12th case. 8th case. 15th case. 14th case. A = AB A = A*b* *a* = *a*B *a* = *ab* *b* = *ab* B = *a*B *b* = A*b* B = AB The reader of the preceding sections will see that each proposition in the lower line is logically equivalent to, and is in fact the contrapositive of, that above it (p. 83). Thus the propositions A = A*b* and B = *a*B both give the same combinations, shown in the eighth column of the table, and trial shows that the twelfth, eighth, fifteenth and fourteenth columns are thus accounted for. We come to this conclusion then--*The general form of proposition* A = AB *admits of four logically distinct varieties, each capable of expression in two modes*. In two columns of the table, namely the seventh and tenth, we observe that two combinations are missing. Now a simple identity A = B renders impossible both A*b* and *a*B, 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* (p. 115). 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 or a simple identity; 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 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 objects 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.[85] [85] 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. *The Inverse Logical Problem involving Three Classes.*