Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T09:20:18.902Z Has data issue: false hasContentIssue false
  • English
  • Français

Hereditarily structurally complete modal logics

Published online by Cambridge University Press:  12 March 2014

V. V. Rybakov
V. V. Rybakov*
Affiliation:
Department of Mathematics, University of Krasnoyarsk, Av. Svobodnyi 79, Krasnoyarsk, 660-062, Russia, E-mail: rybakov@kgu.krasnoyarsk.su
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider structural completeness in modal logics. The main result is the necessary and sufficient condition for modal logics over K4 to be hereditarily structurally complete: a modal logic λ is hereditarily structurally complete iff λ is not included in any logic from the list of twenty special tabular logics. Hence there are exactly twenty maximal structurally incomplete modal logics above K4 and they are all tabular.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 1 , March 1995 , pp. 266 - 288
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]Bellissima, F., Finitely generated free Heyting algebras, this Journal, vol. 51 (1986), pp. 152–165.Google Scholar
[2]Blok, W. J., Pretabular varieties of modal algebras, Studia Logica, vol. 39 (1980), pp. 101–124.CrossRefGoogle Scholar
[3]Citkin, A. I., On admissible rules of intuitionistic propositional logic, Mathematics USSR Sbornik, vol.31 (1977), pp. 279–288.CrossRefGoogle Scholar
[4]Citkin, A. I., On structurally complete superintuitionistic logics, Soviet Mathematics Doklady, vol. 19 (1978), pp. 816–819.Google Scholar
[5]de Jong, D. H. and Troelstra, A. S., On the connection of partially ordered sets with some pseudo-Boolean algebras, Indagationes Mathematicae, vol. 28 (1966), pp. 317–329.CrossRefGoogle Scholar
[6]Dziobiak, W., Structural completeness of modal logics containing k4, Manuscript, 1989.Google Scholar
[7]Fagin, R., Halpern, J. Y., and Vardi, M. Y., What is an inference rule, this Journal, vol. 57 (1992), pp. 1018–1045.Google Scholar
[8]Fine, K., Logics containing S4.3, Zeitschrift für Mathematische Logic und Grundlagen der Mathematik, vol. 17 (1971), pp. 371–376.CrossRefGoogle Scholar
[9]Fine, K., Logic containing K4, part I, this Journal, vol. 39 (1974), pp. 229–237.Google Scholar
[10]Gabbay, D. M., Investigations in modal and tense logics with applications to problems in philosophy and linguistics, D. Reidel Publishing Company, Dordrecht-Holland/Boston-USA, 1976.CrossRefGoogle Scholar
[11]Goldblatt, R. I., Metamathematics of modal logics (part 1), Reports on Mathematical Logic, vol. 6 (1976), pp. 41–78.Google Scholar
[12]Goldblatt, R. I., Metamathematics of modal logics (part 2), Reports on Mathematical Logic, vol. 7 (1976), pp. 21–52.Google Scholar
[13]Kracht, M., Prefinitely axiomatizable modal and intermediate logics, Preprint No. 73, Department of Philosophy, University of Utrecht, The Netherlands, 1992.Google Scholar
[14]Kracht, M., Splittings and the finite model property, this Journal, vol. 58 (1993), pp. 139–154.Google Scholar
[15]Makinson, D., A characterization of structural completeness of a structural consequence operation, Reports on Mathematical Logic, vol. 6 (1976), pp. 99–102.Google Scholar
[16]Maksimova, L. L., Pretabular super-intuitionistic logics, Algebra and Logic, vol. 11 (1972), pp. 558–570, In Russian, English Translation in Algebra and Logic, Translations of the AMS.CrossRefGoogle Scholar
[17]Maksimova, L. L. and Rybakov, V. V., A lattice of normal modal logics, Algebra and Logic, vol. 13 (1974), pp. 105–122.CrossRefGoogle Scholar
[18]Mints, G. E., Derivability of admissible rules, Journal of Soviet Mathematics, vol. 6 (1976), pp. 417–421.CrossRefGoogle Scholar
[19]Prucnal, T., Structural completeness of Medvedev's propositional calculus, Reports on Mathematical Logic, vol. 6 (1976), pp. 103–105.Google Scholar
[20]Rautenberg, W., Der verband der normal verzweigten modallogiken, Mathematische Zeitschrift, vol. 156 (1977), pp. 123–140.CrossRefGoogle Scholar
[21]Rautenberg, W., Klassische und nichtklassische aussagenlogik, Braunschweig/Wiesbaden, 1979.CrossRefGoogle Scholar
[22]Rybakov, V. V., A criterion for admissibility of rules in the modal system S4 and the intuitionistic logic, Algebra and Logic, vol. 23 (1984), pp. 369–384, (English translation).CrossRefGoogle Scholar
[23]Rybakov, V. V., Problems of substitution and admissibility in the modal system Grz and intuitionistic calculus, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 71–106.CrossRefGoogle Scholar
[24]Rybakov, V. V., Rules of inference with parameters for intuitionistic logic, this Journal, vol. 57 (1992), pp. 912–923.Google Scholar
[25]Segerberg, K., An essay in classical modal logic, vol. 1–3, Filosofiska Studier, Uppsala, 1971.Google Scholar
[26]van Benthem, J. F. A. K., Modal and classical logic, Bibliopolis, 1983.Google Scholar