204 MR. JOSEPH JOHN MURPHY ON ADDITION 



relatives. But the most important law of the addition of 

 relatives is that 



L + L^L, 



whatever relative L may be. The same is true of the ad- 

 dition of absolute terms in the logical systems of Jevons, 

 MacCoU, and Pierce. 



So far as I see, we have exhausted the subject of the 

 formal relations between the terms in the simplest possible 

 universe. The possible interpretations, however, are not 

 confined to those given above ; the relative i may mean 

 agreement in possessing or not possessing any character 

 whatever, and — i the corresponding difierence. Let the 

 universe, for instance, be a village of one street ; let A, B, 

 C, &c. be the houses on one side of the street, and X, Y, 

 Z, &c. the houses on the other side ; let i be the relation 

 between any two houses on the same side, and — i the 

 relation between any two houses on opposite sides j then 

 the interpretation of the four syllogistic forms will be : — 



Same side with same side Opposite of same side is 



is same side; opposite; 



Same side with opposite Opposite of opposite is 



is opposite. same side. 



Now let us imitate the order of nature, and evolve the 

 more complex out of the simpler. The first step we have 

 to take in the direction of greater complexity consists in 

 presenting a case exactly similar to the last, except that 

 the sides of the street which constitutes our universe are 

 distinguished as north and south. The relations now 

 arising cannot be expressed by numerical coefficients ; so 

 let us use L (L being the symbol used by De Morgan for 

 relation in general) to indicate the relation of any house 



