Consequence

We take mathematical logic to be the mathematical study of consequence and proof by means of formalised languages or calculi. A single-conclusion calculus is constituted by a universe of formulae and a relation of consequence, construed as a binary relation between a set of premisses and a conclusion. We make no stipulations about the sort of entity that can be a formula or what sort of internal structure the formulae may possess. It is natural to assume that there is at least one formula, though no result of importance appears to depend on this. We make no other stipulation about the number of formulae, though since we use the axiom of choice we shall be assuming that they can be well-ordered. We write V for the relevant universe of formulae, and use T, U, W, X, Y, Z to stand for sets of formulae (i.e. subsets of V), and A, B, C, D for formulae. We write L for a calculus and ⊢ for its consequence relation, adding a prime or suffix when necessary, so that for example ⊢1 stands for consequence in L1. By a single-conclusion relation we mean any binary relation between sets of formulae on the one hand and individual formulae on the other. If R is a single-conclusion relation we write X R B to indicate that R holds between the set X and the formula B, and whenever X R B we say that the pair 〈X,B〉 is an instance of R.