A multiple-conclusion proof can have a number of conclusions, say B1,…,Bn. It is not to be confused with a conventional proof whose conclusion is some one of the Bj, nor is it a bundle of conventional proofs having the various Bj for their respective conclusions: none of the Bj need be ‘the’ conclusion in the ordinary sense. This fact led Kneale to speak of the ‘limits’ of a ‘development’ of the premisses instead of the conclusions of a proof from them. We prefer to extend the sense of the existing terms, but hope to lessen one chance of misconstruction by speaking of a proof from A1,…,Am to B1,…,Bn instead of a proof of B1,…,Bn from A1,…,Am.
The behaviour of multiple conclusions can best be understood by analogy with that of premisses. Premisses function collectively: a proof from A1,…,Am is quite different from a bundle of proofs, one from A1, another from A2 and so on. Moreover they function together in a conjunctive way: to say that B follows from A1,…,Am is to say that B must be true if A1 and … and Am are true. Multiple conclusions also function collectively, but they do so in a disjunctive way: to say that B1,…,Bn follow from A1,…,Am is to say that B1 or … or Bn must be true if all the Ai are true.