vi.J THE INDIRECT METHOD OF INFERENCE. 117 



A = AB ) ( A = AC 



B = BC j ' I B = A -I- BC 



Although ia 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. 1 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. 



1 Syllabus of a proposed system of Logic, 57, 121, &c. Formal 

 Logic, p. 66. 



