Intuitionistic modal logic based on neighborhood semantics without superset axiom

TLDR: This paper investigates certain systems of propositional intuitionistic modal logic defined semantically in terms of neighborhood structures and proves soundness and completeness of proposed axiomatization by means of canonical model and shows that finite model property holds.

Abstract: In this paper we investigate certain systems of propositional intuitionistic modal logic defined semantically in terms of neighborhood structures. We discuss various restrictions imposed on those frames but our constant approach is to discard superset axiom. Such assumption allows us to think about specific modalities $\Delta, \nabla$ and new functor $\rightsquigarrow$ depending on the notion of maximal neighborhood. We show how it is possible to treat our models as bi-relational ones. We prove soundness and completeness of proposed axiomatization by means of canonical model. Moreover, we show that finite model property holds. Then we describe properties of bounded morphism, behavioral equivalence, bisimulation and n-bisimulation. Finally, we discuss further researches (like some interesting classical cases and particular topological issues).