On the Lambek Calculus with an Exchange Modality

Authors: Jiaming Jiang, Harley Eades Iii and Valeria de Paiva

Paper Information

Title:On the Lambek Calculus with an Exchange Modality
Authors:Jiaming Jiang, Harley Eades Iii and Valeria de Paiva
Proceedings:Linearity/TLLA Pre-proceedings
Editors: Maribel Fernandez, Valeria de Paiva, Thomas Ehrhard and Lorenzo Tortora De Falco
Keywords:adjoint model, lambek calculus, categorical models, dialectica category, linear/non-linear logic, monad, adjoint logic, non-commutative, exchange, modalities, linear logic

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.

Talk:Jul 07 09:20 (Session 23I)