On the Lambek Calculus with an Exchange Modality

TL;DR

This paper introduces a new logic called CNC logic, combining intuitionistic linear logic and a mixed Lambek calculus, with categorical models and an exchange modality derived via monoidal adjunctions.

Contribution

It presents the first formalization of CNC logic, integrating non-commutative and commutative aspects with categorical models and an exchange modality.

Findings

Categorical models for CNC logic are developed.

An exchange modality is derived using monoidal adjunctions.

Concrete dialectica Lambek space models are provided.

Abstract

In this paper we introduce Commutative/Non-Commutative Logic (CNC logic) and two categorical models for CNC logic. This work abstracts Benton's Linear/Non-Linear Logic by removing the existence of the exchange structural rule. One should view this logic as composed of two logics; one sitting to the left of the other. On the left, there is intuitionistic linear logic, and on the right is a mixed commutative/non-commutative formalization of the Lambek calculus. Then both of these logics are connected via a pair of monoidal adjoint functors. An exchange modality is then derivable within the logic using the adjunction between both sides. Thus, the adjoint functors allow one to pull the exchange structural rule from the left side to the right side. We then give a categorical model in terms of a monoidal adjunction, and then a concrete model in terms of dialectica Lambek spaces.