126 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
slowly with that of s, diminishes with increase of o or p, and varies 
between the limits 0 and 1. Skill in making false predictions is 
indicated by a negative value of cs — op; but the same degree of 
causal relation exists as when equal skill is employed in making 
true predictions ; and a negative value of 7 can never occur. When 
: : 0. : ; 
s = either p or 0,1 = 9? but the apparent indeterminateness van- 
ishes when we consider that 7 is the product of two factors, of which 
one = 0-and the other is indeterminate within limits. And the 
value of 2 is unaltered when predictions of non-occurrences are 
substituted for those of occurrences, and vice versa. In the latter 
case, write s — o for 0,s — p for p, and s—o—p+e fore; and 
the formula reduces to its original form. 
In addition to Mr. Gilbert’s tests, two others may be | 
In the case of predictions all falsely reported, we may write s — p 
for p and o — ¢ for c; and the formula becomes 
4s 4? Lop — ca)? 
~ op (s— 0) (8 —p)’ 
with a proper reversal of signs in the quantity under the exponent 
and no change in the value of 7. 
If occurrences always appear whenever they are not predicted, 
and never appear when they are predicted, we put c= 0 and 
p= s — 0, with the result 
tA; 
or the logical connection is perfect. 
In order that the general formula shall be properly applicable, 
care must be taken that the predictions are fairly homogeneous in 
definiteness of time and space. For illustration: if predictions 
that phenomena will occur in given months are examined indis- 
criminately with those that they will occur on given days, the result 
will be manifestly worthless. 
It has been proposed to extend the problem so as to include three 
or more classes of events of which one must happen and only one 
can happen in any case. It seems clear to me that no single 
numerical expression can be a proper solution of such a problem. 
Suppose the three classes of events, A, B, and C. By the method 
above given A and Not A may be examined; and all instances 
