vi.] THE INDIRECT METHOD OF INFERENCE. 115 



The Equivalence of Propositions 



One great advantage which arises from the study of 

 this Indirect Method of Inference consists in the clear 

 notion which we gain of the Equivalence of Propositions. 

 The older logicians showed how from certain simple 

 premises we might draw an inference, but they failed to 

 point out whether that inference contained the whole, or 

 only a part, of the information embodied in the premises. 

 Any one proposition' or group of propositions may be 

 classed with respect to another proposition or group of 

 propositions, as 



1. Equivalent, 



2. Inferrible, 



3. Consistent, 



4. Contradictory. 



Taking the proposition " All men are mortals " as the 

 original, then " All immortals are not men " is its equiva- 

 lent ; " Some mortals are men " is inferrible, or capable of 

 inference, but is not equivalent ; " All not-men are not 

 mortals " cannot be inferred, but is consistent, that is, 

 may be true at the same time ; " All men are immortals " 

 is of course contradictory. 



One sufficient test of equivalence is capability of mutual 

 inference. Thus from 



All electrics = all non-conductors, 

 I can infer 



All non-electrics = all conductors, 



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

 former. In short, A = B is equivalent to a = 5. 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 in fact 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. 



A T, 7 A = AB ) a = ab\ 



A = B a = b B = ABJ b = ab\ 



The Indirect Method of Inference furnishes a universal 

 and clear criterion as to the relationship of propositions. 

 The import of a statement is always to be measured by 



I 2 



