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  1. Strong completeness of s4 for any dense-in-itself metric space.Philip Kremer - 2013 - Review of Symbolic Logic 6 (3):545-570.
    In the topological semantics for modal logic, S4 is well-known to be complete for the rational line, for the real line, and for Cantor space: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but also strongly complete, for the rational line. But no similarly easy amendment is available for the real line or for Cantor space and the question (...)
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  • (1 other version)Propositional quantifiers in modal logic.Kit Fine - 1970 - Theoria 36 (3):336-346.
    In this paper I shall present some of the results I have obtained on modal theories which contain quantifiers for propositions. The paper is in two parts: in the first part I consider theories whose non-quantificational part is S5; in the second part I consider theories whose non-quantificational part is weaker than or not contained in S5. Unless otherwise stated, each theory has the same language L. This consists of a countable set V of propositional variables pl, pa, ... , (...)
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  • On second order intuitionistic propositional logic without a universal quantifier.Konrad Zdanowski - 2009 - Journal of Symbolic Logic 74 (1):157-167.
    We examine second order intuitionistic propositional logic, IPC². Let $F_\exists $ be the set of formulas with no universal quantification. We prove Glivenko's theorem for formulas in $F_\exists $ that is, for φ € $F_\exists $ φ is a classical tautology if and only if ¬¬φ is a tautology of IPC². We show that for each sentence φ € $F_\exists $ (without free variables), φ is a classical tautology if and only if φ is an intuitionistic tautology. As a corollary (...)
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  • Propositional Quantification in the Topological Semantics for S.Philip Kremer - 1997 - Notre Dame Journal of Formal Logic 38 (2):295-313.
    Fine and Kripke extended S5, S4, S4.2 and such to produce propositionally quantified systems , , : given a Kripke frame, the quantifiers range over all the sets of possible worlds. is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to second-order logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub , (...)
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  • On the complexity of propositional quantification in intuitionistic logic.Philip Kremer - 1997 - Journal of Symbolic Logic 62 (2):529-544.
    We define a propositionally quantified intuitionistic logic Hπ + by a natural extension of Kripke's semantics for propositional intutionistic logic. We then show that Hπ+ is recursively isomorphic to full second order classical logic. Hπ+ is the intuitionistic analogue of the modal systems S5π +, S4π +, S4.2π +, K4π +, Tπ +, Kπ + and Bπ +, studied by Fine.
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  • (1 other version)On Some Completeness Theorems in Modal Logic.D. Makinson - 1966 - Mathematical Logic Quarterly 12 (1):379-384.
    Gives the first published adaptation of the Lindenbaum/Henkin method of maximal consistent sets for establishing the completeness of modal propositional logics with respect to the relational models of Kripke.
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  • Quantified modal logic on the rational line.Philip Kremer - 2014 - Review of Symbolic Logic 7 (3):439-454.
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  • (2 other versions)The Mathematics of Metamathematics.Donald Monk - 1963 - Journal of Symbolic Logic 32 (2):274-275.
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  • On 2nd order intuitionistic propositional calculus with full comprehension.Dov M. Gabbay - 1974 - Archive for Mathematical Logic 16 (3-4):177-186.
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  • The Expressive Power of Second-Order Propositional Modal Logic.Michael Kaminski & Michael Tiomkin - 1996 - Notre Dame Journal of Formal Logic 37 (1):35-43.
    It is shown that the expressive power of second-order propositional modal logic whose modalities are S4.2 or weaker is the same as that of second-order predicate logic.
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