From his essay on evidence.
Extraordinary Claims and Extraordinary Evidence
Another common slogan, also popularized by Sagan, is that Extraordinary claims require extraordinary evidence. Much depends, of course, on what counts as extraordinary, both in a claim and in evidence. It cannot be simply that a claim is unprecedented. At a certain level of detail, almost any claim is unprecedented; but this does not necessarily mean that it requires evidence out of the ordinary to establish it. Consider this claim: “Aunt Matilda won a game of Scrabble Thursday night with a score of 438 while sipping a cup of mint tea.” Each successive modifying phrase renders the claim less likely to have occurred before; yet there is nothing particularly unbelievable about the claim, and the evidence of a single credible eyewitness might well persuade us that it is true.
The case is more difficult with respect to types of events that are deemed to be improbable or rare in principle, such as miracles. It is generally agreed in such discussions that such events cannot be common and that it requires more evidence to render them credible than is required in ordinary cases. (Sherlock 1769) David Hume famously advanced the maxim that No testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavours to establish (Beauchamp 2000, p. 87), which may have been the original inspiration for the slogan about extraordinary evidence. The proper interpretation of Hume’s maxim has been a source of some debate among Hume scholars, but one plausible formulation in probabilistic terms is that
P(MT) > P(~MT) only if P(M) > P(T~M),
where M is the proposition that a miracle has occurred and T is the proposition describing testimonial evidence that it has occurred. This conditional statement is not a consequence of Bayes’s Theorem, but the terms of the latter inequality are good approximations for the terms of the exact inequality.
P(M) P(TM) > P(~M) P(T~M) when both P(~M) and P(TM) are close to 1. There is, then, a plausible Bayesian rationale for Hume’s maxim so long as we understand it to be an approximation.
It does not follow that the maxim will do the work that Hume (arguably) and many of his followers (unquestionably) have hoped it would. Hume appears to have thought that his maxim would place certain antecedently very improbable events beyond the reach of evidence. But as John Earman has argued (Earman 2000), an event that is antecedently extremely improbable, and in this sense extraordinary, may be rendered probable under the right evidential circumstances, since it is possible in principle that
P(TM)/P(T~M) > P(~M)/P(M),
a condition sufficient to satisfy the rigorous condition underlying Hume’s maxim and the slogan about extraordinary events. The maxim is therefore less useful as a dialectical weapon than is often supposed. It may help to focus disagreements over extraordinary events, but it cannot resolve them.
Extraordinary Claims and Extraordinary Evidence
Another common slogan, also popularized by Sagan, is that Extraordinary claims require extraordinary evidence. Much depends, of course, on what counts as extraordinary, both in a claim and in evidence. It cannot be simply that a claim is unprecedented. At a certain level of detail, almost any claim is unprecedented; but this does not necessarily mean that it requires evidence out of the ordinary to establish it. Consider this claim: “Aunt Matilda won a game of Scrabble Thursday night with a score of 438 while sipping a cup of mint tea.” Each successive modifying phrase renders the claim less likely to have occurred before; yet there is nothing particularly unbelievable about the claim, and the evidence of a single credible eyewitness might well persuade us that it is true.
The case is more difficult with respect to types of events that are deemed to be improbable or rare in principle, such as miracles. It is generally agreed in such discussions that such events cannot be common and that it requires more evidence to render them credible than is required in ordinary cases. (Sherlock 1769) David Hume famously advanced the maxim that No testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavours to establish (Beauchamp 2000, p. 87), which may have been the original inspiration for the slogan about extraordinary evidence. The proper interpretation of Hume’s maxim has been a source of some debate among Hume scholars, but one plausible formulation in probabilistic terms is that
P(MT) > P(~MT) only if P(M) > P(T~M),
where M is the proposition that a miracle has occurred and T is the proposition describing testimonial evidence that it has occurred. This conditional statement is not a consequence of Bayes’s Theorem, but the terms of the latter inequality are good approximations for the terms of the exact inequality.
P(M) P(TM) > P(~M) P(T~M) when both P(~M) and P(TM) are close to 1. There is, then, a plausible Bayesian rationale for Hume’s maxim so long as we understand it to be an approximation.
It does not follow that the maxim will do the work that Hume (arguably) and many of his followers (unquestionably) have hoped it would. Hume appears to have thought that his maxim would place certain antecedently very improbable events beyond the reach of evidence. But as John Earman has argued (Earman 2000), an event that is antecedently extremely improbable, and in this sense extraordinary, may be rendered probable under the right evidential circumstances, since it is possible in principle that
P(TM)/P(T~M) > P(~M)/P(M),
a condition sufficient to satisfy the rigorous condition underlying Hume’s maxim and the slogan about extraordinary events. The maxim is therefore less useful as a dialectical weapon than is often supposed. It may help to focus disagreements over extraordinary events, but it cannot resolve them.
In each of these cases, one can back up to a far more minimal conception of what is required. But then it is very difficult to go anywhere with the argument in its subsequent steps. If B can be accepted only in a fairly minimal sense, then it is not at all obvious that D is true. Conversely, in the sense in which D is obviously true, A2 and B are just as obviously false. So the argument gains no traction.
To say this is not to say that it would not be desirable for Christians to have better, fuller knowledge on some of these points; nor is it to say that such knowledge is not available. But the hinge of the argument is the claim in A2 that Christians would need a set of G-beliefs in order to have a personal relationship with God. And Drange gives no good reason to think that this claim is both (a) true and (b) substantive enough to support his subsequent chain of reasoning.