The first axiomatization of relevant logic

This is a review. with historical and critical comments, of a paper b) I. E. Orlov from 1928, which eivcs the oldest known axiomatization of the implicationnegation fragment of the rclcvant logic R. Orlov’s paper also l’omshadow the modal translation of systems with an intuitionistic negation into %-type extensions of systems with a classical. involutive. negation. Orlov introduces the modal postulates of S4 before Becker. Lewis and Giidcl. Orlov’s work. which seems to bc nearly completely ignored, is related to the contemporaneous work on the axiomatization of intuitionistic logic. In (1928) I. E. Orlov gave an axiomatization of the implicationnegation fragment of the relevcnzt loyic R. and this may well be the very first axiomatization of relevant logic. It is perhaps not fortuitous that the first axiomatization of relevant logic should appear at the same time when the first axiomatizations of intuition&tic logic were being produced. This is not the only achievement of Orlov’s paper. He also foreshadows the ~nodul tratdation of systems with an intuitionistic negation into S4-type extensions of systems with a classical, involutive, negation. By “modal translation“ we mean a translation that prefixes the necessity operator to subformulae of a nonmodal formula. That there is such a modal translation from Heyting’s logic into the modal logic S4, which is based on classical logic, has been claimed by Godcl (1933): and has become a very well known fact about Heyting’s logic. In my paper (1992), it is shown that there are analogous modal translations from intuitionistic variants of linear logic, relevant logic and BCK logic into S4-type extensions of the respective systems with a negation like classical negation. Metaphorically speaking, the S4 necessity operator subdues classical negation and transforms it into an intuitionistic one. Such a modal translation for relevant logic is only foreshadowed in Orlov’s paper. But he introduces quite explicitly the characteristic modal postulates of S4, in the form they have in Journal of I~hilosophical Logic 21: 339-356. 1992. CI 1992 Kluwer Academic Yuhli.shars. Prinkd in the ~Ve~herlantls.