Characterizing intermediate tense logics in terms of Galois connections

Abstract

We propose a uniform way of defining for every logic $\mathsf{L}$ intermediate between intuitionistic and classical logics, the corresponding intermediate minimal tense logic $\mathsf{L}{\mathsf{K}}_{\mathsf{t}}$. This is done by building the fusion of two copies of intermediate logic with a Galois connection $\mathsf{LGC}$, and then interlinking their operators by two Fischer Servi axioms. The resulting system is called here $L2GC+\mathsf{FS}$. In the cases of intuitionistic logic $\mathsf{Int}$ and classical logic $\mathsf{Cl}$, it is noted that $Int2GC+\mathsf{FS}$ is syntactically equivalent to intuitionistic minimal tense logic $\mathsf{I}{\mathsf{K}}_{\mathsf{t}}$ by W. B. Ewald and $Cl2GC+\mathsf{FS}$ equals classical minimal tense logic ${\mathsf{K}}_{\mathsf{t}}$. This justifies to consider $L2GC+\mathsf{FS}$ as minimal $\mathsf{L}$-tense logic $\mathsf{L}{\mathsf{K}}_{\mathsf{t}}$ for any intermediate logic $\mathsf{L}$. We define H2GC+FS-algebras as expansions of HK1-algebras, introduced by E. Orlowska and I. Rewitzky. For each intermediate logic $\mathsf{L}$, we show algebraic completeness of $L2GC+\mathsf{FS}$ and its conservativeness over $\mathsf{L}$. We prove relational completeness of $Int2GC+\mathsf{FS}$ with respect to the models defined on $\mathsf{IK}$-frames introduced by G. Fischer Servi. We also prove a representation theorem stating that every H2GC+FS-algebra can be embedded into the complex algebra of its canonical $\mathsf{IK}$-frame.

28 pages, 1 figure

References in corpus (1)

• Representing distributive lattices with Galois connections in terms of rough sets