252 PROCEEDINGS OF THE AMERICAN ACADEMY 



Let -r be the sign of logical subtraction ; so defined that 



(10.) If b -jr- x == a x =p a -r b. 



Here it will be observed that x is not completely determinate. It 

 may vary from a to a with b taken away. This minimum may be 

 • denoted by a — b. It is also to be observed that if the sphere of b 

 reaches at all beyond a, the expression a — b is uninterpretable. If 

 then we denote the contradictory negative of a class by the letter 

 which denotes the class itself, with a line above it,* if we denote by v 

 a wholly indeterminate class, and if we allow [0 -r 1] to be a wholly 

 uninterpretable symbol, we have 



(11.) a-rb=pv,a,b-\-a,b-\-[0 — l~],a,b 



which is uninterpretable unless 



a , b == 0. 



If we define zero by the following identities, in which x may be any 

 class whatever, 



(12.) == x — x == x — x 



then, zero denotes the class which does not go beyond any class, that 

 is nothing! or nonentity. 



Let a ; b be read a logically divided by b, and be defined by the 

 condition that 

 (13.) I(b,x==a x==a;b 



x is not fully determined by this condition. It will vary from a 

 to a -J- b and will be uninterpretable if a is not wholly contained 

 under b. Hence, allowing [1 ; 0] to be some uninterpretable symbol, 



(14.) a;b = a,b-\-v ,a,b -\-[l;0~\a,b 



which is uninterpretable unless 



a , b == 0. 



Unity may be defined by the following identities in which x may be 

 any class whatever. 

 (15.) 1 == x ; x == x : x. 



Then unity denotes the class of which any class is a part ; that is, 

 what is or ens. 



* So that, for example, a. denotes not-a. 



