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

TL;DR

This paper analyzes the proof-theoretic strength of nonclassical versus classical theories of truth, showing that a nonclassical logic with a suitable conditional can match classical strength in fixed-point semantics.

Contribution

It demonstrates that extending HYPE with a conditional yields a nonclassical theory proof-theoretically equivalent to classical theories like KF.

Findings

PKF over HYPE is sound for fixed-point models

PKF over HYPE is proof-theoretically equivalent to KF

The schematic extension of PKF matches the strength of predicative analysis

Abstract

Questions concerning the proof-theoretic strength of classical versus non-classical 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.