Abstract
With each sentential logic C, identified with a structural consequence operation in a sentential language, the class Matr * (C) of factorial matrices which validate C is associated. The paper, which is a continuation of [2], concerns the connection between the purely syntactic property imposed on C, referred to as Maehara Interpolation Property (MIP), and three diagrammatic properties of the class Matr* (C): the Amalgamation Property (AP), the (deductive) Filter Extension Property (FEP) and Injections Transferable (IT). The main theorem of the paper (Theorem 2.2) is analogous to the Wroński's result for equational classes of algebras [13]. It reads that for a large class of logics the conjunction of (AP) and (FEP) is equivalent to (IT) and that the latter property is equivalent to (MIP).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Czelakowski, Equivalential logics, Part I. Studia, Logica, Vol. XL, No. 3 (1981), pp. 227–236; Part II. ibidem, Vol. XL, No. 4 (1981), pp. 353–370.
J. Czelakowski, Logical matrices and the amalgamation property, Studia Logica, Vol. XLI, No. 4 (1982), pp. 329–341.
J. Czelakowski, Logical matrices, primitive satisfaction and finitely based logics, Studia Logica, Vol. XLII, No. 1 (1983) pp. 89–104.
J. Czelakowski, Algebraic aspects of deduction theorems, to appear.
S. Maehara, Craig's interpolation theorem (in Japanese), Sŭgaku (1961), pp. 235–237.
L. Maximova, The Craig's theorem in superintuitionistic logics and amalgamated properties of varieties of pseudo-Boolean algebras (in Russian), Algebra i Logika, Vol. 16, No. 6 (1977), pp. 643–681.
D. Pigozzi, Amalgamation, congruence-extension, and interpolation properties in algebras, Algebra Universalis, Vol. 1 (1971), pp. 269–349.
T. Prucnal and A. Wroński, An algebraic characterization of the notion of structural completeness, Bulletin of the Section of Logic 3 (1974), pp. 30–33.
H. Rasiowa, An Algebraic Approach to Non-Classical Logics, North-Holland and PWN, Amsterdam/Warszawa 1974.
D. J. Shoesmith and T. J. Smiley, Deducibility and many-valuedness, Journal of Symbolic Logic, Vol. 36, No. 4 (1971), pp. 610–622.
G. Takeuti, Proof Theory, North-Holland, Amsterdam-London 1975.
W. Taylor, Appendix 4. Equational Logic, in: G. Grätzer, Universal Algebra (second edition), Van Nostrand, Princeton, New Jersey 1978.
A. Wroński, Maehara-style equational interpolation property, Abstracts of the 7th International Congress of Logic, Methodology and Philosophy of Science, Salzburg 1983, Vol. 1, p. 57. (An extended version entitled On a form of equational interpolation property will appear in Foundations in Logic and Linguistic. Problems and Solutions. Selected Contributions to the 7 th International Congress, Plenum Press, London 1984.)
Author information
Authors and Affiliations
Additional information
The author is indebted to the referee for several suggestions which have helped to simplify the original exposition to its present form; and for a careful reading of the manuscript which uncovered certain minor errors.
Rights and permissions
About this article
Cite this article
Czelakowski, J. Sentential logics and Maehara Interpolation Property. Stud Logica 44, 265–283 (1985). https://doi.org/10.1007/BF00394446
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00394446