Journal of Philosophical Logic

, Volume 22, Issue 6, pp 607–636 | Cite as

Modal logic with names

  • George Gargov
  • Valentin Goranko


We investigate an enrichment of the propositional modal language with a “universal” modality ▪ having semanticsx ⊧ ▪ϕ iff āy(y ⊧ ϕ), and a countable set of “names” — a special kind of propositional variables ranging over singleton sets of worlds. The obtained language ℒc proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment ℒ(
) of, where
is an additional modality with the semanticsx ⊧
ϕ iff āy(y⊧ x → y ⊧ϕ). Model-theoretic characterizations of modal definability in these languages are obtained. Further we consider deductive systems in ℒc. Strong completeness of the normal ℒc-logics is proved with respect to models in which all worlds are named. Every ℒc-logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties ([in]completeness, filtration, finite model property etc.) from to ℒcare discussed. Finally, further perspectives for names in multimodal environment are briefly sketched.


Modal Logic Special Kind Model Property Additional Modality Expressive Power 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. van Benthem, J. F. A. K. (1979), Canonical modal logics and ultrafilter extensions.JSL 44(1), 1–8.Google Scholar
  2. van Benthem, J. F. A. K. (1984), Correspondence theory. In:Handbook of Philosophical Logic, D. Gabbay and F. Guenthner (eds.), 167–247, vol. II, Reidel, Dordrecht.Google Scholar
  3. van Benthem, J. F. A. K. (1986),Modal Logic and Classical Logic, Bibliopolis, Napoli.Google Scholar
  4. Blackburn, P. (1989), Nominal tense logic. Dissertation, Centre for Cognitive Science, University of Edinburgh, 1989.Google Scholar
  5. Bull, R. (1970), An approach to tense logic.Theoria 36(3), 282–300.Google Scholar
  6. Burgess, J. (1982), Axioms for tense logic I: “Since” and “Until”,Notre Dame Journal of Formal Logic 23(2), 367–374.Google Scholar
  7. Chagrova, L. (1989), Undecidable problems connected with first-order definability of intuitionistic formulae,Proceedings of Kalinin State University, p. 42.Google Scholar
  8. Fine, K. (1975), Some connections between the elementary and modal logic. In:Proceedings of the Third Scandinavian Logic Symposium, Uppsala 1973, S. Kanger (ed.), North-Holland, Amsterdam, 15–31.Google Scholar
  9. Gabbay, D. (1981), An irreflexivity lemma with applications to axiomatizations of conditions on tense frames. In:Aspects of Philosophical Logic, U. Monnich (ed.).Google Scholar
  10. Gabbay, D. & I. Hodkinson (1990), An axiomatization of the temporal logic with Since and Until over the real numbers.Journal of Logic and Computation, Vol. 1.Google Scholar
  11. Gargov, G., S. Passy & T. Tinchev (1987), Modal environment for Boolean speculations. In:Mathematical Logic and Its Applications (ed. D. Skordev), 253–263, Plenum Press, New York.Google Scholar
  12. Gargov, G. &. S. Passy (1988), Determinism and looping in Combinatory PDL.Theoretical Computer Science,61, 259–277.Google Scholar
  13. Gargov, G. & S. Passy (1990), A note on Boolean modal logic. In:Mathematical Logic, Proceedings of the Summer School and Conference HEYTING'88, P. Petkov (ed.), Plenum Press, New York and London.Google Scholar
  14. Goldblatt, R. I. (1976), Metamathematics of modal logic.Reports on Math. Logic 6, 41–77 and 7, 21–52.Google Scholar
  15. Goldblatt, R. I. (1982),Axiomatizing the Logic of Computer Programming, Springer LNCS130. Google Scholar
  16. Goldblatt, R. I. & S. K. Thomason (1974), Axiomatic classes in prepositional modal logic. In: J. Crossley (ed.),Algebra and Logic, 163–173, Springer LNM450, Berlin.Google Scholar
  17. Goranko, V. (1990), Modal definability in enriched languages.Notre Dame Journal of Formal Logic 31(1), 81–105.Google Scholar
  18. Goranko, V. & S. Passy (1990), Using the universal modality: gains and questions,Journal of Logic and Computation 2(1), 5–30.Google Scholar
  19. Hughes, G. E. & M. J. Cresswell (1984),A Companion to Modal Logic, Methuen, London.Google Scholar
  20. Koymans, R. (1989), Specifying Message Passing and Time-Critical Systems with Temporal Logic. Dissertation, Eindhoven.Google Scholar
  21. Makinson, D. (1971), Some embedding theorems for modal logic.Notre Dame Journal of Formal Logic,12, 252–254.Google Scholar
  22. Passy, S. & T. Tinchev (1991), An Essay in Combinatory Dynamic Logic.Information and Computation 93(2), 263–332.Google Scholar
  23. Prior, A. (1956), Modality and quantification in S5.Journal of Symbolic Logic 21(1), 60–62.Google Scholar
  24. de Rijke, M. (1989), The modal theory of inequality,Journal of Symbolic Logic 57(2), 566–584.Google Scholar
  25. Sahlqvist, H. (1975), Completeness and correspondence in first and second order semantics for modal logic. In:Proceedings of the Third Scandinavian Logic Symposium, Uppsala 1973, S. Kanger (ed.), North-Holland, Amsterdam.Google Scholar
  26. Segerberg, K. (1971),An Essay in Classical Modal Logic, Filosofiska Studier13, Uppsala.Google Scholar
  27. Venema, Y. (1991), Personal correspondence.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • George Gargov
    • 1
  • Valentin Goranko
    • 2
  1. 1.ShrewsburyUSA
  2. 2.Department of MathematicsUniversity of the North, QwaQwa BranchPhuthaditjhabaRepublic of South Africa

Personalised recommendations