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Provability logic

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Title: Provability logic  
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Subject: Modal logic, Gödel's completeness theorem, Hilbert–Bernays provability conditions, Saul Kripke, List of mathematical logic topics
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Provability logic

Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic.


There are a number of provability logics, some of which are covered in the literature mentioned in the References section. The basic system is generally referred to as GL (for Gödel-Löb) or L or K4W. It can be obtained by adding the modal version of Löb's theorem to the logic K (or K4).


The GL model was pioneered by Dick de Jongh, Franco Montagna, Vladimir Shavrukov, Albert Visser and others.


Interpretability logics present natural extensions of provability logic.

See also


  • George Boolos, The Logic of Provability. Cambridge University Press, 1993.
  • Giorgi Japaridze and Dick de Jongh, The logic of provability. In: Handbook of Proof Theory, S. Buss, ed. Elsevier, 1998, pp. 475-546.
  • Sergei Artemov and Lev Beklemishev, Provability logic. In: Handbook of Philosophical Logic, D. Gabbay and F. Guenthner, eds., vol. 13, 2nd ed., pp. 189-360. Springer, 2005.
  • Per Lindström, Provability logic - a short introduction. Theoria 62 (1996), pp. 19-61.
  • Craig Smoryński, Self-reference and modal logic. Springer, Berlin, 1985.
  • Robert M. Solovay, ``Provability Interpretations of Modal Logic``, Israel Journal of Mathematics, Vol. 25 (1976): 287-304.
  • Provability logic, from the Stanford Encyclopedia of Philosophy.

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