A Characterization of the Classes of Finite Tree Frames Which are Adequate for the Intuitionistic Logic

This article provides necessary and sufficient conditions for a class of Kripke frames consisting of finite trees, to be characteristic for the intuitionistic propositional logic I . In particular, Theorem 1 gives an algebraic proof of the adequacy of the Jaskowski and related classes of frames for I . Prel iminaries . By a Kripke frame we understand a triple '$1 = ( A , 2 , O ) where A is a non-empty set and 2 is a partial ordering of A in which 0 is the unique minimal element. An assignment in a Kripke frame a is a function assigning to each propositional formula q and a E A an element of (0, 1}, denoted [q],, satisfying the conditions: