On Partial and Paraconsistent Logics

Abstract

In this paper we consider the theory of predicate logics in which the principle of bivalence or the principle of noncontradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove model existence. For $ \bf L_4$, the most general logic under consideration, we also prove a version of the Craig-Lyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalize to partial and paraconsistent logics once the right setup is chosen. Our logic $ \bf L_4$ has a semantics that also underlies Belnap's logic and is related to the logic of bilattices. $ \bf L_4$ is in focus most of the time, but it is also shown how results obtained for $ \bf L_4$ can be transferred to several variants.