VII.] INDUCTION. 137 



must be expressed in one of these six forms of law, 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 olDJects can be 

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

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

 it, or a like relation must exist between one class and the 

 negative of the other. We have thus completely solved 

 the inverse logical problem concerning two terms.^ 



Tlie Inverse Logical Problem involving Three Classes. 



No sooner do we introduce into the problem a third term 

 C, than the investigation assumes a far more complex 

 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 

 combinations, and the effect of laws or logical conditions 

 is to destroy any one or more of these combinations. Now 

 we- may make selections from eight things in 2^ or 256 

 ways; so that we have no less than 256 different cases to 

 treat, and the complete solution is at least fifty times as 

 troublesome as with two terms. Many series of com- 

 binations, indeed, are contradictory, as in the simpler 

 problem, and may be passed over, the test of consistency 

 being that each of the letters A, B, C, a, h, c, shall appear 

 soniewhere in the series of combinations. 



My mode of solving the problem was as follows : — 

 Having written out the whole of the 256 series of com- 

 binations, I examined them separately and struck out such 

 as did not fulfil the test of consistency. I then chose 

 some form of proposition involving two or three terms, 

 and vari(!d it in every possible manner, both by the 

 circular interchange of letters (A, V>, C into B, C, A and 

 then into C, A, B), and by the substitution for any one or 

 more of tlie terms of the corresponding negative terms. 



^ The contents of tliis and the following section nearly correspond 

 with those of a paper read before tlie Manchester Literary and 

 Philosojiliical Society on Deceiiiljcr 26t,h, 1871. See Proceediiigs of 

 the Society, vol. xi. j^p. 65—68, and Memoirs, Third Series, vol. v. 

 pp. 119-130. 



