Two types of multiple-conclusion systems

Logic Journal of the IGPL 6 (5):695-718 (1998)
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Hypersequents are finite sets of ordinary sequents. We show that multiple-conclusion sequents and single-conclusion hypersequents represent two different natural methods of switching from a single-conclusion calculus to a multiple-conclusion one. The use of multiple-conclusion sequents corresponds to using a multiplicative disjunction, while the use of single-conclusion hypersequents corresponds to using an additive one. Moreover: each of the two methods is usually based on a different natural semantic idea and accordingly leads to a different 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 surprisingly, the use of multiple-conclusion hypersequents corresponds to the use of structures which are both symmetrical and with a single nondesignated element



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Arnon Avron
Tel Aviv University

Citations of this work

Replacing Modus Ponens With One-Premiss Rules.Lloyd Humberstone - 2008 - Logic Journal of the IGPL 16 (5):431-451.
Hypersequent calculi for intuitionistic logic with classical atoms.Hidenori Kurokawa - 2010 - Annals of Pure and Applied Logic 161 (3):427-446.

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References found in this work

Investigations into Logical Deduction.Gerhard Gentzen - 1964 - American Philosophical Quarterly 1 (4):288 - 306.
The method of hypersequents in the proof theory of propositional non-classical logics.Arnon Avron - 1996 - In Wilfrid Hodges (ed.), Logic: Foundations to Applications. Oxford: pp. 1-32.
Relevance and paraconsistency—a new approach.Arnon Avron - 1990 - Journal of Symbolic Logic 55 (2):707-732.
Multiplicative Conjunction as an Extensional Conjunction.Arnon Avron - 1997 - Logic Journal of the IGPL 5 (2):181-208.

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