Semantic Analysis of Subexponential Modalities in Distributive Non-commutative Linear Logic

TL;DR

This paper explores the semantic properties of subexponential modalities within the distributive Lambek calculus, establishing completeness via Kripke frames and interpreting modalities as S4-like with reflexive and transitive relations.

Contribution

It introduces a Kripke semantics for the distributive Lambek calculus with subexponentials, demonstrating completeness and interpreting modalities as S4-like.

Findings

Completeness of the calculus with respect to Kripke frames.

Subexponentials interpreted as S4-like modalities.

Kripke semantics for distributive Lambek calculus established.

Abstract

In this paper, we consider the full Lambek calculus enriched with subexponential modalities in a distributive setting. We show that the distributive Lambek calculus with subexponentials is complete with respect to its Kripke frames via canonical extensions. In this approach, we consider subexponentials as S4-like modalities and each modality is interpreted with a reflexive and transitive relation similarly to usual Kripke semantics.