Here is a simple representation using Bayes’ Theorem of how a miracle claim would be assessed, where m is the claim that a miracle has occurred, e is the evidence for the claim, and k is background knowledge:

                     p(e/m & k) × p(m/k)

p(m/e & k) = --------------------------------

                         p(e/k)

So, the credibility of a miracle claim given certain evidence and background comes down to three factors:

(1) p(e/m & k), the likelihood that we would have the evidence e given that the miracle did take place and given our relevant background knowledge.

(2) p(m/k), the prior probability of the occurrence of the miracle, that is its probability given only background knowledge and independently of the particular evidence e that we are now considering.

(3) p(e/k), the likelihood of having the evidence e given only background knowledge. This is equivalent to the total probability of e: p(e/m & k) × p(m/k) + p(e/~m & k) × p(~m/k), that is, the probability that we would have evidence e whether or not m took place.

Since the publication of Hume’s famous Section X, “Of Miracles,” of his Enquiry Concerning Human Understanding, debates over the credibility of miracles claims have centered on factor (2), the prior probability of a miracle. Hume focused on this, arguing that a violation of a law of nature would be an event of such low prior probability, that the evidence required to establish its occurrence would have to be impossibly strong (Hume restricted his study to testimonial evidence). Actually, the impossibility is not an impossibility in principle, but it practice. Hume does admit that, in principle, it could be an even greater miracle that certain evidence were false than that a claimed miracle had occurred. For instance, testimony for a miraculous event might be found from a source (or, better, multiple independent sources) so absolutely reliable that it would have an infinitesimal probability of being wrong.

As a case in point, Hume imagines that every nation on earth had recorded eight days of total darkness over the whole world beginning January 1, 1600. In such a case it would be even more marvelous if all those independent records were false than that such an event actually took place, and therefore the testimony should be believed. However, Hume, changing hats from the philosopher to the historian, shows that the actual circumstances of historical miracle reports have provided evidence far weaker than that required to establish a violation of natural law. So, while evidence strong enough to overcome the extremely low prior probability of a miracle is conceivable, all historical miracle claims fall far short of this ideal.

I think that the focus on factor (2) by skeptics is somewhat misplaced and distracts from the most serious issue concerning miracle claims, namely, that p(e/k) is in very many cases going to be quite high. That is, as I see it, the most serious problem with miracle claims is that testimony for miracles is so often generated by non-miraculous causes. When we are confronted with a miracle claim, we have to ask how likely it would be that we would have the cited evidence for the claim even if the miracle did not occur. In very many circumstances, we would have to say that it would be quite likely, and this will greatly lower the credibility of the claim. More on this below.

Why not judge miracle reports on the basis of prior probabilities? The problem is that when we are assessing the personal probability of miracles, as we are here, the priors will vary very considerably—often radically—from person to person. So, much comes down to what we are tacitly including in “k,” our presumed background knowledge. A skeptic such as Hume, who does not presuppose the existence of God, will, of course, put p(m/k) very low, not far from zero. On the other hand, a Christian, one who believes in a God who can and on occasion will perform miracles, will often have a very different prior probability for p(m/k) for a given m. In other words, if k_s is the presumed background knowledge of the skeptic, and k_c is the presumed background knowledge of the Christian, then, for many purported miracles, p(m/k_c) ≫ p(m/k_s).

Given this great divergence of initial priors, it is a rather mundane and expected result when, for a given miracle, the evidence will look great to Christians and poor to skeptics. This is hardly surprising since evidence quite sufficient to overcome a moderate burden of proof will be woefully insufficient to overcome a very heavy burden. If a friend tells me he has a pet goat, then I might think this a bit odd, but his testimony would probably be sufficient to convince me. If he told me that he has a pet Tyrannosaurus rex, his testimony would only make me think that he was joking or delusional. In short, given their greatly divergent priors, debates between skeptics and believers over miracle claims usually wind up just where they began. Even scientific debates can be perennially intractable when the priors are too far apart.

Now Hume does not claim that his argument decisively debunks all miracle claims. He only presents it as providing the “wise and the learned” with a “check to all kinds of superstitious delusion.” In other words, he is confident that the well-girded skeptic, one who has wisely placed his priors low enough, need not worry that apologists will browbeat him with miracle stories. In effect, by placing his priors very low, the skeptic makes the apologist’s burden of proof so heavy that he is insulated from the “impertinent solicitations” of miracle-mongers.

Hume’s argument is logically sound but concedes a rhetorical victory to religious apologists, and even in philosophical debates rhetoric is important. True, if you only want to insulate yourself from the “impertinent solicitations” of pesky apologists, all you have to do is put your priors for miracles close to zero and damn anyone to lift that stupendous burden of proof. While this strategy is logically impeccable, it draws the charge of narrow-mindedness, the accusation that you hermetically seal yourself in ideological presuppositions and simply dismiss the evidence of the miraculous. We live in a highly balkanized intellectual milieu, and strategies that effectively silence debate only contribute to that polarization.

Theists, of course, sometimes make their own contributions to debate-stifling balkanization. In his first version of “Reformed Epistemology” written in the 1980s, Alvin Plantinga recommended that Christians identify their properly basic beliefs with reference to their own paradigms of proper basicality. For instance, Christians could just take the existence of God as paradigmatically properly basic. Plantinga realizes that nonbelievers will disagree, but his response is the philosophical equivalent of thumbing his nose and emitting a Bronx cheer. Christians have to be true to their own paradigms, not those of unbelievers, Plantinga says. So nyaah, nyaah.

A more effective strategy for the skeptical assessment of miracle claims is to focus on the “Bayes Factor.” It is a consequence of Bayes’ Theorem that the evidential force of any evidence e for any claim h is proportional to the ratio

p(e/~h)

———-

p(e/h)

The smaller this ratio, then the more probable it will be that we will have the evidence if the claim is true and not have the evidence if the claim is false. So, the lower the ratio, the greater is the confirmation of h by e (see Colin Howson and Peter Urbach, Scientific Reasoning: The Bayesian Approach, p.97). This makes intuitive sense. For the fingerprints on the ornamental dagger to be evidence that the butler killed Colonel Mustard, it has to be likely that the fingerprints would be there if the butler did kill the Colonel and unlikely that they would be there if he did not.

Where h is m, the claim that a miracle has occurred, the evidence strongly confirms the miracle when p(e/~m) is low and p(e/m) is high. That is, the miracle claim is more strongly confirmed the less likely it is that we would have the evidence if the miracle did not occur and the more likely it is that we would have the evidence if the miracle did occur. Now, it is surely quite likely that if a miracle did occur before witnesses who wanted it to occur, that they would report that it occurred. So, the crucial factor is p(e/~m & k), how likely we are to have e, the evidence supporting a miracle claim, if, in fact, m did not occur and given our other background knowledge.

We now have many reasons, many more than Hume could have known, for regarding it as very likely that we will have miracle reports when no miracle has occurred. Here are some things that more recent research has shown about the human tendency to report the extraordinary when nothing extraordinary has happened:

1) Much psychological research has shown the extent to which perception is constructive. All too often we see what we want to see or what we think we should see rather than what is actually there. All trial lawyers know how easily eyewitness testimony can go wrong. Witnesses will swear in court that they have seen what it is demonstrably physically impossible for them to have seen. It is not that they are committing perjury. All too often they are absolutely convinced that they saw what they claim to have seen. Strong desires or expectations seriously bias our judgments as well as our perceptions. Further, in many circumstances, such as when we are part of large, emotionally excited groups, our perceptions can be even less reliable and our judgments even more impaired. We have known at least since the 19^th Century about mass popular delusions and the “madness of crowds.”

This, among other reasons, is why Paul’s claim about the “500” that supposedly saw the risen Jesus (I Corinthians 15: 6) is not reliable. Supposing that a crowd of 500 persons did assemble at some point shortly after Jesus’ crucifixion, how can we know what really happened? Did Jesus appear on a hilltop or on a stage so that everybody could clearly see him? 500 is a pretty big crowd. Could everybody get close enough to get a good look and independently ascertain that the person was Jesus? Did each one know Jesus by sight? Had the crowd been emotionally aroused so that they were expecting to see something extraordinary? If so, can we be sure that the purported sighting was not a case of the “madness of crowds?”

2) We now know a great deal about false memories and how they are implanted. Researcher Elizabeth Loftus has shown how easily people can be led to misremember all sorts of facts and details and even invent whole episodes. In the 1980s there were tragic incidents of innocent people being accused of having sexually molested their children. Often, these accusations arose when quack “therapists” would convince their clients that they had repressed memories of childhood sexual abuse. With the “aid” of these quacks people would experience what seemed to them to be recovered memories of childhood sexual abuse. In fact, these “memories” were whole-cloth fantasies constructed by the “therapists.” Families were destroyed and some innocent people went to prison.

Like perception, memory is largely a construction. We remember things as they should have been or as how we want them to have been rather than how they were. Experiments have shown that when we form memories we are highly susceptible to suggestions about what we have seen. When events occur in emotionally charged settings, memories will be affected by the emotional ambience. Memories passed on by oral tradition can acquire yet another layer of fiction, as we see below.

3) Hallucinations and other sensory delusions are now known to be much more common, even among psychologically healthy people than was previously believed. Oliver Sacks’ recent book Hallucinations shows that this is so. All sorts of factors can lead psychologically normal people to hallucinate—grief, emotional duress, sensory deprivation or monotony, and exhaustion, for instance. A friend of mine used to work for the Nuclear Regulatory Commission. Once, a vindictive supervisor ordered her to perform multiple inspections of various widely separated sites during a short time frame. Driving late at night from one site to the other with no sleep, she hallucinated what she described as a “cartoon dog” running alongside her car.

Hypnagogic and hypnopompic hallucinations are well-known phenomena that sometimes occur just as people are going to sleep or waking. They have been known for centuries and probably account for many reported experiences of demons, witches, or ghosts. In the 1980s many people, including author Whitley Strieber, reported that they had been abducted by aliens, taken on board spacecraft, and subjected to what were apparently medical probes. These experiences seemed very real to the people that endured them, yet they were in all probability due to hypnagogic or hypnopompic hallucinations.

4) Folklorists now know how stories can grow and spread through a community and how rapidly they can take on fantastic or miraculous content. Even in an era of electronic communications, and even when eyewitnesses are alive and vigorous, false stories can and do spread widely. Consider the famous case of Flight Nineteen: In December 1945 a flight of TBF Avenger dive bombers took off for a training mission from Ft. Lauderdale, Florida and subsequently disappeared. Within thirty years, written accounts told weird stories of how the flight had met its allegedly mysterious end in the “Bermuda Triangle.” Magazine stories and books recounted that the control tower in Ft. Lauderdale had received disturbing and bizarre communications from the flight, messages indicating paranormal occurrences and inexplicable events. Writers speculated that the whole flight had been abducted by aliens. Steven Spielberg played along and had the aircraft and their crews returned by the ETs in Close Encounters of the Third Kind.

In the mid-70s, the PBS science show Nova did a debunking of the Flight Nineteen “mystery.” In the program they assembled personnel from the control tower that had monitored all of the communications from the flight. They unanimously and emphatically insisted that none of the weird and bizarre messages had been received and that the whole story was a fabrication. In an era of modern communications, and when eyewitnesses were alive and communicative, a wild and false story still took root and spread. In general, we now know how the oral transmission of information can be corrupted and influenced by all sorts of factors. Human individual and collective memory is a highly fallible process and stories transmitted down generations are protean in detail. When recounting events we tend to recall gist rather than specifics and imagination and wishful thinking are always ready to impact the story.

In summary, we now know much than was known in Hume’s day about human psychology and how bizarre stories can start and grow. While skeptics certainly have rights to their own priors, the debate should center not on what we presume but on the objective, known ways that tales of the marvelous can grow and spread.