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The expressive power of second-order propositional modal logic. (English) Zbl 0895.03005
The paper considers modal logics with propositional quantifiers (such a logic is briefly called SOPML), with “platonistic” semantics, defined according to K. Fine [Theoria 36, 336-346 (1970; Zbl 0302.02005)]. In the latter paper it was proved that second-order arithmetic is interpretable in SOPML, provided the basic modal logic is S4.2 or weaker. The authors prove a stronger result, namely that in these cases SOPML is mutually interpretable with classical second-order predicate logic. Also they reproduce an unpublished proof of a result by H. Kamp (1977) that SOPML (for the same cases) is embeddable in Thomason’s modal first-order logic Q2, with quantification over individual concepts. Therefore classical second-order predicate logic also is embeddable in Q2.

MSC:
 03B45 Modal logic (including the logic of norms) 03B15 Higher-order logic; type theory (MSC2010)
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References:
 [1] Fine, K., “Propositional quantifiers in modal logic,” Theoria , vol. 36 (1970), pp. 336–346. · Zbl 0302.02005 [2] Fine, K., “Model theory for modal logic, part I: the de re/de dicto distinction,” Journal of Philosophical Logic , vol. 7 (1978), pp. 125–156. · Zbl 0375.02008 [3] Fine, K., “Model theory for modal logic, part II: the elimination of de re modality,” Journal of Philosophical Logic , vol. 7 (1978), pp. 277–306. · Zbl 0409.03007 [4] Garson, J. W., “Quantification in modal logic,” pp. 249–307 in Handbook of Philosophical Logic , edited by D. Gabbay and F. Guenther, Reidel, Dordrecht, 1984. · Zbl 0875.03050 [5] Gurevich, Y., and S. Shelah, “Monadic theory of order and topology in ZFC,” Annals of Mathematical Logic , vol. 23 (1983), pp. 179–198. · Zbl 0516.03007 [6] Gurevich, Y., and S. Shelah, “Interpreting second-order logic in the monadic theory of order,” The Journal of Symbolic Logic vol. 48 (1983), pp. 816-828. JSTOR: · Zbl 0559.03008 [7] Hughes, G. E., and M. J. Cresswell, A Companion to Modal Logic , Methuen, London, 1984. Mathematical Reviews (MathSciNet): · Zbl 0625.03005 [8] Kamp, H., “Two related theorems by D. Scott and S. Kripke,” unpublished manuscript, 1977. [9] Shelah, S., “The monadic theory of order,” Annals of Mathematics vol. 102 (1975), pp. 379–419. JSTOR: · Zbl 0345.02034 [10] Thomason, R. H., “Modal logic and metaphysics,” pp. 119–146 in The Logical Way of Doing Things , edited by K. Lambert, Yale University Press, New Haven, 1969. · Zbl 0188.31705
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