In this paper some parts of the model theory for logics based on generalised Kripke semantics are developed. Löwenheim-Skolem theorems and some applications of ultraproduct constructions for generalised Kripke models with variable universe are investigated using similar theorems of the model theory for classical logic. The results are generalizations of the theorems of [4].

It is demonstrated how Kripke models for intuitionistic predicate logic can be applied in order to prove classical theorems. As examples proofs of the independence of the axiom of constructibility, of the omitting types theorem and of Shelah's ultrapower theorem are sketched.

A game for testing the equivalence of Kripke models with respect to finitary and infinitary intuitionistic predicate logic is introduced and applied to discuss a concept of categoricity for intuitionistic theories.