Provability logic with operations on proofs

  • Tatiana Sidon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1234)


We present a natural axiomatization for propositional logic with a modal operator for formal provability (Solovay, [6]) and labeled modalities for individual proofs with operations on them (Artemov, [2]). For this purpose, the language has to be extended by two new operations. The obtained system \(\mathcal{M}\mathcal{L}\mathcal{P}\)naturally includes both Solovay's provability logic GL and Artemov's operational modal logic \(\mathcal{L}\mathcal{P}\). We prove that the system \(\mathcal{M}\mathcal{L}\mathcal{P}\)is decidable and arithmetically complete. We also show that \(\mathcal{M}\mathcal{L}\mathcal{P}\)realizes all operations on proofs admitting description in the modal propositional language.


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  1. 1.
    S.Artemov, “Logic of Proofs', Annals of Pure and Applied Logic, v.67 (1994), pp. 29–59.Google Scholar
  2. 2.
    S.Artemov, “Operational modal logic”, Techn. Rep. No 95-29, Mathematical Science Institute, Cornell University, December 1995.Google Scholar
  3. 3.
    S.Artemov, “Proof Realization of Intuitionistic and Modal Logic”, Techn. Rep. No 96-06, Mathematical Science Institute, Cornell University, December 1996.Google Scholar
  4. 4.
    D.Hilbert, P.Bernays, “Grundlagen der Mathematik”, I, Springer-Verlag, 1968.Google Scholar
  5. 5.
    C.Smorynski, Self-reference and modal logic, New York, Berlin, Heidelberg, Tokio: Springer Verlag, 1985.Google Scholar
  6. 6.
    R.M.Solovay, “Provability interpretation of modal logic”, Israel Journal of Mathematics, v.25 (1976), pp. 287–304.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Tatiana Sidon
    • 1
  1. 1.Department of Mathematical Logic, Faculty of Mathematics and MechanicsMoscow State UniversityMoscowRussia

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