Studia Logica

, Volume 61, Issue 2, pp 179–197 | Cite as

Axiomatizations with Context Rules of Inference in Modal Logic

  • Valentin Goranko


A certain type of inference rules in (multi-) modal logics, generalizing Gabbay's Irreflexivity rule, is introduced and some general completeness results about modal logics axiomatized with such rules are proved.

modal logic inference rules axiomatizations completeness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    van Benthem, J., Modal Logic and Classical Logic, Bibliopolis, Napoli, 1985.Google Scholar
  2. [2]
    van Benthem, J., ‘Correspondence theory’, in: D. Gabbay, F. Guenthner (eds.), Handbook on Philosophical Logic, vol. II, Reidel, Dordrecht, 1984, 167–247.Google Scholar
  3. [3]
    Fine, K., ‘Some connections between modal and elementary logic’, in: S. Kanger (ed.), Proc. Third Scandinavian Logic Symposium, North-Holland, Amsterdam, 1975, 15–31.Google Scholar
  4. [4]
    Gabbay, D., ‘An Irreflexivity lemma with applications to axiomatizations of conditions on tense frames’, in: U. Monnich (ed.), Aspects of Philosophical Logic, Reidel, Dordrecht, 1981, 67–89.Google Scholar
  5. [5]
    Gabbay, D., & I. Hodkinson, ‘An axiomatization of the temporal logic with Since and Until over the real numbers’, J. of Logic and Computation 1, 1990, 229–259.Google Scholar
  6. [6]
    Gargov, G., & V. Goranko, ‘Modal logic with names’, J. of Philosophical Logic 22(6), 1993, 607–636.Google Scholar
  7. [7]
    Goldblatt, R., ‘Metamathematics of modal logic’, Reports on Mathematical Logic 6, 41–78 (Part I); 7, 21–52 (Part II).Google Scholar
  8. [8]
    Goldblatt, R. I., Axiomatizing the Logic of Computer Programming, Springer LNCS 130, 1982.Google Scholar
  9. [9]
    Goranko, V., ‘Applications of quasi-structural rules to axiomatizations in modal logic’, in: Abstracts of the 9th Intern. Congress of Logic, Methodology and Philosophy of Science, Uppsala, 1991, Vol. 1: Logic, p. 119.Google Scholar
  10. [10]
    Goranko, V., ‘A note on derivation rules in modal logic’, Bull. of the Sect. of Logic, Univ. of Lódź, 24(2), 1995, 98–104.Google Scholar
  11. [11]
    Goranko, V., ‘Hierarchies of modal and temporal logics with reference pointers’, J. of Logic, Language and Information, 5(1), 1996, 1–24.Google Scholar
  12. [12]
    Hollenberg, M., ‘Negative definability in modal logic’, manuscript, Department of Philosophy, Utrecht University, 1994, to appear in Studia Logica.Google Scholar
  13. [13]
    Hughes, G., & M. Cresswell, A New Introduction to Modal Logic, Routledge, London and New York, 1996.Google Scholar
  14. [14]
    Kracht, M., Internal Definability and Completeness in Modal Logic, Ph. D. Thesis, Free University, Berlin, 1990.Google Scholar
  15. [15]
    Kracht, M., ‘How completeness and correspondence theory got married’, in: M. de Rijke (ed.), Colloquium on Modal Logic, Dutch Network for Language, Logic and Information, 1991, 161–186.Google Scholar
  16. [16]
    Passy, S., & T. Tinchev, ‘PDL with data constants’, Information Processing Letters 20, 1985, 35–41.Google Scholar
  17. [17]
    de Rijke, M., ‘The modal logic of inequality’, J. of Symbolic Logic 57(2), 1992, 566–584.Google Scholar
  18. [18]
    Sambin, J., & V. Vaccaro, ‘Topology and duality in modal logic’, Annals of Pure and Applied Logic 44, 1988, 173–242.Google Scholar
  19. [19]
    Thomason, S. K., ‘Semantic analysis of tense logic’, J. of Symbolic Logic 37, 1972, 150–158.Google Scholar
  20. [20]
    Vakarelov, D., ‘Modal rules for intersection’, Bull. of Symbolic Logic 1(2), 1995, 264–265.Google Scholar
  21. [21]
    Venema, Y., ‘Expressiveness and completeness of an interval tense logic’, Notre Dame J. of Formal Logic 31, 1990, 529–547.Google Scholar
  22. [22]
    Venema, Y., ‘A modal logic for chopping intervals’, J. of Logic and Computation 1, 1991, 453–476.Google Scholar
  23. [23]
    Venema, Y., Many-dimensional Modal Logic, Ph. D. Dissertation, Univ. of Amsterdam, 1991.Google Scholar
  24. [24]
    Venema, Y., ‘Derivation rules as anti-axioms in modal logic’, J. of Symbolic Logic 58(3), 1993, 1003–1034.Google Scholar
  25. [25]
    Zanardo, A., ‘A complete deductive system for Since-Until branching-time logic’, J. of Philosophical Logic 20, 1991, 131–148.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Valentin Goranko
    • 1
  1. 1.Department of MathematicsRand Afrikaans UniversityAukland ParkSouth Africa

Personalised recommendations