Classical propositional logic can be characterized, indirectly, by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit “in the negative”. More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization of (...) its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized. (shrink)

A complementary system for a given logic is a proof system whose theorems are exactly the formulas that are not valid according to the logic in question. This article is a contribution to the complementary proof theory of classical propositional logic. In particular, we present a complementary proof-net system, $$\textsf{CPN}$$ CPN, that is sound and complete with respect to the set of all classically invalid (one-side) sequents. We also show that cut elimination in $$\textsf{CPN}$$ CPN enjoys strong normalization along with (...) strong confluence (and, hence, uniqueness of normal forms). (shrink)