22 Mr. T. K. Abbott on the Probability of 



It is important, however, to consider that testimony, true or 

 false, is not given without motives ; and therefore it may seem 

 more proper that the formula should represent the operation of 

 these. Let the probability of the event be p\ the probability 

 that its occurrence would move the witness to announce it truly 

 = /. The probability of the event not happening is 1— p; let 

 the chances that its non-occurrence would move the witness to 

 deny it truly = d. This may not be the same as t. 



Now let the chances that a motive to lie exists be u, and the 

 chances of A's yielding to it be </>, i. e. the chances of his lying 

 = ct(j) = v. Then, supposing that A announces the event p, which 

 is one of the possibilities : 



Either the event occurs and moves him to speak the truth, 

 the motives to lying or to mistake not operating =pt(l— v). 



Or the event does not occur, which non-occurrence has no effect 

 on the witness, the motives to lying operating = (1 — p) (1—d) v ; 

 but this must be multiplied by the chances of the hypothesis, 

 i. e. of A fixing on this particular event, on this supposition, 



namely — . This gives 



m 



\ -r/v / m 



Resulting probability that the event announced has occurred 

 _ pt(l-v) 



^(l-iO + a-iOa-rf) 



If there are r witnesses, and the quantities t, d, v the same for 

 all, we have 



pt r {l-v) r 



pr(i-vy+(i- P )(i-dy^ r 



In this formula t may represent, for example, the interest 

 which the witness would have in truly reporting the event if it 

 happened, and 6? his interest in truly reporting its non-occurrence. 



The case of arguments conspiring or opposed is somewhat 

 different from that of testimony; since an argument may be 

 fallacious, and yet the conclusion true. In the case of argu- 

 ments establishing the same conclusion with the probability a, b, 

 &c. respectively, the resulting probability is clearly 



1-(1-«)(1-S)... 



If two arguments are opposed, we have these cases : 



