A = A*b* . . . . . . . B = *a*B A = *b* . . . . . . . . *a* = B A = BC . . . . . . . . *a* = *b* ꖌ *c* A = AB ꖌ AC . . . . . . *b* = *ab* ꖌ A*b*C A ꖌB = C ꖌ D . . . . . . . *ab* = *cd* A ꖌ *c* = B ꖌ *d* . . . . . . *a*C = *b*D A = AB*c* ꖌ A*b*C . . .{ A = AB ꖌ AC { AB = AB*c* A = B } { A = B B = C } . . . . . . . . . { A = C A = AB } { A = AC B = BC }. . . . . . . . . { B = A ꖌ *a*BC Although in these and many other cases the equivalents of certain propositions can readily be given, yet I believe that no uniform and infallible process can be pointed out by which the exact equivalents of premises can be ascertained. Ordinary deductive inference usually gives us only a portion of the contained information. It is true that the combinations consistent with a set of premises may always be thrown into the form of a proposition which must be logically equivalent to those premises; but the difficulty consists in detecting the other forms of propositions which will be equivalent to the premises. The task is here of a different character from any which we have yet attempted. It is in reality an inverse process, and is just as much more troublesome and uncertain than the direct process, as seeking is compared with hiding. Not only may several different answers equally apply, but there is no method of discovering any of those answers except by repeated trial. The problem which we have here met is really that of induction, the inverse of deduction; and, as I shall soon show, induction is always tentative, and, unless conducted with peculiar skill and insight, must be exceedingly laborious in cases of complexity. De Morgan was unfortunately led by this equivalence of propositions into the most serious error of his ingenious system of Logic. He held that because the proposition “All A’s are all B’s,” is but another expression for the two propositions “All A’s are B’s” and “All B’s are A’s,” it must be a composite and not really an elementary form of proposition.[82] But on taking a general view of the equivalence of propositions such an objection seems to have no weight. Logicians have, with few exceptions, persistently upheld the original error of Aristotle in rejecting from their science the one simple relation of identity on which all more complex logical relations must really rest. [82] *Syllabus of a proposed system of Logic*, §§ 57, 121, &c. *Formal Logic*, p. 66. *The Nature of Inference.*