always some form of disjunction. The most common interpretations are as a simple disjunction

for intermediate logics, or as a disjunction of boxes

for modal logics.

In line with the disjunctive interpretation of the hypersequent bar, essentially all hypersequent calculi include the external structural rules, in particular the external weakening rule

and the external contraction rule

The additional expressivity of the hypersequent framework is provided by rules manipulating the hypersequent

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