Subreflexive Logic: Completeness without Identity

This paper shows that the substructural logic without the identity principle A->A (i.e., subreflexive logic) has principled sound and complete semantics and supports a variety of applications. This decidable generalization of propositional logic naturally interprets implication as robust consequence. Subreflexive logic is proved to admit syntactic cut elimination. Heyting and Boolean semialgebras are introduced as generalizations of Heyting and Boolean algebras and are shown to provide complete algebraic semantics without inadvertently reintroducing reflexivity. Semi-adjunctions on semi-categories and (identity-free) (co-)units are defined to give complete semi-categorical semantics. In the classical case, denotational set semantics that interpret implication as robust material implication are proved complete for subreflexive logic.