Sequent-systems for modal logic

The purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels. Sequents of level 1 are like ordinary sequents, sequents of level 2 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1. A restriction on Thinning on the right of level 2, which when applied to Thinning on the right of level 1 produces intuitionistic out of classical logic (without changing anything else), produces S4 out of S5 (without changing anything else). This characterization of modal constants with sequents of level 2 is unique in the following sense. If constants which differ only graphically are given a formally identical characterization, they can be shown inter-replaceable (not only uniformly) with the original constants salva provability. Customary characterizations of modal constants with sequents of level 1, as well as characterizations in Hilbert-style axiomatizations, are not unique in this sense. This parallels the case with implication, which is not uniquely characterized in Hilbert-style axiomatizations, but can be uniquely characterized with sequents of level 1. These results bear upon theories of philosophical logic which attempt to characterize logical constants syntactically. They also provide an illustration of how alternative logics differ only in their structural rules, whereas their rules for logical constants are identical. ?0. Introduction. The aim of this work is to present sequent formulations of the modal logics S5 and S4 based on sequents of higher levels. Sequents of level 1 have collections of formulae of a given formal language on the left and right of the turnstile, sequents of level 2 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants will involve sequents of level 2, whereas rules for other customary logical constants of first-order logic (with identity) will involve only sequents of level 1. We shall show how a restriction on Thinning of level 2, which when applied to Thinning of level 1 produces intuitionistic out of classical logic, produces in this case S4 out of S5. Both in passing from classical to intuitionistic logic and in passing from S5 to S4, only Thinning is changed-all the other assumptions are unchanged. In particular, this means that S5 and S4 will be formulated with identical assumptions for the necessity operator. We shall also show in what sense our characterization of the necessity operator is Received January 5, 1982; revised December 3, 1983. 1980 Mathematics Subject Classification. Primary 03B45, 03F99. (? 1985, Association for Symbolic Logic 0022-4812/85/5001-001 5/$03.00