Fitting's Heyting-valued modal logic and Heyting-valued logic have previously been examined from an algebraic perspective. Topological duality theorems have been developed in addition to algebraic axiomatizations with the completeness of Fitting's logic and modal logic. Recently, bitopological techniques have been used to study duality for Heyting-valued logic. But the development of duality for Heyting-valued modal logic noticeably lacks bitopology and biVietoris-coalgebra techniques. We are trying to bridge this gap in this paper. We establish a bitopological duality for algebras of Fitting's Heyting-valued modal logic. We build a bi-Vietoris functor on the category of Heyting-valued pairwise Boolean spaces, denoted by $PBS_{\mathcal{L}}$. In the end, we derive a dual equivalence between algebras of Fitting's Heyting-valued modal logic and categories of bi-Vietoris coalgebras. We thus conclude that, with respect to the coalgebras of a bi-Vietoris functor, Fitting's many-valued modal logic is sound and complete.

翻译：菲廷的海廷赋值模态逻辑及海廷赋值逻辑此前已从代数角度进行了研究。除了通过代数公理化证明菲廷逻辑及模态逻辑的完全性外，拓扑对偶定理也相继被发展。最近，双拓扑方法被用于研究海廷赋值逻辑的对偶性。然而，海廷赋值模态逻辑的对偶性研究中仍明显缺乏双拓扑与双Vietoris余代数技术的应用。本文旨在填补这一空白。我们为菲廷的海廷赋值模态逻辑的代数建立了双拓扑对偶性，并在海廷赋值成对布尔空间范畴$PBS_{\mathcal{L}}$上构造了一个双Vietoris函子。最终，我们推导出菲廷的海廷赋值模态逻辑代数与双Vietoris余代数范畴之间的对偶等价关系。因此我们得出结论：对于双Vietoris函子的余代数而言，菲廷的多值模态逻辑是可靠且完全的。