*This is the weekly Q & A blog post by our Research Professor in Philosophy, Dr. William Lane Craig.*

## Question

Hi Dr Craig,

You have written that the skeptical maxim "extraordinary claims require extraordinary evidence" (ECREE) is demonstrably false. And you have appealed to Bayes' theorem to explain why. However, every single Bayesian analysis (besides yours) that I've read about ECREE seems to support it. Hence, your interpretation of Bayes' theorem in regards to ECREE seems to be confused or idiosyncratic.

Let's clarify the point. On Bayesian terms, an "extraordinary claim" would be one whose prior probability is very low (Pr (h/b) <<< 0.5). It means that in order to render the claim more probable than not, we need strong evidence in order to counterbalance the claim's low prior probability (Pr (h/e) >>>> 0.5), which is just to say that we need "extraordinary evidence" for such a claim. Extraordinary claims require extraordinary evidence in order to overcome the overwhelming prior probability against such claims being true.

You could reply that crucial factor in the calculus is the probability that we should have the evidence we do if the extraordinary event had not occurred. In Bayesian terms it would be (Pr (e | b & ~h), that is, the probability of the evidence given the background information and thefalsehood of the extraordinary claim, which just means evidence that is even more improbable on the claim's being false than the claim's being true is already improbable given the background information. But this, again, is equivalent to say that extraordinary claims require extraordinary evidence in order to overcome the prior improbability of the claim.

Can you please clarify your Bayesian understanding of ECREE?

Mauro

## Dr. William Lane Craig's Response

I hesitate to take your question, Mauro, because it is of a technical nature that may be off-putting to many readers. But the issue raised is so important that it deserves to be addressed.

I fear that you have been misled by sceptical thinkers who are trying to misuse Bayes’ Theorem to support their scepticism about miracles.

Let h = the hypothesis that some miracle has occurred; b = our background information about the world; and e = the specific evidence in support of the miracle claim. The so-called “odds form” of Bayes’ Theorem allows us to calculate the relative probability of h and not-h on the total evidence e&b as follows:

Pr(h | e&b) Pr(h | b) Pr(e | h&b)

_____________ = __________ X _____________

Pr(not-h | e&b) Pr(not-h | b) Pr(e | not-h&b)

You’ll notice that Bayes’ Theorem doesn’t require you to assign a probability to (h½e) as a means of determining Pr(h½e&b). Therefore it is plainly false that “*in order to render the claim more probable than not, we need strong evidence in order to counterbalance the claim's low prior probability (Pr (h/e) >>>> 0.5), which is just to say that we need ‘extraordinary evidence’ for such a claim*.”

What is needed to counterbalance any low prior probability of (h½b) is that e be much more probable on h than on not-h. You correctly note this factor in your penultimate paragraph. But then you erroneously infer, “*But this. . . is equivalent to say*[*ing*]* that extraordinary claims require extraordinary evidence in order to overcome the prior improbability of the claim*.” That is plainly wrong, unless one begs the question by simply *defining* “extraordinary” to mean “more probable on h than on not-h”—or as my friend Lydia McGrew prefers to put it, “the claim is either false or it is trivially true.” What the sceptic means by “extraordinary” is something like “enormous or unusual.” In that sense the evidence need not be extraordinary. Otherwise we’d be forced to deny many non-miraculous events which we know to have occurred despite their high improbability. For example, I recently heard on the news that a particular person in South Carolina had won a powerball jackpot of over a billion dollars. I’m told that the odds of winning were something like one chance out of 320 million. So do I need extraordinary evidence to believe the report on the evening news that that particular person had won? Of course not! The evidence that it takes to counterbalance the low prior probability of that person’s winning needn’t be enormous or unusual at all but just more probable given the truth of the hypothesis than its falsehood.

There is no dispute here about Bayes’ Theorem. What is really at issue is the meaning of the term “extraordinary.” Sceptics seem to equivocate on its meaning in order to mislead people into thinking that an enormous or unusual amount of evidence is required in order to establish an event which has a high prior improbability.

*This Q&A and other resources are available on Dr. William Lane Craig's website*.