Simple example of weak modal logic based on intuitionistic core

Abstract: In this paper we present simple example of propositional logic which has one modal operator and is based on intuitionistic core. This system is very weak in modal sense - e.g. rules of regularity or monotonicity do not hold. It has complete semantics composed of possible worlds equipped with neighborhoods and pre-order relation. We discuss certain restrictions imposed on those structures. Also, we present characterization of axiom 4 known from logic S4.