On the predicate logics of finite Kripke frames

Abstract

In [Ono 1987] H. Ono put the question about axiomatizing the intermediate predicate logicLFin characterized by the class of all finite Kripke frames (Problem 4,P41). It was established in [Skvortsov 1988] thatLFin is not recursively axiomatizable. One can easily show that for any finite posetM, the predicate logic characterized byM is recursively axiomatizable, and its axiomatization can be constructed effectively fromM. Namely, the set of formulas belonging to this logic is recursively enumerable, since it is embeddable in the two-sorted classical predicate calculusCPC ₂ (the definition of the truth in a Kripke model may be expressed by a formula ofCPC ₂). Thus the logicLFin is II ⁰₂ -arithmetical.

Here we give a more explicit II ⁰₂ -description ofLFin: it is presented as the intersection of a denumerable sequence of finitely axiomatizable Kripke-complete logics. Namely, we give an axiomatization of the logicLB _n P ⁺ _m characterized by the class of all posets of the finite height ⩽m and the finite branching ⩽n. A finite axiomatization of the predicate logicLP ⁺ _m characterized by the class of all posets of the height ⩽m is known from [Yokota 1989] (this axiomatics is essentially first-order; the standard propositional axiom of the height ⩽m is not sufficient [Ono 1983]). We prove thatLB _n P ⁺ _m =(LP ⁺ _m +B _n),B _n being the propositional axiom of the branching ⩽n (see [Gabbay, de Jongh 1974]).

Our terminology and notations mainly correspond to [Ono 1987].