Yablo-Like Paradoxes and Co-induction

Abstract

Abstract. We review three pairwise similar paradoxes, the modest liar paradox, McGee's paradox and Yablo's paradox, which imply the ω-inconsistency. We show that is caused by the fact that co-inductive definitions of formulae are possible because of the existence of the truth predicate.

Truth without standard models: some conceptual problems reloaded

“…We introduce Lukasiewicz's predicate logic in a semantic fashion, as usual (cf. Hàjek [12], Yatabe [32]) 11 . The set of semantic values is the real interval [0,1] and 1 is the only designated one.…”

“…Recently, Yatabe [32] shows that there is an interesting relation between ω-inconsistent theories of truth and co-induction. In his own words: ω-inconsistent truth theories are intrinsically equipped the machinery, co-induction, which is widely used in computer science and enables to represent infinite processes.…”

“…Although Leitgeb [21], Field [7] and [anonymized] [1] have criticized ω-inconsistent theories of truth, this issue continues to be controversial. In fact, for instance, Yatabe [32] [33], among others, has recently argued in favor of ω-inconsistent first-order theories of truth. In his papers, he focuses on the technical side of the matter and argues that rejecting ω-inconsistent theories of truth would involve to reject co-induction, which is a mathematical principle (dual to induction) useful to prove properties of infinite structures, e.g.…”

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A theory of truth is usually demanded to be consistent, but ωconsistency is less frequently requested. Recently, Yatabe [32] has argued in favor of ω-inconsistent first-order theories of truth, minimizing their odd consequences. In view of this fact, in this paper we present five arguments against ω-inconsistent theories of truth.

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Sorites, Curry and Suitable Models

“…Although, usually, the notion of Û-inconsistency is syntactic, here we define it semantically.11 However, there are some authors that still defend Û-inconsistent theories of truth, such as Yatabe[25], and Gupta[11].…”