THE THEORY OF SYLLOGISM, ETC. 119 



going to make one : and yet he has ventured an assertion against the coming truth of which it 

 was 51 to 1. But a common person says at once, Why not the seven of spades as well as any 

 other ? against which the student of the theory is tempted perhaps to retort, Yes, but why the 

 seven of spades rather than some one or another out of the 51 others ? 



The observation, Why not the seven of spades as well as another, is a sound one : it 

 reminds us, that in our absence of all knowledge of motive or bias, it is as hard to believe 

 in error having fallen exactly on the seven of spades, as it is to believe in the seven of 

 spades having been actually drawn : if I may speak so chemically, these difficulties combine 

 and neutralize each other, and disengage our original belief in the witness. The reply is 

 fallacious: it rubs out the distinctive marks from the other 51 cards, and writes on each of 

 them ' not the seven of spades ' as its only exponent. 



The difference of these two cases is admirably elucidated by Laplace, in two successive 

 problems (Th. des Prob. 3rd edit. pp. 446 — 451,) but the effect of the contrast is destroyed 

 by a strange remark. First, there are ri counters, each of which is marked with a number ; 

 and a witness of veracity p and judgment r announces that n° i was drawn. The probability 

 that it was so drawn is made to be 



(l-p)(l-r) 



r n-\ 



in which it will be observed that how great soever the number of counters, that is, how improbable 

 soever the event announced, a priori, the probability which the testimony gives cannot be less 

 than pr. In the second problem, n — 1 balls are black and one white, and the same witness 

 announces that the white ball has been drawn. The result is, q representing pr + (l — p)(l — r), 

 that the probability of the event is 



9 + (l - 9) (» - ' 



so that the best witness living might be incredible, if n were great enough. 



On this last problem Laplace remarks as follows ; — ' Ainsi Ton voit comment les faits extra- 

 ordinaires affaiblissent la croyance due aux temoins ; le mensonge ou l'erreur devenant d'autant 

 plus vraisemblable, que le fait atteste est plus extraordinaire en lui-meme.' 



Without denying all the conclusion, we may see that a comparison of these two problems 

 shows it by no means sufficiently arrived at. If the counters were 10 10 in number, all differently 

 marked, the production of, say n° 5000, is just as improbable beforehand, as the production 

 of the white ball when one only is white, and 10 10 - 1 black. And the first case more 

 nearly represents our mode of primary distribution than the second. If an astronomer were to 

 tell us that he had seen in the telescope a dragon fly off the moon, we should certainly never 

 think, pro hac vice, of dividing all possible events into that of a dragon flying from the 

 moon — and others. From among other events, we should select and give prominence to the 

 possibilities of a dream, a defect in the object glass, an atmospheric phenomenon, a fly 

 in the telescope, &c. &c. &c. 



The case in which error of judgment is distinguished from wilful falsehood, need hardly 



