192 THE PRINCIPLES OF SCIENCE. 



and exceeds it by the number of persons in the house who 

 are neither men nor aged (Abe). 



It should be understood that this solution applies only 

 to the terms of the example quoted above, and not to the 

 general problem for which De Morgan intended it to serve 

 as an illustration. 



As a second instance, let us take the following ques- 

 tion : The whole number of voters in a borough is a ; the 

 number against whom objections have been lodged by 

 liberals is b ; and the number against whom objections 

 have been lodged by conservatives is c; required the 

 number, if any, who have been objected to on both sides. 

 Taking 



A = voter, 



B = objected to by liberals, 

 C = objected to by conservatives, 



then we require the value of (ABC). Now the following 

 equation in identically true 



(ABC) = (AB) + (AC) + (AJbc) - (A). (i) 

 For if we develop all the terms on the second side we 

 obtain 



(ABC) = (ABC) + (ABc) + (ABC) + (A5C) + (Abe) 



- (ABC) - (ABc) - (AfeC) - (Ale) ; 



and striking out the corresponding positive and negative 

 terms, we have only left (ABC) = (ABC). Since then (i) is 

 necessarily true, we have only to insert the known values, 

 and we have 



(ABC) = b + c - a + (Abe). 



Hence the number who have received objections from both 

 sides is equal to the excess, if any, of the whole number 

 of objections over the number of voters together with the 

 numbers of voters who have received no objections (Abe). 



In many cases classes of objects may exist under special 

 logical conditions, and we must consider how these con- 

 ditions must be interpreted numerically. 



