OF ARTS AND SCIENCES : SEPTEMBER 10, 1867. 403 



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

 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^, A^, A^, &c., and 

 ^2> -^2 J -^3 5 <^c., such that the following conditions will be satisfied. 

 (Here x denotes any individual or class, not nothing; ^,„, A^, 

 B„^, 5„, any members of the two series of terms, and 2!! A, 2^B, 

 2^ {A, B) logical sums of some of the A„'s, the B^'s, and the 

 (^„, i?„)'s respectively). 



No ^ni is A,^. 

 No B^^ is .Z?„. 

 x = 2:{A,B) 



a==2^A. 



b=2:B. 



Some A^ is B^^. 



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. 

 li a =.b, then b =p a. 



II. 

 If a == 5, and b == c, then a =^ c. 



III. 



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



IV. 



If a -|r S =F ^ ^i^d b -\r c=p n and a -\r n z=:z x, then m -fr c = x. 

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

 cal addition. 



V. 



If a -f- i = c and a' -\- 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 05 is 6 (supposing there is any a) 



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



