On the Lambek Calculus with an Exchange Modality
Abstract
In this paper we introduce Commutative/NonCommutative Logic (CNC logic) and two categorical models for CNC logic. This work abstracts Benton's Linear/NonLinear 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/noncommutative 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.06847
 Bibcode:
 2019arXiv190406847J
 Keywords:

 Computer Science  Logic in Computer Science;
 Computer Science  Programming Languages;
 F.3.2;
 F.4.1
 EPrint:
 In Proceedings LinearityTLLA 2018, arXiv:1904.06159