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

TL;DR

This work is the first exploration of proof-theoretic semantics for a substructural logic, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established.

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.

Figures (5)

• Figure 1: Sandqvist's Support in a Base
• Figure 2: Atomic System $\mathscr{N}$
• Figure 3: The Sequential Natural Deduction System $\mathsf{NIMLL}$ for IMLL
• Figure 4: Base-extension Semantics for $\rm IMLL$
• Figure 5: Atomic System $\mathscr{M}$

Theorems & Definitions (50)

• definition 1: Derivability in a Base
• definition 2: Sandqvist's Support in a Base
• theorem 1: Sandqvist Sandqvist2015base
• proof : Theorem \ref{['thm:Sandqvist']} --- Completeness.
• definition 3
• theorem 2
• proof
• corollary 1
• definition 4: Formula
• definition 5: Sequent