What follows is the Appendix to my anthology The Case against Miracles (pp. 551-560). I consider several parts of that book to be a major defense of David Hume. I know there is some debate on Hume, but what Hume said on miracles withstands the criticisms leveled at him. They come from both Christian apologists and philosophers (as one would expect), but also from some atheist philosophers, like Michael Martin (Atheism: A Philosophical Justification, pp. 194-196), Michael Levine (The Cambridge Companion to Miracles, pp. 291-308), and Graham Oppy (Arguing About Gods, pp. 376-382), who strangely says "Hume's argument against belief in miracle reports fails no less surely than do the various arguments from miracle reports to the existence of an orthodoxy conceived monotheistic god" (p. 381). Agnostic/atheist John Earman thinks Hume's argument is an Abject Failure (as seen in his book by that title). And while J.L. Mackie defends Hume against some objections, even he thinks Hume's argument needs "improvement" (p. 25) by being "tidied up and restated" (p. 17) due to "inaccuracies" (p. 27), with one part he calls "very unsatisfactory" (p. 23).

Here's a brief introduction to the debate on miracles LINK. Now for my Appendix:

Hume On Proof and Mathematical Probability by John W. Loftus.

As mentioned in my chapter 3, a major defense of Hume on miracles was to be published in 2019, written by Humean scholar William L. Vanderburgh, titled, David Hume On Miracles, Evidence, and Probability (Lanham: Lexington, 2019). This present anthology on miracles was already in the production stages when I received a copy of it. So, I’m thankful my publisher allowed me to add this Appendix where I’ll briefly review what Vanderburgh writes, since after reading it I think he gets Hume right.

It’s easy to criticize Hume if one seeks to do so. Just nitpick, for one thing. Or uncharitably foster upon his one lone chapter on miracles— of about 8700 words—a worst possible interpretation. With so many uncharitable criticisms of Hume out there, the hard part is to buck the trend by trying to understand Hume.

As I mentioned in my chapter 3 there are some atheists who have agreed with the unfair Christian criticisms of Hume. In my book Unapologetic I mention one possible atheist motive, which is to placate Christian apologists into thinking well of them. By agreeing with the Christian criticisms of Hume, Christians in turn will reward them by quoting from them, asking them to write for their anthologies, and even debate them, which brings them a bigger audience and more respect. But if atheists treat the Christian faith as it deserves, with no more respect than any other delusional faith, or if they defend Hume, Christians will ignore you if they can. That’s the strategy of William Lane Craig, my former professor at Trinity Evangelical Divinity School. He won’t acknowledge my work as important, or debate me. He dislikes the fact that I’m blunt about his faith, by saying it’s delusional. He is embarrassed that I exist, hoping others will follow his example by ignoring me. To do so, he has to ignore my work and the fact that other Christian scholars have acknowledged its importance, like Norman Geisler, Chad Meister, Dale Allison, Gary Habermas, Karl Giberson, James Sennett, Victor Reppert, Randal Rauser, David Marshall, and several others.

Since Christians are convinced Hume’s arguments against miracles have been “refuted” (per William Lane Craig in a personal conversation), some atheists are trying to reach out to Christians by avoiding Hume entirely. Benjamin Blake Speed Watkins, who Hosts and Produces Real Atheology – A Philosophy of Religion Podcast, is one of them. He said, “I see arguments against miracles as separate from Hume. We can have substantive discussion without ever appealing to him.” Now this can be done, I don’t doubt. But I don’t see the need to do it. I also think we should give credit where credit is due. I can’t imagine ignoring David Hume, the philosopher who originally put the issue of miracles up for discussion, like no other had done before. That would be an injustice. I understand that Christians identify the best arguments against miracles with Hume’s arguments. I also understand they think Hume’s arguments fail. So I’m not oblivious they conclude that with Hume’s demise so also goes the case against miracles. Just the same, it makes sense to defend Hume if he can be defended, and I think he can. When we do so successfully it doubly undercuts Christian apologetics. For not only do we show Hume’s arguments defeat the believability of miracles. By showing Christians are wrong about Hume, it also shows how badly they argue their case on behalf of miracles.

With so much at stake, I’m going to argue that Vanderburgh’s book is indispensable reading for understanding why Hume’s case against miracles succeeds. His book is a treasure trove of arguments in Hume’s defense. He argues that “Hume’s account of evidential probability is solid and that when it is understood on its own terms instead of being reinterpreted in a way that Hume did not intend, his conclusion about the incredibility of miracles will be seen to be correct” (p. 4). One specific hope of his is that after considering what he says, “commentators will now stop trying to give Bayesian or other mathematical analyses of Hume on probably, which cannot possibly correctly represent Hume’s own view” (p. 167).

First things first. Hume’s critics object that by defining a miracle as a “violation of the laws of nature” he begged the question of whether a miracle could occur by setting up an impossible standard to overcome. Such an objection gets Hume wrong, Vanderburgh tells us. Hume did not consider it impossible that the laws of nature could be violated by a miracle, since for him the “laws of nature are at most probable, never certain.” When we put miracle claims in the same context of questions about the existence of cause and effect, or the existence of the self, or the existence of god, they are all “epistemological rather than ontological categories.” This means the real question for Hume “is not what sorts of events do and don’t occur, but rather what events it is rational, given the available evidence, to believe to have occurred.” (p. 56) So Vanderburg answers this mistaken objection as follows: “For Hume’s purposes, only miracles conceived as violations of laws of nature deserve the name, since only miracles in that sense could possibly provide an additional, independent kind of evidence for religious hypotheses” (p. 24). That is, if miracles were events that could be completely explained within the world of nature without any supernatural interference, then they lose their capability to provide any reasonable support for religious hypotheses. This would leave the appearance of design to do all the supportive work by itself, something Hume excoriates in his Dialogues concerning Natural Religion.

Vanderburgh sums up Hume’s main argument against miracles in several places. The briefest summations are in his Prologue, where he says, “Some stories are too improbable to be believed, no matter the source of the testimony” (p. 1). Then being more specific, “there is never sufficient evidence for rational belief in the occurrence of miracles” (p. 3). His most detailed summation is as follows:
Hume’s argument here is relatively straightforward. A miracle is a violation of the laws of nature. We construct laws of nature on the dual basis of ‘a firm and unalterable experience’—that is, from an observed constant conjunction of event types, an exceptionless regularity—plus an expectation of the mind that the future resembles the past. The depth and breadth of the exceptionless regularity of past experience gives the strongest kind of warrant possible to the belief that the law will continue to hold in the same way in the future. It is not that the evidence demonstrates with certainty that the law is true, it is just that no empirical claim can possibly have stronger evidence than what we have with regard to those things we call laws of nature. Testimony, the evidence offered in opposition to the exceptionless regularity, is known to be fallible and is especially suspect in cases of reports of miracles because of the likelihood of deception or misperception. Thus, the weight of evidence derived from testimony about a purported exception to a law of nature in fact will never come close to the weight of evidence from experience that the law will be regular in all cases (italics mine, p. 50).
On this point Vanderburgh offers a helpful analogy to answer John Earman’s claim that Hume’s argument is an a priori one (i.e., independent of experience, as opposed to a posteriori, dependent on experience), by arguing that never doesn’t mean logically never. He asks us to consider the claim that, “A human will never bench press 1,500 pounds.” The current record, as he tells us, is 735.5 pounds set by Kirill Sarychev in 2015."
Given what we know about …human physiology, and the laws of physics (breaking strength of bones, etc.), it is utterly unbelievable that a human (as we currently understand the reference class) could complete a 1,500-pound raw bench press. It isn’t logically impossible, just impossible-given-what-we-know. There is a sense in which it is possible that this claim is wrong, but you still should not believe a report that someone has raw benched 1,500 pounds if you hear one. (p. 50).
Of course, an extraordinary claim that someone bench pressed 1,500 pounds would still be more likely than the miraculous claim that someone walked on water or rose from the dead. So he goes on to say, “If you would not believe the bench press claim, then on pain of inconsistency you should not believe the miracle claim either.” (p. 50–51).

Let me turn now to what I consider the main thrust of Vanderburgh’ s defense of Hume, having to do with what Hume means by “proof” and Hume’s non-mathematical kind of probability.


Hume makes a key distinction between what he calls matters of fact and relations of ideas. Relations of ideas are things that are known to be certain, because we see that their contraries contain logical contradictions. So, for example, “All bachelors are unmarried” and “2+5=7” are relations of ideas since statements like “some bachelor is married” and “2+5=6” are contradictory. Matters of fact, by contrast, are about factual matters whereby if we conceive of their opposites we do not thereby conceive a contradiction, like “this is a chair” and “the sun rises every 24 hours.” Although direct sensation and induction are typically reliable, we can make mistakes about them: this means that matters of fact cannot be certain. They have “higher and lower degrees of probability, depending on the kind and strength of the evidence available,” Vanderburgh explains. But relations of ideas, according to Hume, “provide us with certainty, effectively because they are analytic” (p. 27). For Hume, it’s a kind of category mistake to put certainty and probability on the same scale, since they apply to different kinds of ideas.

So far this is pretty much standard stuff with Hume. As I see this, it seems that most all of his critics just aren’t paying attention, for if they understood Hume’s distinction between these two categories of ideas, they could pretty much derive everything important to correctly understand what he says about miracles.

Take for instance Hume’s claim to have produced a “full proof” against miracles. Vanderburgh informs us, “[W]hether or not a miracle has occurred is a question regarding a matter of fact, and for Hume a degree of belief for or against a matter of fact can never reach the level of perfect certainty or logical necessity (demonstration), only moral certainty (proof)” (p. 106). For Hume, “a proof is a category of probability and not a certainty of the sort we have in the case of relations of ideas” (pp. 87–88). So Hume cannot be properly understood to mean that “there is zero chance that miracles can happen” (italics his, p. 87). “A proof against the existence of any miracle is still an epistemic rather than an ontological claim for Hume, because proof is an epistemic category” (italics his, p. 88). So “Contrary to what many of his critics have suggested, Hume does not think that his proof against miracles establishes the impossibility of the existence of miracles. Rather, Hume thinks that the available evidence gives us such a high degree of probability to the laws of nature that belief in the existence of miracles can never be rational— that is, sufficiently well-grounded epistemically” (italics his, p. 7).

"[T]he whole structure of Hume’s argument against miracles is a posteriori (and hence cannot lead to logically necessary propositions). Hume argues that as a matter of fact, and given what we know about human psychology and the facts of history there has never been and probably never will be an instance in which the probability that a miracle has occurred rises to the level greater than the probability that the reporter is mistaken, has been deceived, or is a deceiver. It is true that Hume puts this point very strongly but, in the context of Hume’s thought, it is easy to see that despite his rhetoric he does not really mean anything stronger than this” (italics his, p. 106).

Vanderburgh shows us that Hume stood in a long tradition of nonmathematical reasoning stretching back through Francis Bacon to the Roman world. The fact that so many of Hume’s critics have misunderstood and objected to Hume’s case against miracles is because they have interpreted him as if he was arguing for a mathematical probability. To see this as a mistake in action take a closer look at Hume’s general maxim:
“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; and even in that case there is a mutual destruction of arguments, and the superior only gives us an assurance suitable to that degree of force, which remains, after deducting the inferior.” [italics mine, Hume, “Of Miracles” #91]
It looks like Hume is using mathematical reasoning at the very end. But if this were the case then as John Earman easily objects, Hume is “double counting.” For as Vanderburgh explains, the
“degree to which one should believe that the next roll of a die will come up six is not calculated by subtracting the probabilities that the event will not occur from the probability it will occur: 5/6th minus 1/6th = 4/6th against, or 2/6th in favor. Clearly the correct degree of belief is 1/6th in favor” (p. 67).
In my view Hume surely would not make such a huge math mistake, so he was not making a mathematical argument! It should be that simple. Hume was trained in law though, and such an argument does make good sense in a criminal trial. Vanderburgh suggests we think instead of a “balance beam” where we pile evidence on one side or another to see which side has the weight of evidence for it. Vanderburgh concludes that mathematical means of understanding Hume “are not appropriately analogous to Hume’s reasoning about evidential probability” (p. 68). Vanderburgh:
“In Hume’s epistemology, mathematical calculations are relations of ideas, things that can be known with absolute certainty, whereas empirical matters of fact are not the sorts of things that can be known with certainty” (italics his, p. 29).
When it comes to Bayes’ Theorem, I’ve already discussed it in chapter 3 of this volume, where I also put a link in footnote #61 expressing some reservations on using it to investigate miracles. I stand by what I said. It’s nice Vanderburgh seems to share my reservations about it. I’ll not repeat myself here. Vanderburgh makes two claims about it. Firstly, using precise numbers to express probabilities (including but not limited to Bayes Theorem) is not the proper way to understand Hume. We’ve already seen this to be the case, and Vanderburgh has a lot more interesting things to say on that topic. Secondly, Hume’s non-mathematical approach to evidential probability is “perfectly adequate” (p. 163) and “might even be correct” (p. 112). Vanderburgh argues that “mathematical probability is not the only viable approach to problems regarding weight of evidence.” Furthermore, “in many situations the numerical approach to probability is simply inappropriate” (p. 114).

Vanderburgh argues that subjective values are not always (or easily) quantifiable (p. 127), especially when the issues are deeply complex (pp. 113–114, 126, 164). “Bayesianism,” he notes, “does not solve the problem of induction” either, “it just sidesteps or ignores that problem” (p. 165). Furthermore, Bayesianism “has not shown itself to be able to improve upon the conclusions of the non-numerical tradition in many spheres of analysis” (p. 166). On this he notes the many different conflicting Bayesian interpretations of Hume’s argument itself (pp. 122, 124 & 128). He informs us that “the central problem of determining the probability of propositions supported by exceptionless past experience has resisted all attempts at mathematical analysis” (p. 114).

There is a time to use math and a time not to use it though: “Mathematical probability applies perfectly in stochastic setups with known alternatives; in other contexts, it is an artificial tool that does not accurately reflect how humans do or should reason about evidence; using it can mislead us” (p.127). Too often however, “The appearance of precision is nothing but an illusion.” So, “it can be potentially misleading, and is even positively harmful” (p. 163). “Numbers are impressively powerful in a great many contexts” he says. But “We should expect only the degree of precision appropriate to the subject matter” (p. 127).

We’ve already seen an instance of this dangerous illusion in my chapter 3, where I quote William Lane Craig in a debate with Bart Ehrman, saying that his arguments against miracles are “mathematically fallacious.” *cough*

Non-mathematical reasoning isn’t as precise, but some issues don’t lead to precision. It’s good enough in those cases. It’s even better when evaluating miracle claims according to the strength of the evidence. Sound non-mathematical reasoning, Baconian reasoning, legal reasoning according to the standard of reasonable doubt, reasoning on a continuum from an extremely high degree of probability to an extremely low degree of probability, and every point in between, can be described without mathematical precision. What’s the difference in saying x is highly probable or saying x is 92% or 93% or 94% or 95% or 96% or 97% or 98% probable? Very little if anything. We’re not counting spoons here. We’re weighing evidence. One can even translate Bayes’ Theorem into language and use it without the math, by asking and answering the same kinds of questions it demands. We can start by putting off all mathematical reasoning as little more than pure guesswork and conjecture, until there is some objective evidence for an extraordinary miraculous claim. On this point Christopher Hitchens is dead on when he said, “What can be asserted without evidence can also be dismissed without evidence.” (God Is Not Great, p. 150).

Like myself and many others, Vanderburgh agrees with the ECREE principle I defended earlier in this book, saying “The more extraordinary claim, the more extraordinary must be the evidence that would be required in order to establish belief in the claim as reasonable” (p. 109). As both Hume, Vanderburgh and I argue, there is no escaping this reasonable requirement, which just happens to be the demise of all miracles. It didn’t have to turn out this way. It’s just that miracles never meet this reasonable requirement. The very fact that believing apologists must reject the ECREE principle is evidence, all on its own, that they’re trying to defend that which cannot be defended by reason.


John W. Loftus is a philosopher and counter-apologist credited with 12 critically acclaimed books, including The Case against Miracles, God and Horrendous Suffering, and Varieties of Jesus Mythicism. Please support DC by sharing our posts, or by subscribing, donating, or buying our books at Amazon. Thank you so much!