CHAP. VI.] OF INTERPRETATION. 81 



PROPOSITION I. 



2. The constituents of the expansion of any function of the logi- 

 cal symbols x, y, fyc., are inter pretable, and represent the several 

 exclusive divisions of the universe of discourse, formed by the predica- 

 tion and denial in every possible way of the qualities denoted by the 

 symbols x, y, fyc. 



For greater distinctness of conception, let it be supposed that 

 the function expanded involves two symbols x and y, with re- 

 ference to which the expansion has been effected. We have then 

 the following constituents, viz. : 



xy, x(l-y\ (l-x)y, (l-x)(l-y). 



Of these it is evident, that the first xy represents that class 

 of objects which at the same time possess both the elementary 

 qualities expressed by x and ?/, and that the second x ( 1 - y) re- 

 presents the class possessing the property # , but not the property 

 y. In like manner the third constituent represents the class of 

 objects which possess the property represented by y, but not 

 that represented by x ; and the fourth constituent (1- x) (1 - y), 

 represents that class of objects, the members of which possess nei- 

 ther of the qualities in question. 



Thus the constituents in the case just considered represent 

 all the four classes of objects which can be described by affirma- 

 tion and denial of the properties expressed by x and y. Those 

 classes are distinct from each other. No member of one is a mem- 

 ber of another, for each class possesses some property or quality 

 contrary to a property or quality possessed by any other class. 

 Again, these classes together make up the universe, for there is 

 no object which may not be described by the presence or the 

 absence of a proposed quality, and thus each individual thing in 

 the universe may be referred to some one or other of the four 

 classes made by the possible combination of the two given 

 classes x and y, and their contraries. 



The remarks which have here been made with reference to the 

 constituents of/ (x, y) are perfectly general in character. The 

 constituents of any expansion represent classes those classes 



