Journal of Symbolic Logic 71 (2):677 - 712 (2006)

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Abstract
We investigate axiomatizations of Kripke's 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 that can be proved to be 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 the system RA< ωω ramified analysis or a system of Tarskian ramified truth up to ωω. Thus any such axiomatization is much weaker than Feferman's axiomatization KF in classical logic, which is equivalent to the system RA<ε₀ of ramified analysis up to ε₀
Keywords Truth   Kripke-Feferman theory   axiomatic theories of truth   reflective closure   Strong Kleene scheme   partial logic   liar paradox
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DOI 10.2178/jsl/1146620166
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References found in this work BETA

Outline of a Theory of Truth.Saul Kripke - 1975 - Journal of Philosophy 72 (19):690-716.
The Revision Theory of Truth.Vann McGee - 1996 - Philosophy and Phenomenological Research 56 (3):727-730.
Recent Essays on Truth and the Liar Paradox.Robert L. Martin (ed.) - 1984 - Oxford University Press.
Basic Proof Theory.Roy Dyckhoff - 2001 - Bulletin of Symbolic Logic 7 (2):280-280.

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