Abstract
We investigate an enrichment of the propositional modal languageℒ 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(
) ofℒ, 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.) fromℒ to ℒc are discussed. Finally, further perspectives for names in multimodal environment are briefly sketched.
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References
van Benthem, J. F. A. K. (1979), Canonical modal logics and ultrafilter extensions.JSL 44(1), 1–8.
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.
van Benthem, J. F. A. K. (1986),Modal Logic and Classical Logic, Bibliopolis, Napoli.
Blackburn, P. (1989), Nominal tense logic. Dissertation, Centre for Cognitive Science, University of Edinburgh, 1989.
Bull, R. (1970), An approach to tense logic.Theoria 36(3), 282–300.
Burgess, J. (1982), Axioms for tense logic I: “Since” and “Until”,Notre Dame Journal of Formal Logic 23(2), 367–374.
Chagrova, L. (1989), Undecidable problems connected with first-order definability of intuitionistic formulae,Proceedings of Kalinin State University, p. 42.
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.
Gabbay, D. (1981), An irreflexivity lemma with applications to axiomatizations of conditions on tense frames. In:Aspects of Philosophical Logic, U. Monnich (ed.).
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.
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.
Gargov, G. &. S. Passy (1988), Determinism and looping in Combinatory PDL.Theoretical Computer Science,61, 259–277.
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.
Goldblatt, R. I. (1976), Metamathematics of modal logic.Reports on Math. Logic 6, 41–77 and 7, 21–52.
Goldblatt, R. I. (1982),Axiomatizing the Logic of Computer Programming, Springer LNCS130.
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.
Goranko, V. (1990), Modal definability in enriched languages.Notre Dame Journal of Formal Logic 31(1), 81–105.
Goranko, V. & S. Passy (1990), Using the universal modality: gains and questions,Journal of Logic and Computation 2(1), 5–30.
Hughes, G. E. & M. J. Cresswell (1984),A Companion to Modal Logic, Methuen, London.
Koymans, R. (1989), Specifying Message Passing and Time-Critical Systems with Temporal Logic. Dissertation, Eindhoven.
Makinson, D. (1971), Some embedding theorems for modal logic.Notre Dame Journal of Formal Logic,12, 252–254.
Passy, S. & T. Tinchev (1991), An Essay in Combinatory Dynamic Logic.Information and Computation 93(2), 263–332.
Prior, A. (1956), Modality and quantification in S5.Journal of Symbolic Logic 21(1), 60–62.
de Rijke, M. (1989), The modal theory of inequality,Journal of Symbolic Logic 57(2), 566–584.
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.
Segerberg, K. (1971),An Essay in Classical Modal Logic, Filosofiska Studier13, Uppsala.
Venema, Y. (1991), Personal correspondence.
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Gargov, G., Goranko, V. Modal logic with names. J Philos Logic 22, 607–636 (1993). https://doi.org/10.1007/BF01054038
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DOI: https://doi.org/10.1007/BF01054038