THE EQUIVALENCY OF PROPOSITIONS. 135 



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 ascer 

 tained. Ordinary deductive inference usually gives us 

 only a portion of the contained information. It is true 

 that the combinations consistent with a set of propositions 

 are logically equivalent to them, but the difficulty consists 

 in passing back from the combinations to a new set of 

 propositions. 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 com 

 pared 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 

 any considerable complexity. 



The late Professor de Morgan was unfortunately led 

 by this equivalency 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, was but 

 another expression for the two propositions All A s are 

 B s and All B s are As, it must be a composite and not 

 really an elementary form of proposition 1 . But on taking 

 a general view of the equivalency 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 



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

 Logic, p. 66. 



