Intuitionistic logic, dual intuitionistic logic, and modality

TL;DR

This paper investigates the semantic foundations of dual intuitionistic logic through algebraic and topological frameworks, revealing connections to modal logic S4.

Contribution

It extends the understanding of dual intuitionistic logic by linking co-Heyting algebras, topological spaces, and Kripke semantics, highlighting their relationship with modal logic S4.

Findings

Established a connection between co-Heyting algebras and topological spaces.

Proved properties linking dual intuitionistic logic to modal logic S4.

Extended Heyting algebra ideas to dual intuitionistic setting.

Abstract

We explore various semantic understandings of dual intuitionistic logic by exploring the relationship between co-Heyting algebras and topological spaces. First, we discuss the relevant ideas in the setting of Heyting algebras and intuitionistic logic, showing organically the progression from the primordial example of lattices of open sets of topological spaces to more general ways of thinking about Heyting algebras. Then, we adapt the ideas to the dual intuitionistic setting, and use them to prove a number of interesting properties, including a deep relationship both intuitionistic and dual intuitionistic logic share through Kripke semantics to the modal logic $\mathsf{S4}$.