35 



it, "attribute a; is a mark of the attribute y," we do not 

 quantify the predicate, and the appropriate notation is 

 (retaining the form of an equation) 



X = xy. 

 In the former of these two contrasted notations the 

 relation between the whole and the part is symbolized by 

 the addition and subtraction of terms ; in the second, the 

 co-existence of attributes is symbolized by the combination 

 of terms. They consequently show to the eye how the 

 attributes become more numerous as the class becomes 

 smaller. For instance, the following syllogism, " Man is an 

 animal; an animal is an organism; therefore man is an 

 organism " will be thus expressed in the two notations : — 

 x = y-p x = xy 



y = z-y y = yx 



x = z-q-p x — xyz = xz 



These relations may be further illustrated by considering 

 the interpretation of Boole's operation of abstraction, or 

 logical division. Abstraction is the inverse of combination, 

 or logical multiplication, and it consists in removing part of 

 a definition. 



Let us express the proposition "Man is the rational 



animal " by 



m = ra 



To every multiplication correspond two divisions, which 



here give 



m , m 



— = r and — = a 

 a r 



whereof the meaning is that " Man without the animal 



attributes is a being of pure reason," and " Man without 



reason is a mere animal." These abstractions relate only 



