A Cointuitionistic Adjoint Logic

Abstract

Bi-intuitionistic logic (BINT) is a conservative extension of intuitionistic logic to include the duals of each logical connective. One leading question with respect to BINT is, what does BINT look like across the three arcs -logic, typed λ-calculi, and category theory -of the Curry-Howard-Lambek correspondence? Categorically, BINT can be seen as a mixing of two worlds: the first being intuitionistic logic (IL), which is modeled by a cartesian closed category, and the second being the dual to intuitionistic logic called cointuitionistic logic (coIL), which is modeled by a cocartesian coclosed category. Crolard [11] showed that combining these two categories into the same category results in it degenerating to a poset. However, this degeneration does not occur when both logics are linear. We propose that IL and coIL need to be separated, and then mixed in a controlled way using the modalities from linear logic. This separation can be ultimately achieved by an adjoint formalization of bi-intuitionistic logic. This formalization consists of three worlds instead of two: the first is intuitionistic logic, the second is linear bi-intuitionistic (Bi-ILL), and the third is cointuitionistic logic. They are then related via two adjunctions. The adjunction between IL and ILL is known as a Linear/Non-linear model (LNL model) of ILL, and is due to Benton [4]. However, the dual to LNL models which would amount to the adjunction between coILL and coIL has yet to appear in the literature. In this paper we fill this gap by studying the dual to LNL models which we call dual LNL models. We conduct a similar analysis to that of Benton for dual LNL models by showing that dual LNL models correspond to dual linear categories, the dual to Bierman's [5] linear categories proposed by Bellin [3]. Following this we give the definition of bi-LNL models by combining our dual LNL models with Benton's LNL models to obtain a categorical model of bi-intuitionistic logic, but we leave its analysis and corresponding logic to a future paper. Finally, we give a corresponding sequent calculus, natural deduction, and term assignment for dual LNL models.