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 q ; 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 q ; 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 x 

 and y represent the testimonies of A and B, w and z the facts to which these 

 testimonies respectively relate. Observe that no hypothesis is here made as 

 to the connection, by sameness or difference, of iv 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 z are identical, we must write 



as a connecting logical equation, 



w=z 



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. x=c, Prob. xw = cp, ) 

 Given n u , D _ , ... (13) 



Prob. y = c, Prob. yz = c'q, ) 



Prob. ocywz n .^ 



" Prob. xy wz + Vvoh. xyw z 



First, we will seek the value of Prob. xywz. 



Let xw = s, yz=t, xywz — v 



From these logical equations we must now determine v as a developed logical 

 function of as, y, s, and t. The result is 



v = xyst + 0(xyst +xyts+xyst + xsyt +x y s t 



+ ytxs+yxst+xyst) 



+ terms whose coefficients are ^. 



Let u be the value of Prob. v. Then, by the simplification of Art. 21, we 

 have 



