48 DERIVATION OF THE LAWS. [CHAP. III. 



which 1 and y represent classes, it appears that the symbol 1 

 must represent such a class that all the individuals which are 

 found in any proposed class y are also all the individuals \y that 

 are common to that class y and the class represented by 1. A 

 little consideration will here show that the class represented by 1 

 must be " the Universe," since this is the only class in which 

 are found all the individuals that exist in any class. Hence the 

 respective interpretations of the symbols and 1 in the system 

 of Logic are Nothing and Universe. 



14. As with the idea of any class of objects as "men," there 

 is suggested to the mind the idea of the contrary class of beings 

 which are not men ; and as the whole Universe is made up of 

 these two classes together, since of every individual which it 

 comprehends we may affirm either that it is a man, or that it is 

 not a man, it becomes important to inquire how such contrary 

 names are to be expressed. Such is the object of the following 

 Proposition. 



PROPOSITION III. 



If x represent any class of objects, then will 1 - x represent the 

 contrary or supplementary class of objects, i. e. the class including 

 all objects which are not comprehended in the class x. 



For greater distinctness of conception let x represent the class 

 men, and let us express, according to the last Proposition, the 

 Universe by 1 ; now if from the conception of the Universe, as 

 consisting of " men" and " not-men," we exclude the conception 

 of " men," the resulting conception is that of the contrary class, 

 " not-men." Hence the class " not-men" will be represented by 

 1 - x. And, in general, whatever class of objects is represented 

 by the symbol x, the contrary class will be expressed by 1 - x. 



15. Although the following Proposition belongs in strictness 

 to a future chapter of this work, devoted to the subject of 

 maxims or necessary truths, yet, on account of the great impor- 

 tance of that law of thought to which it relates, it has been 

 thought proper to introduce it here. 



