Abstract
Predicate modal formulas are considered as schemata of arithmetical formulas, where □ is interpreted as the standard formula of provability in a fixed “sufficiently rich” theory T in the language of arithmetic. QL T(T) and QL T are the sets of schemata of T-provable and true formulas, correspondingly. Solovay's well-known result — construction an arithmetical counterinterpretation by Kripke countermodel — is generalized on the predicate modal language; axiomatizations of the restrictions of QL T(T) and QL T by formulas, which contain no variables different from x, are given by means of decidable prepositional bimodal systems; under the assumption that T is Π 1-complete, there is established the enumerability of the restrictions of QL T(T) and QL T by: 1) formulas in which the domains of different occurrences of □ don't intersect and 2) formulas of the form □ n ⊥ → A.
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Dzhaparidze, G. Decidable and enumerable predicate logics of provability. Stud Logica 49, 7–21 (1990). https://doi.org/10.1007/BF00401550
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DOI: https://doi.org/10.1007/BF00401550