Friday, January 20, 2012

A reply to a question

A reader has attempted to post an entirely off-topic question in a recent post. The question concerns probability theory and the resurrection. After some consideration, I've decided to recast the question a bit more clearly and answer it here rather than either ignoring it or publishing it in a thread where it does not belong. (Had the reader left an e-mail address, I might have replied that way, but he didn't.)

The reader's question, reworded by me, runs approximately like this:
It seems to the reader that the prior probability of the resurrection is an exception to the law of total probability. The reader asserts that P(R|~G) = 0. He also correctly points out that, on the assumption that P(R|~G) = 0, we should calculate P(R) = P(G) x P(R|G). The problem, the reader claims, is that multiplying the prior probability of God's existence by the probability that the resurrection takes place given God's existence appears to produce a probabilistic error. The reader produces a modus ponens argument:

p1= If God doesn't exist, then the resurrection is impossible.(The reader takes this to be analytically true.)

p2= God doesn't exist.

c= Therefore, the resurrection is impossible.

If premise 1 is analytic, one must deny premise 2 to deny the conclusion. But, says the reader, premise 2 need only be more plausible than not to be assertable. That would seem to mean that if P(G)<50%, the probability of the resurrection is 0, which, however, is not what we would get if we calculated the prior probability of the resurrection as we should using the law of total probability--that is P(R) = P(G) x P(R|G).

I'm going to waive the question of whether it is analytically true that the resurrection is impossible if God doesn't exist, because that either is simply a definitional matter (e.g., if you define "the resurrection" as an act of God) or involves a near-zero probability of a naturalistic resurrection.

The error in the reader's reasoning arises from his putting the wrong kind of weight--specifically, probabilistic weight--on the claim that one is justified in asserting that God doesn't exist if the probability of God's existence is less than .5. Even supposing that we grant that, that has no weight whatsoever for calculating the prior probability of the resurrection. You cannot go from, "'God does not exist' can be asserted justifiably if it is more probable than not" to "We should do our calculations of the probability of other propositions based on treating the probability of God's existence as 0 whenever the probability of God's existence is less than .5."

In essence, the above argument is a completely confused attempt to combine deductive and probabilistic reasoning. There would be no probabilistic inconsistency if the atheist were to say, viewing the prior probabilities, that probably the resurrection could not happen, or something like that. But that would have to be carefully spelled out by adding the word "probably" after "therefore" in the conclusion. (Compare "If John [defined by some definite description] doesn't exist, it is impossible for John to speak to me. John [defined by that definite description] doesn't exist. Therefore, it is impossible for John to speak to me.) The prior probability of R just is what it is. Nothing magical happens if the prior probability of G is below .5. Whatever the prior probability of G might be, you just plug that into the total probability calculation for the prior of R, and that's it. The modus ponens argument given simply doesn't tell us what the prior probability of R is.

Another way to put this is that you have already taken into account the assumption that the resurrection is impossible if God does not exist in the very act of reducing the prior probability of R to P(G) x P(R|G). Nothing more is required to take that assumption into account. The attempt to take it into account (somehow) more seriously by the modus ponens argument and the worry about what happens if the prior for G is less than .5 only darkens counsel.

The moral of the story: Don't mix apples and oranges, at least unless you're well-trained in the art of making apple-orange preserves. When you do probability, do probability. When you do deductive logic, do deductive logic. If you insist on mixing them, be verry, verry careful, or you could get yourself very, very confused.


William Luse said...

Two questions:

1. How does one come up with a probablility (e.g., <50%) about God's existence?

2. Suppose one begins (for whatever reason) with a very low probability for God's existence, but then finds that the rational evidence for the resurrection is very good indeed, such that natural explanations are found wanting. Wouldn't that then force one retroactively to grant an increase in the probability for God's existence?

Lydia McGrew said...

Excellent questions, Bill.

To the first, it depends on what evidence is available to the particular person who is trying to evaluate the probability. This is what we call a prior probability, and the idea is that we're looking at the probability prior to taking into account some other specific evidence. Since the context is the resurrection, this would be all the evidence available to the person other than (that is, “prior to”) the specific evidence (e.g., testimony of the apostles, origin of Christianity) for the resurrection of Jesus. So a prior for the existence of God in that context could be influenced by the evidence of natural theology, such as philosophical arguments for God's existence. One would include any evidence for design in nature. It could be influenced by arguments from the fulfillment of Old Testament prophecy. It could even be influenced by reports of Jesus' other miracles during his lifetime. Some people would include personal religious experience in the mix here, but I think that has to be handled with care. Some philosophers would consider simplicity considerations that they take to be relevant to an “absolute prior” independent of all empirical evidence. Anyway, you get the picture.

As to the second, when one is now going to take into account the new (or more specific) evidence for something like the resurrection, the person is supposed to _update_ his probability for the existence of God. That isn't considered to be a retroactive change. The prior was the prior. That was the probability relative to a different and smaller set of evidence. The posterior, the probability after updating, is the probability relative to that prior evidence and also the additional evidence for, say, the resurrection.

If that specific evidence is very strongly confirmatory of an event like the resurrection, there can be a dramatic shift between the prior and the posterior probabilities, both for that event and for some background requirement such as the existence of God.

Truth Unites... and Divides said...

Hi Lydia,

What are your thoughts on Pascal's Wager? I think or thought it's pretty good, but I've been informed that it's not really that strong.

What do you think?

(Sorry, if slightly off-topic. It's just since this was a math/logic/probability post, then I could ask something about the famous French mathematician.)

Lydia McGrew said...

I think I shd. resign myself to the fact that some people want to talk about OT topics on my posts here at Extra Thoughts.

My problem with Pascal's wager is that it is usually cast in terms of belief. But this confuses utilities--decision theory--with probabilities--epistemic considerations. I know it's extremely common, indeed, centrally Bayesian, to unite these, but I think that's a mistake. You should decide how much confidence to have in a proposition entirely based on your evidence, entirely separately from decision-theoretic considerations about what you have to gain or lose by believing it. Belief is not a token that can be handed out because you stand to gain something for it.

If the wager is understood instead in terms of _doing_ things solely, not in terms of belief, then it depends upon what one is doing. If it's taking the sacraments, I'd say, bad idea. St. Paul warns pretty strictly about taking sacraments unworthily, so if you don't believe in God, you shouldn't be doing it. There are also real problems with trying to "gin up" belief in something where you already believe that the evidence for it is insufficient.

On the other hand, it would be entirely reasonable to think of a different, sideways-related wager: Since so much is at stake here, wager your time and effort that it is worth investigating thoroughly. Make sure you aren't becoming an atheist, or staying an atheist, without due investigation of the best evidence that can be given _for_ theism and Christianity. There, it seems to me, decision theoretic considerations about infinite utilities of eternal gain in heaven or eternal loss in hell are exceedingly relevant. Too many people are lazy atheists. Bad idea. Stupid idea. As C.S. Lewis has one character say to another in The Great Divorce, "When did we ever put up any actual resistance to the loss of our faith?"

It happens too often (and once is too often) that a young person, a college student or a graduate student, raised Christian will lose his faith, and I'll hear about it only secondhand and think, "Why did this person not come to us and ask if we knew of anywhere he could get answers to his questions? He must have had some idea we might be able to help, but he never asked. He let us find it out from a third party."

So that wager, that's good. By all means, be willing to risk "wasting" your time looking into the evidences of Christianity, because much is at stake. Bet that it is worth taking the time to look into it. But that isn't really an epistemic wager directly, is it?