Abstract
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 logics for which the class Matr (C) prime of C-prime matrices (models) is axiomatizable. In Section V, the attention is focused on axiomatic strengthenings of filter distributive logics. The theorems placed there may be regarded, to some extent, as the matrix counterparts of Baker's well-known theorem from universal algebra [9, § 62, Theorem 2].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. L. Bloom, Some theorems on structural consequence operations, Studia Logica, Vol. XXXIV, No. 1 (1975), pp. 1–9.
C. C. Chang and H. J. Keisler, Model Theory, North-Holland and American Elsevier, Amsterdam-London-New York, 1973.
P. Crawley and R. P. Dilworth, Algebraic Theory of Lattices, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973.
J. Czelakowski, Reduced products of logical matrices, Studia Logica, Vol. XXXIX, No. 1 (1980), pp. 19–43.
—, Equivalential logics, Part I, Studia Logica, Vol. XL, No. 3 (1981), pp. 227–236, Part II, ibidem, Vol. XL, No. 4 (1981), pp. 355–372.
—, Matrices, primitive satisfaction and finitely based logics, Studia Logica, Vol. XLII, No. 1 (1983), pp. 89–104.
-, Algebraic aspects of deduction theorems, to appear in Studia Logica.
W. Dzik and R. Suszko, On distributivity of closure systems, Bulletin of the Section of Logic, Vol. 6, No. 2 (1977), pp. 64–66.
B. Jónsson, Appendix: 3 — Congruence varieties, in: G. Grätzer, Universal Algebra, Second edition, Van Nostrand, Princeton, 1978.
J. Łoś and R. Suszko, Remarks on sentential logics, Indagationes Mathematicae 20 (1958), pp. 177–183.
D. J. Shoesmith and T. J. Smiley, Deducibility and many-valuedness, Journal of Symbolic Locic, Vol. 36, No. 4 (1971), pp. 610–622.
A. Tarski, Fundamental concepts of the methodology of the deductive sciences, in: Logic, Semantics and Metamathematics, Oxford, 1956.
R. Wójcicki, Matrix approach in methodology of sentential calculi, Studia Logica, Vol. XXXII (1973), pp. 7–37.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Czelakowski, J. Filter distributive logics. Stud Logica 43, 353–377 (1984). https://doi.org/10.1007/BF00370507
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00370507