Recovery operators, paraconsistency and duality
There are two foundational but not properly developed ideas in da Costa’s approach to paraconsistency: the ‘well-behavedness’ operator and the duality between paraconsistent and intuitionistic logics. The aim of this paper is to present how these two ideas can be developed by Logics of Formal Inconsistency (LFIs) and Logics of Formal Undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, and LFUs recover the validity of the excluded middle in a paracomplete scenario. We will present two formal systems, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core – in the case studied here, classical positive propositional logic (CPL+). mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. Then, we combine these two logics obtaining logics of formal inconsistency and undeterminedness (LFIUs), called mbCD and mbCDE. The swap structures semantics framework for LFIs, is adapted here for LFUs and LFIUs. This semantics allows us to prove the decidability of the proposed systems by means of finite non-deterministic matrices.