Abstract
In this paper we investigate some basic semantic and syntactic conditions characterizing the equivalence connective. In particular we define three basic classes of algebras: the class of weak equivalential algebras, the class of equivalential algebras and the class of regular equivalential algebras (see [12]).
Weak equivalential algebras can be used to study purely equivalential fragments of relevant logics and strict equivalential fragments of some modal logics (for investigations of strict implicational fragments of modal logics see [20]). Equivalential algebras are suitable to study purely equivalential fragment of BCI and BCK logic (see [21], p. 316). A subclass of the class of regular equivalential algebras is suitable to study equivalential fragments of Łukasiewicz logics. Some subvarieties of the class of regular equivalential algebras provide natural semantics for equivalential fragments of the intuitionistic prepositional logic and various intermediate logics (see [13]).
The last chapter is a selection of research problems which in the author's opinion are worth to be solved.
Similar content being viewed by others
References
A.R. Anderson, N.D. Belnap Jr., Entailment. The Logic of Relevance and Necessity, Vol. I. Princeton and London, Princeton University Press, 1975, xxxii + 542 pp.
M. Dummett, A propositional calculus with denumerable matrix, The Journal of Symbolic Logic 24 (1959), pp. 97–106.
G. Grätzer, Universal Algebra. Second Edition. New York, Heidelberg and Berlin, Springer-Verlag, 1979, xviii + 581 pp.
A. Heyting, Die Formalen Regeln der intuitionistischen Logik, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, 1930, pp. 42–56.
J.K. Kabziński, On problems of definability of propositional connectives, Bulletin of the Section of Logic, Polish Academy of Sciences, 2 (1973), pp. 127–130.
J. K. Kabziński, Algebry równoważnościowe, Ph. D. dissertation, Jagiellonian University, Cracow, 1974, 54 pp.
J. K. Kabziński, Some problems connected with equivalential formalization of classical sentential calculus, Zeszyty Naukowe Wyższej Szkoly Pedagogicznej im. Powstańców Śląskich w Opolu, Matematyka, Logika matematyczna 15 (1975), pp. 29–36.
J. K. Kabziński, An axiomatization of the variety of equivalential algebras by a single identity, Bulletin of the Section of Logic, Polish Academy of Sciences, 6 (1977), pp. 102 –106.
J. K. Kabziński, On equivalential fragment of the three-valued logic of Łukasiewicz, Bulletin of the Section of Logic, Polish Academy of Sciences, 8 (1979), pp. 182–187.
J. K. Kabziński, Towards the source of the notion of implication, Bulletin of the Section of Logic, Polish Academy of Sciences, 9 (1980), pp. 180–183.
J. K. Kabziński, What is the equivalence connective, Bulletin of the Section of Logic, Polish Academy of Sciences, 9 (1980), pp. 184–188.
J. K. Kabziński, Investigations into the equivalence connective, Uniwersytet Jagielloński, Rozprawy Habilitacyjne, Nr 48, Kraków, 1980, 113 pp.
J. K. Kabziński, A. Wroński, On equivalential algebras, Proceedings of the 1975 International Symposium on Multiple-Valued Logic, Indiana University, Bloomington, May 13–16 (1975), pp. 419–428.
Y. Komori, The separation theorem of the ℵ 0-valued Łukasiewicz propositional logic, Reports of Faculty of Science, Shizouka University, 12 (1978), pp. 1–5.
J. Kotas, Logical systems with implications, Studia Logica 28 (1971), pp. 101–115.
S. Leśniewski, Grundzüge eines neuen Systems der Grundlagen der Mathematik, Fundamenta Mathematicae 14 (1929), pp. 15–30.
J. Łukasiewicz, O logice trójwartościowej (On three-valued logic), Ruch Filozoficzny 5 (1920), pp. 170–171.
G. Malinowski, J. Zygmunt, Sprawozdanie z Jesiennej Szkoly Logicznej, Międzygórze, 21–29.XI.1977 r. (Proceedings of the Autumn School on Strongly Finite Sentential Calculi), 1977, 35 pp. (unpublished).
E.P. Martin, The P-W problem, Bulletin of the Australian Mathematical Society 20 (1979), pp. 157–158.
C. A. Meredith, A. N. Prior, Investigations into implicational S5, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 10 (1964), pp. 203–220.
A. N. Prior, Formal Logic, Second Edition. Oxford, Clarendon Press, 1962.
H. Rasiowa, An Algebraic Approach to Non-Classical Logics, Studies in Logic and the Foundations of Mathematics, Volume 78, PWN — Polish Scientific Publishers — Warszawa, North-Holland Publishing Company — Amsterdam — London, 1974, xv + 403 pp.
H. Rasiowa, R. Sikorski, The Mathematics of Metamathematics, Warszawa, PWN — Polish Scientific Publishers, 1968, 519 pp.
K. Suszko, equational logic and theories in sentential languages, Colloquium Mathematicum 29 (1974), pp. 19–23.
R.E. Tax, On the intuitionistic equivalential calculus, Notre Dame Journal of Formal Logic 14 (1973), pp. 448–456.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kabziński, J.K. Basic properties of the equivalence. Stud Logica 41, 17–40 (1982). https://doi.org/10.1007/BF00373491
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00373491