90 . OF INTERPRETATION. [CHAP. VI. 



remainder of " not men" is implied by the premiss. It might 

 happen that the remainder included all the beings who are not 

 men, or it might include only some of them, and not others, or it 

 might include none, and any one of these assumptions would be 

 in perfect accordance with our premiss. In other words, whether 

 those beings which are not men are all, or some, or none, of them 

 mortal, the truth of the premiss which virtually asserts that all 

 men are mortal, will be equally unaffected, and therefore the 



expression here indicates that all, some, or none of the class to 



whose expression it is affixed must be taken. 



Although the above determination of the significance of the 



symbol - is founded only upon the examination of a particular 



case, yet the principle involved in the demonstration is general, 

 and there are no circumstances under which the symbol can pre- 

 sent itself to which the same mode of analysis is inapplicable. 



We may properly term - an indefinite class symbol, and may, if 



convenience should require, replace it by an uncompounded sym- 

 bol v, subject to the fundamental law, v (1 - v) = 0. 



4th. It may happen that the coefficient of a constituent in an 

 expansion does not belong to any of the previous cases. To as- 

 certain its true interpretation when this happens, it will be ne- 

 cessary to premise the following theorem : 



11. THEOREM. If a function V, intended to represent any 

 class or collection of objects, w, be expanded, and if the numerical 

 coefficient, a, of any constituent in its development, do not satisfy 

 the law. 



then the constituent in question must be made equal to 0. 



To prove the theorem generally, let us represent the expan- 

 sion given, under the form 



iv - aiti + a z t z + 3^ 3 + &c., (1) 



in which H 2 , t 39 &c. represent the constituents, and a lt 2) 3> &c- 

 the coefficients ; let us also suppose that j and a z do not satisfy 



the law 



il - i)= 0, 2 (1 - 2 ) = 0; 



