Truth in a Logic of Formal Inconsistency: How classical can it get?

Weakening classical logic is one of the most popular ways of dealing with semantic paradoxes. Their advocates often claim that such weakening does not affect non-semantic reasoning. Recently, however, Halbach and Horsten (2006) have shown that this is actually not the case for Kripke’s fixed-point theory based on the Strong Kleene evaluation scheme. Feferman’s axiomatization KF in classical logic is much stronger than its paracomplete counterpart PKF⁠, not only in terms of semantic but also in arithmetical content. This paper compares the proof-theoretic strength of an axiomatization of Kripke’s construction based on the paraconsistent evaluation scheme of LP⁠, formulated in classical logic with that of an axiomatization directly formulated in LP⁠, extended with a consistency operator. The ultimate goal is to find out whether paraconsistent solutions to the paradoxes that employ consistency operators fare better in this respect than paracomplete ones