Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-13T00:47:29.501Z Has data issue: false hasContentIssue false
  • English
  • Français

Cylindric modal logic

Published online by Cambridge University Press:  12 March 2014

Yde Venema
Yde Venema*
Affiliation:
Department of Mathematics and Computer Science, Free University, de Boelelaan 1081, 1081 HV Amsterdam, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Treating the existential quantification ∃νi as a diamond ♢i and the identity νi = νj as a constant δij, we study restricted versions of first order logic as if they were modal formalisms. This approach is closely related to algebraic logic, as the Kripke frames of our system have the type of the atom structures of cylindric algebras; the full cylindric set algebras are the complex algebras of the intended multidimensional frames called cubes.

The main contribution of the paper is a characterization of these cube frames for the finite-dimensional case and, as a consequence of the special form of this characterization, a completeness theorem for this class. These results lead to finite, though unorthodox, derivation systems for several related formalisms, e.g. for the valid n-variable first order formulas, for type-free valid formulas and for the equational theory of representable cylindric algebras. The result for type-free valid formulas indicates a positive solution to Problem 4.16 of Henkin, Monk and Tarski [16].

Keywords

Primary 03B2003B4503G15.Algebraic logicmodal logiclogic with finitely many variablescompletenessderivation rules
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 2 , June 1995 , pp. 591 - 623
Copyright
Copyright © Association for Symbolic Logic 1995

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

[1] Andréka, H., Complexity of equations valid in algebras of relations, Third Doctoral Dissertation, Budapest, 1991.Google Scholar
[2] Andréka, H., Gergely, T., and Németi, I., A simple purely algebraic proof of the completeness of some first-order logics, Algebra Universalis, vol. 5, (1975), pp. 8–15.CrossRefGoogle Scholar
[3] Andréka, H., Monk, J. D., and Németi, I. (editors), Algebraic logic: proceedings of the 1988 Budapest conference on algebraic logic, North-Holland, Amsterdam, 1991.Google Scholar
[4] Andréka, H. And Németi, I., Dimension-complemented and locally finite-dimensional cylindric algebras are elementary equivalent, Algebra Universalis, vol. 13, (1981), pp. 157–163.CrossRefGoogle Scholar
[5] Andréka, H., Németi, I., and Sain, I., Abstract model theoretic approach to algebraic logic, CCSOM Working Paper 92–92, CCSOM, University of Amsterdam, Amsterdam, 1992, 45 pp.Google Scholar
[6] Andréka, H., Németi, I., and Sain, I., personal communication.Google Scholar
[7] Van Benthem, J. F. A. K., Correspondence theory, in [13], pp. 167–247.CrossRefGoogle Scholar
[8] Bergman, C. H., Maddux, R. D., and Pigozzi, D. L. (editors), Algebraic logic and universal algebra in computer science, Lecture Notes in Computer Science, vol. 425, Springer-Verlag, Berlin, 1990.CrossRefGoogle Scholar
[9] Biró, B., Non-finite-axiomatizability results in algebraic logic, this Journal, vol. 57 (1992), pp. 832–845.Google Scholar
[10] Blok, W. and Pigozzi, D., Algebraizable logics, Memoirs of the American Mathematical Society, vol. 77, (1989), no. 396.CrossRefGoogle Scholar
[11] Burgess, J., Basic tense logic, in [13], pp. 89–133.CrossRefGoogle Scholar
[12] Gabbay, D. M., An irreflexivity lemma with applications to axiomatizations of conditions on linear frames, in [25], pp. 67–89.CrossRefGoogle Scholar
[13] Gabbay, D. M. and Guenthner, F. (editors), Handbook of philosophical logic, Vol. II, Reidel, Dordrecht, 1984.CrossRefGoogle Scholar
[14] Goldblatt, R., Varieties of complex algebras, Annals of Pure and Applied Logic, vol. 44, (1989), pp. 173–242.CrossRefGoogle Scholar
[15] Henkin, L., Logical systems containing only a finite number of symbols, Presses de l’ Université de Montréal, Montréal, 1967.Google Scholar
[16] Henkin, L., Monk, J. D., and Tarski, A., Cylindric algebras, Parts I, II, North-Holland, Amsterdam, 1971, 1985.Google Scholar
[17] Immerman, N. and Kozen, D., Definability with bounded number of bound variables, in [20], pp. 236–244.Google Scholar
[18] Kanger, S. (editor), Proceedings of the third Scandinavian logic symposium, Uppsala 1973, North-Holland, Amsterdam, 1975.Google Scholar
[19] Kuhn, S., Quantifiers as modal operators, Studia Logica, vol. 39, (1980), pp. 145–158.CrossRefGoogle Scholar
[20] LICS87: Proceedings of the symposium on logic in computer science 1987, Ithaca, New York, IEEE Computer Society Press, Washington, D.C., 1987.Google Scholar
[21] Maddux, R., Nonfinite axiomatizability results for cylindric and relation algebras, this Journal, vol. 54 (1989), pp. 951–974.Google Scholar
[22] Mikulás, Sz., Gabbay-style calculi, Bulletin of the Section of Logic, University of Lódź, Department of Logic, vol. 22, (1993), pp. 50–60.Google Scholar
[23] Monk, J. D., Nonfinitizability of classes of representable cylindric algebras, this Journal, vol. 34 (1969), pp. 331–343.Google Scholar
[24] Monk, J. D., Provability with finitely many variables, Proceedings of the American Mathematical Society, vol. 27, (1971), pp. 353–358.CrossRefGoogle Scholar
[25] Mönnich, U. (editor), Aspects of philosophical logic, Reidel, Dordrecht, 1981.CrossRefGoogle Scholar
[26] Németi, I., On varieties of cylindric algebras with applications to logic, Annals of Pure and Applied Logic, vol. 36, (1987), pp. 235–277.CrossRefGoogle Scholar
[27] Németi, I., Algebraizations of quantifier logics: an introductory overview, Studia Logica, vol. 50, (1991), pp. 485–570.CrossRefGoogle Scholar
[28] Pigozzi, D., Amalgamation, congruence-extension, and interpolation properties in algebras, Algebra Universalis, vol. 1, (1972), pp. 269–349.CrossRefGoogle Scholar
[29] Prijatelj, A., Existence theorems for models of dynamic intensional logic, University of Ljubljana, Department of Mathematics, Preprint Series, vol. 28, (1990), no. 318.Google Scholar
[30] De Rijke, M., The modal logic of inequality, this Journal, vol. 57 (1992), pp. 566–584.Google Scholar
[31] De Rijke, M. and Venema, Y., Sahlqvist's theorem for Boolean algebras with operators, Studia Logica (to appear).Google Scholar
[32] Rybakov, V., Metatheories of the first-order theories, Abstracts of International Congress of Mathematicians (Kyoto, Japan, August 21–29, 1990), Section 8.Google Scholar
[33] Sahlqvist, H., Completeness and correspondence in the first and second order semantics for modal logic, in [18], pp. 110–143.CrossRefGoogle Scholar
[34] Sain, I., Finitary logics of infinitary structures are compact, Abstracts of papers presented to the American Mathematical Society, vol. 3, (1982), p. 252, (abstract no. 82T-03-197).Google Scholar
[35] Sain, I., Searching for a finitizable algebraization of first order logic, Preprint, Mathematical Institute of the Hungarian Academy of Sciences, Budapest, 1987.Google Scholar
[36] Sain, I., Positive results related to the Jónsson-Tarski-Givant representation problem, Expanded version of Preprint, Mathematical Institute of the Hungarian Academy of Sciences, Budapest, 1987.Google Scholar
[37] Sain, I., Beth's and Craig's properties via epimorphisms and amalgamation in algebraic logic, in [8], pp. 209–226.CrossRefGoogle Scholar
[38] Sambin, G. and Vaccaro, V., A new proof of Sahlqvist's theorem on modal definability and completeness, this Journal, vol. 54 (1989), pp. 992–999.Google Scholar
[39] Simon, A., Finite schema completeness for typeless logic and representable cylindric algebras, in [3], pp. 665–671.Google Scholar
[40] Tarski, A. and Givant, S., A formalization of set theory without variables, American Mathematical Society, Providence, Rhode Island, 1987.CrossRefGoogle Scholar
[41] Venema, Y., Two-dimensional modal logic for relation algebras and temporal logic of intervals, ITLI-Prepublication Series L-89-03, University of Amsterdam, Amsterdam, 1989.Google Scholar
[42] Venema, Y., Many-dimensional modal logic, Doctoral dissertation, University of Amsterdam, Amsterdam, 1991.Google Scholar
[43] Venema, Y., Derivation rules as anti-axioms, this Journal, vol. 58 (1993), pp. 1003–1034.Google Scholar
[44] Venema, Y., A modal logic of quantification and substitution, manuscript, Department of Philosophy, Utrecht University, Utrecht, 1993.Google Scholar