AND MULTIPLICATION OF LOGICAL RELATIVES. 203 



given above show how syllogism is analogous to multipli- 

 cation : if the relative terms are numerical coefiBcients, the 

 process is multiplication ; if they are logical relatives, it is 

 syllogism. The problem of syllogism may be thus stated : — 

 Given the relations of two terms to a third, to find the 

 resultant relation of the first two to each other. The 

 problem of what I propose to call the addition of relatives 

 is this : — Given two relations between two terms, to find 

 their resultant. 



In the case before us the solution is as follows : — 



i+(_i)=o. 



This is true both in common algebra and in logic; its logical 

 interpretation is that these two relations cannot coexist; 

 and it is the expression of the law of contradiction within 

 the limits of the present case. 



As a further illustration of the relation between the ad- 

 dition and the multiplication of logical relatives, let L 

 signify the relation of teacher and M that of brother ; 

 then 



means that A is a teacher and a brother of B ; while 



A=(i.xM)B 



means that A is a teacher of a brother of B. 



The order of addition is a matter of indifference, that is 

 to say 



L + M=M+L, 



whatever the meaning assigned to L and M. This, as we 

 shall see, is not generally true of the multiplication of 



