Tarski's theorem and liar-like paradoxes

Philosophical Studies

Tarski's hierarchical solution to the Liar paradox is widely viewed as ad hoc. In this paper I show that, on the contrary, Tarski's solution is justified by a sound philosophical principle that concerns the inner structure of truth. This principle provides a common philosophical basis to a number of solutions to the Liar paradox, including Tarski's and Kripke's. Tarski himself may not have been aware of this principle, but by providing a philosophical basis to his hierarchical solution to the paradox, it undermines the ad-hocness objection to this solution. Indeed, it contributes to the defense of Tarski's theory against other objections as well.

2013

It is proved that Yablo's paradox and the Liar paradox are equiparadoxical, in the sense that their paradoxicality is based upon exactly the same circularity condition---for any frame K, the following are equivalent: (1) Yablo's sequence leads to paradox in K;(2) the Liar sentence leads to paradox in K;(3) K contains odd cycles.This result does not conflict with Yablo's claim that his sequence is non-self-referential. Rather, it gives Yablo's paradox a new significance: his construction contributes a method by which we can eliminate the self-reference of a paradox without changing its circularity condition.

A relativized version of Tarski’s T-scheme is introduced as a new principle of the truth predicate. Under the relativized T-scheme, the paradoxical objects, such as the Liar sentence and Jourdain’s card sequence, are found to have certain relative contradictoriness. That is, they are contradictory only in some frames in the sense that any valuation admissible for them in these frames will lead to a contradiction. It is proved that for any positive integer n, the n-jump liar sentence is contradictory in and only in those frames containing at least an n-jump odd cycle. In particular, the Liar sentence is contradictory in and only in those frames containing at least an odd cycle. The Liar sentence is also proved to be less contradictory than Jourdain’s card sequence: the latter must be contradictory in those frames where the former is so, but not vice versa. Generally, the relative contradictoriness is the common characteristic of the paradoxical objects, but different paradoxical objects may have different relative contradictoriness.

This paper gives a definition of self-reference on the basis of the dependence relation given by Leitgeb (2005), and the dependence digraph by Beringer & Schindler (2015). Unlike the usual discussion about self-reference of paradoxes centering around Yablo's paradox and its variants, I focus on the paradoxes of finitary characteristic, which are given again by use of Leitgeb's dependence relation. They are called 'locally finite paradoxes', satisfying that any sentence in these paradoxes can depend on finitely many sentences. I prove that all locally finite paradoxes are self-referential in the sense that there is a directed cycle in their dependence digraphs. This paper also studies the 'circularity dependence' of paradoxes, which was introduced by Hsiung (2014). I prove that the locally finite paradoxes have circularity dependence in the sense that they are paradoxical only in the digraph containing a proper cycle. The proofs of the two results are based directly on König's infinity lemma. In contrast, this paper also shows that Yablo's paradox and its ∀∃-unwinding variant are non-self-referential, and neither McGee's paradox nor the ω-cycle liar has circularity dependence.

few.vu.nl

2018

We can classify the (truth-theoretic) paradoxes according to their degrees of paradoxicality. Roughly speaking, two paradoxes have the same degrees of paradoxicality, if they lead to a contradiction under the same conditions, and one paradox has a (non-strictly) lower degree of paradoxicality than another, if whenever the former leads to a contradiction under a condition, the latter does so under the very condition. This paper aims at setting forth the theoretical framework of the theory of paradoxicality degree, and putting forward some basic open questions about paradoxes around the notion of paradoxicality degree.

2018

SU M M A RY: Our approach to the liar paradox is based on the Wittgensteinian approach to semantic and logical paradoxes. The main aim of this article is to point out that the liar sentence is only seemingly intelligible, and that it has not been given any sense. First, we will present the traditional solutions of the paradox, especially those which we call modificational. Then we will determine what the defects of these solutions are. Our main objection is that the modificational approaches assume that we can express in languages certain senses which are improper. Next, we will explain why we think that the liar sentence is a mere nonsense. This sentence does not have any role in any language game – it is completely useless. We will also respond to several objections to our approach. 1. That it is not consistent with the principle of compositionality of sense. 2. According to the Quineian philosophy of logic, paradoxical sentences can be conceived as false assumptions leading to cr...

Truth and Paradox, 2004

Reviews the standard semantic paradoxes, and constructs a simple formal language in which the paradoxical reasoning can be reconstructed. Particular attention is paid to Löb's paradox, which allows for the derivation of any sentence in the language as a theorem. The advantages of a natural deduction system over an axiomatic logic is discussed.

Special issue on Logic: Consistency, Contradiction, and Consequence. Principia 22(1), pp. 59 - 85. , 2018

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, or else 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 Paradox (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, internal 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 resources 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. As a result, the formal account suffers from severe limitations, which make it unable to represent the contradiction obtained in the conclusion of each of the paradoxes. _____________________________________________________________________________ A final version of this paper is published in a special issue ("Logic: Consistency, Contradiction, and Consequence") of Principia 22(1), pp. 59 - 85, 2018. https://periodicos.ufsc.br/index.php/principia/article/view/1808-1711.2018v22n1p59

History and Philosophy of Logic , 2021

In Formale Logik, published in 1956, J. M. Bocheński presented his first proposal for the solution to the liar paradox, which he related to Paul of Venice's argumentation from Logica Magna. A formalized version of this solution was then presented in Formalisierung einer scholastischen Lösung der Paradoxie des 'Lügners' in 1959. The historical references of the resulting formalism turn out to be closer to Albert de Saxon's argument and the later solution by John Buridan. Bocheński did not pose the question of the consistency of his theory. The case was taken up by B. Sobociński in his private letter to Bocheński from August 12, 1954. Sobociński used a smart translation of the language of Bocheński's theory into the classical propositional language with the notions of truth and falsehood inverted. The translation preserves the structure of the formalized solution. We explore Sobociński's idea and reconstruct his original proof.