  We investigate a recent proposal for modal hypersequent calculi. The
interpretation of relational hypersequents incorporates an accessibility
relation along the hypersequent. These systems give the same interpretation of
hypersequents as Lellman's linear nested sequents, but were developed
independently by Restall for S5 and extended to other normal modal logics by
Parisi. The resulting systems obey Dosen's principle: the modal rules are the
same across different modal logics. Different modal systems only differ in the
presence or absence of external structural rules. With the exception of S5, the
systems are modular in the sense that different structural rules capture
different properties of the accessibility relation. We provide the first direct
semantical cut-free completeness proofs for K, T, and D, and show how this
method fails in the case of B and S4.
