Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-22T20:27:46.069Z Has data issue: false hasContentIssue false
  • English
  • Français

The contributions of Alfred Tarski to algebraic logic

Published online by Cambridge University Press:  12 March 2014

J. Donald Monk
J. Donald Monk*
Affiliation:
Department of Mathematics, University of Colorado, Boulder, Colorado 80309
Get access Rights & Permissions [Opens in a new window]

Extract

One of the most extensive parts of Tarski's contributions to logic is his work on the algebraization of the subject. His work here involves Boolean algebras, relation algebras, cylindric algebras, Boolean algebras with operators, Brouwerian algebras, and closure algebras. The last two are less developed in his work, although his contributions are basic to other work in those subjects. At any rate, not being conversant with the latest developments in those fields, we shall concentrate on an exposition of Tarski's work in the first four areas, trying to put them in the perspective of present-day developments.

For useful comments, criticisms, and suggestions, the author is indebted to Steven Givant, Leon Henkin, Wilfrid Hodges, Bjarni Jónsson, Roger Lyndon, and Robert Vaught.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 4 , December 1986 , pp. 899 - 906
Copyright
Copyright © Association for Symbolic Logic 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andréka, H., Gergely, T. and Németi, I. [1977] On universal algebraic constructions of logics, Studio Logica, vol. 36, pp. 9–47.CrossRefGoogle Scholar
Andréka, H. and Németi, I. [1975] A simple, purely algebraic proof of the completeness of some first order logics, Algebra Universalis, vol. 5, pp. 8–15.CrossRefGoogle Scholar
Baumgartner, J., Taylor, A. and Wagon, S. [1982] Structural properties of ideals, Dissertationes Mathematicae (Rozprawy Matematyczne), vol. 197.Google Scholar
Everett, C. and Ulam, S. [1946] Projective algebra. I, American Journal of Mathematics, vol. 68, pp. 77–88.CrossRefGoogle Scholar
Halmos, P. [1962] Algebraic logic, Chelsea, New York.Google Scholar
Henkin, L. [1956] The representation theorem for cylindrical algebras, Mathematical interpretation of formal systems, North-Holland, Amsterdam, pp. 85–97. [1966] Logical systems containing only a finite number of symbols, Séminaire de Mathématiques Supérieures, no. 21, Les Presses de l'Université de Montréal, Montréal.Google Scholar
Imieliński, T. and Lipski, W. [1984] The relational model of data and cylindric algebras, Journal of Computer and System Sciences, vol. 28, pp. 80–103.CrossRefGoogle Scholar
Jech, T. [198-] Distributive laws, Handbook of Boolean algebras, North-Holland, Amsterdam (to appear).Google Scholar
Jönsson, B. [1959] Representation of modular lattices and of relation algebras, Transactions of the American Mathematical Society, vol. 92, pp. 449–464.CrossRefGoogle Scholar
Jönsson, B. [19??] The theory of binary relations (preprint).Google Scholar
Kozen, D. [1979] On the duality of dynamic algebras and Kripke models, Logic of programs (papers from a workshop, Zürich, 1979), Lecture Notes in Computer Science, vol. 125, Springer-Verlag, Berlin, 1981, pp. 1–11.Google Scholar
Lyndon, R. [1950] The representation of relational algebras, Annals of Mathematics, ser. 2, vol. 51, pp. 707–729.CrossRefGoogle Scholar
Lyndon, R. [1956] The representation of relational algebras. II, Annals of Mathematics, ser. 2, vol. 63, pp. 294–307.CrossRefGoogle Scholar
Lyndon, R. [1961] Relation algebras and projective geometries, Michigan Mathematical Journal, vol. 8, pp. 21–28.CrossRefGoogle Scholar
Maddux, R. [1978] Topics in relation algebras, Ph.D. Thesis, University of California, Berkeley, California.Google Scholar
Maddux, R. [1980] The equational theory of CA3 is undecidable, this Journal, vol. 45, pp. 311–316.Google Scholar
Mayer, R. D. and Pierce, R. S. [1980] Boolean algebras with ordered bases, Pacific Journal of Mathematics, vol. 10, pp. 925–942.CrossRefGoogle Scholar
McKenzie, R. [1970] Representations of integral relation algebras, Michigan Mathematical Journal, vol. 17, pp. 279–287.CrossRefGoogle Scholar
Monk, J. D. [1961] Studies in cylindric algebras, Ph.D. Thesis, University of California, Berkeley, California.Google Scholar
Monk, J. D. [1964] On representable relation algebras, Michigan Mathematical Journal, vol. 11, pp. 207–210.CrossRefGoogle Scholar
Monk, J. D. [1969] Nonfinitizability of classes of representable cylindric algebras, this Journal, vol. 34, pp. 331–343.Google Scholar
Myers, D. [1976] Cylindric algebras of first-order languages, Transactions of the American Mathematical Society, vol. 216, pp. 189–202.CrossRefGoogle Scholar
Plotkin, B. [1984] Databases and Halmos algebras, preprint, Institute of Aviation, Riga.Google Scholar
Rabin, M. [1969] Decidability of second-order theories and automata on infinite trees, Transactions of the American Mathematical Society, vol. 141, pp. 1–35.Google Scholar
Rotman, B. [1972] Boolean algebras with ordered bases, Fundamenta Mathematicae, vol. 75, pp. 187–197.CrossRefGoogle Scholar
Rubin, M. [1976] The theory of Boolean algebras with a distinguished subalgebra is undecidable, Anuales Scientifiques de l'Université de Clermont, no. 60 (ser. math. no. 13), pp. 129–134.Google Scholar
Rubin, M. [1983] A Boolean algebra with few subalgebras, interval Boolean algebras, and retractiveness, Transactions of the American Mathematical Society, vol. 278, pp. 65–89.CrossRefGoogle Scholar
Schröder, E. [1890] Vorlesungen über die Algebra der Logik (exakte Logik), Vol. I, Teubner, Leipzig; reprint, Chelsea, New York, 1966.Google Scholar
Sikorski, R. [1964] Boolean algebras, Springer-Verlag, Berlin.Google Scholar
Surma, S. J. [1982] On the origin and subsequent applications of the concept of the Lindenbaum algebra, Logic, methodology and philosophy of science, VI (Hannover, 1979), North-Holland, Amsterdam, 1982, pp. 719–734.Google Scholar
Weese, M. [198–] Decidable extensions of the theory of Boolean algebras, Handbook of Boolean algebras, North-Holland, Amsterdam (to appear).Google Scholar
Weese, M. [198–a] Undecidable extensions of the theory of Boolean algebras, Handbook of Boolean algebras, North-Holland, Amsterdam (to appear).Google Scholar