PRINCIPLES OF NUMBER. 193 



Every logical proposition or equation now gives rise 

 to a corresponding numerical equation. Sameness of 

 qualities occasions sameness of numbers. Hence if 



A = B 



denotes the identity of the qualities of A and B, we may 

 conclude that 



(A) = (B). 



It is evident that exactly those objects, and those objects 

 only, which are comprehended under A must be compre 

 hended under B. It follows that wherever we can draw 

 an equation of qualities, we can draw a similar equation of 

 numbers. Thus, from 



A = B = C 

 we infer 



A = C; 

 and similarly from 



(A) = (B) = (C), 



meaning the numbers of A s and C s are equal to the 

 number of B s, we can infer 



(A) = (C). 



But, curiously enough, this does not apply to negative 

 propositions and inequalities. For if 



A = B - D 



means that A is identical with B, which differs from D, it 

 does not follow that 



(A) = (B) , (D). 



Two classes of objects may differ in qualities, and yet they 

 may agree in number. This is a point which strongly 

 confirms me in the opinion I have already expressed, 

 that all inference really depends upon equations, not 

 differences (p. 186). 



The Logical Abecedarium thus enables us to make a 

 complete analysis of any numerical problem, and though 

 the symbolical statement may sometimes seem prolix, I 

 conceive that it really represents the course which the 



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