Provably True Sentences Across Axiomatizations of Kripke’s Theory of Truth

Abstract

We study the relationships between two clusters of axiomatizations of Kripke’s fixed-point models for languages containing a self-applicable truth predicate. The first cluster is represented by what we will call ‘ $$\fancyscript{PKF}$$ -like’ theories, originating in recent work by Halbach and Horsten, whose axioms and rules (in Basic De Morgan Logic) are all valid in fixed-point models; the second by ‘ $$\fancyscript{KF}$$ -like’ theories first introduced by Solomon Feferman, that lose this property but reflect the classicality of the metatheory in which Kripke’s construction is carried out. We show that to any natural system in one cluster—corresponding to natural variations on induction schemata—there is a corresponding system in the other proving the same sentences true, addressing a problem left open by Halbach and Horsten and accomplishing a suitably modified version of the project sketched by Reinhardt aiming at an instrumental reading of classical theories of self-applicable truth.