Modal logics are studied in their algebraic disguise of varieties of so-called modal algebras. This enables us to apply strong results of a universal algebraic nature, notably those obtained by B. Jonsson. It is shown that the degree of incompleteness with respect to Kripke semantics of any modal logic containing the axiom □ p → p or containing an axiom of the form $\square^mp \leftrightarrow\square^{m + 1}p$ for some natural number m is 2 ℵ 0 . Furthermore, we show that (...) there exists an immediate predecessor of classical logic (axiomatized by $p \leftrightarrow \square p$ ) which is not characterized by any finite algebra. The existence of modal logics having 2 ℵ 0 immediate predecessors is established. In contrast with these results we prove that the lattice of extensions of S4 behaves much better: a logic extending S4 is characterized by a finite algebra iff it has finitely many extensions and any such logic has only finitely many immediate predecessors, all of which are characterized by a finite algebra. (shrink)

In this paper we define the notion of frame based formulas. We show that the well-known examples of formulas arising from a finite frame, such as the Jankov-de Jongh formulas, subframe formulas and cofinal subframe formulas, are all particular cases of the frame based formulas. We give a criterion for an intermediate logic to be axiomatizable by frame based formulas and use this criterion to obtain a simple proof that every locally tabular intermediate logic is axiomatizable by Jankov-de Jongh formulas. (...) We also show that not every intermediate logic is axiomatizable by frame based formulas. (shrink)

We prove that if a modal formula is refuted on a wK4-algebra ( B ,□), then it is refuted on a finite wK4-algebra which is isomorphic to a subalgebra of a relativization of ( B ,□). As an immediate consequence, we obtain that each subframe and cofinal subframe logic over wK4 has the finite model property. On the one hand, this provides a purely algebraic proof of the results of Fine and Zakharyaschev for K4 . On the other hand, it (...) extends the Fine-Zakharyaschev results to wK4. (shrink)

We introduce relativized modal algebra homomorphisms and show that the category of modal algebras and relativized modal algebra homomorphisms is dually equivalent to the category of modal spaces and partial continuous p-morphisms, thus extending the standard duality between the category of modal algebras and modal algebra homomorphisms and the category of modal spaces and continuous p-morphisms. In the transitive case, this yields an algebraic characterization of Zakharyaschev’s subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we (...) give an algebraic description of canonical, subframe, and cofinal subframe formulas, and provide a new algebraic proof of Zakharyaschev’s theorem that each logic over K4 is axiomatizable by canonical formulas. (shrink)

We develop duality between nuclei on Heyting algebras and certain binary relations on Heyting spaces. We show that these binary relations are in 1–1 correspondence with subframes of Heyting spaces. We introduce the notions of nuclear and dense nuclear varieties of Heyting algebras, and prove that a variety of Heyting algebras is nuclear iff it is a subframe variety, and that it is dense nuclear iff it is a cofinal subframe variety. We give an alternative proof that every subframe variety (...) of Heyting algebras is generated by its finite members. (shrink)

We introduce partial Esakia morphisms, well partial Esakia morphisms, and strong partial Esakia morphisms between Esakia spaces and show that they provide the dual description of (∧, →) homomorphisms, (∧, →, 0) homomorphisms, and (∧, →, ∨) homomorphisms between Heyting algebras, thus establishing a generalization of Esakia duality. This yields an algebraic characterization of Zakharyaschev’s subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we obtain a new simplified proof (which is algebraic in nature) of Zakharyaschev’s (...) theorem that each intermediate logic can be axiomatized by canonical formulas. (shrink)

We use the apparatus of the canonical formulas introduced by Zakharyaschev [10] to prove that all finitely axiomatizable normal modal logics containing K4.3 are decidable, though possibly not characterized by classes of finite frames. Our method is purely frame-theoretic. Roughly, given a normal logic L above K4.3, we enumerate effectively a class of frames with respect to which L is complete, show how to check effectively whether a frame in the class validates a given formula, and then apply a Harropstyle (...) argument to establish the decidability of L, provided of course that it has finitely many axioms. (shrink)

Related Works: Part I: Michael Zakharyaschev. Canonical Formulas for $K4$. Part I: Basic Results. J. Symbolic Logic, Volume 57, Issue 4 , 1377--1402. Project Euclid: euclid.jsl/1183744119 Part III: Michael Zakharyaschev. Canonical Formulas for K4. Part III: The Finite Model Property. J. Symbolic Logic, Volume 62, Issue 3 , 950--975. Project Euclid: euclid.jsl/1183745306.

This paper investigates the structure of lattices of normal mono- and polymodal subframelogics, i.e., those modal logics whose frames are closed under a certain type of substructures. Nearly all basic modal logics belong to this class. The main lattice theoretic tool applied is the notion of a splitting of a complete lattice which turns out to be connected with the “geometry” and “topology” of frames, with Kripke completeness and with axiomatization problems. We investigate in detail subframe logics containing K4, those (...) containing polymodal Altn and subframe logics which are tense logics. (shrink)

An old conjecture of modal logics states that every splitting of the major systems K4, S4, G and Grz has the finite model property. In this paper we will prove that all iterated splittings of G have fmp, whereas in the other cases we will give explicit counterexamples. We also introduce a proof technique which will give a positive answer for large classes of splitting frames. The proof works by establishing a rather strong property of these splitting frames namely that (...) they preserve the finite model property in the following sense. Whenever an extension Λ has fmp so does the splitting Λ/f of Λ by f. Although we will also see that this method has its limitations because there are frames lacking this property, it has several desirable side effects. For example, properties such as compactness, decidability and others can be shown to be preserved in a similar way and effective bounds for the size of models can be given. Moreover, all methods and proofs are constructive. (shrink)

A logic Λ bounds a property P if all proper extensions of Λ have P while Λ itself does not. We construct logics bounding finite axiomatizability and logics bounding finite model property in the lattice of intermediate logics and in the lattice of normal extensions of K4.3. MSC: 03B45, 03B55.

In the following we will use the well-known correspondence between modal logics and varieties of modal algebras in our investigation of the function which assigns to a modal logic its degree of incompleteness. A modal algebra is an algebra A = where is a Boolean algebra and is a unary operation satisfying 1 = 1 and = x y ; is called a modal operator. A variety of algebras is a class of algebras closed under the operations of forming homomorphic (...) images, subalgebras as and direct products, and if K is a class of algebras then V denotes the smallest variety containing K. The variety of modal algebras is denoted by M, the subvariety of M dened by the equation x x = x by MR and the subvariety of M dened by the equation x n = x n1 , n a natural number, by Mn . Here x 0 = x, x n = , n a natural number. We write for the lattice of subvarieties of a variety K. If K, K 0 are varieties such that K $ K 0 but for no variety K 00 K $ K 00 $ K 0 then we say that K 0 is a cover of K. The smallest normal modal logic { containing the axiom ! and closed under the inference rules of modus ponens, substitution and necessitation { will be denoted by K; the lattice of modal logics which are extensions of K by. (shrink)