PRINCIPLES OF NUMBER. 191 



For every man in the house there is a person who is 

 aged ; some of the men are not aged. It follows that 

 some persons in the house are not men?/ 

 Now let A = person in house, 



B = male, 

 C = aged. 



By enclosing logical symbols in brackets, let us denote 

 the number of objects belonging to the class indicated by 

 the symbol. Thus let 



(A) = number of persons in house, 

 (AB) number of male persons in house, 

 (ABC) = number of aged male persons in house, 

 and so on. Now if we use w and w to denote unknown 

 and indefinite numbers, the conditions of the problem may 

 be thus stated according to my interpretation of the 

 words 



(AB) = (AC) -w, (i) 



that is to say, the number of persons in the house who are 

 aged is at least equal to, and may exceed, the number of 

 male persons in the house ; 



(ABc) - u/, (2) 



that is to say, the number of male persons in the house 

 who are not aged is some unknown positive quantity. 



If we develop the terms in (i) by the Law of Duality 

 (pp. 87, 95, 97), we obtain 



(ABC) + (ABc) = (ABC) + (A6C) - w. 

 Subtracting the common term (ABC) from each side and 

 substituting for (ABc) its value as given in (2), we get at 

 once 



(A&C) - 10 + u/, 

 and adding (Abe) to each side, we have 



(Ab) = Abe + 10 4 wf. 



The meaning of this result is that the number of persons 

 in the house who are not men is at least equal to w + iu , 



S Syllabus of a proposed System of Logic, p. 29. 



