We investigate axiomatizations of Kripke’s theory of truthbased on the Strong Kleene evaluation scheme for treating sentenceslacking a truth value. Feferman’s axiomatization KF formulatedin classical logic is an indirect approach, because it is not soundwith respect to Kripke’s semantics in the straightforwardsense; only the sentences that can be proved to be true in KF arevalid in Kripke’s partial models. Reinhardt proposed to focusjust on the sentences that can be proved to be true in KF andconjectured that the detour through classical logic in KF isdispensable. We refute Reinhardt’s Conjecture, and provide adirect axiomatization PKF of Kripke’s theory in partiallogic. We argue that any natural axiomatization of Kripke’stheory in Strong Kleene logic has the same proof-theoretic strength asPKF, namely the strength of the systemRA<ωω ramified analysis ora system of Tarskian ramified truth up toωω. Thus any such axiomatization is muchweaker than Feferman’s axiomatization KF in classical logic,which is equivalent to the systemRA<ε₀ of ramified analysis up toε₀.
"Axiomatizing Kripke’s Theory of Truth." J. Symbolic Logic 71 (2) 677 - 712, June 2006. https://doi.org/10.2178/jsl/1146620166