Finite and finitely separable intermediate propositional logics

Abstract

§0. Introduction and material background. The present paper is devoted to the study of intermediate propositional logics, and it is based on [Be, §§1 and 2].§2 (§§0 and 1 are introductory) concerns the axiomatization of finite logics. In the literature several effective procedures to axiomatize finite logics are present (cf., for instance, [MK] and [Wr]), but, in each case, the number of propositional variables which are used is redundant. In this direction, Theorem 2.2 provides (a) a criterion to determine, given a finite logic L, the least n such that L is axiomatizable by formulas in n variables, and (b) an effective axiomatization by an n-formula. As a corollary we obtain a negative answer to Problem 7.10 of [Ho/On], showing that there is no connection between the slice to which L belongs and the number of propositional variables necessary to axiomatize L.The principal results of the paper are in §3. In fact, a great deal of research has been done on the correspondence between conditions on the relation of Kripke-structures from one side, and axioms added to Int from the other. In this section we (a) introduce the concept of finitely separable class of Kripke-frames, and show, by means of several examples, that this concept is “wide”, in the sense that all the most studied classes of frames determined by semantical conditions are finitely separable; (b) show that each finitely separable class is axiomatizable, and that the axioms can be found by means of semantical considerations only; and (c) establish the finite model property for all the finitely separable logics.