Doctrine of Creation (Part 17): Hume’s Abject Failure

Hume’s Abject Failure

Today we want to look at David Hume’s “in principle” argument against miracles. Despite its influence, Hume’s argument is generally recognized by philosophers today, in the words of the philosopher of science John Earman, as an “abject failure.”[1] Earman is a Professor of History and Philosophy of Science at the University of Pittsburgh and not a Christian (not even a theist), and yet he recognizes that Hume’s argument against miracles is, as he puts it, an abject failure. What Earman means by that is that it's not just a minor mistake – this argument is demonstrably, irremediably a failure. Even Hume’s admirers today try at most to salvage some insightful nugget from Hume’s convoluted discussion, typically Hume’s maxim that “no testimony . . . is sufficient to establish a miracle, unless this testimony is of such a kind that . . . its falsehood would be more miraculous, than the fact which it endeavors to establish.” But, as we'll see, even that maxim requires re-interpretation.

Hume’s “in principle” argument actually involves two more or less independent claims. First, on the one hand, there is his claim that miracles are by definition utterly improbable. Secondly, on the other hand, there is his claim that no amount of evidence could ever serve to overcome that intrinsic improbability. So, on the one hand, miracles are intrinsically, utterly improbable; and secondly, no amount of evidence could possibly overcome that improbability and establish the probability of a miracle. Well, as it turns out, both of these claims are mistaken.

Let's look first at the second claim that no amount of evidence could ever serve to establish a miracle. Stimulated by Hume’s argument against miracles, there arose a discussion among probability theorists from Nicolas de Condorcet in the 18th century to John Stuart Mill in the 19th century over how much evidence it would take to establish the occurrence of a highly improbable event.[2] It was soon realized by probability theorists that if you simply weigh the probability of the event over against the reliability of the witnesses to the event, then we would be led into denying the occurrence of events which, though highly improbable, we reasonably know to have actually occurred. To give an example, suppose on the morning news you hear that the pick in last night's lottery was 7-4-9-2-8-7-1. This is a report of an event-- that that number would be picked--that is extraordinarily improbable, one out of several million, and even if the morning news' accuracy is known to be 99.99% reliable, nevertheless the improbability of the event will swamp the probability of the witness’s reliability, so that we should never believe such a report. Even the lottery winner should never believe that, in fact, the report is accurate. In order to believe the report, Hume would require us to have enough evidence in favor of the morning news’ reliability to counterbalance the intrinsic improbability of the event itself, which is just absurd.

What probability theorists came to see is that what also needs to be considered is not just the intrinsic probability of the event or the reliability of the witness, but you also need to consider the probability that if the reported event had not occurred, then the witness's testimony would be just as it is. You need to weigh the probability that if the event had not occurred then the witness's testimony would be just as it is. As John Stuart Mill wrote,

So you've got to weigh the probability that the evidence would be just as it is if in fact the event had not taken place.

To return to our example of the morning news, the probability that the morning news would announce the pick as 7-4-9-2-8-7-1 if some other number had in fact been chosen is incredibly small given that the newscasters had no preference for that announced number. On the other hand, the announcement is much more probable if 7-4-9-2-8-7-1 were the actual number chosen. This comparative likelihood easily counterbalances the high improbability of the event reported. So, even though the event itself is highly improbable, nevertheless the improbability that the evidence would be just as it is if the event had not occurred can counterbalance that high intrinsic improbability.

Let’s proceed to look at this more closely.

The realization on the part of probability theorists that other factors need to be included in the correct calculation of the probability of some event comes to expression in a formula of probability theory known as Bayes’ Theorem. Let’s let R represent some miraculous event, say the resurrection of Jesus. Let's let E represent the specific evidence for that event. In the case of the resurrection, in my analysis this would be the facts of the empty tomb, the postmortem appearances of Jesus, and the very origin of the Christian faith itself. Those would be comprised in the specific evidence E. Finally, let B represent our general background information of the world apart from the specific evidence E. So you take our basic knowledge of the world and abstract from that E (take E out of it) and that will leave you with B – the background knowledge of the world.

Bayes’ Theorem allows us to calculate the probability of R in a so-called “odds form” which is one of the simplest forms of Bayes’ Theorem. But before I put this on the board, I recognize that many of us suffer from what my friend Lydia McGrew calls lurking math-o-phobia; that is to say, when we see an equation our eyes sort of glaze over, and it's difficult even to take it in. But in this case I'm going to go through it slowly, and I think make it quite comprehensible. So stick with me and we will examine it together.

Pr(R|E&B)             Pr(R|B)                 Pr(E|R&B)

_____________  =  __________    ☓   _____________

Pr(not-R|E&B)       Pr(not-R|B)           Pr(E|not-R&B)

We want to consider what is the probability (which we represent by Pr) of the resurrection of Jesus on the evidence and the background information. So Pr is probability, R is the resurrection hypothesis, and the straight line [ | ] indicates that we're going to consider the probability of R given E and B, or on the assumption of E and B, or relative to E and B. So what is the probability of R given the specific evidence and the background information? We're going to compare that to the probability of not-R on E and B – that is to say, what is the probability that the resurrection did not occur given the evidence and the background information? This ratio expresses to us the probability of the resurrection on the total evidence E and B – the background information and the specific evidence.

This ratio will enable us to determine the odds of the resurrection being true on E and B. If the number in the numerator is smaller than the number in the denominator, then it will turn out that the resurrection is improbable. What Hume wants to argue is that the numerator in this case is always inevitably going to be less than the denominator, and therefore it can never be rational to believe in the resurrection. If the ratio were 1-to-1, then that would mean that they have an equal chance of occurring and so the odds of the resurrection occurring would be 50/50 or 50%. If you have a 1-to-1 ratio, you've got odds of 50/50 for the resurrection occurring. But if the numerator is smaller than the denominator then the odds of the resurrection occurring are less than 50%. What Hume wants to show is that in principle the numerator is always smaller than the denominator, and therefore given the odds no rational person should ever believe (no matter what the evidence is) that the resurrection has taken place.

Whether or not the resurrection is more probable than not is going to depend upon two other ratios on the right-hand side of the equation. In the first ratio, we consider the probability of the resurrection on the background information alone [Pr(R|B)]. Leaving aside the specific evidence, what is the probability of the resurrection just given the background information? And then we consider the probability that the resurrection did not take place on the background information [Pr(not-R|B)]. So, what is the probability of the resurrection or not given the background information and leaving aside the specific evidence for the resurrection? This ratio gives us the intrinsic probability of the resurrection. It is the prior probability of the resurrection before you look at the specific evidence. Before you look at any evidence, this is the probability of the resurrection – just the intrinsic probability of the resurrection. So we're simply asking: given our background information of the world without any specific evidence, which is more probable? R or not-R?

In the second ratio, which is multiplied by the first, we consider what is the probability of the evidence given the resurrection and the background information [Pr(E|R&B)], and we contrast that with the probability of the evidence given that the resurrection did not occur [Pr(E|not-R&B)]. So, what is the probability that the evidence would be as it is if the resurrection did take place, and what is the probability that the evidence would be as it is if, in fact, the resurrection had not taken place? This is called the explanatory power of the hypothesis. How well does the event or hypothesis explain the evidence? Is the evidence more probable on the hypothesis than on the negation of the hypothesis? That's the explanatory power. What we have in the right hand side of the equation is the intrinsic probability of the resurrection multiplied by the explanatory power of the resurrection.

Notice that even if the intrinsic probability of the resurrection is very low – suppose relative to the background information, not-R is vastly more probable than R – that doesn't mean that the resurrection is improbable on the total evidence because that improbability could be counterbalanced by the higher explanatory power of the resurrection hypothesis. Even if this ratio [Pr(R|B) / Pr(not-R|B)] is very low, this one [Pr(E|R&B) / Pr(E|not-R&B)] could be very high and counterbalancing. For example, suppose that the intrinsic probability of the resurrection is 1-to-90. Nevertheless, suppose that the explanatory power of the resurrection is 90-to-1. In that case, you multiply these together and you get 1-over-1 which means the resurrection has a 50% chance of being true. So you can see that even if the intrinsic probability of the resurrection is extremely low, so long as the second ratio is extremely high it can counterbalance any improbability intrinsically in the resurrection itself. That was the factor that Mill and others identified as being critical. What is going to be the probability if the event had not occurred that the evidence would be just as it is?

Hume, in his argument, never discusses the second ratio. He focuses entirely on the intrinsic probability of a miracle and argues that because this value is so low, therefore the probability of the resurrection on the evidence and the background information is comparably low. He clearly overlooked the explanatory power of the resurrection hypothesis, so that his argument is demonstrably a failure. As I said, even if this ratio is incredibly low, so long as this ratio is comparably high it can counterbalance it. So it's just demonstrably mathematically false that the intrinsic probability of a miracle can never be overcome. So much for Hume’s vaunted in principle argument!

There is a slogan beloved in the free thought culture: “Extraordinary events require extraordinary evidence.” I don't know how many times I've heard this said as an excuse for not believing in the miracles and the resurrection of Jesus. “Extraordinary events require extraordinary evidence.” But what we can now see is that this seemingly commonsensical slogan is, in fact, false as usually understood. In order to establish the occurrence of a highly improbable event, you don't need to have lots of evidence. What the skeptic seems to be saying by his slogan is that in order for us to believe in a miraculous event you've got to have a tremendous amount of evidence. But why think that that's the case? Because a miracle is so improbable, the skeptic will say. But Bayes’ Theorem shows that rationally believing in a highly improbable event doesn't require an enormous amount of evidence. All that is crucial is that the evidence is far more probable given the occurrence of the event than it would be if the event had not taken place. The bottom line is that it doesn't always take a huge amount of evidence to establish the occurrence of a miracle.