome these difficulties, and is based on a detailed analysis of some of the linguistic means by which the Strengthened Liar can be expressed in English. In particular, we propose to take literally the quantificational form of the negative expression something other than true. Like other quantifiers, it may have different implicit domain restrictions, which give rise to a variety of negations of different strengths (e.g., something other than true among the values {0, 1}, or among {0, 1, 2}, etc). This analysis naturally leads to a logic with as many truth values as there are ordinals—a conclusion reached independently by Cook (2008a). We develop the theory within a generalization of the Strong Kleene Logic, augmented with negations that each have a nonmonotonic semantics. We show that fixed points can be constructed for our logic, and that it enjoys a limited form of ‘expressive completeness’. Finally, we discuss the relation between our theory and various alternatives, including one in which the word true (rather than negation) is semantically ambiguous, and gives rise to a hierarchy of truth predicates of increasing strength.

scholarly journals A GRAPH-THEORETIC ANALYSIS OF THE SEMANTIC PARADOXES

2017 ◽  
Vol 23 (4) ◽  
pp. 442-492 ◽  
Author(s):  
TIMO BERINGER ◽  
THOMAS SCHINDLER
Keyword(s):  
Truth Predicate ◽  
Theoretic Analysis ◽  
Semantic Paradoxes ◽  
Truth Values ◽  
First Order ◽  
Graph Theoretic ◽  
Fine Grained ◽  
Stage Process ◽  
Order Arithmetic ◽  
The Liar

AbstractWe introduce a framework for a graph-theoretic analysis of the semantic paradoxes. Similar frameworks have been recently developed for infinitary propositional languages by Cook [5, 6] and Rabern, Rabern, and Macauley [16]. Our focus, however, will be on the language of first-order arithmetic augmented with a primitive truth predicate. Using Leitgeb’s [14] notion of semantic dependence, we assign reference graphs (rfgs) to the sentences of this language and define a notion of paradoxicality in terms of acceptable decorations of rfgs with truth values. It is shown that this notion of paradoxicality coincides with that of Kripke [13]. In order to track down the structural components of an rfg that are responsible for paradoxicality, we show that any decoration can be obtained in a three-stage process: first, the rfg is unfolded into a tree, second, the tree is decorated with truth values (yielding a dependence tree in the sense of Yablo [21]), and third, the decorated tree is re-collapsed onto the rfg. We show that paradoxicality enters the picture only at stage three. Due to this we can isolate two basic patterns necessary for paradoxicality. Moreover, we conjecture a solution to the characterization problem for dangerous rfgs that amounts to the claim that basically the Liar- and the Yablo graph are the only paradoxical rfgs. Furthermore, we develop signed rfgs that allow us to distinguish between ‘positive’ and ‘negative’ reference and obtain more fine-grained versions of our results for unsigned rfgs.

The Theory at Work

2018 ◽  
Author(s):  
Keith Simmons
Keyword(s):  
Liar Paradox ◽  
Singularity Theory ◽  
Intrinsic Interest ◽  
Semantic Paradoxes ◽  
Yablo’S Paradox ◽  
Russell's Paradox ◽  
Curry Paradox ◽  
Truth Teller ◽  
The Liar ◽  
The Liar Paradox

Chapter 7 puts the singularity theory to work on a number of semantic paradoxes that have intrinsic interest of their own. These include a transfinite paradox of denotation, and variations on the Liar paradox, including the Truth-Teller, Curry’s paradox, and paradoxical Liar loops. The transfinite paradox of denotation shows the need to accommodate limit ordinals. The Truth-Teller, like the Liar, exhibits semantic pathology-but, unlike the Liar, it does not produce a contradiction. The distinctive challenge of the Curry paradox is that it seems to allow us to prove any claim we like (for example, the claim that 2+2=5). Paradoxical Liar loops, such as the Open Pair paradox, extend the Liar paradox beyond single self-referential sentences. The chapter closes with the resolution of paradoxes that do not exhibit circularity yet still generate contradictions. These include novel versions of the definability paradoxes and Russell’s paradox, and Yablo’s paradox about truth.

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.

Tarski

Truth ◽  
2011 ◽  
Author(s):  
Alexis G. Burgess ◽  
John P. Burgess
Keyword(s):  
Mathematical Logic ◽  
Model Theory ◽  
The State ◽  
Truth Predicate ◽  
Object Language ◽  
French And English ◽  
Direct Definition ◽  
Definition Of ◽  
Basic Features ◽  
Self Reference

This chapter offers a simplified account of the most basic features of Alfred Tarski's model theory. Tarski foresaw important applications for a notion of truth in mathematics, but also saw that mathematicians were suspicious of that notion, and rightly so given the state of understanding of it circa 1930. In a series of papers in Polish, German, French, and English from the 1930s onward, Tarski attempted to rehabilitate the notion for use in mathematics, and his efforts had by the 1950s resulted in the creation of a branch of mathematical logic known as model theory. The chapter first considers Tarski's notion of truth, which he calls “semantic” truth, before discussing his views on object language and metalanguage, recursive versus direct definition of the truth predicate, and self-reference.

A theory of properties

1987 ◽  
Vol 52 (2) ◽  
pp. 455-472 ◽  
Author(s):  
Ray Turner
Keyword(s):  
Recent Work ◽  
Set Theory ◽  
Semantic Theory ◽  
Truth Predicate ◽  
Minimal Logic ◽  
History Of ◽  
Concept Of Truth ◽  
Almost All ◽  
The Liar ◽  
Self Reference

Frege's attempts to formulate a theory of properties to serve as a foundation for logic, mathematics and semantics all dissolved under the weight of the logicial paradoxes. The language of Frege's theory permitted the representation of the property which holds of everything which does not hold of itself. Minimal logic, plus Frege's principle of abstraction, leads immediately to a contradiction. The subsequent history of foundational studies was dominated by attempts to formulate theories of properties and sets which would not succumb to the Russell argument. Among such are Russell's simple theory of types and the development of various iterative conceptions of set. All of these theories ban, in one way or another, the self-reference responsible for the paradoxes; in this sense they are all “typed” theories. The semantical paradoxes, involving the concept of truth, induced similar nightmares among philosophers and logicians involved in semantic theory. The early work of Tarski demonstrated that no language that contained enough formal machinery to respresent the various versions of the Liar could contain a truth-predicate satisfying all the Tarski biconditionals. However, recent work in both disciplines has led to a re-evaluation of the limitations imposed by the paradoxes.In the foundations of set theory, the work of Gilmore [1974], Feferman [1975], [1979], [1984], and Aczel [1980] has clearly demonstrated that elegant and useful type-free theories of classes are feasible. Work on the semantic paradoxes was given new life by Kripke's contribution (Kripke [1975]). This inspired the recent work of Gupta [1982] and Herzberger [1982]. These papers demonstrate that much room is available for the development of theories of truth which meet almost all of Tarski's desiderata.

Contextualism, minimalism, and situationalism

2007 ◽  
Vol 15 (1) ◽  
pp. 115-137 ◽  
Author(s):  
Eros Corazza
Keyword(s):  
Semantic Content ◽  
Truth Values ◽  
Context Dependent ◽  
Truth Value ◽  
True False

After discussing some difficulties that contextualism and minimalism face, this paper presents a new account of the linguistic exploitation of context, situationalism. Unlike the former accounts, situationalism captures the idea that the main intuitions underlying the debate concern not the identity of propositions expressed but rather how truth-values are situation-dependent. The truth-value of an utterance depends on the situation in which the proposition expressed is evaluated. Hence, like in minimalism, the proposition expressed can be truth-evaluable without being enriched or expanded. Along with contextualism, it is argued that an utterance’s truth-value is context dependent. But, unlike contextualism and minimalism, situationalism embraces a form of relativism in so far as it maintains that semantic content must be evaluated vis-à-vis a given situation and, therefore, that a proposition cannot be said to be true/false eternally.

Construction of Truth Predicates: Approximation Versus Revision

1998 ◽  
Vol 4 (4) ◽  
pp. 399-417 ◽  
Author(s):  
Juan Barba
Keyword(s):  
Liar Paradox ◽  
Predicate Symbol ◽  
Main Idea ◽  
Truth Predicate ◽  
Order Language ◽  
Fixed Model ◽  
Contextual Dependence ◽  
The Subject ◽  
Concept Of Truth ◽  
The Liar

§1. Introduction. The problem raised by the liar paradox has long been an intriguing challenge for all those interested in the concept of truth. Many “solutions” have been proposed to solve or avoid the paradox, either prescribing some linguistical restriction, or giving up the classical true-false bivalence or assuming some kind of contextual dependence of truth, among other possibilities. We shall not discuss these different approaches to the subject in this paper, but we shall concentrate on a kind of formal construction which was originated by Kripke's paper “Outline of a theory of truth” [11] and which, in different forms, reappears in later papers by various authors.The main idea can be presented as follows: assume a first order language ℒ containing, among other unspecified symbols, a predicate symbol T intended to represent the truth predicate for ℒ. Assume, also, a fixed model M = 〈D, I〉 (the base model)where D contains all sentences of ℒ and I interprets all non-logical symbols of ℒ except T in the usual way. In general, D might contain many objects other than sentences of ℒ but as that would raise the problem of the meaning of sentences in which T is applied to one of these objects, we shall assume that this is not the case.

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