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). It was pioneered by Robert M. Solovay in 1976. Since then until his death in 1996 the prime inspirer of the field was George Boolos. Significant contributions to the field have been made by Sergei Artemov, Lev Beklemishev, Giorgi Japaridze, Dick de Jongh, Franco Montagna, Vladimir Shavrukov, Albert Visser and others. Interpretability logics present natural extensions of provability logic.
- 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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