Modal logics of provability predicates

Hilbert and Bernays (1939) proposed some conditions (derivability conditions) of provability predicates that are sufficient for the second incompleteness theorem to hold. Subsequently, the proof of the second incompleteness theorem using some conditions of provability predicates established by L"ob (1955) became widely known. Other derivability conditions sufficient for the second incompleteness theorem were also given by Jeroslow (1973) etc. On the other hand, derivability conditions for provability predicates for which the second incompleteness theorem does not hold, such as Rosser's provability predicate, have also been studied. In this talk, I will present some known results on the study of derivability conditions by means of modal logic.