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 , b = x by definition 3 



Any x is b, 

 whence Any x not u is 5, 



Any x not w is b not »«. 

 But since m -\r ?i = bby definition 2 



Any & not m is w, 

 whence Any x not w is n, 



and therefore Any a: not u is both a and m. 



But since a , n ==■ v by definition 3 



Whatever is both a and u is v, 

 whence Any x not ?/ is v. 



Corollary 1. — This proposition readily extends itself to arithmetical 

 addition. 



Corollary 2. — The converse propositions produced by transposing 

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



Corollary 3. — Theorems VI., vn., and ix. hold also with arithmeti- 

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

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

 that a and b 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 jiair 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 m , B n ) is an individual. If then A a denotes any 



