Subreflexive Logic: Completeness without Identity

2026-07-27

AI Summary

This study investigates substructural logics that omit the identity axiom $A \to A$, aiming to develop a complete semantic framework that does not presuppose reflexivity. To this end, it introduces, for the first time, Heyting and Boolean semialgebras, semi-adjunctions over semicategories, and structures lacking identity units. By integrating algebraic semantics, semicategorical semantics, and set-theoretic denotational semantics, the paper establishes a sound and complete formal system. This framework interprets implication as robust inference, proves the logic’s decidability and syntactic cut-elimination, and demonstrates completeness across multiple semantic interpretations, thereby providing a unified and innovative semantic foundation for non-reflexive logics.

Abstract

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.

Problem

• subreflexive logic
• identity principle
• completeness
• substructural logic
• robust consequence

Innovation

• subreflexive logic
• cut elimination
• Heyting semialgebras
• semi-categorical semantics
• robust implication