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There exist important deductive systems, such as the non-normal modal logics, that are not proper subjects of classical algebraic logic in the sense that their metatheory cannot be reduced to the equational metatheory of any particular class of algebras. Nevertheless, most of these systems are amenable to the methods of universal algebra when applied to the matrix models of the system. In the present paper we consider a wide class of deductive systems of this kind called protoalgebraic logics. These include almost all (non-pathological) systems of prepositional logic that have occurred in the literature. The relationship between the metatheory of a protoalgebraic logic and its matrix models is studied. The following results are obtained for any finite matrix model U of a filter-distributive protoalgebraic logic Λ: (I) The extension Λ of Λ is finitely axiomatized (provided Λ has only finitely many inference rules); (II) Λ has only finitely many extensions. [ABSTRACT FROM AUTHOR]

Let S denote the variety of Sugihara algebras. We prove that the lattice Λ ( K) of subquasivarieties of a given quasivariety K $$ \subseteq $$ S is finite if and only if K is generated by a finite set of finite algebras. This settles a conjecture by Tokarz [6]. We also show that the lattice Λ ( S) is not modular. [ABSTRACT FROM AUTHOR]

We study modal logics in the setting of varieties of modal algebras. Any variety of modal algebras generated by a finite algebra - such, a variety is called tabular - has only finitely many subvarieties, i.e. is of finite height. The converse does not hold in general. It is shown that the converse does hold in the lattice of varieties of K4-algebras. Hence the lower part of this lattice consists of tabular varieties only. We proceed to show that there is a continuum of pretabular varieties of K4-algebras - those are the non-tabular varieties all of whose proper subvarieties are tabular - in contrast with Maksimova's result that there are only five pretabular varieties of S4-algebras. [ABSTRACT FROM AUTHOR]

The veiled recession frame has served several times in the literature to provide examples of modal logics failing to have certain desirable properties. Makinson [4] was the first to use it in his presentation of a modal logic without the finite model property. Thomason [5] constructed a (rather complicated) logic whose Kripke frames have an accessibility relation which is reflexive and transitive, but which is satisfied by the (non-transitive) veiled recession frame, and hence incomplete. In Van Benthem [2] the frame was an essential tool to find simple examples of incomplete logics, axiomatized by a formula in two proposition letters of degree 2, or by a formula in one proposition letter of degree 4 (the degree of a modal formula ϕ is the maximal number of nested occurrences of the necessity operator in ϕ). In [3] we showed that the modal logic determined by the veiled recession frame is incomplete, and besides that, is an immediate predecessor of classical logic (or, more precisely, the modal logic axiomatized by the formula p↔□ p), and hence is a logic, maximal among the incomplete ones. Considering the importance of the modal logic determined by the veiled recession frame, it seems worthwhile to ask for an axiomatization, and in particular, to answer the question if it is finitely axiomatizable. In the present paper we find a finite axiomatization of the logic, and in fact, a rather simple one consisting of formulas in at most two proposition letters and of degree at most three. [ABSTRACT FROM AUTHOR]