448 INDUCTION. 



sand tickets. Or instead of the balls suppose a lottery, with 1000 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 cases, as in Laplace's example ; yet 

 he surely would not say that if the witness answered 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 announcing its occurrence he shall in- 

 crease 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 concerned, there will be 999 times as many 

 chances of his announcing black falsely as white. 



Or suppose a regiment of 1000 men, 999 Englishmen and one French- 

 man, and that of these one man has been killed, and it is not known 

 which. I ask the question, and the witness answers, the Frenchman. This 

 was not only as improbable a 'priori, but is in itself as singular a circum- 

 stance, as remarkable a coincidence, as the drawing of the white ball; yet 

 we 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 differed from the Frenchman, they were not, like the 

 999 black balls, undistinguishable in every other respect ; but being all dif- 

 ferent, 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 selected by D'Alembert, that of sixes 

 thrown on a pair of dice ten times in succession, belongs to this sort of 

 cases rather than to such as Laplace's. The coincidence is here far more 

 remarkable, because of far rarer occurrence, than the drawing of the white 

 ball. But though the improbability of its really occurring is greater, the 

 superior probability of its being announced falsely can not be establish- 

 ed with the same evidence. The announcement "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 a 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 instance, to first 

 principles, and estimate afresh what is the probability that the given testi- 

 mony would have been delivered in that instance, supposing the fact which 

 it asserts not to be true. 



With these remarks we close the discussion of the Grounds of Disbelief ; 

 and along with it, such exposition as space admits, and as the writer has it 

 in his power to furnish, of the Logic of Induction. 



