OF TESTIMONIES OR JUDGMENTS. 643 
In the second place, it may be shown, that 7 approaches more nearly to that’ 
one of the two values p and g, which most nearly approaches either of the limits 
0 and 1. To show this, let us suppose g greater than p, and let us first inquire, 
under what circumstances 7 approaches more nearly to g than to p. 
We must then assume 
Gate <ot —p 
Substituting the values of these members from (9) and (10) we have 
vq =p)—VPU=) , yoga VG) —Ve =) 
——#/ x l—q) == Se — = 
Vien) nV ggiV 2) (lag 
Now, qg being by hypothesis greater than p, it is evident that / q (1—p)—Vp (1—4q) 
will be positive. Rejecting, then, the common positive factor on both sides of the 
inequation we have 
V4 (1—q) <S Vp (1—p) 
q-F <pP-p* 
q-P <¢-p’ 
and dividing both sides by the positive factor g—p 
l<ptq 
“l-q<p 
a condition which shows that g 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 7 may approach more nearly to p than 
to g, is that p may be nearer to 0 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 g 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, I 
have determined the general value of 7, when the number of judgments is 7, and 
