David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Journal of Symbolic Logic 71 (2):677 - 712 (2006)
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.
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Hannes Leitgeb (2007). What Theories of Truth Should Be Like (but Cannot Be). Philosophy Compass 2 (2):276–290.
Leon Horsten (2009). Levity. Mind 118 (471):555-581.
Alan Weir (2013). A Robust Non-Transitive Logic. Topoi 34 (1):1-9.
Colin R. Caret & Zach Weber (2015). A Note on Contraction-Free Logic for Validity. Topoi 34 (1):63-74.
Leon Horsten (forthcoming). One Hundred Years of Semantic Paradox. Journal of Philosophical Logic:1-15.
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