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5 - Counterparts
Published online by Cambridge University Press: 06 July 2010
Summary
The range of counterparts
We have so far treated multiple- and single-conclusion calculi separately, though on parallel lines. We now try to establish some direct connections between them. For this purpose let L stand for an arbitrary single-conclusion calculus and L′ for an arbitrary multiple-conclusion one. We have seen how a set of partitions can be used to characterise a calculus of either kind, and when L and L′ are characterised by the same set of partitions we shall say that they are counterparts of each other. Since any partition satisfies 〈X,B〉 iff it satisfies 〈X,{B}〉 an alternative criterion for two calculi to be counterparts can be formulated directly in terms of their consequence relations:
Theorem 5.1 A necessary and sufficient condition for a single-conclusion calculus L and a multiple-conclusion calculus L′ to be counterparts is that X ⊢ B iff X ⊢′ {B}.
Theorem 5.1 shows that there is only one counterpart L of each L′ and that even though ⊢ is not literally a subrelation of ⊢′ it is the image of one, namely the subrelation comprising just those instances of ⊢′ whose conclusions are singletons. (This subrelation is not itself a consequence relation at all since it does not permit dilution.) We therefore call L the single-conclusion part of L′. But although a multiple-conclusion calculus has a unique single-conclusion counter part the converse is not true.
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- Multiple-Conclusion Logic , pp. 72 - 94Publisher: Cambridge University PressPrint publication year: 1978