Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T06:10:11.213Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Kripke models and predicate logics of provability

Published online by Cambridge University Press:  12 March 2014

Sergei Artemov  and
Giorgie Dzhaparidze
Sergei Artemov
Affiliation:
Steklov Mathematical Institute, Academy of Sciences of the USSR, 117966 Moscow, USSR
Giorgie Dzhaparidze
Affiliation:
Institute of Philosophy, Academy of Sciences of the Georgian SSR, 380009 Tbilisi, USSR
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper proves a predicate version of Solovay's well-known theorem on provability interpretations of modal logic:

If a closed modal predicate-logical formula R is not valid in some finite Kripke model, then there exists an arithmetical interpretation f such that PA ⊬ fR.

This result implies the arithmetical completeness of arithmetically correct modal predicate logics with the finite model property (including the one-variable fragments of QGL and QS). The proof was obtained by adding “the predicate part” as a specific addition to the standard Solovay construction.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 3 , September 1990 , pp. 1090 - 1098
Copyright
Copyright © Association for Symbolic Logic 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Gödel, K., Eine Interpretation des intuitionistischen Aussagenkalküls, Ergebnisse eines Mathematischen Kolloquiums, vol. 4 (1933), pp. 39–40.Google Scholar
[2]Solovay, R. M., Provability interpretations of modal logic, Israel Journal of Mathematics, vol. 25 (1976), pp. 287–304.CrossRefGoogle Scholar
[3]Boolos, G., The unprovability of consistency: an essay in modal logic, Cambridge University Press, Cambridge, 1979.Google Scholar
[4]Montagna, F., The predicate modal logic of provability, Notre Dame Journal of Formal Logic, vol. 25(1984), pp. 179–189.CrossRefGoogle Scholar
[5]Artemov, S. N., Nonarithmeticity of truth predicate logics of provability, Doklady Akademii Nauk SSSR, vol. 284 (1985), pp. 270–271; English translation in Soviet Mathematics Doklady, vol. 33 (1986), pp. 403–405.Google Scholar
[6]Artemov, S. N., On logics having a provability interpretation, Complexity of calculations and algorithms (Adian, S. I., editor) (= Voprosy Kibernetiki, vyp. 134), "Nauka", Moscow, 1988, pp. 5–22. (Russian)Google Scholar
[7]Boolos, G. and McGee, V., The degree of the set of predicate provability logics that are true under every interpretation, this Journal, vol. 52 (1987), pp. 165–171.Google Scholar
[8]Vardanyan, V. A., Arithmetic complexity of predicate logics of provability and their fragments, Doklady Akademii Nauk SSSR, vol. 288 (1986), pp. 11–14; English translation in Soviet Mathematics Doklady, vol. 33 (1986), pp. 569–572.Google Scholar
[9]Vardanyan, V. A., Bounds on the arithmetical complexity of predicate logics of provability, Complexity of calculations and algorithms (Adian, S. I., editor) (= Voprosy Kibernetiki, vyp. 134), "Nauka", Moscow, 1988, pp. 46–72. (Russian)Google Scholar
[10]Artemov, S. N. and Dzhaparidze, G. K., On effective predicate logics of provability, Doklady Akademii Nauk SSSR, vol. 297 (1987), pp. 521–523; English translation in Soviet Mathematics Doklady, vol. 36(1988), pp. 478–480.Google Scholar
[11]Wajsberg, M., Ein erweiterter Klassenkalkül, Monatshefte für Mathematik und Physik, vol. 40 (1933), pp. 113–126.CrossRefGoogle Scholar
[12]Dzhaparidze, G. K., Arithmetical completeness of provability logics with quantifier modalities, Soobshcheniya Akademii Nauk Gruzinskoĭ SSR, vol. 132 (1988), pp. 265–268. (Russian)Google Scholar