Advertisement

Studia Logica

, Volume 49, Issue 1, pp 7–21 | Cite as

Decidable and enumerable predicate logics of provability

  • Giorgie Dzhaparidze
Article

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. QLT(T) and QLT 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 QLT(T) and QLT by formulas, which contain no variables different from x, are given by means of decidable prepositional bimodal systems; under the assumption that T is Π1-complete, there is established the enumerability of the restrictions of QLT(T) and QLT by: 1) formulas in which the domains of different occurrences of □ don't intersect and 2) formulas of the form n ⊥ → A.

Keywords

Mathematical Logic Computational Linguistic Standard Formula Predicate Logic Modal Language 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    С. Н. Артëмов, Неарифметичность истинностных предикатных логик доказуемости, ДАН СССР, vol. 284 (1985), 12, pp. 270–271.Google Scholar
  2. [2]
    С. H. Артëмов and Г. К. Джапаридзе, Об. эффективных предикатных логиках доказуемости, ДAH СССР, vol. 297 (1987), No. 3, pp. 521–523.Google Scholar
  3. [3]
    G. Boolos, The Unprovability of Consistency: An Essay in Modal Logic, Cambridge University Press, 1979.Google Scholar
  4. [4]
    G. Boolos and V. McGee, The degree of the set of sentences of predicate provability logic that are true under every interpretation, Journal of Symbolic Logic 52 (1987), N∘ 1, pp. 165–171.Google Scholar
  5. [5]
    Л. Л. Эсакиа, Логика доказуемости с кванторными модальностями, в сб.: Интенсиональные логики и логическая структура теорий, Тбилиси, Мецниереба, 1988, pp. 11–19.Google Scholar
  6. [6]
    J. Barwise (ed.), Handbook of Mathematical Logic, 1977, part IV.Google Scholar
  7. [7]
    Г. К. Джапаридзе, Арифметическая полнота логики доказуемости с кванторными модальностями, Сообщения АН ГССР, vol. 132 (1988), No 2, pp. 265–268.Google Scholar
  8. [8]
    R. Solovay, Provability interpretations of modal logic, Israel Journal of Mathematics, v. 25 (1976), pp. 287–304.Google Scholar
  9. [9]
    В. А. Варданян, Арифметическая сложность предикатных логик доказуемости, ДАН СССР, 288 (1986), No 1, с. 11–14.Google Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • Giorgie Dzhaparidze
    • 1
  1. 1.Academy of Sciences of Georgian SSRInstitute of PhilosophyTbilisiUSSR

Personalised recommendations