Two Is Enough - Bisequent Calculus for S5

We present a generalised sequent calculus based on the use of pairs of ordinary sequents called bisequents. It may be treated as the weakest kind of system in the rich family of systems operating on items being some collections of ordinary sequents. This family covers hypersequent and nested sequent calculi introduced for several non-classical logics. It seems that for many such logics, including some many-valued and modal ones, a reasonably modest generalization of standard sequents is sufficient. In this paper we provide a proof theoretic examination of S5 in the framework of bisequent calculus. Two versions of cut-free calculus are provided. The first version is more flexible for proof search but admits only indirect proof of cut elimination. The second version is syntactically more constrained but admits constructive proof of cut elimination. This result is extended to several versions of first-order S5.

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