Topological Semantics and Bisimulations for Intuitionistic Modal Logics and Their Classical Companion Logics ⋆

Abstract. We take the well-known intuitionistic modal logic of Fischer Servi with semantics in bi-relational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic pre-order (or partial-order) with the modal accessibility relation generalise to the requirement that the relation and its inverse be lower semi-continuous with respect to the topology. We then investigate the notion of topological bisimulation relations between topological Kripke frames, as introduced by Aiello and van Benthem, and show that their topology-preserving conditions are equivalent to the properties that the inverse-relation and the relation are lower semicontinuous with respect to the topologies on the two models. Our first main result is that this notion of topological bisimulation yields semantic preservation w.r.t. topological Kripke models for both intuitionistic tense logics, and for their classical companion multi-modal logics in the setting of the Gödel translation. After giving canonical topological Kripke models for the Hilbert-style axiomatizations of the Fischer Servi logic and its classical multi-modal companion logic, we show that the syntactic Gödel translation induces a natural semantic map from the intuitionistic canonical model into the canonical model of the classical companion logic, and this map is itself a topological bisimulation.

Modele otoczeniowe i topologiczne dla klasycznych i intuicjonistycznych logik modalnych

We may speak about syntax. From this point of view any logic can be considered as as the set of axioms and rules. Here we are interested in formal proofs and deduction systems. Second, we can also think about semantics, namely, about some models in which it is possible to de ne the notions of truth and falsity. As for the logical calculi, we are working with propositional logics. Thus, we are not so much interested in quanti ers. Our logics are non-classical. Of course, there are many kinds of non-classical logic and many reasons for which certain system can be considered as nonclassical. In our case, there are two main ways which are notoriously combined. On the one hand, we are interested in intuitionistic, superintuitionistic and subintuitionistic systems. This means that we narrow down the set of axioms and rules of classical logic. On the other hand, we use modal operators to de ne and analyse the ideas of necessity and possibility. As a result, we often obtain classical and intuitionistic modal logics. Our semantic models are mostly neighborhood, topological and relational. These three approaches are also combined. For this reason, we may speak about bi-relational and relational-neighborhood structures. Moreover, we go beyond the standard notion of topology in order to study its various generalizations. Finally, our aim is to investigate several non-classical calculi using all the tools mentioned above. We are interested in the issues of completeness (axiomatization), nite model property, bisimulation and decidability. Moreover, we analyse some purely topological properties of the structures in question. The philosophical aspect is also important