P[M/(T&K)] = P[T/(M&K)] P[M/K] / P[T/(M&K)] P[M/K] + P[T/(-M&K)] P[-M/K]The more reliable the witness, the greater the credibility of the testimony. But also, the more unlikely the event to which the witness is testifying, the smaller the credibility of the testimony.
Before considering numerical values, let's simplify a bit. Let cK , credibility in context K, represent P[M/T&K)], and let pK , the prior probability of a miracle occurring in K, represent P[M/K]. Let's assign it the value 10-m. Let lK , the probability that the witness giving a miracle report is lying in K, represent P[T/(-M&K)], and assume it has the value 10-r. Then, assuming that the probability that someone witnessing a miracle will report it is relatively high, and that lK is significantly greater than pK , Bayes's formula allows us to approximate the credibility of a witness in context K as follows:
cK ≈ pK / lK = 10r - m.Now, to get actual probabilities out of Bayes's theorem, we need to have values for the prior probability of the miraculous event occurring and values for the reliability of the witness or witnesses. Assessing these is of course immensely difficult. But let's make a rough estimate for a report claiming someone to have been raised from the dead.
First, we need to estimate the prior probability of such an event. The Bible contains several such reports, but their veracity is in question. Since the beginning of time there have been, within an order of magnitude or so, about ten billion human beings on the planet. And there have been only a few scattered reports of resurrections, whose credibility is in question. So, let's estimate the probability of resurrection, given the available evidence, at 1 in 10 billion: 10-10.
The reliability of witnesses is perhaps easier to estimate. People are generally reliable, especially on matters such as whether someone is walking on or through the water, whether someone is alive or dead, etcetera. Indeed, as Donald Davidson has argued, the possibility of linguistic communication depends on such reliability. In the case of a miracle report, we must estimate the probability that someone, a disciple of Jesus, say, will report a miracle if it occurs. Presumably the probability is very high, though it is not 1, as Peter's denial of Jesus illustrates. So, let's estimate the probability, cautiously, at .99. What about the probability that someone will report a miracle even if one does not occur? David Owen and others have assumed, given the values we've estimated so far, that this will be unlikely, having probability .01. Hume clearly thinks it is higher; disciples having a tendency to inflate the reputation of their leader. Still, very few spiritual leaders have been alleged to have the ability raise people from the dead. (No such reports are associated with Confucius, Laozi, the Buddha, Zoroaster, or Mohammed, for example.) So, let's estimate this, cautiously, at .1.
Now, given these estimates, Bayes's theorem tells us that the probability of someone's being raised from the dead, given testimony to such an event, is approximate 10-9: one in a billion. Hume appears to be vindicated. The probability we should rationally assign to someone's being raised from the dead, even given testimony that it has occurred, is very low. Even if we abandon our cautious estimates above, raising the witness's reliability to .999 and lowering the likelihood of a false report .01, the odds of the report's being correct are approximately 10-8, one in a hundred million. Hume is right that "no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavours to establish" (90) -- so long as the miracle in question is isolated, violates a law of nature (or at least has an extremely low prior probability), and is attested by a single witness.
But now, suppose that we have not one witness but several. As John Earman and Rodney Holder have observed, having multiple witnesses changes the outcome of our assessment of miracle reports dramatically. Oddly, few other philosophers have thought the number of witnesses makes any difference. Dawid, Gillies, and Sobel, for instance, speak simply of "a witness or group of witnesses." Yet an analogy to law should suggest that this is absurd. It matters how many independent witnesses testify similarly. One witness who identifies the perpetrator has some effect on the probability of guilt or innocence; a dozen who independently do so have a much more powerful effect.
If we were to take Hume's argument as showing that testimony can never establish the likelihood of a miracle, as he wants us to, it would prove too much. Hume's argument depends solely on the thought that miracles are extremely unlikely. So, his argument should apply to anything that has a very low priori probability. It thus, if successful, will imply that we can never be rationally justified in believing that an extremely unlikely event has actually occurred. But that is outrageous.
Consider a situation that might be represented by similar calculations: a case of medical diagnosis. Suppose that a highly reliable test diagnoses you as having an exceptionally rare disease. Say that the reliability of the test is .999; it is wrong in only one case in a thousand. And suppose the disease is very rare, afflicting only one person in a million. What is the probability that you actually have the disease? According to Bayes's theorem, only about 103 - 6 = 10-3, that is, about one in a thousand! Although the test is right 999 times out of a thousand, its positive result in your case will be a false positive 999 times out of a thousand.
This result is surprising. But think of how the test might function applied to all the roughly 300 million residents of the United states. About 300 would have the disease, and the test would accurate give a positive result for (nearly) all of them. But 299,999,700 people would not have the disease, and the test, wrong only one time in a thousand, would nevertheless produce about 300,000 false positives. So, the test, applied to the population of the U.S., would come up positive 300,300 times, and be right in only 300 of them. We tend to ignore base rates (that is, low prior probabilities) in our thinking, something some psychologists have dubbed a "cognitive illusion." So, one test, even if it is highly reliable, is not very good evidence that any particular person has a rare disease.
But it would be absurd to conclude from this that we can never have good reason to believe that any particular person has a rare disease. True, any single test, taken by itself, is poor evidence. But, faced with a positive result, what might we do? We might repeat the same test. We might administer additional tests. We might look for symptoms. In short, we might gather additional evidence.
Analogously, faced with a miracle report, we ought rationally to gather additional evidence. Just as we might seek additional tests, we might, for example, seek testimony of additional and independent witnesses. Suppose we have n independent witnesses, all of equal reliability. Then Bayes's theorem tells us that the credibility of their reports, taken together, is:
P[M/(T&K)] = P[T/(M&K)]n P[M/K] / P[T/(M&K)]n P[M/K] + P[T/(-M&K)]n P[-M/K]
Approximating as before, we get something that breaks down as nr approaches m:
cK ≈ pK / lKn = 10nr - m
Let's apply this to the medical diagnosis case. We have 300,000 false positives and only 300 true positives. Suppose we apply a second medical test the reliability of which equals that of our first test, getting it right 999 times out of a thousand, but the errors of which are probabilistically independent of those of the first test. The second test will give a positive result in (almost) all the 300 true positive cases. It will also give a false positive result in 300 of the 300,000 false positives from the original test. So, we end up with 600 positives, of which half are real. The probability of having the disease, given positive results on both tests, is about .5. The second test, or "witness," if you like, raises the probability from one in a thousand to an even bet.
The same principle applies to the case of Biblical miracles. Given our cautious estimates, it would take ten witnesses to make the miracle have close to .5 probability (actually .4749), and twelve (!) to make it highly likely (.9888). Give our incautious estimates -- appropriate for the most trusted disciples, such as Peter, James, and John, say -- these levels are reached much more quickly. Five independent witnesses give the miracle an even chance of occurring; six make it highly probable.
One might object that the disciples are not independent witnesses, but very much under one another's influence; that the four Gospels are not entirely independent, but depend on many of the same sources; that many miracle reports were recorded long after the miracles are supposed to have taken place; and so on. There is something to these objections, though less, perhaps, than many think. Minor differences in the Gospel accounts offer evidence of independence. All the Apostles who faced imprisonment, beatings, and martyrdom for their testimony had strong incentive to recant anything for which they did not have overwhelming independent evidence. The Gospels appear to have been written within the lifespans of those who knew Jesus and witnessed the events recorded in them. But return to Paul's argument concerning the Resurrection. Writing perhaps just twenty to twenty-five years after the event, he points to hundreds of witnesses. Not all are independent, but many are. The credibility he attaches to the Resurrection is thus, reasonably, very high, even setting aside his own experience on the road to Damascus.
Paul was in a far better epistemic state with respect to Christ's Resurrection than we are, say, with respect to the attack on a canoeing President Carter by a crazed, swimming rabbit in 1980. That was surely an improbable event, a little further removed in time, witnessed by only a few government employees whose reliability may not compare very well with that of the disciples. Yet most of us -- rationally -- believe that it occurred. If we are to throw out belief in Biblical miracles on Humean grounds, we should throw out many of our historical beliefs on those very same grounds, for they would fail Hume's test too, and for the very same reasons.
"The Argument from Miracles"
Oxford Studies in Philosophy of Religion, volume 3.