Axiomatizing Kripke's Theory of Truth

We investigate axiomatizations of Kripke’s [18] theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman’s axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke’s semantics in the straightforward sense; only the sentences provably true in KF are valid in Kripke’s partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt’s Conjecture, and provide a direct axiomatization PKF of Kripke’s theory in partial logic. We argue that any natural axiomatization of Kripke’s theory in Strong Kleene logic has the same proof-theoretic strength as PKF, namely the strength of ramified analysis RA<ωω or a system of Tarskian ramified truth up to ω and thus any such axiomatization is much weaker than Feferman’s axiomatization KF in classical logic, which is equivalent to ramified analysis RA< 0 up to 0. Wovon man nicht sprechen kann, darüber muß man schweigen. adapted from Wittgenstein, Tractatus logico-philosophicus 7 §