Abstract
We study several modal languages in which some (sets of) generalized quantifiers can be represented; the main language we consider is suitable for defining any first order definable quantifier, but we also consider a sublanguage thereof, as well as a language for dealing with the modal counterparts of some higher order quantifiers. These languages are studied both from a modal logic perspective and from a quantifier perspective. Thus the issues addressed include normal forms, expressive power, completeness both of modal systems and of systems in the quantifier tradition, complexity as well as syntactic characterizations of special semantic constraints. Throughout the paper several techniques current in the theory of generalized quantifiers are used to obtain results in modal logic, and conversely.
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References
Ackermann, W.: 1954,Solvable Cases of the Decision Problem. Amsterdam: North-Holland.
Atzeni, P. and Parker, D.S.: 1988, “Set containment inference and syllogisms”,Theoretical Computer Science 62, 39–65.
van Benthem, J.F.A.K.: 1984, “Questions about quantifiers”,The Journal of Symbolic Logic 49, 443–466.
van Benthem, J.F.A.K.: 1986,Essays in Logical Semantics. Dordrecht: Reidel.
Fattorosi-Barnaba, F. and Cerrato, C.: 1988, “Graded modalities. III”,Studia Logica 47, 99–110.
Fine, K.: 1972, “In so many possible worlds”,Notre Dame Journal of Formal Logic 13, 516–520.
Gärdenfors, P.: 1975, “Qualitative probability as an intensional logic”,Journal of Philosophical Logic 4, 171–185.
Goranko, V. and Passy, S.: 1992, “Using the universal modality: gains and questions”,Journal of Logic and Computation 2, 5–30.
van der Hoek, W.: 1991, “Qualitative modalities”, pp. 322–327 inProceedings of SCAI '91, Roskilde, Denmark, B. Mayoh, ed., Amsterdam: IOS Press.
van der Hoek, W.: 1992, “On the Semantics of Graded Modalities”,Journal of Applied Non-Classical Logics 2, 81–123.
Ladner, R.: 1977, “The computational complexity of provability in systems of modal propositional logic”,SIAM J. Comput. 6, 467–480.
de Rijke, M., 1992, “The modal logic of inequality”,The Journal of Symbolic Logic 57, 566–584.
Westerståhl, D., 1989, “Quantifiers in Formal and Natural Languages”, pp. 1–131 inHandbook of Philosophical Logic, Vol. IV, D. Gabbay and F. Guenthner, eds., Dordrecht: Reidel.
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This author was supported by the Foundation for Philosophical Research (SWON), which is subsidized by the Netherlands Organization for Scientific Research (NWO).
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Van Der Hoek, W., De Rijke, M. Generalized quantifiers and modal logic. J Logic Lang Inf 2, 19–58 (1993). https://doi.org/10.1007/BF01051767
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DOI: https://doi.org/10.1007/BF01051767