162 PROFESSOR BOOLK S MATHEMATICAL THEOHY 



tlie sign of identity, =. The sign + is used to express the mental 

 operation by which parts (of extensive quantity) are collected into a 

 whole. For instance, if a? represent animals, and y vegetables, oc -\- y 

 will represent the class made up of animals and vegetables together. 

 On the other hand, the sign — is used to express the mental operation 

 of separating a whole (of extensive quantity) into its parts. Thus, 

 X representing human beings, and y representing negroes, x — y will 

 represent all human beings except negroes. With regard to the sign 

 X , a? X y OY X y (as it may be written) is used to denote those ob- 

 jects which belong at once to the class x and to the class y ; just as, 

 in common language, the expression dark waters denotes those objects 

 which are at once dark and waters. Hence we obtain a method of 

 representing a concept taken particularly. For, if x denote men, 

 then, since some men may be viewed as those who besides belonging 

 to the class 'x belong also to some other class v, some men will be 

 denoted by v x. In general, 



V X =: some x (1) 



It can easily be shown, that, as in Algebra, so in the logical sys- 

 tem which we are describing, the literal symbols, oc, y, &c., are com- 

 mutative ; that is, 



^y = y»; (2) 



and that they are also distributive ; that is, 



z {x +y) = z X + z y. (3) 



Another relation between Algebra and the Logical System under 

 consideration is, that, in the latter as well as in the former, a literal 

 symbol may be transposed from one side of an equation to the other 

 by changing the sign of operation, + or — ■. But there is an im- 

 portant relation which subsists in the science of Thought, and not 

 generally in Algebra, namely, 



x^ =^ X (4) 



That this is true in the Logical system, is plain ; for x^, which is 

 another form of x x, denotes (by definition) those things which belong 

 at once to the class x and to the class x ; that is, it denotes simply 

 those things which belong to the class x ; and it is therefore identi- 

 cal with X. But though the equation (4) does not generally subsist 

 in Algebra, it subsists when x is unity or zero. If, therefore, we 

 take the science of Algebra with the limitation that its unknown 



