Predicate extensions of the intermediate logic of the weak excluded middle and of the modal logic S4.2 are introduced and investigated. In particular it is shown that some of them are characterized by subclasses of the class of directed frames with either constant or nested domains.

In [Ono 1987] H. Ono put the question about axiomatizing the intermediate predicate logicLFin characterized by the class of all finite Kripke frames. 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. Thus the logicLFin is II 2 0 -arithmetical.Here we give a more explicit II 2 0 -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]. We prove thatLB n P m + =,B n being the propositional axiom of the branching n. (shrink)

An intermediate predicate logic L is called finite iff it is characterized by a finite partially ordered set M, i.e., iff L is the logic of the class of all predicate Kripke frames based on M. In this paper we study axiomatizability of logics of this kind. Namely, we consider logics characterized by finite trees M of a certain type (levelwise uniform trees) and establish the finite axiomatizability criterion for this case.

Some properties of Kripke-sheaf semantics for super-intuitionistic predicate logics are shown. The concept ofp-morphisms between Kripke sheaves is introduced. It is shown that if there exists ap-morphism from a Kripke sheaf 1 into 2 then the logic characterized by 1 is contained in the logic characterized by 2. Examples of Kripke-sheaf complete and finitely axiomatizable super-intuitionistic (and intermediate) predicate logics each of which is Kripke-frame incomplete are given. A correction to the author's previous paper Kripke bundles for intermediate predicate logics (...) and Kripke frames for intuitionistic modal logics (Studia Logica, 49(1990), pp. 289–306 ) is stated. (shrink)

The logicD-J of the weak exluded middle with constant domains is proved to be incomplete with respect to Kripke semantics, by introducing models in presheaves on an arbitrary category. Additional incompleteness results are obtained for the modal systems with nested domains extendingQ-S4.1.

We propose a new, rather simple and short proof of Kripke-completeness for the predicate variant of Dummett's logic. Also a family of Kripke-incomplete extensions of this logic that are complete w.r.t. Kripke frames with equality (or equivalently, w.r.t. Kripke sheaves [8]), is described.