We study the relations of being substructure and elementary substructure between Kripke models of intuitionistic predicate logic with the same arbitrary frame. We prove analogues of Tarski's test and Löwenheim-Skolem's theorems as determined by our definitions. The relations between corresponding worlds of two Kripke models [MATHEMATICAL SCRIPT CAPITAL K] ⪯ [MATHEMATICAL SCRIPT CAPITAL K]′ are studied.

A value space is a topological algebra equipped with a non-empty family of continuous quantifiers . We will describe first-order logic on the basis of . Operations of are used as connectives and its relations are used to define statements. We prove under some normality conditions on the value space that any theory in the new setting can be represented by a classical first-order theory.

In this paper, some basic properties of automorphisms of first order languages and some examples are presented. In particular, an omitting types theorem is proved.

We prove some preservation theorems concerning inductive and model-complete theories in the framework of semi-classical logic introduced in [1].