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Earlier, the authors introduced the logic IntGC, which is an extension ofintuitionistic propositional logic by two rules of inference mimicking theperformance of Galois connections (Logic J. of the IGPL, 18:837-858, 2010). Inthis paper, the extensions Int2GC and Int2GC+FS of IntGC are studied. Int2GCcan be seen as a fusion of two IntGC logics, and Int2GC+FS is obtained fromInt2GC by adding instances of duality-like connections $\Diamond(A \to\ B) \to(\Box A \to \Diamond B)$ and $(\Diamond A \to \Box B) \to \Box(A \to B)$,introduced by G. Fischer Servi (Rend. Sem. Mat. Univers. Politecn. Torino,42:179-194, 1984), for interlinking the two Galois connections of Int2GC. BothKripke-style and algebraic semantics are presented for Int2GC and Int2GC+FS,and the logics are proved to be complete with respect to both of thesesemantics. We show that rough lattice-valued fuzzy sets defined on completeHeyting algebras are proper algebraic models for Int2GC+FS. We also prove thatInt2GC+FS is equivalent to the intuitionistic tense logic IKt, and anaxiomatisation of IKt with the number of axioms reduced to the half of thenumber of axioms given by W. B. Ewald (J. Symb. Log, 51:166-179, 1986) ispresented.

Intuitionistic logic with two Galois connections combined with Fischer Servi axioms