Fitting’s Heyting-valued modal logic and Heyting-valued logic have been extensively examined from an algebraic perspective. The development of topological duality theorems and algebraic axiomatizations has shed light on the completeness of Fitting’s logic and modal logic. However, until now, there has been a noticeable lack of bitopology and biVietoris-coalgebra techniques in the study of duality for Heyting-valued modal logic.
This paper aims to bridge this gap by establishing a bitopological duality for algebras of Fitting’s Heyting-valued modal logic. To achieve this, the authors introduce a bi-Vietoris functor on the category of Heyting-valued pairwise Boolean spaces, denoted as $PBS_{mathcal{L}}$. This functor allows for a deeper understanding of the relationships between algebras of Fitting’s logic and categories of bi-Vietoris coalgebras.
The key result derived from this study is a dual equivalence between algebras of Fitting’s Heyting-valued modal logic and categories of bi-Vietoris coalgebras. This finding demonstrates that, in relation to the coalgebras of a bi-Vietoris functor, Fitting’s many-valued modal logic is both sound and complete.
This research contributes significantly to the field of modal logic by not only expanding the understanding of Heyting-valued modal logic but also incorporating bitopology and biVietoris-coalgebra techniques into the analysis. This sets the stage for further exploration and potential advancements in the study of duality for Heyting-valued modal logic.