Okay, I've been playing with this post way too long now - time to just hit the "publish" button! I previously mentioned a debate between Bart Ehrman and William Lane Craig about whether the miracle of the resurrection happened. Ehrman basically said that the resurrection is unlikely to have happened based on our knowledge of how the world works today. He says “Historians cannot establish miracle as the most probable occurrence because miracles, by their very nature are the least probable occurrence.” Craig says that Ehrman is just presenting a “warmed over” version of Hume’s well-known argument against miracles in which he memorably said, “No testimony is sufficient to establish a miracle unless… its falsehood would be more miraculous than the fact which it endeavors to establish.”
1. Craig’s presentation of Baye’s Rule
Craig argues that Ehrman and Hume are confusing two different probabilities. The heart of Craig’s argument is a form of Baye’s Rule that he presents for the probability that the resurrection happened. Craig defines the following variables:
Pr(X) = probability of X
Pr(R/B) = the probability of the resurrection given our general knowledge of the world (our Background knowledge)
Pr(E/B&R) = the probability of the specific evidence we see for the resurrection existing given our general knowledge of the world and assuming the resurrection happened.
Pr(~R/B) = the probability that that resurrection did not happen, given our general knowledge of the world.
Pr(E/B&~R) = the probability of the specific evidence we see for the resurrection existing given our general knowledge of the world and assuming the resurrection did not happen.
Pr(R/B&E) = the probability of the resurrection having happened given our general knowledge of the world and the specific evidence for it.
He then applies Baye’s Rule and gives us an expression for Pr(R/B&E):
Pr(R/B&E) = Pr(R/B) Pr(E/B&R) / (Pr(R/B) Pr(E/B&R) + Pr(~R/B) Pr(E/B&~R)).
2. Looking at this equation in another way.
I will assume this relation is correct and that no term in it is zero. Then it can also be expressed (after dividing both numerator and denominator on the right side of the equation by the product Pr(R/B) Pr(E/B&R)) as:
Pr(R/B&E) = 1/ {1 + ( [Pr(~R/B) Pr(E/B&~R)] / [Pr(R/B) Pr(E/B&R)])}
So it really all depends on that ratio: [Pr(~R/B) Pr(E/B&~R)] / [Pr(R/B) Pr(E/B&R)]. If it is nearly zero, the probably of the resurrection becomes about 1 (very likely). If it is very large, the probability of the resurrection becomes nearly zero.
3. Reasonable Assumptions
Now comes the real problem. I am very suspect about assigning numbers to something like “the probability of the resurrection having happened given our general knowledge of the world.” How in the world would one calculate that? But since that is the game we are playing, can we say anything reasonable about it? It seems to me that one would have to grant that given just In fact Craig seems to acknowledge this in the debate. His argument is that Ehrman confuses Pr(R/B), which he agrees may be a small number, with Pr(R/B&E), which Craig says is not necessarily small just because Pr(R/B) is small.) Pr(R/B) is the probability of a miraculous event happening given our general knowledge of the world. our general knowledge of the world the resurrection (or any miracle) seems highly unlikely.
Almost by definition a miracle would have to have a low probability given our general knowledge of the world because otherwise it wouldn’t be miraculous. For example, think about the probability that any major league hitter will end his season this year with a normal number of at bats and a greater than .500 average. It is extremely unlikely – no one has ever done it. But if someone did it, would it be considered miraculous? No, clearly not. Very improbable is still not improbable enough to designate something a miracle. That would mean, at the least, that Pr(R/B) would be very small and the Pr(~R/B) would be nearly 1. So we are really dealing with a small number (lets call Pr(~R/B) = ε, where ε is a very small number). We may not know what it is, but I think it reasonable to assume it is very small.
4. The Crucial Ratio (A Guess?)
So the crucial ratio is actually = [Pr(E/B&~R)] / [ε (Pr(E/B&R)] or
[Pr(E/B&~R) / Pr(E/B&R)] times 1/ ε.
Now since ε is a very small number, 1/ ε is a very large number. So, for this product to be small (making the probability of the resurrection close to 1) the ratio Pr(E/B&~R)/ Pr(E/B&R) (let’s call this number Cr for “Crucial ratio”) must be very small.
Remembering what these represent, what this is essentially saying is that the probability of the specific evidence for the resurrection existing as we see it now if the resurrection actually didn’t happen must be small compared to the probability of the evidence as we see if the resurrection did. How small? Small enough for Cr to overwhelm the largeness of 1/ ε. How large is 1/ ε? Here we get into the problem of assigning numbers to probabilities in ways. Remember that 1/ ε is essentially the ratio of the probability that that resurrection did not happen, given our general knowledge of the world and the probability of the resurrection given our general knowledge of the world. Just given our general knowledge of the world, one would have to say that is very small, but how small? We have billions of examples of cases where people have died and not been resurrected. We have no cases (excluding for the moment the one we are considering) where it has been definitively observed to happen. In addition it violates many of our well-understood principles of biology. I would think then that 1/ ε would have to be very large – but is it large like 10000000 or large like 100000000000000000000000000? Who knows?
5. Estimating Cr compared to 1/ ε
But what about Cr? Again, I have no idea how one would estimate such a thing. But the important question is, Is it small compared to 1/ ε? Cr represents the ratio of how likely it is for the specific evidence we see for the resurrection (or in a more general case, any miracle) to exist if the resurrection /miracle didn't occur and the probability of the specific evidence existing if it did.
I think Ehrman and Hume’s point would be that whatever this ratio is, if the evidence it is relating to is a few eyewitness accounts, it is highly unlikely to be small enough to make Cr times 1/ ε small. Why not? If E amounts to eyewitness accounts then we know that the eyewitnesses could be (1) lying, (2) mistaken, (3) fooled, (4) misinterpreting what they see, etc. And what is on the other side? 1/ ε is based on our background information of the way the world works, essentially. And what is that based on other than reason applied to the collective observations of humanity over the centuries? That simply has to overwhelm an eyewitness account, no matter how reliable.
Say a fellow swears that he saw a guru levitate, and you feel he is an honest guy and you know no reason that he would lie. You may estimate that the probability of him saying it if it didn't happen is less than the probability of him saying it if it did - at least then you would have a motive for him to make the statement, namely that it is a true statement. If that case Cr would be less than one. But how much less than one? Certainly not enormously less than one. Because people we think are honest, can still lie. People who actually are honest are sometimes mistaken, etc.
On the other hand, the evidence that gravity cannot be overcome in such a way is so overwhelming that, though I don’t know exactly what Cr would be for such a case, and I don’t know how we would calculate 1/ ε, I have to imagine that the product of the two would be very large anyway and the probability given by Baye’s Rule would have to be very small. For most anything that we would normally define as a “miracle” I would think this would have to be true, including the resurrection.
6. No Error
We don’t have a way of getting any real numbers out of something like this. However, it seems plausible that Cr times 1/ ε is very large and that the probability of the Pr(R/B&E) is small. Certainly Craig’s argument doesn’t seem to me to provide much support for saying that Erhman made an “egregious error” or that Hume was far off with his reasoning. Craig may say that there is additional evidence for the resurrection and that Cr should be very small. I am unconvinced of that - anonymous accounts of other peoples experiences written a generation after the fact seems not terribly weighty to me. And "evidence" based on anonymous accounts of how people acted, based on their reported beliefs which may or may not have been obtained by first hand knowledge, doesn't really strike me as being much stronger. I may be missing something here (let me know if you think I am) but it still seems to me that Hume and Erhman are on firm ground here.