THE THEORY OF SYLLOGISM, ETC. 123 



here to stamp the weights : but only to show that the application of the theory explains the 

 apparent inconsistency of the three phaenomena above noted. 



Let us now suppose that the statement of the witness is a denial that A k happened, and 

 nothing more. Then either A k happened and he denies it, of which the previous probability 

 is v k (1 - k k ) : or A r (k not being = 1) happened, and he denies A k , of which the previous 

 probability is v\ (l — &i), &c. So that, from his assertion, 



" k C 1 - K) and 1 ~ "* - s^A 



are the particular probabilities that A k did and did not happen. The first is the probability 

 of his inaccuracy after the statement ; and the previous probability that he shall inaccurately 

 deny Ak is 1 — k k . So that, looking at the previous force of ~Zv s k s when equal to v k , we see 

 that his previous reputation for inaccuracy as to A k , is increased, unaltered, or diminished, 

 according as the probability of the event denied is greater than, equal to, or less than, the 

 probability against the event, the affirmation of which would have left his reputation for 

 accuracy unaltered. 



The particular proposition being considered as more probable than the universal, the 

 denial of the universal A k , which is the affirmation of the contrary particular, generally 

 makes the universal still more improbable. But we can hardly suppose a formal affirmation 

 of the particular, except in opposition to some amount of belief gained for the universal, of a 

 larger amount than its natural probability. About such an hypothesis there is nothing parti- 

 cular to examine. 



The following is the most plausible supposition as to the bias of inaccuracy. The receiver 

 of the testimony supposes that the probabilities of the n events are Xj, X 2 , ... X„ in the mind of 

 the witness ; and, k k being still the previous probability of accuracy when k happens, he 

 divides 1 — k k , the probability of inaccuracy, among the events which may be inaccurately 

 stated, in proportion to the presumed probabilities of these events in the mind of the witness. 

 That is, since 1„, + 2 m + ... + n m m 1, he makes p m = \ p (l - m m )-i-(l - X m ). On these sup- 

 positions, the particular credibility of the assertion A k is 





v k K + X t 2' 



&) 



Here, for a given probability in the receiver's mind, both as to the event and the witness stating 

 it if it happen, the more incredible he is supposed to think it, the more must the receiver be 

 inclined to believe it. 



If X s = s s , for all values of s, that is, if the previous probability of the witness asserting an 

 event when it happens, be exactly the previous probability in his own mind that it shall 

 happen, then P k = v k , and his testimony is nothing either way. According to the point of view 

 from which this result is looked at, it appears exceedingly rational or exceedingly absurd. 

 That the more extraordinary he thinks an event, the less likely is he to state it, even if it 

 happen, would by itself make us trust him : but then, on the other hand, the more likely he 

 thinks an event, the more likely is he to state it when it does not happen ; which by itself 



16—2 



