Semantics
Given a Heyting algebra H, the set of propositional formulas that are valid in H is an intermediate logic. Conversely, given an intermediate logic it is possible to construct its Lindenbaum algebra which is a Heyting algebra.
An intuitionistic Kripke frame F is a partially ordered set, and a Kripke model M is a Kripke frame with valuation such that is an upper subset of F. The set of propositional formulas that are valid in F is an intermediate logic. Given an intermediate logic L it is possible to construct a Kripke model M such that the logic of M is L (this construction is called canonical model). A Kripke frame with this property may not exist, but a general frame always does.
Read more about this topic: Intermediate Logic