Consequence

A multiple-conclusion calculus is constituted by a universe V of formulae and a relation ⊢ of multiple-conclusion consequence. As in Section 1.1 we assume that V is not empty, though curiously enough with an empty universe of formulae there would still be two distinct multiple-conclusion calculi, in one of which ∧ ⊢ ∧ and in the other ∧ ⊬ ∧. By a multiple-conclusion relation we mean a binary relation between sets of formulae, and when such a relation R holds between sets X and Y we write X R Y and call the pair 〈X,Y〉 an instance of R. We shall generally take the adjective ‘multiple-conclusion’ for granted and write simply ‘calculus’ to contrast with ‘single-conclusion calculus’, and similarly with ‘relation’ and other terms.

Our task in this and the succeeding chapter is to develop the appropriate multiple-conclusion analogues of the various definitions of consequence that we have produced for single conclusions. First we say that a partition 〈T,U〉 satisfies 〈X,Y〉 if X overlaps U or Y overlaps T. Otherwise, i.e. if X ⊂ T and Y ⊂ U, we say that 〈T,U〉 invalidates 〈X,Y〉. If 〈X,Y〉 is satisfied by every partition in the set I we say it is valid in I and write X ⊢I Y. We say that ⊢ is a consequence relation if ⊢ = ⊢I for some I, in which case we say too that ⊢ is characterised by I.