Nonclassical truth with classical strength. A proof-theoretic analysis of compositional truth over hype.

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Abstract

Questions concerning the proof-theoretic strength of classical versus nonclassical theories of truth have received some attention recently. A particularly convenient case study concerns classical and nonclassical axiomatizations of fixed-point semantics. It is known that nonclassical axiomatizations in four- or three-valued logics are substantially weaker than their classical counterparts. In this paper we consider the addition of a suitable conditional to First- Degree Entailment – a logic recently studied by Hannes Leitgeb under the label HYPE. We show in particular that, by formulating the theory PKF over HYPE, one obtains a theory that is sound with respect to fixed-point models, while being proof-theoretically on a par with its classical counterpart KF. Moreover, we establish that also its schematic extension – in the sense of Feferman – is as strong as the schematic extension of KF, thus matching the strength of predicative analysis.

Original languageEnglish
Pages (from-to)1-26
Number of pages26
JournalReview Of Symbolic Logic
DOIs
Publication statusPublished - 25 Mar 2021

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