Characterizing intermediate tense logics in terms of Galois connections

Abstract

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