338 MEMOIRS OF THE AMERICAN ACADEMY 



We may divide all relatives into limited and unlimited. Limited relatives express 

 such relations as nothing has to everything. For example, nothing is knower of 

 everything- Unlimited relatives express relations such as something has to every- 

 thing. For example, something is as good as anything. For limited relatives, then, 

 we may write 



pi = • 



The* converse of an unlimited relative expresses a relation which everything has to 

 something. Thus, everything is as bad as something. Denoting such a relative by q , 



yl = I. 



These formulae remind one a little of the logical algebra of Boole; because one of 

 them holds good in arithmetic only for zero, and the other only for unity. 



We have by (10) l x = (q ^ == q(°*) = f = 1 , 



or 1* = 1 . 



We have by (4) lx = {a -\j- l)x = ax -fr lx , 



or by (21) ax —^ lx . 



But everything is somehow related to x unless x is ; hence, unless x is 0, 



Lr=I. 



If a denotes "what possesses," and y u character of what is denoted by x" 



x 



or 



av =*= aiifO ±s (avy = x* , 



• x? = x . 



I 



Since / means "identical with," l,Yw denotes whatever is both a lover of and iden- 

 tical with a woman, or a woman who is a lover of herself. And thus, in general, 



x,Y = xo, . 



Nothing is identical with every one of a class ; and therefore Y x is zero, unless x 

 denotes only an individual when /* becomes equal to x. But equations founded on 



interpretation may not hold in cases in which the symbols have no rational inter- 

 pretation. 



Collecting together all the formulae relating to zero and unity, we have 



(34.) x-^0 = x 



(35.) *-fcl = l 



(36.) x0 = 0. 



(37.) 0z=0. 



(Jevons.) 

 (Jevons.) 



