OF ARTS AND SCIENCES : SEPTEMBER 10, 1867. 403 



7. Arithmetical Multiplication, a b represents an event when a and b 

 are events only if these events are independent of each other, in 

 which case a b===a ,b. By the events being independent is meant 

 that it is possible to take two series of terms, A x , A 2 , A 3 , &c, and 

 B. 2 , B 2 , B 3 , &c, such that the following conditions will be satisfied. 

 (Here x denotes any individual or class, not nothing; A m , A n , 

 B m , B n , any members of the two series of terms, and 21 A, J?B, 

 2]{A,B) logical sums of some of the A n 's, the B n 's, and the 

 (A n , B a )'s respectively). 



From these definitions a series of theorems follow syllogistically, the 

 proofs of most of which are omitted on account of their ease and want 

 of interest. 



Theorems. 



i. 

 If a == b, then b = a. 



ii. 

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



in. 



If a -\r b = c, then b -fr a == c. 



IV. 



If a -fr- b = m and b -|r c == n and a -jr n = x, then m -fp c = x. 



Corollary. — These last two theorems hold good also for arithmeti- 

 cal addition. 



v. 



If a -\- b == c and a' -f- b = c, then a ==. a', or else there is nothing 

 not b. 



This theorem does not hold with logical addition. But from defini- 

 tion 6 it follows that 



No a is b (supposing there is any a) 



No a' is b (supposing there is any a') 



