NUMERICALLY DEFINITE REASONING. 339 



the few quantitative expressions used in ordinary language. 

 We can easily represent the two premises in the form 



(AB)= ® +«> (I) 



(BC)= (|)+,«' (2) 



To deduce the conclusion, we must add these equations to- 

 gether, thus, 



{AB)-\- {BC) = {B) +W + W. 



Developing the logical terms on each side, we have 



(ABC) 4- (ABc) + (ABC) + (aBC)= (ABC) -f (ABc) + (aBC) 

 H- {aBc) -\-w-j- w', 



Subtracting the common terms, there remains 



{A'BC)=w + w'+{aBc). 



The meaning of this conclusion is, that there must be 

 some C^s which are A^s, amounting to at least the sum of 

 the quantities w and w', the unknown excesses beyond half 

 the B's which are A^s and C's. The number {oBc) is wholly 

 undetermined by the premises, but it cannot be negative ; 

 in proportion as its amount is greater, so is the number 

 of the ABC^s. The conclusion, in short, is that w + io' 

 is the lower limit of (ABC). 



14. The above problem is only one case of a more general 

 problem, which may be stated as follows: — Given the 

 numbers of three classes of objects. A, B, and C, to deter- 

 mine what circumstances or conditions will necessitate 

 the existence of a class ABC. 



This may be solved very simply 



(B) + (C) - (A) = (ABC) + (ABc) + (ABC) + (A^'C) - (A) 

 = (ABC)-(A^c), 

 (ABC) = (B) + (C)-(A) + (A^c). 



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