OF TESTIMONIES OR JUDGMENTS. 639 
which is substituted for the true one is the following. The probability that A 
speaks truth is p, that B speaks truth is ¢; what is the probability that, if 
they both make assertions, and these assertions are both true or both false, they 
are both true? Whether A and B make the same assertion or not is assumed 
to be a matter of indifference. But this assumption is, in point of fact, as erro- 
neous as it is unwarranted. The problem which we have solved in the preceding 
sections, interpreted in relation to testimony, is the following. Two witnesses, A 
and B, assert a fact. The probability of that fact, if we only knew of A’s state- 
ment, would be p, if we only knew of B’s, would be 7; what is its probability 
when we know of both? The formal expression of this problem will be seen in 
Art. 34. The most complete formal expression of the problem which has been 
substituted for it, taking into account all its elements, is as follows. Let a 
and y represent the testimonies of A and B, w and < the facts to which these 
testimonies respectively relate. Observe that no hypothesis is here made as 
to the connection, by sameness or difference, of w and z. And the simple ab- 
sence of any such hypothesis is properly signified by expressing the events by 
different symbols, unaccompanied by any logical equation connecting these sym- 
bols. 
If we wish to indicate that the events w and ¢ are identical, we must write 
as a connecting logical equation, 
W=2 
though it must be simpler to express the identity by the employment of a single 
symbol as before. Any other definite relation may be expressed in a similar way. 
The Problem now stands thus :— 
‘ Prob. z=c, Prob. ew=cp, 
Given A } (13) 
Prob. y=c’, Prob. yz=e'q, 
: Prob. xywz 
Required - (14) 
Prob. aywz+ Prob. xyw z 
First, we will seek the value of Prob. vywz. 
Let 2w=s, Ya—t, 2ywz=v 
From these logical equations we must now determine v as a developed logical 
function of x, y, s, and ¢ The result is 
; v=ayst+O(wystt+aytstaysttasyttayst 
+ytastyast+xyst) 
vt. 1 
+ terms whose coefficients are 0 
Let u be the value of Prob. v. Then, by the simplification of Art. 21, we 
have 
