Proof-Theoretic Semantics for Intuitionistic Multiplicative Linear Logic

Abstract

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic ($$\mathrm IMLL$$ ). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for $$\mathrm IMLL$$, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established.