Gaps, gluts, and theoretical equivalence

Abstract

When are two formal theories of broadly logical concepts, such as truth, equivalent? The paper investigates a case study, involving two well-known variants of Kripke–Feferman truth. The first, $$\mathtt {KF}+\mathtt {CONS}$$ KF + CONS , features a consistent but partial truth predicate. The second, $$\mathtt {KF}+\mathtt {COMP}$$ KF + COMP , an inconsistent but complete truth predicate. It is known that the two truth predicates are dual to each other. We show that this duality reveals a much stricter correspondence between the two theories: they are intertraslatable. Intertranslatability, under natural assumptions, coincides with definitional equivalence, and is arguably the strictest notion of theoretical equivalence different from logical equivalence. The case of $$\mathtt {KF}+\mathtt {CONS}$$ KF + CONS and $$\mathtt {KF}+\mathtt {COMP}$$ KF + COMP raises a puzzle: the two theories can be proved to be strictly related, yet they appear to embody remarkably different conceptions of truth. We discuss the significance of the result for the broader debate on formal criteria of conceptual reducibility for theories of truth.