the premise, there is not a word to be said for such a step. But if we look at the process with the more indulgent eye of Induction or Probability we see that a very fair case may sometimes be made out for it. The mere fact that ‘Some B is A’ raises a certain presumption that any particular B taken at random will be an A. There is some reason, at any rate, for the belief, though in the absence of statistics as to the relative frequency of A and B we are unable to assign a value to this belief. I suspect that there may be many cases in which a man has inferred that some particular B is an A on the ground that All A is B, who might justly plead in his behalf that he never meant it to be a necessary, but only a probable inference. The same remarks will of course apply also to the logical fallacy of Undistributed Middle.
Now for a case of the opposite kind, i.e.[ TN: space] one in which Probability fails us, whereas the circumstances seem closely analogous to those in which ordinary inference would be able to make a stand. Suppose that I know that one letter in a million is lost when in charge of the post. I write to a friend and get no answer. Have I any reason to suppose that the fault lies with him? Here is an event (viz.[ TN: space] the loss of the letter) which has certainly happened; and we suppose that, of the only two causes to which it can be assigned, the ‘value,’ i.e.[** TN: space] statistical frequency, of one is accurately assigned, does it not seem natural to suppose that something can be inferred as to the likelihood that the other cause had been operative? To say that nothing can be known about its adequacy under these circumstances looks at first sight like asserting that an equation in which there is only one unknown term is theoretically insoluble.
As examples of this kind have been amply discussed in the chapter upon Inverse rules of Probability I need do no more here than remind the reader that no conclusion whatever can be drawn as to the likelihood that the fault lay with my friend rather than with the Post Office. Unless we either know, or make some assumption about, the frequency with which he neglects to answer the letters he receives, the problem remains insoluble.
The reason why the apparent analogy, indicated above, to an equation with only one unknown quantity, fails to hold good, is that for the purposes of Probability there are really two unknown quantities. What we deal with are proportional or statistical propositions. Now we are only told that
in the instance in question the letter was lost, not that they were found to be lost in such and such a proportion of cases. Had this latter information been given to us we should really have had but one unknown quantity to determine, viz.[** TN: space] the relative frequency with which my correspondent neglects to answer his letters, and we could then have determined this with the greatest ease.
1 Discussed by Mr F. Y. Edgeworth, in the Phil.[ TN: space] Mag.[** TN: space] for April, 1887.
2 Journal of the Statistical Soc.[ TN: space] (Vol. XLII.[ TN: space] p. 328) Dare one suspect a joke?
3 It appears to have been long known to gamblers under the name of the Martingale. There is a paper by Babbage (Trans.[ TN: space] of Royal Soc.[ TN: space] of Edinburgh, for 1823) which discusses certain points connected with it, but scarcely touches on the subject of the sections which follow.
4 Attention will be further directed to this distinction in the chapter on Insurance and Gambling.