One Connection between Standard Invariance Conditions on Modal Formulas and Generalized Quantifiers
The language of standard propositional modal logic has one operator (□ or ♦), that can be thought of as being determined by the quantifiers ∀ or ∃, respectively: for example, a formula of the form □Φ is true at a point s just in case all the immediate successors of s verify Φ.This paper uses a propositional modal language with one operator determined by a generalized quantifier to discuss a simple connection between standard invariance conditions on modal formulas and generalized quantifiers: the combined generalized quantifier conditions of conservativity and extension correspond to the modal condition of invariance under generated submodels, and the modal condition of invariance under bisimulations corresponds to the generalized quantifier being a Boolean combination of ∀ and ∃.
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- Goldblatt, R., 1992, Logics of Time and Computation, Stanford, CA: CSLI Publications.Google Scholar
- Keenan, E.L. and Westerståhl, D., 1997, “Generalized quantifiers in linguistics and logic,” pp. 837–893 in Handbook of Logic and Language, J. van Benthem and A. ter Meulen, eds, Amsterdam: Elsevier.Google Scholar
- van Benthem, J., 1986, Essays in Logical Semantics, Dordrecht: Reidel.Google Scholar
- van der Hoek, W., 1992, “Modalities for reasoning about knowledge and quantities,” Dissertation, Free University of Amsterdam.Google Scholar
- Westerståhl, D., 1989, “Quantifiers in formal and natural languages,” pp. 1–131 in Handbook of Philosophical Logic, D.M. Gabbay and F. Guenthner, eds, Dordrecht: Reidel.Google Scholar