 Two strands of research have led to the birth of provability logic. The first one stems from a paper by  K. Gödel  (1933), where he introduces translations from intuitionistic propositional logic into modal logic (more precisely, into the system nowadays called S4), and briefly mentions that provability can be viewed as a modal operator. Even earlier, C.I. Lewis started the modern study of modal logic by introducing strict implication as a kind of deducibility, where he may have meant deducibility in a formal system like  Principia Mathematica,  but this is not clear from his writings.