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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 Also they reproduce an unpublished proof of a result by H. Kamp (1977) that SOPML (for the same cases) is embeddable in Thomason’s modal first-order logic Q2, with quantification over individual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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[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
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