Cornered-circuit proofs

We know from Theorems 8.12 and 8.13 that an adequate class of abstract proofs must permit the formation of both corners and circuits. With this in mind we offer a third solution to the problem of junction and cut posed in Section 9.1, by invoking the operation of identifying or coalescing different occurrences of the same formula in a proof.

We say that π′ is obtained from a graph argument π by identification if it is the result of omitting vertices a2,…,an from π and replacing every edge to or from any of them by a corresponding edge to or from the vertex a1, where all the ai are occurrences of the same formula; provided that if any ai is a premiss (conclusion) then a1 is a premiss (conclusion) and all the ai are initial (final). For example, the pattern shown in Figure 10.1 is obtained by identifying the same-numbered vertices in the pattern of Figure 9.18. By prohibiting the identification of premisses with non-initial vertices (or conclusions with non-final ones), the proviso in the definition ensures that the result of identification is always a graph argument; and by prohibiting the identification of a premiss ai with a non-premiss a1 (or a conclusion with a non-conclusion a1) it ensures that Xπ′ = Xπ and Yπ, = Yπ and hence, since evidently Rπ, = Rπ, that identification preserves validity.