Greg Cavin on Bayes' Theorem and Miracles

I wrote most of this post several months ago when my friend Calum Miller came to southern California for a semester abroad. Unfortunately, I simply never got around to finishing it up. Hence, this post comes about five or six months late. However, I still think it's worth posting, in case someone watches the video or comes upon the type of fallacy that I suspect goes into the argument. Here's the post:

A couple weeks ago I went to a debate between my friend Calum Miller and philosopher Greg Cavin on the Resurrection of Jesus. The video can be found here. Cavin's opening speech on Bayes starts at (6:00). He gets into his arguments again at (13:40). In this post I'll discuss a small part of Cavin's opening speech.

At the beginning, Cavin claims that he will show that it is "virtually 100% certain that no miracles ever occur."

Greg Cavin formulates the argument in terms of an assessment of a comparison between probabilities. While Cavin goes into a ton of mathematical detail that I suspect could be simplified to get to the main point, a little bit of it is probably necessary. He formulates the argument in terms of the Odds Form of Bayes' Theorem.

In general, the Odds Form of Bayes' Theorem is as follows. For any events A, B, and D:

P(A|D)/P(B|D) = P(D|A)/P(D|B) * P(A)/P(B)

Cavin comes up with a partition of probability space which is exhaustive and exclusive. In other words, at least one of the following hypotheses holds and if one holds then the others do not.

• M: At least one miracle has, had, or will occur in the universe.
• L: The laws of the sciences as these are currently formulated in standard reference works, without any supernatural non-interference proviso, are true and are laws of nature in their restricted domains.
• (¬M & ¬L): Neither M nor L hold. 

With respect to L, what it is saying is that if you have a law of science 'S,' then a statement of the law will just be of the form: "For all times, all places, S," rather than "Except for the intervention of some supernatural force, for all times, all places, S." In other words, laws of science lack the underlined "proviso."

M and L are taken to be incompatible because if a miracle occurs, i.e. if M is true, then that entails the failure of at least one "un-provisoed" law of science at some time and place, whereas L entails all "un-provisoed" laws of science hold at all times and places. Cavin defines the evidence E with respect to which we will evaluate these probabilites as follows (27:00):

• E: The total evidence, which is a combination of T & C, where T and C are understood as follows:
• T: All of the traces (call them Ti) of miracles. These are all of the pieces of evidence people could take to provide evidence for a miracle.
• C: All of the confirmation instances (call them Ci) of the laws of science. These are all of the pieces of evidence people could take to provide evidence for the various scientific laws.

The partitioning of probability space.

Applying Bayes' Theorem to the argument at hand, this is the Ratio of Posterior Probabilities of L vs M:

P(L|E)/P(M|E) = P(E|L)/P(E|M) * P(L)/P(M)

In other words, the left hand side compares the likelihood of L given the evidence with the likelihood of M given the evidence.

Now, a crucial part of Cavin's argument is in calculating the ratio P(E|L)/P(E|M). This is done by calculating probabilities of all of the Ti given L and M and calculating the probabilities of all the Ci given L and M. If these are lower on M than they are on L, then P(E|L) will be higher than P(E|M). His official argument here is from (31:00) to (34:00), but I asked him a question later that gets to the same point.

After the talk, I asked Cavin why he thought M could not explain C and T as well as L could. In other words, why are, say, the confirmation instances of science less likely given that miracles have occurred than if L holds? He said, "Well, if I told you, 'This is a desk,'  what would that explain? Not much. How can you make any predictions from that? So, likewise, how can the proposition that at least one miracle holds explain anything? It could hardly have any predictive or explanatory power." Of course, that seems true. If the only sentence you knew to be true were "At least one miracle occurs," then you wouldn't be able to predict much, just as you couldn't predict much from just knowing "This is a desk." Hence, the argument goes, P(E|M) is very low.

However, it's a little bit misleading to put things this way. P(E|M), strictly speaking, isn't defined in terms of how much you can predict from the single proposition that at least one miracle occurs. This is clear after considering some very basic probability theory.

First, note that we can always define P(M) as P(M∩A) + P(M∩¬A) for any event A. This can clearly be seen by the following diagram:

P(M) = P(M∩A) + P(M∩¬A)

A is marked out in dark blue.

¬A is marked out in light blue.

Suppose that 'A' denotes some hypothesis, maybe the hypothesis 'The laws of nature almost always but not always hold.' Then the probability that some miracle happens is equal to the probability that some miracle happens and A holds plus the probability that some miracle happens and A does not hold. Again, P(M) = P(M∩A) + P(M∩¬A).

From this we can infer: P(E|M) = P(E|M∩A) + P(E|M∩¬A). Now, you might still think that this is lower than P(E|L) for various reasons. But you certainly couldn't infer it from the type of argument I sketched above. That would be much too easy.

Maybe I am misrepresenting what Cavin said. I hope I'm not. But if I am, let's just say that if someone were to argue in the way I represented Cavin as arguing, then they would be committing a fallacy.

There were many other interesting issues that came up during the debate, such as the likelihood of the laws of nature holding most of the time given theism, and these deserve attention. But for now I think it's worth noting that Cavin's argument doesn't go through as easily as it might have seemed.