Bayes’ Theorem And Hume’s Treatment Of Miracles -- By: George I. Mavrodes
TrinJ 1:1 (Spring 1980) p. 47
Bayes’ Theorem And Hume’s Treatment Of Miracles
University Of Michigan
“I should not believe such a story were it told me by Cato.”1 Hume quotes this “proverbial saying in Rome” in his celebrated chapter on miracles, and it serves to encapsulate one strand of his argument in a memorable way. There may be other important and provocative suggestions in this chapter-maybe some which are not entirely consistent with this line-but in this paper I will abstract from them as far as possible to concentrate on the argument which I think this quotation embodies.
This argument, I think, invites interpretation in terms of probabilities. Hume, indeed, often uses the word “probability” in expounding it. C. S. Peirce says of Hume’s treatment of miracles that “the argument is based upon a misunderstanding of the doctrine of probabilities, of which some of the early treatises had appeared in his day.”2 In this paper I will try to clarify and evaluate this argument by applying to it one version of Bayes’ Theorem; but first I should like to sketch out the argument in an informal way.
It begins with what we might call the “Wise Man’s Principle.” “A wise man,” Hume says, “proportions his belief to the evidence.”3 There is some difficulty in producing a full interpretation of this principle which is plausible. In this argument, however, all of the work can be done by a more restricted version, which is presumably meant to be entailed by the full principle. It can be stated as follows:
For any two propositions, p and q, if a wise man has more evidence for p than for q, and if he believes exactly one of them, then he believes p.
It is this restricted version which I shall call the “Wise Man’s Principle.”
Even in this restricted form the principle is highly controversial. There is a powerful utilitarian argument against it, based on the fact that the benefit to be gained from believing a given proposition, if it is true, may greatly outweigh the loss to be suffered from believing it, if it is false. Such a proposition would be a good bet because, as we might say, it pays such good odds, even if its chances of being true are lower than those of its negation.4 In turn, there is a
TrinJ 1:1 (Spring 1980) p. 48
powerful objection to the utilitarian argument, based on the fact that its principles, in the extreme case, would seem to justify our believing something even against absolutely ...
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