Nonclassical truth with classical strength : a proof-theoretic analysis of compositional truth over HYPE

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.