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# Provability logic

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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.

## Examples

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).

## History

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

## Generalizations

Interpretability logics present natural extensions of provability logic.

## References

• 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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