THE THEORY OF SYLLOGISM, ETC. 121 



assertion. The strangeness of the assertion is, as it were, balanced by the strangeness of 

 his making it. In the formula v k = ~2v s k s , if v k be very small, and Ar 15 k 2 ...k n also very 

 small, these circumstances may produce the balance mentioned, so that P k = k k . 



If l, = 2 2 &c. = fx, the particular' credibility is 



v k ix + Zv s k s 



If there be no particular bias towards inaccuracy, then, 2's m being 1 - m m , s m is 



(1 - m m ) -T- (n - 1), or (1 - ju) ^ (n - 1), 



except only when s = m. Hence, when the chance of accuracy is the same for every event, 

 and no bias whatever towards one inaccuracy rather than another, the particular credibility, 

 after the assertion of A k , is (2'^ being 1 — v k ) 



p _ »*M 



"AM + (» - 1)"' (1 - v h ) (1 - M) 



Here P k = n when v k = 1 -5- n, the mean probability. So that, when we know nothing of any 

 particular bias, the particular credibility exceeds, equals, or falls short of the general credibility, 

 according as the previous probability of the assertion exceeds, equals, or falls short of, the 

 mean probability. 



When n is very great, the preceding is very near to unity, unless v k fi be of the cor- 

 responding order of smallness. For a given value of /x, the supposition of absence of particular 

 bias will make a very bad witness almost an infallible authority. Practically, we allot more 

 than his general credibility to the particular statements of any witness, when we see no reason 

 to suppose a particular bias. But experiments upon extreme cases cannot be made : for in 

 fact, the want of particular bias is almost the sufficient reason for a growth of general habit 

 of accuracy : so that the preceding case is almost always one in which /u. is not small. 



In human affairs it generally happens that a great majority of the cases have probabilities 

 of the same order of magnitude as 1 -~ n ; and these are the ordinary events. Cases of a 

 probability much differing from 1 H- n are comparatively few. Hence, when we disbelieve the 

 dragon flying from the moon, above supposed as an assertion, it is not because the probability 

 is small, for so, generally speaking, is that of an ordinary event. But the probability of the 

 asserted event is small compared with I -~ n; or n v k is small. 



Any one event which we were not expecting, and for reasons, will be such as, a priori, we 

 should call improbable. And in the common run of occurrences, things improbable (but still 

 ordinary) are happening one after another. D'Alembert pronounced the occurrence of 100 suc- 

 cessive tosses of head to be metaphysically possible, and physically impossible. In our day, 

 we should translate his phrases into subjectively possible, and objectively impossible; conceivable, 

 but unattainable. 



In the remarks made upon this assertion, whether with or without reference to D'Alembert, 

 there are several different points to notice; and some matters irrelevant to my main subject 

 must be touched upon, to clear the way for the consideration of the manner in which the 

 question of primary distribution enters. 



Vol. IX. Part I. 16 



