134 MR. JOSEPH JOHN MURPHY ON THE 



which is true in arithmetic, is not generally true here ; but 



it is true when 



R~ l =R. 



If R, for instance, means class-fellow, the truth of one of 

 the two equations 



X=iRY and X=i~ l R- l Y 



will imply the truth of the other. That is to say, if X is 

 the only class-fellow of Y, then X is the class-fellow of Y 

 only, or of none but Y. Of course it is here understood 

 that neither X nor Y belongs to more than one class. 



When the absolute terms X and Y are not the names of 

 individuals but of classes, as is usually the case in the 

 common logic, we shall use as the copula the sign of in- 

 clusion (<) instead of the sign of equality (=). When 

 only one of the two terms X and Y is quantified by the 

 coefficient i or i — r , the proposition is singly total ; when 

 both are so quantified, the proposition is doubly total. 



Thus, 



iX<RY 



means that every X is a teacher of a Y ; and 



iX<RiY 



means that every X is a teacher of every Y. 



The contrary of a singly total proposition is doubly 

 total, and vice versa. Thus the two following are con- 

 traries : — 



iX<RY; iX<riY. 



That is to say, " Every X is teacher of a Y (or Y's) " has 

 for its contrary, " Every X is not-teacher of every Y" or, 

 in commoner language, e< No X is teacher of any Y" 



A singly total proposition of the above form admits of 

 the following four forms, whereof the truth or falsehood 



