INDUCTION. 157 



accounting for the tenth case ; and if we change B into b 

 the identity A = b accounts for the seventh case. There 

 may indeed be two other varieties of the simple identity, 

 namely a = b and a = B ; but it has already been shown 

 repeatedly that these are equivalent respectively to A = B 

 and A.~b (pp. 133, 134). As the sixteenth column has 

 already been accounted for as governed by no special 

 conditions, we come to the following general conclusion : 

 The laws governing the combinations of two terms must 

 be capable of expression either in a partial identity 

 (A AB), or a simple identity (A = B) ; the partial 

 identity is capable of only four logically distinct varieties, 

 and the simple identity of two. Every logical relation 

 between two terms must be expressed in one of these 

 six laws, or must be logically equivalent to one of them. 



In short, we may conclude that in treating of partial 

 and complete identity, we have exhaustively treated the 

 modes in which two terms or classes of objects can be 

 related. Of any two classes it may be said that one must 

 either be included in the other, or must be identical with 

 it, or some similar relation must exist between one class 

 and the negative of the other. We have thus completely 

 solved the inverse logical problem concerning two terms d . 



The Inverse Logical Problem involving Three Terms. 



No sooner do we introduce into the problem a third 

 term C, than the investigation assumes a far more com 

 plex character, so that some readers may prefer to pass 

 over this section. Three terms and their negatives may be 

 combined, as we have frequently seen, in eight different 



d The contents of this and the following section nearly correspond 

 with those of a paper read before the Manchester Literary and Philosophical 

 Society on December 26th, 1871. See Proceedings of the Society, vol. xi. 

 pp. 65-68, and Memoirs, Third Series, vol. v. pp. 119-130. 



