OF ARTS AND SCIENCES .* SEPTEMBER 10, 1867. 405 



1st. That any u is x. 

 2d. That any v is x. 

 3d. That any' a; not u is v. 



First Proposition. 

 Since m == a , w by definition 3 



Any u is m, 

 and since m -fr ^ = ^ by definition 2 



Any m is b, 

 whence Any m is ^, 



But since u^=z a ,m by definition 3 



Any u is a, 

 whence Any u is both a and h. 



But since a , & = x by definition 3 



Whatevei" is both a and 6 is a; 

 whence Any m is a?. 



Second Proposition. 

 This is proved like the first. 



Third Proposition. 

 Since a ,m =: ic by definition 3, 



Whatever is both a and m is m. 



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 ic and is a is not m. 



But since a ,b == x hj definition 3 



