24 Mr. Murphy on a 



will mean that A is a teacher of B ; or let B mean boys, 

 then its meaning will be that A is a teacher of a boy or 

 boys. According to Boole's plan of indicating the co- 

 existence of attributes by the juxtaposition of their literal 

 symbols, 7?B means a teacher who is a boy. 



The conversion of such a proposition as the above is 

 effected by transposing the relative term with change of 

 index, when, if both A and B are the names of individuals, 



the transposed form 



B = i?-ix A 



means that B is a pupil of A. Let A : B mean the relation 

 of A to B, then the following four propositions are mutually 

 equivalent ; 



A:B = 7? B:A = i2-^ 



A = 7?xB B = i?-A. 



The same is true if i? be a numerical ratio, and A:B means 

 the ratio of A to B. 



In converting a compound relative, the order of the 

 terms is reversed, thus 



{R X S)-^ - S-^ X R~^. 

 For instance : if R means husband and 5 daughter, 



will be the symbol for son-in-law, and its converse 



S-^ X i?-i 

 for father or mother-in-law. This rule for conversion is 

 well known, but we have to show that it is true of our 

 coefficients of quantity as well as of symbols of relation. If 

 A and B are individuals as before, and R means teacher, 

 then " A is the only teacher of B " (or, as it might be 

 expressed, logically though not quite grammatically, " A is 

 all the teacher of B ") will be written in symbols 



A=ii?xB 

 and the converse of this is 



B = A'-ix i-iA 



