In my first post in this series, I offered a Bayesian interpretation of the principle, “extraordinary claims require extraordinary evidence” (ECREE). Greg Koukl, however, disagrees with ECREE. He recently explained why on his radio show (click here for audio); also, Melinda Penner, a member of Koukl’s staff, has written on the issue here and here. In this post, I want to explain why I think Koukl’s and Penner’s objections to ECREE, like those of William Lane Craig and T. Kurt Jaros, are misguided.

Penner’s First Rebuttal: “The nature of the “extraordinary evidence” required can be understood in two ways: extraordinary with respect to quality or extraordinary with respect to quantity.” So how should “extraordinary evidence” be understood in ECREE?

This statement is a perfect example of why a formal, Bayesian interpretation of ECREE can be helpful. Please bear with me as I review some basic probabilistic notation.

B: background evidence

E: the evidence to be explained

H: an explanatory hypothesis

~H_i: the rival explanatory hypotheses to H

Pr(x): the probability of x Pr(x | y): the probability of x conditional upon y

Next, let us define the following conditional probabilities.

Pr(H |B)= the prior probability of H with respect to B—a measure of how likely H is to occur at all, whether or not E is true.

Pr(~H_i | B) = the prior probability of ~H_i with respect to B—a measure of how likely ~H_i is to occur at all, whether or not E is true.

Pr(E | H & B) = the explanatory power of H—a measure of the degree to which the hypothesis H predicts the data E given B.

Pr(E | ~H_i & B ) = the explanatory power of ~H_i —a measure of the degree to which ~H_i predicts E given B.

Pr(H | E & B) = the final probability that H is true conditional upon the total evidence B and E.

As I explained elsewhere, an “extraordinary claim” can be interpreted as Pr(H|B ) <<< 0.5. And “extraordinary evidence” can be interpreted as the requirement that a hypothesis’s explanatory power is proportionally high enough to offset its prior improbability (the “extraordinary claim”). This could be due to a single item of evidence which has an extremely high degree of explanatory power, i.e., Pr(E | B & H). In Penner’s terms, this would be”extraordinary with respect to quality.” This could also be due to multiple, independent items of evidence which, taken together, have an extremely high degree of explanatory power, i.e., Pr(E1 | B & H) x Pr(E2 | B & H) x … x Pr(En | B & H).[1] In Penner’s terms, this would be “extraordinary with respect to quantity.”

Let us now return to Penner’s first objection. What is the problem?

If the former (quality), then the evidence produced is itself extraordinary, and it will also need to meet the requirement of having extraordinary evidence, and a vicious regress ensues.

And perhaps that is the point of the requirement because it presupposes naturalism, precluding the possibility of offering evidence that will justify a supernatural claim.

If the requirement is for an extraordinary quantity of evidence, then the next question is, how much ordinary evidence is necessary for the total quantity to be considered extraordinary?  This is perhaps a “problem of heaps” – how much is enough?  There is no determinate solution (at least epistemically, if not metaphysically determinate).  So once again, it’s begging the question to ask for extraordinary evidence.

I think it will be easiest to sort through this by considering a specific example of an extraordinary claim, the alleged resurrection of Jesus. Let R be the hypothesis that God resurrected Jesus from the dead. Let ~R be the hypothesis that God did not resurrect Jesus from the dead. Finally, let us define B, the relevant background evidence, as containing the following propositions.

B1: Approximately 107,702,707,791 humans have ever lived.

B2: God, if He exists, has resurrected from the dead at most only one person (Jesus).

For simplicity, let us round down the number in B1 to 100 billion, i.e., 10¹¹. Then the prior probability of R may be defined as follows.

Pr(R | B) <= 0.00000000001, i.e., 10^-11.[2]

Thus, the alleged resurrection of Jesus is an “extraordinary claim” in the sense that it has an extremely low prior probability, i.e., Pr(R | B) <= 10^-11. In other words, even if God exists, R has an extremely low prior probability for the simple reason that God has an extremely weak tendency to resurrect people from the dead.[3] To be precise, He resurrects from the dead less than one human in every 100 billion.

With this context in mind, let us now review Penner’s objections in detail.

If the former (quality), then the evidence produced is itself extraordinary, and it will also need to meet the requirement of having extraordinary evidence, and a vicious regress ensues.(italics mine)

Why should anyone believe that the evidence produced must itself be extraordinary?  Not only is there no reason to think this is true, but it seems to me that this is obviously false. For the sake of argument, let us assume the following statement (about evidence) is true.

E1. Jesus’ tomb was found empty by a group of his women followers three days after His death and burial.

Let us further assume that E1 is evidence for the resurrection in the following sense:

Pr(E1 | R & B) > Pr(E1 | ~R & B).

With those assumptions, why would E1 have to be extraordinary? As we’ve seen, an extraordinary claim can be interpreted as any claim where Pr(H|B ) <<< 0.5. So E1 would be “extraordinary” just in case Pr(E1 | B) <<< 0.5. But there is no way to go from this alone:

Pr(E1 | R & B) > Pr(E1 | ~R & B)

to this:

Pr(E1 | B) <<< 0.5.

Thus, there is no reason to think that the evidence produced must itself be extraordinary.

Let us now move onto the next part of Penner’s comments regarding her first rebuttal.

And perhaps that is the point of the requirement [ECREE] because it presupposes naturalism, precluding the possibility of offering evidence that will justify a supernatural claim.

Let us define “metaphysical naturalism” (N) as the hypothesis that the universe is a closed system, which means that nothing that is not part of the natural world affects it.

The definition of N should make it obvious that ECREE does not “presuppose naturalism.” It is impossible to show, by substituting synonyms for synonyms, that ECREE implies N. In fact, I’ve already given an example (B2) that is logically compatible with both theism and ~N. This example alone proves that ECREE does not presuppose naturalism.

Penner’s Second Rebuttal: “If the requirement is for an extraordinary quantity of evidence, then the next question is, how much ordinary evidence is necessary for the total quantity to be considered extraordinary?”

The Bayesian interpretation of ECREE illustrates why this objection is false. It follows from Bayes’s Theorem that the final probability of any hypothesis (extraordinary or not) is a function of both its prior probability and its explanatory power.  If we are able to quantify the prior probability of a hypothesis, Pr(R|B), it is trivial to calculate the minimum explanatory power, Pr(E|R&B;), required to show that the hypothesis is probably true. All we need is the value of the explanatory power of R’s denial. Just for the sake of argument, let’s set Pr(E|~R&B;) equal to 0.1. If Pr(R|E&B;) > 0.5, it follows from Bayes’s Theorem that

Pr(E|R&B;) > ((1-10^-11) x 0.1) / 10^-11

Thus,

Pr(E|R&B;) > 10¹⁰

Just to avoid any misunderstandings, the point of this example is not in any way dependent upon assigning Pr(E|~R&B;) a value of 0.1. If you don’t like that value, feel free to substitute whatever value seems appropriate to you. The point is that Bayes’s Theorem makes it trivial to quantify the amount of explanatory power needed to show that an extraordinary claim is probably true.

Finally, Penner writes:

… So once again, it’s begging the question to ask for extraordinary evidence.

This is false. As we’ve seen, the request for extraordinary evidence is logically compatible with both the denial of N and with the truth of theism.

(to be continued in a future post)

Notes

[1] Another possibility would be to show that the evidence constitutes a cumulative case. (See here for an analysis of cumulative cases.)

[2] I believe this over-estimates the  value of Pr(R | B) even for theists, but I won’t argue for that point here.

[3] I owe this point to Greg Cavin.