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On distributivity of closure systems

Wojciech Dzik & Roman Suszko

Bulletin of the Section of Logic 6 (2):64-66 (1977)

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Matrices, primitive satisfaction and finitely based logics.Janusz Czelakowski - 1983 - Studia Logica 42 (1):89 - 104.details We examine the notion of primitive satisfaction in logical matrices. Theorem II. 1, being the matrix counterpart of Baker's well-known result for congruently distributive varieties of algebras (cf [1], Thm. 1.5), links the notions of primitive and standard satisfaction. As a corollary we give the matrix version of Jónsson's Lemma, proved earlier in [4]. Then we investigate propositional logics with disjunction. The main result, Theorem III. 2, states a necessary and sufficient condition for such logics to be finitely based. Download Export citation Bookmark 11 citations
Finite basis theorem for Filter-distributive protoalgebraic deductive systems and strict universal horn classes.Katarzyna Pałasińska - 2003 - Studia Logica 74 (1-2):233 - 273.details We show that a finitely generated protoalgebraic strict universal Horn class that is filter-distributive is finitely based. Equivalently, every protoalgebraic and filter-distributive multidimensional deductive system determined by a finite set of finite matrices can be presented by finitely many axioms and rules. Download Export citation Bookmark 8 citations
Local deductions theorems.Janusz Czelakowski - 1986 - Studia Logica 45 (4):377 - 391.details The notion of local deduction theorem (which generalizes on the known instances of indeterminate deduction theorems, e.g. for the infinitely-valued ukasiewicz logic C ) is defined. It is then shown that a given finitary non-pathological logic C admits the local deduction theorem iff the class Matr(C) of all matrices validating C has the C-filter extension property (Theorem II.1). Download Export citation Bookmark 31 citations
Filter distributive logics.Janusz Czelakowski - 1984 - Studia Logica 43 (4):353 - 377.details The present paper is thought as a formal study of distributive closure systems which arise in the domain of sentential logics. Special stress is laid on the notion of a C-filter, playing the role analogous to that of a congruence in universal algebra. A sentential logic C is called filter distributive if the lattice of C-filters in every algebra similar to the language of C is distributive. Theorem IV.2 in Section IV gives a method of axiomatization of those filter distributive (...) Download Export citation Bookmark 14 citations
(3 other versions)Equivalential logics (II).Janusz Czelakowski - 1981 - Studia Logica 40 (4):355 - 372.details In the first section logics with an algebraic semantics are investigated. Section 2 is devoted to subdirect products of matrices. There, among others we give the matrix counterpart of a theorem of Jónsson from universal algebra. Some positive results concerning logics with, finite degrees of maximality are presented in Section 3. Download Export citation Bookmark 35 citations

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