Modal Extensions of Sub-classical Logics for Recovering Classical Logic

In this paper we introduce non-normal modal extensions of the sub-classical logics CLoN, CluN and CLaN, in the same way that S0.5 extends classical logic. The first modal system is both paraconsistent and paracomplete, while the second one is paraconsistent and the third is paracomplete. Despite being non-normal, these systems are sound and complete for a suitable Kripke semantics. We also show that these systems are appropriate for interpreting as “is provable in classical logic”. This allows us to recover the theorems of propositional classical logic within three sub-classical modal systems. Mathematics Subject Classification (2010). Primary 03B45; Secondary 03B20, 03B53.