AND MULTIPLICATION OF LOGICAL RELATIVES. 205 



on the north side to any house on the south sidcj and L"' 

 for the inverse relation *. 



In the system here expounded the rue for logical con- 

 version is simply to transpose the relative term, changing 

 the sign of its index. This is also valid in common algebra. 

 When a relative is equal to its own reciprocal, it is not 

 necessary to change the sign of the index, though to do so 

 would not be erroneous. 



Two other relations arise out of L and L~\ The fol- 

 lowing syllogistic reasoning is valid in common algebra for 

 all values of L : — 



, A=LX, 



B=LX or X=Z-'B, 

 A=LxL-'B, 



= L°B; 



and it is also valid in logic, with this interpretation : — 



A is north of X ; 

 B is north of X ; 

 A is fellow -northern of B. 



The following is equally valid : — 



X = L-'A, 



y=L-^A or A = Z,y, 

 X=L-'xLy, 



= {L-ry, 



* The use of the negative index to signify inverse relation is, I believe, 

 the only mathematical expression introduced by De Morgan into logic. The 

 use of the zero index was first, so far as I am aware, introduced into logic iu 

 my paper " On an Extension of the ordinary Logic, connecting it with the 

 Logic of Relatives,"' in the 'Memoirs of the Manchester Literary and Pbi- 

 losophical Society," Session 1879-80 {sHpva, pp. UO-101). 



