OF TESTIMONIES OR JUDGMENTS. 643 



In the second place, it may be shown, that r approaches more nearly to that 

 one of the two values p and q, which most nearly approaches either of the limits 

 and 1. To show this, let us suppose q greater than p, and let us first inquire, 

 under what circumstances r approaches more nearly to q than to p. 



We must then assume 



q — r<r —p 



Substituting the values of these members from (9) and (10) we have 



V q(l-p)-Vp(l-q) / K/ q (l- p )-^ p (l- q ) 



^w ( i-p)(i-< z) gU q)< v P 7wWpTW^) Wp( - 1 ~ p) 



Now, q being by hypothesis greater thanj?, it is evident that Vq (1 — p) — \/p (1 — q) 

 will be positive. Rejecting, then, the common positive factor on both sides of the 

 inequation we have 



Vq (l—q) < *Jp (1— p) 



q — q 2 < p—p 2 

 q-p <q 2 -p 2 



and dividing both sides by the positive factor q-p 



i < p + q 



.: 1-q <p 



a condition which shows that q must be nearer to 1 than p is to 0. 



On the other hand, as would appear from the very same analysis, changing 

 only the signs < into >, the condition that r may approach more nearly to p than 

 to q, is that p may be nearer to than q is to 1. 



Now, 1 and 0, as limiting the measures of probability of the event z, indicate, 

 the one that it certainly will, the other that it certainly will not occur. And the 

 approach of any measure of probability to these limits indicates the approach of 

 the probability to certainty. We see, then, that when p and q are measures of 

 the probability of an event founded on different judgments, the mean between 

 these measures, as determined by (8), will not be the usual arithmetical mean, but 

 will always fall nearer to that one of the two values p and q which expresses a 

 probability the most nearly approaching to certainty. 



Now, this seems to be in accordance with reason. Evidence of any kind 

 which enables us to pronounce a judgment with certainty, entirely preponderates 

 over that which only enables us to affirm a probable judgment, Art. 35. And 

 the more of the character of certainty that is possessed, the greater is the weight 

 which is due to the evidence to which it belongs. 



38. By an analysis similar to that which is applied in the previous sections, 1 

 have determined the general value of r, when the number of judgments is n, and 



