404 PROCEEDINGS OF THE AMERICAN ACADEMY 



neither of which propositions would be implied in the corresponding 

 formulae of logical addition. Now from definitions 2 and 6, 



Any a is c 



.'. Any a is c not b 



But again from definitions 2 and 6 we have 



Any c not b is a 1 (if there is any not b) 



.'. Any a is a' (if there is any not b) 



And in a similar way it could be shown that any a' is a (under the 

 same supposition). Hence by definition 1, 



a == a 1 if there is anything not b. 



Scholium. — In arithmetic this proposition is limited by the suppo- 

 sition that b is finite. The supposition here though similar to that is 

 not quite the same. 



VI. 



If a , b == c, then b ,a = c. 



VII. 



If a , b == m and b,c ==n and a ,n = x, then m, c == x. 



VIII. 



If m , n == b and a -fr m = u and a -fr n == v and a -fr b == x, 

 then u, v = x. 



IX. 



If m -\r n == b and a,m — u and a,n — v and a , b = x, then 

 u -\r v = x. 



The proof of this theorem may be given as an example of the 

 proofs of the rest. 



It is required then (by definition 3) to prove three propositions, 

 viz. 



