Complementary Proof Nets for Classical Logic

Abstract

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}, 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} enjoys strong normalization along with strong confluence (and, hence, uniqueness of normal forms).