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Earlier, the authors introduced the logic IntGC, which is an extension of intuitionistic propositional logic by two rules of inference mimicking the performance of Galois connections (Logic J. of the IGPL, 18:837-858, 2010). In this paper, the extensions Int2GC and Int2GC+FS of IntGC are studied. Int2GC can be seen as a fusion of two IntGC logics, and Int2GC+FS is obtained from Int2GC by adding instances of duality-like connections (A → B) → ( A → B) and ( A → B) →(A → B), introduced by G.arXiv:1208.2971v1 fatcat:lact2xfl4vb5jheyi3juj6nom4