Two Types of Multiple-Conclusion Systems

Hypersequents are nite sets of ordinary sequents. We show that multiple-conclusion sequents and single-conclusion hypersequents represent two di erent natural methods of switching from a singleconclusioncalculusto a multiple-conclusionone. The use of multiple-conclusionsequentscorresponds to using a multiplicative disjunction, while the use of single-conclusionhypersequents corresponds to using an additive one. Moreover: each of the two methods is usually based on a di erent natural semantic idea and accordingly leads to a di erent class of algebraic structures. In the cases we consider here the use of multiple-conclusion sequents corresponds to focusing the attention on structures in which there is a full symmetry between the sets of designated and antidesignated elements. The use of single-conclusion hypersequents, on the other hand, corresponds to the use of structures in which all elements except one are designated. Not surprisingy,the use of multiple-conclusionhypersequents corresponds to the use of structures which are both symmetrical and with a single nondesignated element
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