120 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



be entered upon, except in a manner subsequently mentioned. Let the general credibility 

 of a witness, with reference to a particular event, be the measure of our previous belief 

 that, should that event happen, he will state it to have happened, if he make any state- 

 ment at all. Should his general credibility be the same with reference to each event which 

 can happen, we have the case to which calculation is usually applied. But it is necessary 

 to recognize the distinctions which exist in fact, between our opinion of a witness as to one 

 event, and as to another. We may have reason to think we know beforehand that, according 

 as A t or A 2 shall happen, the narrator will be a willing or unwilling witness, a sagacious or 

 a foolish expositor. 



Let the particular credibility of the witness be the measure of our belief in his statement, 

 after it has been made, and we know what it is. Let J.,, A 2 , ...A a be the names of the events 

 any one of which may have happened, and one of which, A k , the witness asserts to have hap- 

 pened. Let vi, v 2 , •■■"„ be the probabilities of these several events, before the assertion, in the 

 mind of the receiver of the testimony. Let p q signify, in the same mind, the previous proba- 

 bility that if A q should happen, the witness will state A p . Hence the general credibility of the 

 witness, before any statement, is fi = v,l, •+• i> 2 2 2 +...+v n n n ; after statement of A k , it isk k - But 

 the previous probability of his asserting A k is v l k l + v 2 k 2 +... + v n k n . 



The particular credibility, after assertion of A k , is thus found. Either A k has happened, 

 and he has announced it, of which the previous probability was v k k k ; or A l (if k be not = l) 

 has happened and he announces A k , of which the previous probability was v^ ; or A 2 

 {k not being = 2) has happened, &c. &c. Hence the particular credibility of the witness is 



P , v j& . 



* vik 1 + v 2 k 2 + + i',A 



If there be only two cases, then, if 1, - 2 2 , we have n= 1,; and 1, + Z t = 1, 1 2 + 2 2 = 1, 

 give 1 2 = 2 X = 1 — fi. Hence the particular credibility is, for the statement A„ r t /u divided by 

 vin + (l - v t ) (1 — n), which agrees with the result of Laplace's second problem, in which 

 Vl = l -h n, ,u = q. 



The general credibility as to A k remains unaltered by the statement, if v k = ~S,v s k s . That 

 is to say, the witness is unaltered by the statement, with reference to the event stated, if the 

 asserted event, and the assertion of that event, be a priori equally probable ; if what he says 

 be just as likely as that he should have said it. And according as the first is more or less 

 likely than the second, the witness is raised or lowered by his assertion. 



Again, let Hfv,k, stand for ~2v s k s with v k k k omitted. Then v k = "2>'v s k s -H (l - k k ). When 

 k k = ix, particularly when the immediate source of this equation is 1, = 2 2 = ... =w„, this last 

 result is of a well known and admitted character. For 2'i»,&, -r- (l — fi) then represents the 

 probability that of all erroneous announcements any given one shall be A k : and we thus see that 

 the credit of the witness is unaltered by his statement if the previous probability of the event 

 stated be equal to the previous probability of its erroneous statement; that is, its previous pro- 

 bability in the minds of those who know that some error is coming. 



When a witness asserts an extraordinary thing, his credit may be maintained, in spite 

 of his assertion, by our previous feeling of the great unlikelihood of his making such an 



