206 MR. JOSEPH JOHN MURPHY ON ADDITION 



with the interpretation 



X is south of A, 

 Y is south of A, 

 X is fellow-southern of Y. 



And^ generally, if any two terms stand in any relation L to 

 a third term, they stand in the relation L° to each other. 



The canonical equations of the two foregoing syllogisms 

 are the following : — 



LxL-'=L°; 



and „ 



We here see that logical multiplication is not neces- 

 sarily commutative ; that is to say, the product may, as 

 in this case, be changed by changing the order of the 

 factors. 



In logic, as in common algebra, every term with zero 

 index is transitive, 



and invertible, 



properties which are combined in no other terms. 



Every relative term with zero index signifies identity or 

 exact similarity in some respect. Equality means exact 

 similarity of magnitude ; and the axiom that " equals of 

 equals are equal " is a particular case of the more general 

 truth expressed by 



where L means any relation whatever. 



