Page:Mind (Old Series) Volume 9.djvu/242

 230 F. Y. EDGEWOETH : which some prefer to call intuitive knowledge. At any rate the belief is supported by actual experience, the rough ex- perience of anyone who has ever worked sums in arithmetic, the precise experience of Mr. Proctor, who actually counted the digits occurring in the pages of a logarithm table. This ci priori knowledge about the forthcoming digit is perfectly real and substantial ; yet it does not count for much when weighed against special knowledge that the digit in question is in a certain neighbourhood. Suppose, for instance, that a constant is being evaluated, and that special knowledge about a forthcoming place is afforded by actual observation : for instance, that '6 has been very often observed, '7 and '5 not so often, and the rest very rarely. It would be quite correct, I think, according to the method of Bayes, to treat the CL priori probability of each digit as l-10th, the CL pos- teriori probability of say or 9 as indefinitely small. Simi- larly, I submit, the assumption that any probability-constant about which we know nothing in particular is as likely to have one value as another is grounded upon the rough but solid experience that such constants do, as a matter of fact, as often have one value as another. And accordingly such constants afford a basis for that inverse method which is so beautifully illustrated by Donkin in the article referred to. The ridicule which has been heaped upon Bayes's theorem and the inverse method will be found only applicable to the pretence, here deprecated, of eliciting knowledge out of ignorance, something out of nothing. The most formi- dable objection is that which was made by Boole, and is repeated by Mr. Venn, 1 Mr. Peirce, 2 and others with appro- bation. Our procedure in treating one value as a priori not less likely than another is, it is said, of a quite arbitrary character, and apt to lead to different conclusions from the plausible one which we have reached by accident. I will not transcribe the argument of Boole, 3 but will attempt to parody it. Suppose we are considering the probable value of a forthcoming decimal place in the evaluation of some constant. Now this tt priori probability may be considered as thus given. Suppose there are ten urns, A, B,. . . J. A constitution of the system consists in a particular com- bination of the emptiness of some urns with the fulness of others, e.g., ABCDEFGHLJ is a particular constitution, where the superposed negative sign denotes emptiness. Now the forthcoming digit corresponds to the number of full urns 1 Logic of Chance, ch. 6, 13. 2 Studies in Logic. 3 Laws of Thought, 370.