This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia’s original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations (...) to weak Heyting algebras and implicative semilattices. The book also dives into the Blok-Esakia theorem and provides an outline of the intuitionistic modal logic KM which is closely related to the Gödel-Löb provability logic GL. One chapter scrutinizes Esakia’s work interpreting modal diamond as the derivative of a topological space within the setting of point-free topology. The final chapter in the volume is dedicated to the derivational semantics of modal logic and other related issues. (shrink)

We characterize the bimodal provability logics for certain natural (classes of) pairs of recursively enumerable theories, mostly related to fragments of arithmetic. For example, we shall give axiomatizations, decision procedures, and introduce natural Kripke semantics for the provability logics of (IΔ 0 + EXP, PRA); (PRA, IΣ 1 ); (IΣ m , IΣ n ) for $1 \leq m etc. For the case of finitely axiomatized extensions of theories these results are extended to modal logics with propositional constants.