

THE LOGIC OF RELATIVES. 335 



(24.) x-ky = x + y—z$. - 

 The principle of contradiction is 

 (25.) z,n x = , 



where n stands for "not." The principle of excluded middle is 



(26.) x -t a? = 1 . 



It is an identical proposition, that, if cp be determinative, we have 

 (27.) lix—y (p X = <py. 

 The six following are derivable from the formulae already given : 



(28.) (» -t y),(x 4«) = f + w. 



(29.) {x—y)J !T (* — «)=(x + *)-(# 4 ; w)+ ! ,,z,(l-») + x,(\-yu*. 



In the following, <p is a function involving only the commutative operations and 

 the operations inverse to them* 



(30.) <px = (cpl),x + (<pO),(l-x). 



(31.) 9* = ( 9 1 + (1 -*)),(*<) + *) 



(32.) Ify*=0 (yl),(?0) = 0. 



(33.) l{cpx=l yl-ty0 = l. 



(Boole.) 



(Boole.) 



The reader may wish information concerning the proofs of formulas (30) to (33). When involution i* not in- 

 volved in a function nor any multiplication except that for which x,x = x, it is plain that <pz is of the first degree, and 

 therefore, since all the rules of ordinary algebra hold, we have as in that 



cpx = ijpO -j- (<jpl — <p0),x . 

 We shall find, hereafter, that when <p has a still more general character, we have, 



9 ,a: = qp0-f- (qp/ — <jp0)x. 



The former of these equations by a simple transformation gives (30). 

 If we regard (<pl), (g>0) as a function ofx and develop it by (30), we have 



( 9 1),( 9 0) = *,(*1),(90) + (»1)»(»0),(1 - *) ■ 

 Comparing these terms separately with the terms of the second member of (30), we see that 



(fl) v (*0)-<f*. 



This gives at once (32), and it gives (31) after performing the multiplication indicated in the .econd m«*»tf that 



equation and equating cpx to its value as 



Hr 



the factors of the second member are compared with those of the second member of (31), we get 



from which (33) follows immediately. 



<P 



x-< <jpl -jr gpO , 



