212 MS. JOSEPH JOHN MURPHY ON ADDITION 



When we interpret L as inclusion, there is a double re- 

 lation between L and L~^ : they are not only inverse or 

 reciprocal to each other, but also contrapositive. The 

 contrapositive of the relation between any two terms is 

 defined as the relation between the negatives of those terms. 

 Thus, writing a and b for the logical negatives of A and B 

 (that is to say, whatever is not A, and whatever is not B), 

 if any one of the following four propositions is true, the 

 rest are true : — 



A=2:B; B = i:-'A; 



a = L~'b; b = La. 



We now go on to that part of the old logic which deals 

 with the relation of exclusion. 



When A is not B, I propose to say that A and B are 

 excludents of each other ; and to use N as the symbol of 

 exclusion. 



If A=iVB, then B = ArA; 



that is to say, /V is an invertible relative, or 



But it is intransitive, or not equal to its own second power 

 — excludent of excludent is not excludent. 



Let us use M as the symbol of the contrapositive re- 

 lation to Ny so that if either of the following two propo- 

 sitions is true the other is true: — 



A = iVB; a = m. 



That is to say, if nothing is both A and B, then every thing 

 is either not-A or not-B. In other words, if A and B are 



