PROPOSITIONS. 45 



It is plain that any positive term and its corresponding 

 negative divide between them the whole universe of 

 thought : whatever does not fall into one must fall into the 

 other, by the third fundamental Law of Thought, the Law 

 of Duality. It follows at once that there are two modes 

 of representing a difference. Supposing that the things 

 represented by A and B are found to differ, we may indicate 

 (see p. 17) the result of the judgment by the notation 



A~B. 



We may now represent the same judgment by the assertion 

 that A agrees with those things which differ from B, or 

 that A agrees with the not-B's. Using our notation for 

 negative terms (see p. 14), we obtain 



A = AZ> 



as the expression of the ordinary negative proposition. 

 Thus if we take A to mean quicksilver, and B solid, then 

 we have the following proposition : 



Quicksilver = Quicksilver not-solid. 



There may also be several other classes of negative pro- 

 positions, of which no notice was taken in the old logic. 

 We may have cases where all A's are not-B's, and at the 

 same time all not-B's are A's ; there may, in short, be 

 a simple identity between A and not-B, which may be 

 expressed in the form 



A = 1. 

 An example of this form would be 



Conductors of electricity = non-electrics. 



We shall also frequently have to deal as results of de- 

 duction, with simple, partial, or limited identities between 

 negative terms, as in the forms 



a = &, a = ab, aC = &C, etc. 



It would be possible to represent affirmative propositions 

 in the negative form. Thus " Iron is solid," might be ex- 

 pressed as " Iron is not not-solid," or " Iron is not fluid ; " 

 or, taking A and b for the terms " iron," and " not-solid," 

 the form would be A ~ I. 



But there are very strong reasons why we should employ 

 all propositions in their affirmative form. All inference 

 proceeds by the substitution of equivalents, and a proposi- 

 tion expressed in the form of an identity is ready to yield 

 all its consequences in the most direct manner. As will be 

 more fully shown, we can infer in a negative proposition, 



