202 MR. JOSEPH JOHN MURPHY ON ADDITION 



A=iB, 



A=-iX, 



B=iC, 



X=iY, 



A=iC; 



A=-iY; 



A=iB, 



A=-iX, 



B=-iX, 



X=-iB, 



A=-iX; 



A=iB. 



These syllogisms may, however, be more compendiously 

 expressed by means of canonical equations, using the 

 relative terms only, thus : — 



I X 1 = 1 ; (— i)x i = — I ^ 



IX(— l) = — Ij (— l)x— 1 = 1. 



These equations are also true in common algebra. Their 

 logical interpretations are : — 



Identical of identical is Negative of identical is 

 identical ; negative ; 



Identical of negative is Negative of negative is 

 negative. identical. 



Thus I is equal to its own second power, indicating that 

 identity is a transitive relation ; — i is not equal to its 

 own second power, indicating that the relation of the logical 

 negative is intransitive. 



In investigating this simplest possible case, we have now 

 considered the formulae of conversion and syllogism, which 

 are generally regarded as coextensive with the whole o£ 

 elementary logic. But there is in the logic of relatives a 

 third operation, which appears to be related to addition as 

 syllogism is to multiplication *. The syllogistic formulae 



* My logical reading has been by no means extensive; and I am quite 

 prepared to find that my ideas have been anticipated ; but, so far as I know, 

 what I have written on the addition of relatives is original. 



