## Works by W. J. Blok

15 found
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1. Protoalgebraic logics.W. J. Blok & Don Pigozzi - 1986 - Studia Logica 45 (4):337 - 369.
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 (...)

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2. Equivalence of Consequence Operations.W. J. Blok & Bjarni Jónsson - 2006 - Studia Logica 83 (1-3):91-110.
This paper is based on Lectures 1, 2 and 4 in the series of ten lectures titled “Algebraic Structures for Logic” that Professor Blok and I presented at the Twenty Third Holiday Mathematics Symposium held at New Mexico State University in Las Cruces, New Mexico, January 8-12, 1999. These three lectures presented a new approach to the algebraization of deductive systems, and after the symposium we made plans to publish a joint paper, to be written by Blok, further developing these (...)

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3. The lattice of modal logics: An algebraic investigation.W. J. Blok - 1980 - Journal of Symbolic Logic 45 (2):221-236.
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 (...)

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4. Pretabular varieties of modal algebras.W. J. Blok - 1980 - Studia Logica 39 (2-3):101 - 124.
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 (...)

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5. Algebraic semantics for deductive systems.W. J. Blok & J. Rebagliato - 2003 - Studia Logica 74 (1-2):153 - 180.
The notion of an algebraic semantics of a deductive system was proposed in [3], and a preliminary study was begun. The focus of [3] was the definition and investigation of algebraizable deductive systems, i.e., the deductive systems that possess an equivalent algebraic semantics. The present paper explores the more general property of possessing an algebraic semantics. While a deductive system can have at most one equivalent algebraic semantics, it may have numerous different algebraic semantics. All of these give rise to (...)

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6. The Beth Property in Algebraic Logic.W. J. Blok & Eva Hoogland - 2006 - Studia Logica 83 (1-3):49-90.
The present paper is a study in abstract algebraic logic. We investigate the correspondence between the metalogical Beth property and the algebraic property of surjectivity of epimorphisms. It will be shown that this correspondence holds for the large class of equivalential logics. We apply our characterization theorem to relevance logics and many-valued logics.

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7. Transitivity follows from Dummett's axiom.J. F. A. K. Van Benthem & W. J. Blok - 1978 - Theoria 44 (2):117-118.

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8. Algebraic semantics for quasi-classical modal logics.W. J. Blok & P. Köhler - 1983 - Journal of Symbolic Logic 48 (4):941-964.
A well-known result, going back to the twenties, states that, under some reasonable assumptions, any logic can be characterized as the set of formulas satisfied by a matrix 〈,F〉, whereis an algebra of the appropriate type, andFa subset of the domain of, called the set of designated elements. In particular, every quasi-classical modal logic—a set of modal formulas, containing the smallest classical modal logicE, which is closed under the inference rules of substitution and modus ponens—is characterized by such a matrix, (...)

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9. On the lattice of quasivarieties of Sugihara algebras.W. J. Blok & W. Dziobiak - 1986 - Studia Logica 45 (3):275 - 280.
Let S denote the variety of Sugihara algebras. We prove that the lattice (K) of subquasivarieties of a given quasivariety K 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 (S) is not modular.

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10. Free łukasiewicz and hoop residuation algebras.Joel Berman & W. J. Blok - 2004 - Studia Logica 77 (2):153 - 180.
Hoop residuation algebras are the {, 1}-subreducts of hoops; they include Hilbert algebras and the {, 1}-reducts of MV-algebras (also known as Wajsberg algebras). The paper investigates the structure and cardinality of finitely generated free algebras in varieties of k-potent hoop residuation algebras. The assumption of k-potency guarantees local finiteness of the varieties considered. It is shown that the free algebra on n generators in any of these varieties can be represented as a union of n subalgebras, each of which (...)

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11. Fragments of R-Mingle.W. J. Blok & J. G. Raftery - 2004 - Studia Logica 78 (1-2):59-106.
The logic RM and its basic fragments (always with implication) are considered here as entire consequence relations, rather than as sets of theorems. A new observation made here is that the disjunction of RM is definable in terms of its other positive propositional connectives, unlike that of R. The basic fragments of RM therefore fall naturally into two classes, according to whether disjunction is or is not definable. In the equivalent quasivariety semantics of these fragments, which consist of subreducts of (...)

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12. An axiomatization of the modal theory of the veiled recession frame.W. J. Blok - 1979 - Studia Logica 38 (1):37 - 47.
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 (...)

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13. Alfred Tarski's work on general metamathematics.W. J. Blok & Don Pigozzi - 1988 - Journal of Symbolic Logic 53 (1):36-50.
In this essay we discuss Tarski's work on what he calledthe methodology of the deductive sciences, or more briefly, borrowing the terminology of Hilbert,metamathematics, The clearest statement of Tarski's views on this subject can be found in his textbookIntroduction to logic[41m].1Here he describes the tasks of metamathematics as “the detailed analysis and critical evaluation of the fundamental principles that are applied in the construction of logic and mathematics”. He goes on to describe what these fundamental principles are: All the expressions (...)

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14. Introduction.W. J. Blok & D. Pigozzi - 1991 - Studia Logica 50 (3-4):365-374.