Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic (Extended Abstract)

TL;DR

This paper introduces a novel proof-theoretic semantics for intuitionistic multiplicative linear logic, extending base-extension semantics to substructural logics and demonstrating soundness and completeness.

Contribution

It develops the first proof-theoretic semantics for IMLL, defining logical constants via elimination rules and establishing their soundness and completeness.

Findings

Semantics for IMLL are sound and complete.

Alternative treatment of conjunction via elimination rule.

Extension of B-eS to substructural 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 (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 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.