OF ARTS AND SCIENCES : SEPTEMBER 10, 1867. 405 



1st. That any u is x. 

 2d. That any v is x. 

 3d. That any x not u is v. 



First Proposition. 

 Since u = a , m by definition 3 



Any u is m, 

 and since m -\r n ■==. b by definition 2 



Any m is 6, 

 whence Any u is 6, 



But since u = a ,m by definition 3 



Any u is a, 

 whence Any a is both a and #, 



But since a , b = x by definition 3 



Whatever is both a and 6 is x 

 whence Any u is x. 



Second Proposition. 

 This is proved like the first. 



Third Proposition. 

 Since a , m = u by definition 3, 



Whatever is both a and »i is w. 



or Whatever is not u is not both a and m. 



or Whatever is not u is either not a or not m. 



or Whatever is not u and is a is not /». 



But since a , b = x by definition 3 



