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Marginal summary and editorial marks by J. Mill.
2 Nov r 1806
Evidence
Circumstantial
Ch [...?] Phys
§ Price
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What on this occasion I can not help suspecting is that in the statement of this case one circumstance or condition, and that an essential one, has been left out:—viz: that of the several MS alt. ‘disjunctive’ chosen by J. Mill. facts each of which is in the degree in question improbable, the existence of one or other is a fact so far from [im] ‘improbable’ J. Mill correction.probable as to be altogether certain. For this certainty though [...?] seems to be implicitly and necessarily involved in the conception of the case. In strictness of speech no conceivable event is certain, just before any and every drawing, the tickets may be burnt, the boy that should have drawn them may drop down dead or run away: the hall in which they should have been drawn may be swallowed up by an earthquake. But these and all other such contingencies are laid out of the case: and so are all others by which the drawing of the lottery in the ordinary course could have been defeated. Certainly then as that explained being assumed, as it can not but be, what is certain is that of these 50,000 tickets some one will be first drawn: and the chances against each being the same, that it should prove to be the first drawn [is] ‘is’ J. Mill addition. it is just as likely of each one as of every other. That in the case of any one of them there should be an improbability I would not take upon me to deny: whether the term be proper or no, depends upon usage: and in this case the usage seems to be fixed: fixed and by the cultivators of science, by those who have the best chance to the privilege of fixing it. But as to incredibility, I would venture to ask whether a fact of this description be in any the smallest degree incredible: the fact altogether incredible would be that as one ticket of the whole 50,000 should be the first drawn: and of each one, that it should have been the first drawn it is exactly as credible as of every other.
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Leaving aside all consideration as to practical drawing of y e lottery, equally probable & credible y t any one partar ticket sh d be drawn first at y t any other sh d
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Title: [This folio contains red pencil editing marks]Description: This folio contains red pencil editing marks probably in the hand of James Mill. Marginal summary also probably in his hand. 2 Nov r 1806 Evidence Circumstantial Ch. Mend[?] Phys § Price 2 In this argument, the reader, I think, can scarce fail of having before now suspected some fallacy: though in what particular point it lurks may not be quite so easy to pronounce. Before we endeavour to point out the exact seat or cause of the fallacy let us apply to the mathematical Doctor's mathematical argument, a counter argument, which i/ in use with mathematicians—the argumentum ab absurdo: let us pursue the Doctor's through its consequences, and observe to what it leads. 1. Improbable as it is, and in the degree in question, that the reported number should have been first drawn, so is it and in the same degree that any other number should have been the first drawn: and this is equally true of every other. 2. This is as much as to say in every lottery of 50,000 tickets it is 50,000 to one that no one ticket will be drawn first: 3. Whence again it follows that in that same degree it is improbable in a lottery of that number of tickets any ticket should be drawn at all. 4. As it is improbable, and in this degree that a lottery of 50,000 tickets should be drawn at all, at the same time that in London alone lotteries of this same or a greater number of tickets are actually drawn every year, it is in the same way in the power of government to give existence to facts that shall be in any given degree improbable. In this way it shall be in the power of government, to give existence to facts more improbable than the most improbable of any that ever were pretended to have had existence: and those transcendently not to say infinitely improbable facts shall still be capable of being rendered credible by ordinary evidence. 5. In the same way it may be proved, that scarce any fact ever does take place that is not improbable: improbable in an indefinite not to say infinite degree. Look at any one of the boards on the floor on which you are treading: observe the veins in it. If in that same floor there are 20 boards look at them all, there is not one to a given extent of surface, in which the picture made by the veins shall be exactly the same as in the first. Repeat this survey upon all the floors in your house: upon all the houses in your street: upon all the streets in your town: upon all the towns in your province Apply to this argument the argumentum ab absurdo 1. If improbable y t this number sh d be drawn a prize it is equally improbable y t any other sh d. 2. There y t chances in a lottery are t t no one ticket in partar will be drawn first 3. And in y t some degree improbable y t any ticket at all sh d drawn 4. [...?] encreasing y t no. of tickets gov t may give existence to facts in an indefinite degree improbable 5. In this way all [...?] [...?] an infinite degree improbable as for example that any board sh d exist with ye partar configuration with y e one before you.
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Title: [This folio contains red pencil editing marks]Description: This folio contains red pencil editing marks probably in the hand of James Mill. Marginal summary also probably in his hand. 2 Nov r 1806 Evidence Circumstantial Ch Mend[?] Phys 1 § 16 Price A mathematician of considerable celebrity+ has brought forward an argument, by which if it were just, the disprobative force of the species of circumstantial evidence in question would be reduced to little or nothing. Facts highly and indisputably improbable, are what we are accustomed every day to give credence to, and with any practical inconvenience, upon any the slightest evidence. The definition of improbability is to be taken from a source from which it always has been taken—the language—the correct mathematical language of the doctrine of chances. Take any fact whatever let the chances against its happening be to the chances in favour it its happening, as 1000 to one: you can not deny but that this fact is in a very considerable degree improbable. Instead of 1000 to 1, say 10,000 to one, it is now ten times as improbable as before: instead of 10,000 say 50,000, it is now fifty times, it is improbable in the ratio of 50,000 to one. But facts as improbable as this are believed, and turn out to be true every day: and believed and would at any time be believed by any rational man, upon any the slightest evidence: upon the evidence of a paragraph in a newspaper. A lottery is drawn with 50,000 tickets in it: and the first drawn ticket is intitled to a prize. You read that such or such a number has in this way been become intitled to the prize. You make no difficulty in believing it: and yet the chances were 50,000 to 1 against its being the first drawn: so improbable are the facts which the [...?] of men are believing every day without scruple and without error. But suppose instead of 50,000 the tickets were 500,000 or 5,000,000: how prodigious the improbability! But even in this last case would the fact be incredible? incapable of being rendered credible by any number of witnesses? On the contrary you can hardly speak of it as being less credible under these last and highest numbers than it was under the first. In the same way I can make my fact improbable in as high a degree as I please: improbable, if I please in an infinite degree: and still it shall remain not incredible: and in a word perhaps not less credible than at first credible: upon the strength of any ordinary mass of evidence, upon evidence of much less strength than many a mass that is exhibited, and obtain credence, and without prejudice to truth, in truth of justice. + D r Price [...?] 1 D r Price's argument y t facts as improbable in proportion to y e numerical chance ag t the happening not true, since facts in the highest degree improbable not as ag t a particular lottery ticket as drawn a prize are believed upon y e slightest evidence
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Title: [This folio in hand of Colls. 29 July]Description: This folio in hand of Colls. 29 July 1815 Campbel and Price 15 Price 7 (1) Let us apply to the mathematical Doctor's mathematical argument a counter argument which is in use with mathematicians—the argumentum ab absurdo: let us pursue the Doctor's through its consequences, and observe to what it leads. 1. Improbable as it is, and in the degree in question, that the reported number should have been first drawn, so is it and in the same degree that any other number should have been the first drawn: and this is equally true of every other. 2. This is as much as to say in every lottery of 50,000 tickets it is 50,000 to one that no one ticket will be drawn first. 3. Whence again it follows that in that same degree it is improbable that in a lottery of that number of tickets any ticket should be drawn at all. 4. As it is improbable, and in this degree, that a lottery of 50,000 tickets should be drawn at all, at the same time that in London alone lotteries of this same or a greater number of tickets are actually drawn every year, it is in the same way in the power of government to give existence to facts that are in any given degree improbable. In this way it shall be in the power of government to give existence to facts more improbable than the most improbable of any that ever were pretended to have had existence: and those transcendently, not to say infinitely, improbable facts shall still be capable of being rendered credible by ordinary evidence. 5. In the same way it may be proved that scarce any fact ever does take place that is not improbable: improbable in an indefinite not to say infinite degree. Look at any one of the boards on which you are treading: observe the veins in it. If in that same floor that are 20 boards—look at them all, there is not one, to a given extent of surface, in which the picture made to the eyes shall be exactly the same as in the first. Repeat—this survey upon all the floors in your house: upon all the houses in your street; upon all the streets in your town: upon all the towns in your province, till you have surveyed a million of boards or any number of millions that you think fit. Say for shortness one million. In this degree—the degree of a million to one—is it improbable that upon any one board such a set of configurations should exist as those which
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