Introduction of different semantics for intuitionistic logic

Intuitionistic logic is a non classical logic obtained by omitting the axiom of excluded middle from classical logic. This logic was created by philosophical motivation towards the foundation of mathematics. There are several semantics for intuitionistic logic (such as Kripke semantics, neighborhood semantics and topological semantics) that are sound and complete. In this paper, we first present two new neighborhood semantics for propositional intuitionistic logic (IPC). Then we establish soundness and completeness of IPC with respect to these new neighborhood semantics. The relation between neighborhood and topological semantics are also investigated. One of these new neighborhood semantics is introduced with a somewhat more complex definition than the usual neighborhood semantics which was introduced before. This semantics is called NB-neighborhood semantics. In order to establish completeness with respect to NB-neighborhood semantics for IPC, first we need to introduce a system WF of subintuitionistic logic, weaker than Corsi's basic subintuitionistic system F.