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.

Remarks on "Divine Hiddenness"

[Originally posted at the Rational Gang. Please direct any comments there.]

A lot of atheists say that if God were to perform a miracle in front of their eyes, then they would believe. So then, if God is all good, and "wills all men to be saved," then why doesn't he provide such obvious evidence for his existence? Since an all-loving God is expected to reveal himself to all, and there are atheists that have reasonable nonbelief, this is evidence that God does not exist.

1. Coercion and Moral Responsibility:
A first thing to note is that God wants to preserve our moral responsibility and freedom. Now, almost all accept that it is possible to coerce someone into doing something. This can be done, most obviously, by physical force, as when kidnappers pick up a hostage and put him into a van against his will. Clearly in this context the person is not responsible for this event. Another, more interesting case, is where a mother is forced by a cruel killer to choose between her two sons being shot, or else they will both be tortured to death. The mother is not morally responsible, or at least very responsible, if she ultimately chooses one child in order to save at least one; the ultimate responsibility goes to the wicked sadist who is forcing the "choice" upon her. The point here though is that the mother is not literally coerced by force; rather, she is coerced by reasons. She knows that if she doesn't pick one, they will both be lost. In cases of coercion, one loses moral responsibility for one's actions.

But then it's straightforward applying this to the case of God. God wants us to have the freedom to choose him. He doesn't want to coerce us and ultimately remove our responsibility. But if one can be coerced by reasons, and incontrovertible proof would coerce some people into belief, then it's easy to see why God would not provide incontrovertible proof to absolutely everyone in order that they might all have absolute certainty: It removes responsibility on our part.

2. Divine Obviousness:
A reply inspired by the Thomistic tradition is to say that, as a matter of fact, God isn't hidden at all. Rather, God, as the ultimate source of all being, completely and totally permeates throughout nature. This is how God is omni-present, not in the sense that the universe is God (a la pantheism), but rather that the very existence of things is just one step away from God himself. However, being creatures of habit, and being familiar with the world, we take the awful mystery of existence for granted, and fail to see the divinity which sustains it all. But this is no failure on God's part. It's a failure of us to appreciate God's creation.

3. They're Wrong:
While atheists say they know that if God were to provide them a clear sign such as a miracle then they would convert, that might, for all we (or they) know, not be the case. In fact, this is what we find from the Christian perspective in John 11, where the Pharisees, even upon hearing and accepting that Christ performed miracles, refused to acknowledge him: "So the chief priests and the Pharisees gathered the council, and said, 'What are we to do? For this man performs many signs. If we let him go on thus, every one will believe in him, and the Romans will come and destroy both our holy place and our nation.'" So for these people, even the greatest of miracles would not convince them. How would this apply in the case of others?

According to one theory of the relationship between God's foreknowledge and our freedom, Molinism, God knows infallibly what we would freely do in any given circumstance. Supposing this view is coherent, we can apply it in the case of atheists: God does not give incontrovertible evidence to people who he knows that, in any circumstance where they are given incontrovertible evidence, would not convert. This means that atheists cannot say, "Well, if only God had given me more evidence, then I would have converted." Not necessarily.

If this is too strong, we can even weaken the principle: God does not give reasonable evidence to people who he knows that, in any circumstance where they are given reasonable evidence, would not convert. This seems more plausible anyway. For one, it doesn't seem necessary for God to give everyone incontrovertible evidence. If we are given reasonable evidence and fail to see it, then that is our fault. Moreover, this widens the group of people we are dealing with because, though some may accept God given incontrovertible evidence, fewer will accept God given only reasonable and sufficient evidence, which may explain a large number of unbelievers.

What this ultimately shows is that, even if some atheists are not actually given incontrovertible, or even reasonable, evidence, this doesn't make it the case that God is doing something evil, the reason being that they would not accept the evidence in other feasible circumstances anyhow. It would be for God to cast pearls to swine.

4. Rationality and Reasonableness:
This leads to a further question: are atheists, in fact, given reasonable evidence? Their whole argument may be begging the question here. After all, theists believe that there is reasonable evidence for the existence of God. But then if God has provided everyone with reasonable evidence, the argument fails, because this is precisely what is in question, i.e. whether belief in God is reasonable.

The atheist finds himself in a bit of a pickle. In proposing that there is not enough evidence for Christian belief, he must give us a criterion of reasonable belief that is strong enough such that it supports the reasonableness of atheism, but is not so strong that it begs the question against the theist. To put it another way, there is a tension in the claim that "the reasonableness of atheism entails the unreasonableness of theism," because it is possible that, given the atheist's criterion of reasonableness, the theist can turn right around and say, "Well, it is reasonable to believe that God might not reveal himself to everyone with absolutely convincing evidence."

5. Five points.
To sum up, I think there are five reasons we can give as to why God's existence is not incompatible with "divine hiddenness."
(a) Reasons can coerce. So if one were given incontrovertible reasons to believe in God, then one would be coerced into belief in God. But since coercion is incompatible with significant moral responsibility, and God wants to leave us morally responsible, God does not want to coerce us. Therefore, God does not give incontrovertible evidence.
(b) God is not in fact hidden; rather, God is obvious. However, humans fail to recognize the mystery of their own existence and the creation before them, and fail to see the divine in nature.
(c) For all we know, God does not give those who have a reasonable non-belief in God incontrovertible evidence, because he knows that in all circumstances, they will not believe in God.
(e) For all we know, God does not give those who have a reasonable non-belief in God reasonable evidence, because he knows that in all reasonable circumstances, they will not believe in God.
(e) The atheist begs the question against the theist in two ways: first, by saying that his nonbelief in God is reasonable, since the theist believes there are reasonable grounds for belief, and second, by saying that his reasonable nonbelief implies God's nonexistence, since the the theist can say that, by the atheist's own criteria, it is reasonable to believe that God has reasons to stay hidden.

All in all, this provides a strong cumulative case for believing that the argument from divine hiddenness doesn't work, and that theism is left unharmed.

What Does It Take to Be An Atheist?

Consider some valid theistic arguments and what an atheist must do to deny their soundness.

Argument A - A Necessary Concrete Entity:
(1) Possibly a necessary concrete being exists implies a necessary concrete being exists.
(2) Possibly, a necessary concrete being exists.
(3) Therefore, a necessary concrete being exists.

Assume further that it can be shown that the necessary being is God.

What must the atheist do to deny this argument? (1) is just axiom S5. Therefore they must deny (2). This implies that (a) possibly nothing exists or (b) necessarily, there exists at least one contingent entity. This also implies that all concrete entities are contingent.

Atheist Theorem 1 (AT1): It is not possible that a necessary concrete being exists.
Atheist Theorem 2 (AT2): Either (a) possibly nothing exists or (b) necessarily there exists at least one contingent entity.
Atheist Theorem 3 (AT3): For any concrete entity x, x is contingent.

Disproving AT1, AT2 or AT3 is sufficient for refuting atheism.

Argument B - The Scotistic Cosmological Argument:
(1') Whatever is possible is contingent or necessary.
(2') A first cause is possible.
(3') Therefore, a first cause is contingent or necessary.
(4') Any contingent substance is possibly actualized by another substance.
(5') A first cause is not possibly actualized by another substance.
(6') Therefore a first cause is not contingent.
(7') Therefore, a first a cause is necessary.

What must the atheist do to deny this argument? The atheist can only really deny either (2) or (4).

Atheist Theorem 4 (AT4): Either (a) a first cause is impossible or (b) some contingent substance is not possibly actualized by another.

Hence, disproving this disjunction AT4 is sufficient for refuting atheism. Saying that (b) is false also leads to some interesting consequences about infinite regresses.

Argument C – A Modal Cosmological Argument:
(1*) Every contingent entity possibly has an external cause.
(2*) If the sum total of contingent concrete entities C has an external cause, that cause is necessary.
(3*) C is a contingent concrete entity.
(4*) Possibly, C has an external cause.
(5*) Therefore, possibly there is a necessary cause of C.
(6*) Therefore, there is a necessary cause of C.

What must the atheist deny? All premises seem quite strong. He must then deny (1*).

Atheist Theorem 5 (AT5): For some concrete contingent entity x it is impossible that x is caused.

Disproving AT5 is sufficient for disproving atheism.

Thus far we have Atheist Theorems 1-5:

AT1: It is not possible that a necessary concrete being exists.
AT2: Either (a) possibly nothing exists or (b) necessarily at least one contingent entity exists.
AT3: For any concrete entity x, x is contingent.
AT4: Either (a) a first cause is impossible or (b) some contingent substance is not possibly actualized by another.
AT5: For some concrete contingent entity x it is impossible that x is caused.

Disproving one of these theorems is sufficient for disproving atheism. Showing a contradiction between these beliefs and other atheist beliefs is sufficient for showing atheism as such to be incoherent. To me all of these propositions seem quite implausible in their own right.