84 



When we write an ordinary predication with the quantified 



predicate, we may express it in language by "oj is part of y " 



and in notation by 



x = y-p 



where p is so much of y as is not x. Boole sometimes, and 



Jevons always, express the same predication by 



z = xy. 



But though this is in form an equation, it does not in reality 



quantify the predicate; it is only the translation of the 



ordinary predication 



x<y \ 



into a different and for some purposes preferable notation. 



Sir "William Hamilton showed, though he was not the 

 first who discovered, that the propositions of the ordinary 

 logic admit of a twofold interpretation, in extension and in 

 comprehension. For instance, the proposition, " Man is an 

 animal," if interpreted in extension, will be " The class man 

 18 included in the class animal; " but if interpreted in com- 

 prehension it wiU be, "The attributes of the man include 

 the attributes of the animal." When we interpret in 

 extension, x and y mean things or classes of things, and the 

 copula means identity: — when we interpret in comprehension, 

 X and y mean attributes, and the copula means co-existence. 



The foregoing appears to be self-evident; and it appears 

 to foUow, that when we interpret in extension, and assert 

 that "x is included in y," or, as Sir William Hamilton 

 expresses it, " all x is some y" we quantify the predicate, 

 and the appropriate notation is 



x = y-p. 

 But when we interpret in comprehension, and assert that 

 " the attributes of x include those of y," or, as Mill expresses 



