100 ON AN EXTENSION OF THE ORDINARY LOGIC. 



We have now to speak of the relation of exclusion, 

 which is expressed in the common logic by 



A is not B ; 

 whereof the inverse is 



B is not A. 



If we use N as the symbol of exclusion, the above pro- 

 position will be expressed by 



A=iVB, 

 and 



B = iVA. 



This relation is thus seen to be invertible ; or, to express 

 it symbolically, 



an equation which is true of two numbers, namely i and 

 — I , Further, N is not equal to its own second power ; 

 so that it combines these two characters, which are united in 

 negative unity and not in any other number, that it is not 

 equal to its own square, and is equal to its own reciprocal. 

 Consequently the following equations are true in arith- 

 metic as well as in logic, — N meaning in logic excludent, 

 and in arithmetic negative unity; N° meaning in logic 

 co-excludent, and in arithmetic unity, like any other 

 term with zero index : — 



N.N=N°; N.N°=N; 



N°.N=N', N°.N° = N°. 



The following are the logical interpretations : — 



Excludent of excludent Excludent of co-excludent 



is co-excludent. is excludent. 



Co-excludent of excludent Co-excludent of excludent 



is excludent. is co-excludent. 



In this system, when we speak of exclusion and co- 



