By Adler R.J.
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Additional resources for An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes
A good example is found in John Stuart Mill's attempted rescue of Hume in System of Logic (Bk. III. ch. 25) from the Indian prince embarrassment. When Hume says that the stories the Indian prince found marvelous were “not contrary to his experience,” Mill takes him to mean that the facts related are not contrary to any “law of causation” known to the prince. Mill himself employs a straight rule of induction for establishing that A causes B. To be sure, his rule is more complicated than Hume's since the uniform experience needed to instantiate Mill's rule must support the generalizations that correspond to Mill's Methods of Agreement.
Finally, it is worth reflecting on what Hume says about analogy. In the paragraph added to the main text of the 1750 edition, Hume seems to be saying that frozen water bears some positive analogy to the states of nature with which the prince is acquainted, although the analogy is very weak. However, the appended note seems to say that there is no positive analogy: One may sometimes conjecture from analogy what may follow; but still this is conjecture. And it must be confessed, that, in the present case of freezing, the event follows contrary to the rules of analogy, and is such that a rational Indian would not look for.
Suppose further as part of the background knowledge K that the witness is a religious enthusiast who takes the alleged miracle in question to have religious significance. 44 should be very close to 1. Thus, the factor  in (C′) will be close to unity. As a consequence, (C′) will surely fail if Pr(t(M)/¬ M&E&K) is substantially greater than Pr(M/E&K) (= 10−20). And Pr(t(M)/¬ M&E&K) will be non-negligible for religious enthusiasts who have a tendency to testify to the miraculous event when it doesn't occur, either because they have been deceived or because they resort to the use of deceit to win over the unconverted.
An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes by Adler R.J.