NUMERICALLY DEFINITE REASONING. 345 



The condition of the problem may be expressed in the 

 inequality 



(AB) (A6) 

 (B) > (b) ' 



or reciprocally in the inequality 



(B) ^ (b) 



(AB) {Ab) 



Subtracting unity from each side, and simplifying, we have 



(«B) ^ (ab) 



(AB) - (Ab) 

 Multiplying each side of this inequality by -— — we obtain 



(A^)^ (ab) 



(AB) ^ (aB) 

 Restoring unity to each side, and simplifying 



or reciprocally 



(A) , (a) 

 (AB) (aB)' 



(AB) (aB) 

 (A) ^ (a) ' 



which expresses the result to be proved, namely, that B 

 occurs in a larger proportion of the cases where A is than 

 of the cases where A is not. 



20. The examples hitherto considered have been mostly 

 free from logical conditions ; that is to say, the classes of 

 objects have been supposed capable of combination or co- 

 incidence in all conceivable ways. We will briefly examine 

 the effects of certain simple logical conditions. 



If there be two terms A and B, and one condition, all 

 A's are B's, symbolically expressed in the equation 



A=AB, 



