Monthly Archives: March 2013

The Probability of a Miracle

mr jesus no es hereIt’s the time of year when Christians celebrate the central miracle of their faith: the resurrection of Jesus. Of course, the resurrection has always had its share of skeptics, many of them philosophers. Perhaps the most famous is the 18th century philosopher David Hume, who argued that a miracle is, by definition, a highly improbable event. In fact, it is always more probable that eyewitnesses to an alleged miracle are lying or mistaken than that a law of nature has been violated. Therefore, eyewitness testimony cannot provide us with sufficient evidence to overturn our uniform experience of nature’s regularity. According to Hume and like-minded skeptics, a miraculous explanation is always less probable than even the least probable naturalistic explanation. But Hume lived before Bayes’ Theorem and arguably didn’t have the tools to give a rigorous account of the relevant probabilities. So it might be worth revisiting the question: how probable is a miracle? I don’t think there’s any general answer to this question; the probability will depend on the particular claim. For the sake of convenience, then, let’s stick with the alleged resurrection of Jesus of Nazareth.

First, however, a few preliminary words about skepticism. I’m skeptical of miracle claims and regard skepticism as the default position. There are a lot of miracle claims out there, many of which are unimpressive. I don’t regard the image of Christ on a grilled cheese sandwich as anything other than pareidolia. Also, the many alleged cases of miraculous healing are unlikely to convince those of a skeptical disposition. I’m very happy when someone’s cancer goes into remission, but attributing it to divine agency is a bit hasty because we know that cancer sometimes goes into remission spontaneously or in response to conventional treatment. These stories remind me of an anecdote, possibly apocryphal, about Emile Zola of “J’accuse!” fame. Zola, a renowned atheist, upon witnessing the many crutches abandoned at Lourdes observed that there were no wooden legs. The question “Why doesn’t God heal amputees?” has become an internet meme in the atheist community. For reasons I’ve mentioned elsewhere, I don’t think it’s a knock-down atheistic argument, I understand the sentiment behind it.

So, I’m sympathetic to skepticism, at least as a starting point, with respect to miracle claims. To paraphrase Hume, it would take a great deal of evidence to believe in a miracle. This Humean point has been paraphrased as another free-thought mantra: “extraordinary claims require extraordinary evidence.” As a rhetorical device, it does its job, however, it’s seldom defined with any degree of philosophical rigor. Since I don’t want to get too technical, I’ll simply agree that a resurrection qualifies as an extraordinary claim. But what is meant by ‘extraordinary evidence’? As I mentioned, Hume lived before Bayes’ Theorem, so we can sharpen some of this terminology in light of advances in probability theory. If we think along Bayesian lines, the amount of evidence necessary to render an extraordinary claim probable would be the amount of evidence necessary to raise the probability of the claim above .5 (with 1 being certainty). So ‘extraordinary evidence’ for our purposes is simply evidence sufficient to raise the probability of the claim above .5.

But here’s where things get complicated. Probability only makes sense on the basis of background knowledge. For example, we have to assign each claim a prior probability (henceforth ‘prior’) based on our background knowledge. When skeptics allege that a miracle is, by definition, an improbable event, they usually mean that it has a low prior based on our background knowledge. However, the theist and atheist are going to disagree about our background knowledge and, thus, about the prior. Importantly, background knowledge is not worldview neutral. The theist, who takes God’s existence as part of his background knowledge, is going to have a higher prior for miracles (maybe as high as .5). After all, if God created the entire universe out of nothing, a resurrection isn’t such a big deal. The atheist, however, does not take the existence of God as part of his background knowledge and, therefore, will assign a much lower prior (close to 0). Is there a way around this impasse? Well, for my part, I’m happy to assign a low prior (see below) because resurrections, if they happen at all, happen infrequently. In other words, if there is a God, he’s stingy with resurrections. I would agree that an event like a resurrection, even on theism, has a very low prior (but still not 0).

Of course, we have to consider more than the prior. We also have to consider specific evidence that might raise the posterior probability above .5. In other words, although an event may have a very low prior probability (like a man walking on the moon) there might be specific evidence (eyewitnesses, documentation, film footage, etc.) that raises the overall probability above .5. Therefore, determining the prior is only the first step in determining overall probability, not the last. Incidentally, Hume is often accused of considering only the prior probability, i.e. our background knowledge, in his critique of miracles [Pr (R/B) instead of Pr (R/B&E)*]. But we have another problem here: Christians and atheists are going to disagree about the strength of the specific evidence in the case of the resurrection. Pretty much all of the evidence that we have for the resurrection (empty tomb, postmortem appearances) comes from alleged eyewitness testimony in the four gospels and Paul’s letters. Christians typically think this testimony is trustworthy; skeptics do not. Once again, it seems like someone will have to make a concession.

One skeptic who is willing to play along and assume eyewitness testimony for the sake of argument is Stephen Law. However, he’s unimpressed with the power of eyewitness testimony to overcome very low priors. He uses a hypothetical scenario called the “Ted and Sarah case” to make his point. Suppose two close friends, Ted and Sarah, who are generally reliable, not given to practical jokes, etc. tell you that a man named Bert “flew around their sitting room by flapping his arms, died, came back to life again, and finished by temporarily transforming their sofa into a donkey.” Law concludes that he is not justified in believing that his friends have witnessed a miracle. Although his friends’ testimony provides some evidence, it is by no means sufficient. Law then makes the analogy between the Ted and Sarah case and the gospels explicit:

Of course, we should acknowledge there are differences between the historical evidence for the miracles of Jesus and the evidence provided by Ted and Sarah that miracles were performed in their sitting room. For example, we have only two individuals testifying to Bert’s miracles, whereas we have all four Gospels, plus Paul, testifying to the miracles of Jesus. However, even if we learn that Ted and Sarah were joined by three other witnesses whose testimony is then added to their own, surely that would still not raise the credibility of their collective testimony by very much.

Of course, this all sounds intuitively plausible. Unfortunately, our intuitions regarding probability are often unreliable. What does a Bayesian approach say about the number of witnesses needed to raise the probability of an extraordinary claim?

Another philosopher named Daniel Bonevac has addressed this question. Bonevac disagrees with Law’s assertion that increasing the number of witnesses only negligibly raises the probability. He sets the problem up this way: Let’s assume that the probability of a resurrection is 1 in 10 billion. Let’s further assume that the probability that someone would report a miracle if it occurred is .99. Finally, let’s suppose that the probability that someone would report a resurrection if it did not occur is .1. If we only have one witness, on Bayes’ Theorem, the odds that a resurrection occurred are 1 in a billion. So far, so good for the skeptic. But Bonevac contends that a few more witnesses drastically increase the odds. Given the numbers, it only takes 10 witnesses to bring the probability up to .5 and 12 (apostles?) to make it highly likely (.9888). Using less conservative probability estimates (.999 and .01 in place of .99 and .1) he argues that it only takes 5 witnesses (the gospels and Paul, we might say) to bring the probability of the resurrection up to .5 and 6 witnesses to make it a near certainty. If Bonevac is right, his conclusion clearly has a bearing on Law’s dismissal of eyewitness testimony.

Of course, there are some issues one could raise against Bonevac’s methodology. For example, one might raise the problem of dwindling probabilities. Later in the essay, Bonevac suggests that a series of miracles might be more credible than one miracle in isolation. But a series of miracles only raises our credence in subsequent miracles if we already know earlier miracles occurred. In other words, if we already know that Jesus did in fact turn water into wine, feed the five thousand, and raise Lazarus from the dead, then his own resurrection becomes more probable. But we can’t simply assume all of that. Ironically, in the absence of hard evidence for these earlier miracles, the series of reported miracles may serve to decrease our credence in these reports. In other words, if the probability of miracle #1 is .5, the probability of miracles #1 and #2 is .25, and the probability of miracles #1, #2 and #3 is .125. This is the problem of dwindling probabilities. Of course, one might also question the extent to which the gospels are independent and to what extent, if any, they are eyewitness accounts. Nonetheless, Bonevac’s calculations are enough to warrant caution in accepting the intuition behind Law’s thought experiment. When it comes to probability, our intuitions are usually wrong; we have to do the math.

After all that, I’m afraid my conclusion is going to be rather anticlimactic. Obviously, I don’t think we can prove a miracle or even show that it’s more probable than not. I think even the most charitable assessment of the evidence only puts the probability at .5. Strictly speaking, we’re left with agnosticism. However, I do think that we can at least show that Hume’s easy dismissal is premature. At the very least, we might have to modify our intuitions about the value of eyewitness testimony in assessing extraordinary claims. I also think this case is a good reminder that reason is not neutral with respect to one’s worldview, which will largely determine the priors one assigns. In other words, believers are not necessarily irrational in affirming the resurrection. Skeptics often rely upon the prima facie incredulity of such a claim, but, as we’ve seen, it’s quite a bit more complicated. On the basis of the prior alone, the skeptic is quite right: the odds are infinitesimal. However, if one accepts specific evidence (testimony, empty tomb, postmortem appearances), then the belief that something extraordinary happened is not flatly irrational. Naturally, the skeptic will reject this evidence and we may be left with a stalemate. Nevertheless, if a believer can show that he’s not irrational in his belief, that’s no small accomplishment.

*William Lane Craig used the following Bayesian formulation in a debate with Bart Ehrman.

B = Background Knowledge

E = Specific Evidence (testimony, empty tomb, postmortem appearances)

R = Resurrection

Pr (R/B&E)= [Pr (R/B) × Pr (E/B&R)]/ [Pr (R/B) × Pr (E/B&R) + Pr (not-R/B) × Pr (E/B& not-R)]

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Does the US Postal Service Discriminate Against Atheists?

An interesting article from io9.

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Investing in Immortality

Those banking on the Singularity can now put their money where their mouth is. Check out this investment opportunity!

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Re: Transhumanism

He’s a bit of a hipster and could’ve made his point in less than 8 1/2 minutes, but its worth a watch.

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March 14, 2013 · 8:55 pm

A Thomistic Take on Transhumanism

I find transhumanism a fascinating topic. Here’s a Thomistic view on the subject. Although I don’t necessarily agree with the author’s conclusions, it’s an interesting read.

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Is the Consensus on the Logical Problem of Evil Changing?

I’ve been following exapologist’s posts on some new developments regarding the logical problem of evil (LPOE). I especially recommend this post, ‘Plantinga’s Abject Failure?‘ Those who follow philosophy of religion know that, for the last several years, the consensus has been that Plantinga’s Free Will Defense (FWD) defeated the LPOE. Theists and atheists alike have, for the most part, agreed on this and discussion has shifted to the evidential problem. Theists frequently appeal to this consensus whenever the problem of evil comes up in debate. However, a series of new papers — all written by theists as far as I can tell — has begun to question this reigning consensus. You can find the details through the above link. In summary, these papers involve questioning Plantinga’s notion of Trans-world Depravity (TWD), which asserts that in every logically possible world, free creatures go wrong. Critics are now arguing that TWD is necessarily false. If this is true, then the FWD is not the knock-down argument many theists think it is. It would also breathe new life into the LPOE. Perhaps it’s time for us to rethink the consensus on the issue.

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Ode to A Dictator

In the old days, when a dictator died, somebody would write a folk ballad about it. Here’s one for Josef Stalin. I doubt Hugo Chavez will get this treatment.

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