Tin.] PRINCIPLES OP NUMBER. 171 



the Aristotelian Logic ; (3) by the method of numerical 

 logic just expounded ; and then to decide which method 

 is most satisfactory. 



Numerical meaning of Logical Conditions. 



In many cases classes of objects may exist under spe- 

 cial logical conditions, and we must consider how these 

 conditions can be interpreted numerically. Every logical 

 proposition 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 that 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 point strongly confirms me 

 in the opinion which I have already expressed, that all 

 inference really depends upon equations, not differences. 



The Logical Alphabet thus enables us to make a com- 

 plete analysis of any numerical problem, and though the 

 symbolical statement may sometimes seem prolix, I con- 



