406 PROCEEDINGS OF THE AMERICAN ACADEMY 



Any X is a, 

 whence Any x not u is not u and is a, 



whence Any x not u is not m. 



But since a , 5 = x by definition 3 



Any x\%h, 

 whence Any x not u is 5, 



Any x not u is & not m. 

 But since m -jr ?i = ^ by definition 2 



Any 6 not m is n, 

 whence Any x not u is n, 



and therefore Any x not i« is both a and m. 



But since a , re == v by definition 3 



Whatever is both a and u is y, 

 whence Any x not m is r. 



Corollary 1. — This proposition readily extends itself to arithmetical 

 addition. 



Corollary 2. — The converse propositions produced by transposing 

 the last two identities of Theorems viii. and ix. are also true. 



Corollary 3. — Theorems vi., vii., and ix. hold also with arithmeti- 

 cal multiplication. This is sufiiciently evident in the case of theorem 

 VI., because by definition 7 we have an additional premise, namely, 

 that a and h are independent, and an additional conclusion which is 

 the same as that premise. 



In order to show the extension of the other theorems, I shall begin 

 with the following lemma. If a and b are independent, then corre- 

 sponding to every pair of individuals, one of which is both a and b, there 

 is just one pair of individuals one of which is a and the other b ; and 

 conversely, if the pairs of individuals so correspond, a and b are inde- 

 pendent. For, suppose a and b independent, then, by definition 7, con- 

 dition 3, every class {A„, B„) is an individual. If then A^ denotes any 



