Labeling sequents; its motivations and applications

Hacking applied a method involving the labeling of sequent members to the problem of giving a Gentzen-style system for S 5, called L5, formulated with strict implication and no special modal operator. The method was generalized to a labeling of hypersequents by G. Mints. In this paper, we show how this generalization eliminates an “anti-symmetry” in Hacking’s system and greatly simplifies L5. We then suggest a semantic interpretation of the labeling of L5 derived objects and indicate a straightforward completeness result through Shvarts’ LK45 system. We finally show how the labeling method can be used to introduce new strong negation operators in logics intermediate between classical and intuitionistic logic.