404 



PROCEEDINGS OP 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 h is a' (if there is any not h) 



.\ Any a is a' (if there is any not V) 



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

 same supposition). Hence by definition 1, 



a == a' if there is anything not h. 



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. 



If a , 5 =p c, then b ,a = c. 



VI. 



VII. 



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



VIII. 



If m , n == b and a -{r m = u and a -\r n = v and a -\r 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. 



