On the geometry of interaction for classical logic

Abstract

It is well-known that weakening and contraction cause naive categorical models of the classical sequent calculus to collapse to Boolean lattices. We introduce sound and complete models that avoid this collapse by interpreting cut-reduction by a partial order between morphisms. We provide concrete examples of such models by applying the geometry-of-interaction construction to quantaloids with finite biproducts, and show how these models illuminate cut reduction in the presence of weakening and contraction. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish non-deterministic choices of cut-elimination.