Provability logic, or the logic of provability, is a modal logic where the "necessity" operator is interpreted as provability in a reasonably rich formal theory such as Peano arithmetic. It was pioneered by Robert Solovay in 1976. Since then until his passing 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.

References

• Provability logic, from the Stanford Encyclopaedia of Philosophy.
• 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.