THE EQUIVALENCY OF PROPOSITIONS. 133 



be true at the same time ; ' All men are immortals ' is of 

 course contradictory. 



One sufficient test of equivalency is the capability of 

 mutual inference. Thus from 



All electrics = all non-conductors, 

 I can infer 



All non-electrics = all conductors, 



and vice versa from the latter I can pass back to the 

 former. In short A = B is equivalent to a = b. Again, 

 from the union of the two propositions, A AB and 

 B AB, I get A = B, and from this I might as easily 

 deduce the two with which I started. In this case one 

 proposition is equivalent to two other propositions. There 

 are indeed no less than four modes in which we may 

 express the identity of two classes A and B, namely, 



FIRST MODE. SECOND MODE. THIRD MODE. FOURTH MODE. 



7 A = AB1 



a = T> A-n f 



B AB J 



The Indirect Method of Inference furnishes an universal 

 and clear criterion as to the relationship of propositions. 

 The import of a statement is always to be measured by 

 the combinations of terms which it destroys. Hence two 

 propositions are exactly equivalent when they remove 

 exactly the same combinations from the Abecedarium, 

 and neither more nor less. A proposition is inferrible 

 but not equivalent to another when it removes some but 

 not all the combinations which the other removes. Again, 

 propositions are consistent provided that they leave some 

 one combination containing each term, and the negative 

 of each term. If after all the combinations inconsistent 

 with two propositions are struck out, there still appears 

 in the Abecedarium each of the letters A, a, B, I, C, c, D, d, 

 which were there before, then no inconsistency between 

 the propositions exists, although they may not be equiva- 

 lent or even inferrible. Finally, contradictory propositions 



A -p 7 = a = ab\ 



A = D a = A- -L ~L\ 



b = ab J 



