Advertisement

Studia Logica

, Volume 51, Issue 2, pp 249–277 | Cite as

The logic of linear tolerance

  • Giorgie Dzhaparidze
Article

Abstract

A nonempty sequence 〈T1,...,Tn〉 of theories is tolerant, if there are consistent theories T 1 + ,..., T n + such that for each 1 ≤i ≤n, T i + is an extension of Ti in the same language and, if i ≤n, T i + interprets T i+1 + . We consider a propositional language with the modality ◊, the arity of which is not fixed, and axiomatically define in this language the decidable logics TOL and TOLω. It is shown that TOL (resp. TOLω) yields exactly the schemata of PA-provable (resp. true) arithmetical sentences, if ◊(A1,..., An) is understood as (a formalization of) “〈 PA+A1, ..., PA+An〉 is tolerant”.

Keywords

Mathematical Logic Computational Linguistic Decidable Logic Consistent Theory Propositional 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]
    A. Berarducci, The interpretability logic of Peano arithmetic, Entailment: The Journal of Symbolic Logic, 55 (1990), No.3, pp. 1059–1089.Google Scholar
  2. [2]
    G. Boolos, The Unprovability of Consistency, Cambridge Universitety Press, Cambridge, 1979.Google Scholar
  3. [3]
    D. De Jongh, and F. Veltman, Provability logics for relative interpretability. To apper in the Proceedings of Heyting'88 Conference, Chaika, Bulgaria, 1988.Google Scholar
  4. [4]
    G. Dzhaparidze, Provability logic with modalities for arithmetical complexities, Bulletin of the Academy of Sciences of the Georgian SSR 138 (1990), No. 3. pp. 481–484.Google Scholar
  5. [5]
    S. Peferman, Arithmetization of Metamathematics in a general setting, Fundamenta Mathematicae, 49 (1990), pp. 35–92.Google Scholar
  6. [6]
    D. Guaspari, Partially conservative extentions of arithmetic, Transactions of the Amer. Math. Soc. 254 (1979), pp. 47–68.Google Scholar
  7. [7]
    P. Hajek On interpretability in set theories I, II Comm. Math. Univ. Carolinae, 12 (1971), pp. 73–79 and 13 (1972), pp. 445–455.Google Scholar
  8. [8]
    K. Ignatiev, Logic of1 -interpolability over Peano arithmetic, (in Russian), Manuscript, Moscow, September 1990.Google Scholar
  9. [9]
    S. Orey, Relative interpretations, Zeitschrift für Math. Logik und Grundlagen der Mathematik, 7 (1961), pp. 146–153.Google Scholar
  10. [10]
    V. Shavrukow, Logic of relative interpretability over Peano arithmetic, Preprint No. 5, Steklov Mathematical Institute, Academy of Sciences of the USSR, Moscow, December 1988.Google Scholar
  11. [11]
    R. M. Solovay, Provability interpretations of logic, Israel Journal of Mathematics, 25 (1976), pp. 287–304.Google Scholar
  12. [12]
    A. Tarski, in colab with A. Mostowski and R. M. Robinson, Undecidable theories, Amsterdam, 1953.Google Scholar
  13. [13]
    A. Visser Preliminary notes on interpretability logic, Logic group preprint series, No. 29, Department of Philosophy, University of Utrecht, Utrecht, 1988.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • Giorgie Dzhaparidze
    • 1
  1. 1.Institute of PhilosophyGeorgian Academy of SciencesTbilisiGeorgia

Personalised recommendations