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Canonical formulas for K4. Part III: the finite model property

Published online by Cambridge University Press:  12 March 2014

Michael Zakharyaschev
Michael Zakharyaschev*
Affiliation:
II. Mathematisches Institut, Fu Berlin, Arnimallee 3, D-14195 Berlin, Germany, E-mail: mishaz@math.fu-berlin.de
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Extract

This paper, a continuation of the series [22, 24], presents two methods for establishing the finite model property (FMP, for short) of normal modal logics containing K4. The methods are oriented mainly to logics represented by their canonical axioms and yield for such axiomatizations several sufficient conditions of FMP. We use them to obtain solutions to two well known open FMP problems. Namely, we prove that

• every normal extension of K4 with modal reduction principles has FMP and

• every normal extension of S4 with a formula of one variable has FMP.

These results are interesting not only from the technical point of view. Actually, they reveal important properties of a quite natural family of modal logics—formulas of one variable and, in particular, modal reduction principles are typical axioms in modal logic. Unfortunately, the technical apparatus developed in this paper is applicable only to logics with transitive frames, and the situation with FMP of extensions of K by modal reduction principles, even by axioms of the form □np → □mp still remains unclear. I think at present this is one of the major challenges in completeness theory.

The language of the canonical formulas, introduced in [22] (I'll refer to that paper as Part I), is a way of describing the “geometry and topology” of formulas' refutation (general) frames by means of some finite refutation patterns.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 950 - 975
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Anderson, J. G., Superconstructive propositional calculi with extra axiom schemes containing one variable, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 18 (1972), pp, 113–130.CrossRefGoogle Scholar
[2]Bellissima, F., An effective representation for finitely generated free interior algebras, Algebra Universalis, vol. 20 (1985), pp. 302–317.CrossRefGoogle Scholar
[3]Blok, W. J., Varieties of interior algebras, Dissertation, University of Amsterdam, 1976.Google Scholar
[4]Bull, R. and Segerberg, K., Basic modal logic, (Gabbay, D. and Guenthner, F., editors), Reidel, Dordrecht, 1984, pp. 1–88.Google Scholar
[5]Chagrov, A. V., Undecidable properties of superintuitionistic logics, Mathematical questions of cybernetics (Jablonskii, S. V., editor), vol. 5, Nauka, Moscow, 1994, Russian, pp. 62–108.Google Scholar
[6]Fine, K., An ascending chain of S4 logics, Theoria, vol. 40 (1974), pp. 110–116.CrossRefGoogle Scholar
[7]Fine, K., Logics containing K4, part II, this Journal, vol. 50 (1985), pp. 619–651.Google Scholar
[8]Gabbay, D. M. and de Jongh, D. H. J., A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property, this Journal, vol. 39 (1974), pp. 67–78.Google Scholar
[9]Jankov, V. A., Constructing a sequence of strongly independent superintuitionistic calculi, Soviet Mathematics Docladi, vol. 181 (1968), pp. 806–807.Google Scholar
[10]Kracht, M., Splittings and the finite model property, this Journal, vol. 58 (1993), pp. 139–157.Google Scholar
[11]Maksimova, L. L., On maximal intermediate logics with the disjunction property, Studia Logica, vol. 45 (1986), pp. 69–75.CrossRefGoogle Scholar
[12]Ono, H., Some results on the intermediate logics, Publications of the Research Institute for Mathematical Science, vol. 8 (1972), pp. 117–130.CrossRefGoogle Scholar
[13]Rybakov, V. V., A modal analog for Glivenko's theorem and its applications, Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 244–248.CrossRefGoogle Scholar
[14]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
[15]Segerberg, K., An essay in classical modal logic, Philosophical Studies, Uppsala, vol. 13 (1971).Google Scholar
[16]Shehtman, V. B., Rieger-Nishimura lattices, Soviet Mathematics Doklady, vol. 19 (1978), pp. 1014–1018.Google Scholar
[17]Shehtman, V. B., Topological models of propositional logics, Semiotics and Informatics, vol. 15 (1980), pp. 74–98.Google Scholar
[18]Shehtman, V. B., Derived sets in euclidean spaces and modal logic, Preprint X-90-05, 1990.Google Scholar
[19]Sobolev, S. K., On the finite approximability of superintuitionistic logics, Mathematical Sbornik, vol. 102 (1977), pp. 289–301.Google Scholar
[20]van Benthem, J. F. A. K., Modal reduction principles, this Journal, vol. 41 (1976), pp. 301–312.Google Scholar
[21]Zakharyaschev, M. V., Syntax and semantics of intermediate logics, Algebra and Logic, vol. 28 (1989), pp. 402–429.Google Scholar
[22]Zakharyaschev, M. V., Canonical formulas for K4. part I: Basic results, this Journal, vol. 57 (1992), pp. 1377–1402.Google Scholar
[23]Zakharyaschev, M. V., A sufficient condition for the finite model property of modal logics above K4, Bulletin of the IGPL, vol. 1 (1993), pp. 13–21.CrossRefGoogle Scholar
[24]Zakharyaschev, M. V., Canonical formulas for K4. part II: Cofinal subframe logics, this Journal, vol. 61 (1996), pp. 421–449.Google Scholar
[25]Zakharyaschev, M. V., Canonical formulas for modal and superintuitionistic logics: a short outline, Modal logic and its neighbours '92 (de Rijke, M., editor), Kluwer Academic Publishers, 1996, to appear.Google Scholar