A Perspective on Modal Sequent Logic

This paper was drafted in 1981—2 when the first author was lecturing on modal logic for the Philosophy Subfaculty at Oxford University and the second author was visiting Oxford on study leave; it was revised the following year.^ Both of us had been graduate students of D. S. Scott at Oxford in the 1970's and were impressed by his emphasis on the desirability of isolating the structural properties of a (logical) consequence relation — such as are encoded in the principles (jR), (M), and (T) of [11], [12] —from principles relating to specific connectives. Extending this idea to the case of the modal operators, we found that distinctions between several well-known systems of (normal) modal logic could be reflected at the purely structural level, if an appropriate notion of sequent was adopted. Actually, we work with one notion of sequent in §§1—4 and consider a somewhat more refined version in §5. On later finding that sequents of the latter type had already been used by M. Sato, who, in §3.4 of [10], credits the idea to O. Sonobe, we had some misgivings about publishing the material at full length. That anticipation notwithstanding, however, it appears to us still worth proceeding with a somewhat abridged version of the paper, both so as to highlight the original motivation and also because our treatment and Sato's differ on many points of detail. We should mention that K. Dosen, in [4], also advocates a variation on the traditional idea of what a sequent should look like for the case of modal logic. Though the framework he sets up is quite different from our own, he is in part motivated by similar consideratons (e.g., the concern with 'unique characterization' — see §4 below). Some aspects of our own way of proceeding may be seen (again, in retrospect) as steps in the execution of Belnap's "Display Logic' programme (see [2]), in that the rules (a) of §4 serve