DESCRIPTION OF A NOTATION FOR THE LOGIC OF RELATIVES. 3H 



General Method of Working with this Notation. 



Boole's logical algebra contains no operations except onr invertible addition and 

 commutative multiplication, together with the corresponding subtraction and divisic n 

 He has, therefore, only to expand expressions involving division, by moans of (30), so 



free himself from 



all non-determinative operations, in order to be aide to uac 

 ordinary methods of algebra, which are, moreover, greatly simplified by the fact 



that 



x,x = X . 



Mr. Jevons's modification of Boole's algebra involves only 



and commutative multiplication, without the corresponding inverse operations. Ib- 

 is enabled to replace subtraction by multiplication, owing to the principle of contra- 

 diction, and to replace division by addition, owing to the principle of excluded middle. 

 For example, if x be unknown, and we have 



x -fc m as a , 



or what is denoted by x together with men make up animals, we can only conclude, 

 with reference to x, that it denotes (among other things, perhaps ) all animate not 

 men ; that is, that the x's not men are the same as the animals not men. Let m de 

 note non-men : then by multiplication we have 



x,m -fc- m,m = x,m = a,m , 



because, by the principle of contradiction, 



m,m = . 



Or, suppose, x being again unknown, we have given 



a,x = m . 



Then all that W e can conclude is that the * consist of all the m'. and pert-* 

 or all of the non-a's, or that the rt and non-a's together make up the m . and 



together. If, then, 5 denote non-a, add 5 to both sides and we have 



O 



Then by (28) 



But by the principle of excluded middle 



a,x-t a = m-ka. 

 (a-fe-a),(x-) r a) = m-i 7 a. 



} 



and therefore 



VOL. IX. 



a _)_a = 1 



x _|_a = m-^a 



47 



