GROUNDS OF DISBELIEF. 



417 



cause of falsehood or error operating 

 in favour of any of them, which would 

 not operate in the same manner if 

 there were only one. Alter thi8 sup- 

 position, and the whole argument 

 falls to the ground. Let the balls, 

 for instance, be numbered, and let 

 the white ball be No. 79. Considered 

 in respect of their colour, there are 

 but two things which the witness can 

 be interested in asserting, or can have 

 dreanit or hallucinated, or has to 

 choose from if he answers at random, 

 viz. black and white ; but considered 

 in respect of the numbers attached to 

 them, there are a thousand ; and if 

 his interest or error happens to be 

 connected with the numbers, though 

 the only assertion he makes is about 

 the colour, the case becomes precisely 

 assimilated to that of the thousand 

 tickets. Or instead of the balls sup- 

 pose a lottery, with looo tickets and 

 but one prize, and that I hold No. 79, 

 and being interested only in that, ask 

 the witness not what was the number 

 drawn, but whether it was 79 or some 

 other. There are now only two ca.se8, 

 as in Laplace's example ; yet he surely 

 would not say that if the witness an- 

 swered 79, the assertion would be in 

 an enormous proportion less credible 

 than if he made the same answer to 

 the same question asked in the other 

 way. If, for instance, (to put a case 

 supposed by Laplace himself,) he has 

 staked a large sum on one of the 

 chances, and thinks that by announc- 

 ing its occurrence he shall increase 

 his credit ; he is equally likely to 

 have betted on any one of the 999 

 numbers which are attached to black 

 balls, and, so far as the chances of 

 mendacity from this cause are con- 

 cerned, there will be 999 times as 

 many chances of his announcing black 

 falsely as white. 



Or suppose a regiment of lOOO men, 

 999 Englishmen and one Frenchman, 

 and that of these one man has been 

 killed, and it is not known which. I 

 ask the question, and the witness an- 

 swers, the Frenchman. This was not 

 pol^ as improbable d friori, but i» 



in itself as singular a circumstance, 

 as remarkable a coincidence, as tho 

 drawing of the white ball ; yet w» 

 should believe the statement as readily 

 as if the answer had been John 

 Thompson. Because, though the 999 

 Englishmen were all alike in the 

 point in which they diflfeied from the 

 Frenchman, they were not, like the 

 999 black balls, undistinguishable in 

 every other respect ; but being all 

 different, they admitted as many 

 chances of interest or error as if each 

 man had been of a different nation ; 

 and if a lie was told or a mistake 

 made, the misstatement was as likely 

 to fall on any Jones or Thompson of 

 the set as on the Frenchman. 



The example of a coincidence se- 

 lected by D'Alembert, that of sixes 

 thrown on a pair of dice ten times ia 

 succession, belongs to this sort of case* 

 rather than to such as Laplace's. The 

 coincidence is here far more remark- 

 able, because of far rarer occurrence, 

 than the drawing of the white ball. 

 But though the improbability of its 

 really occurring is greater, the supe- 

 rior probability of its being an- 

 nounced falsely cannot be es ablished 

 with the same evidence. The an- 

 nouncement "black" represented 999 

 cases, but the witness may not have 

 known this, and if he did, the 999 

 cases are so exactly alike, that there 

 is really only one set of possible causes 

 of mendacity corresponding to the 

 whole. The announcement, "sixes 

 not drawn ten times," represents, and 

 is known by the witness to represent, 

 a great multitude of contingencies, 

 every one of which being unlike every 

 other, there may be a different and & 

 fresh set of causes of mendacity cor- 

 responding to each. 



It appears to me, therefore, that 

 Laplace's doctrine is not strictly true 

 of any coincidences, and is wholly 

 inapplicable to most ; and that to 

 know whether a coincidence does or 

 does not require more evidence to 

 render it credible than an ordinary 

 event, we must refer, in every in- 

 stance, to first principles, and esti- 



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