vni.] PRINCIPLES OF NUMBER. 169 



further data are required to determine them. As an 

 example of the kind of questions treated in numerical 

 logic, and the mode of treatment, I give the following 

 problem suggested by De Morgan, with my mode of 

 representing its solution. 



" 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." 1 

 Now let A = person in house, 



B = male, 

 C = aged. 



By enclosing a logical symbol 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 

 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) = w', (2) 



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

 who are not aged is some unknown positive quantity. 



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

 (pp. 74, 8 1, 89), 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) = w + w', 

 and adding (Aftc) to each side, we have 



(Aft) = (A6c) + w + w'. 



The meaning of this result is that the number of persons 

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

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

 are neither men nor aged (Aftc). 



1 Syllabus of a Proposed System of Logic, p. 29. 



