Krlpke Models and Intermediate Logics

In [10], Kripke gave a definition of the semantics of the intuitionistic logic. Fitting [2] showed that Kripke's models are equivalent to algebraic models (i.e., pseudo-Boolean models) in a certain sense. As a corollary of this result, we can show that any partially ordered set is regarded as a (characteristic) model of a intermediate logic ̂ We shall study the relations between intermediate logics and partially ordered sets as models of them, in this paper. We call a partially ordered set, a Kripke model.^ At present we don't know whether any intermediate logic 'has a Kripke model. But Kripke models have some interesting properties and are useful when we study the models of intermediate logics. In §2, we shall study general properties of Kripke models. In §3, we shall define the height of a Kripke model and show the close connection between the height and the slice, which is introduced in [7]. In §4, we shall give a model of LP» which is the least element in n-ih slice Sn (see [7]).