A patron asked me to evaluate a video by TMM titled “WLC Misunderstands Hume’s Rejection of Miracles,” in which the host critiques William Lane Craig’s “rebuttal” to David Hume’s argument against miracles—because Craig’s response involves discussion of Bayes’ Theorem, and the TMM video appears to contain confusions on all sides requiring elucidation. I categorize this under Crank Bayesianism because what Craig says about Bayes’ Theorem here is bogus, and it’s another typical example of the abuse of that formula by Christian apologists (see my article Crank Bayesians: Swinburne & Unwin for an overall discussion of this category of epistemic fraud). In the process of following my analysis of this video, you’ll have a better grasp of how to update Hume’s argument in a Bayesian format so that it survives all the criticisms Christian apologists like Craig level at it. Of course this is not the narrative Craig wants you to know about; he wants the narrative to be, “Hume’s argument doesn’t work and therefore fails to disprove miracle claims.”

Of course it’s already a mistake to conflate Hume’s original Argument Against Miracles with a proper Bayesian argument against miracles, because though Hume was getting close to the same insights posthumously published by his contemporary Thomas Bayes, Hume’s argument against miracles isn’t fully Bayesian. He might not have even been aware of Bayes’ Theorem at the time. And yet his argument’s flaws are fully corrected when it is reformulated in a proper Bayesian form, as proved by John Earman in Hume’s Abject Failure: The Argument Against Miracles, which Christian apologists will sometimes cite as having proved Hume’s argument fails—because they only read the title of the book and think that’s what it’s about, when in fact it does exactly the opposite: Earman actually restores Hume’s argument using Bayes’ Theorem. Hume’s “abject failure” was merely failing to arrive at the Bayesian structure he should have used; but once one gets there, Hume’s argument becomes pretty much unassailable (just not, exactly, in the form Hume intended). That’s Earman’s actual thesis; and he’s right. (Sorry, Christian apologists. Wah, wah.)

I’ve discussed the modern, Bayesian case against miracles before, in Naturalism Is Not an Axiom of the Sciences but a Conclusion of Them and Theism & Atheism: Miracles, and also in my talk on Miracles & Historical Method. And for important background see Bayesian Statistics vs. Bayesian Epistemology.

The Problem of Assigning Priors

TMM begins by pausing, before Craig makes his argument, to insist that the probability of a miracle is “incalculable” and therefore “there is no way to put a number to it” and so no arguments for the truth of a miracle claim can ever succeed. This is false. And it misunderstands a crucial distinction between physics and epistemology. I explain this in my book on Bayesian epistemology (Proving History, index, “epistemic probability”), but in short, the epistemic probability that a miracle happened is not the physical probability that a miracle happened. For example, the physical probability that I won the California State Lottery today may be a million to one against; but the epistemic probability that I did could be nearly 100%. Because the truth of the statement “I won the California State Lottery today” is a function of the evidence for that having happened, which can be such as to increase the probability that it did happen quite enormously—easily well above that million to one odds. That’s why we routinely believe lottery winners won. We don’t go around saying to every reported winner of the lottery, “That’s too improbable; so clearly you didn’t really win the lottery.” It would depend on what evidence there is. In other words, the probability that something is true, is different from the mere physical probability that it happened. So when we want to know whether something is true, we need the former, not the latter. The latter can afford us relevant information about the former; but with obscure phenomena they’ll rarely be identical (Ibid., pp. 266-75, with pp. 26 and 51 and accompanying endnotes). For example, no matter how improbable it would be, we would certainly have enough evidence today that the sun went dark for three hours in 1983 to believe that it did, had it happened; ergo it is theoretically possible to have enough evidence of it had it happened in, say, 31 A.D. (Ibid., pp. 41-60). Which entails the prior probability of miracles cannot be inscrutable.

This entails two things with respect to miracles. First, we are asking what the epistemic probability is of a miracle claim being true, given the observed base rate of miracle claims being true—that gets us the “prior probability” that a miracle claim is true. And it is not the case that this is incalculable in the pertinent sense. We might not know any exact probability here; but we do know its upper and lower bound. Unless you have actual evidence otherwise (more on which shortly), the epistemic base rate of a miracle claim being true can’t be lower than is entailed by Laplace’s Rule of Succession: (s+1)/(n+2), which is therefore the upper bound of our margin of error. This can actually be formally proved (see link). Translated into present terms: the prior probability of a miracle claim being true equals one more than the number of miracle claims that have been proved true, divided by two more than the number of miracle claims ever reliably tested.

And yes, tested claims. It can’t be “claims ever made,” because you don’t know of those claims whether they were true or false, so their numbers can’t be assessed; we can only assess reliably vetted miracle claims. Which means if, for example, only a million miracle claims have been properly vetted in human history, and no other information applies (again, more on that qualifier in a moment), then you simply don’t have enough information to claim the base rate of miracles is below 1/1,000,002 (or 0.0000999998%), because it’s not credible that there have been more than a million miracle claims vetted to date (if you were to try gathering examples to comprise your evidence-base, you will quickly see the number you would end up with after a complete survey won’t be that high). So any evidence that presents us a likelihood ratio of more than 1,000,002/1 will lead to a single confirmed case of a miracle being more likely true than false. Yes, that’s all it would take. Given the caveats.

Because it is not usually the case that miracle claims don’t butt up against known agencies and laws of physics. The minimum epistemic base rate I just calculated would encompass all possible kinds of miracle claim, which explains why it can be so high: that would be the sum of priors for all possible models of miraculous explanation, which is always going to be far higher than the prior probability of any single particular model of miraculous explanation (lest you commit the conjunction fallacy). And the qualifier that “no other information applies,” actually doesn’t apply most of the time. You can then get more specific about the upper bound—the highest possible base rate given available data. For example, we have far more than a million observations regarding whether there even is an agent anywhere around Earth who performs miracles or even could (hence Naturalism Is Not an Axiom). So any miracle claim that requires such an agent faces that base rate upper bound. You could avoid this problem by proposing some other agent immune to that observation, but that will then run into problems of theory complexity that will also substantially drop the prior probability (see The Argument from Specified Complexity against Supernaturalism and my discussion of Cartesian Demons). But either way you will still have a definable upper bound. For example, scientists could, if they put in the work, calculate the probability of a quantum mechanical accident producing any effect—like, say, a spontaneous reversal of a corpse into a healthy, living person, or even an incredible Boltzmann event converting a corpse “by chance accident” into an invincible replicant. The probability would be absurdly low. But not zero.

Even Christian apologists have been willing to grant the probability of a miracle could be as low as 1 in 10^43, for example, which is surely still many orders of magnitude higher than a spontaneous quantum probability, but that’s okay, because they are theorizing something more organized than that is happening, which would indeed be more probable—if that “something” existed, which becomes a question of epistemic probability, which will typically be higher than a physical probability when you have evidence. Obviously, as every new physics has not had to pass a hurdle of improbability of anywhere near as low as 1 in 10^43. So 1 in 10^43 is surely a lower-bound probability of a miracle. It might not be the probability of exactly the agent miraculists insist is responsible (they don’t understand that that is actually fantastically less probable than a far less impressive agent). But still. We are only here asking about the probability of “a miracle,” not of a specific Being causing it.

Once we start talking the latter, things get way worse for the apologist (see John Loftus, “Christianity Is Wildly Improbable,” in The End of Christianity, combined with my discussion in The Argument from Specified Complexity against Supernaturalism). Which is generally why apologists want to talk about “merely” the probability of a miracle occurring. They want to fallaciously leverage that into “my exact specific God therefore exists,” which is a non sequitur. But we all know that. They aren’t actually doing science here; thus they aren’t really formulating a causal model and then testing it. They are just using the rhetoric of (hand wave hand wave hand wave) “a miracle happened” therefore (hand wave hand wave hand wave) “my God exists.” I don’t fall for that. But to illustrate my point about miracles in general I will skip the latter part and just focus on the former: that “a miracle happened,” regardless of what caused it.

The Prior Probability of Just Any Magic

Contrary to apologetic efforts to redefine magic in ways less embarrassing to their primitive superstitions, “magic” in everyday discourse actually just refers to any effect brought about directly by an intelligent will in some degree—like wishing making it so. Magic in this sense might involve component rituals and procedures (magic words, binding demons, enchanted objects, or whatever) or not (“gods” are typically imagined to not require any), but ultimately, either way, there will be no physical causal model that connects the desired effect with the actual outcome. If there was, it would then be just another technology. Even a Star Trek transporter works by mechanism, not magic. Whereas if you can “cast a spell” and teleport without any such machinery, that’s magic. As such, the Star Trek character “Q” appears more or less to be in fact magical (unless their powers were maintained by some hidden cosmic machinery). I lay out these distinctions in Defining the Supernatural. There are far more logically possible regimes of magic than involve gods. So rather than specifying a specific kind of miraculous agent, I shall ask, what’s the prior probability of there being just any magic whatever?

A lower bound for this probability isn’t what we want; so that 1 in 10^43 I’ll set aside. We want the upper bound—the highest reasonable probability of a miracle, which on standard “apologetical” theorizing entails a prior probability of some (just any) “magic user” being around who is both keen and able to “make things happen” of the kind in question; like, say, reverse the entropy in a human corpse from dead to alive. So how do we charitably find the highest prior odds that such a miracle really occurred? The answer can come from answering a different question altogether: how strong would the evidence have to be to convince us such a miracle occurred? Because the “strength” of evidence is a function of how much more likely it is on an explanation of it being true than on any other explanation of it, and that entails the prior cannot be lower than the converse probability to that (see Advice on Probabilistic Reasoning and Bayesian Epistemology vs. Susan Haack). So we can reverse engineer what a prior probability must be.

Let’s start with something just short of magic. We’ll take a famous fictional example to illustrate: the resurrection of Klaatu in the 1951 film The Day the Earth Stood Still. The original script simply had his ship’s technology resurrect him; the Christian-controlled Hays Commission compelled the producers to not allow that and instead insert a line that only God can do that. We now know the original script was right: suitably advanced, perfectly natural machinery could reverse the entropy in a corpse and restore it to life (hence Clarke’s Laws). The only question is whether such machinery is present and applied. This amounts to the same question for magic, only instead of “the requisite machinery,” we are asking about “the requisite magical agent.” But start with the question of “is the machinery present.” If the sequence of events depicted in that film were real, would the Earthling Helen be forced to deny what she’d seen—and insist Klaatu must never have really been dead and just faking his wounds? Of course not. She has by that point been presented ample evidence that these people have absurdly advanced technologies and are now employing them to effect a resurrection, and just did so before her eyes. There is a nonzero probability everything was faked; but it’s extremely improbable. Importantly, it’s far less probable than that what she is seeing is all not real. And even more importantly, that’s so much less probable as to reverse any prior probability against it. Which means the probability against it cannot have been 1 in 10^43. In no way is the evidence she has 10^43 times more likely on it all being real than it all being fake. And we intuitively know it doesn’t have to be. We would agree we’d be rightly convinced by such evidence even if it were only thousands of times more likely on it being real than it being fake. Ergo the prior has to be no more than thousands to one against.

For example, the highest standards in science for satisfying conditions for belief are found in “high-energy physics,” wherein “the threshold for ‘evidence of a particle'” is “0.003” (lower than one in three hundred) but “the standard for ‘discovery'” is “0.0000003” (lower than one in three million). Thus, to be warranted in suspecting a conclusion is true, you need evidence good enough to beat just hundreds to one prior odds against; but to be warranted in confidence that a conclusion is true, you need evidence good enough to beat a prior odds of millions to one against. And that’s the highest scientific standard. In between those two extremes is, indeed, thousands to one against. Thus, belief in a “miracle-like” encounter or discovery should only need evidence good enough to beat thousands of to one prior odds against. But that is because such beliefs rest on a mountain of background evidence establishing the plausibility of any machinery in question. The evidence is already vast that there are as-yet-undiscovered subatomic particles, and likely of certain properties; just as the evidence for Helen, even before a UFO landed in the middle of the United States capital, was already vast that machines could, in principle, mechanically fix a corpse. By contrast, the same kind of background evidence of spells working does not exist. So the prior odds of magic must be substantially lower.

So if Klaatu were a sorcerer instead of a spaceman, and walked out of the invisible feywild to effect the same deeds and mission as depicted in the film, Helen might be justified in suspecting he was really a spaceman secretly using technologies to emulate magic (a la Star Trek’s Ardra), but she would not be justified in concluding everything was being faked (the army surrounding him all crisis actors, his wounds a Penn & Teller routine, and the like). The prior probability that it was really magic would be substantially lower than that it was secretly all technology (see, again, my discussion of the distinction in Defining the Supernatural), because the latter benefits from a lift in base rate by its depending on components that are all established to exist (every technological step in effecting a resurrection existed even in Helen’s time, differing only in respect to various aspects of scale, e.g. the ability to build machines that move objects is just a scaled version of the ability to build machines that move atoms, a point used by Captain Picard to explain to Nuria why he was not a god). In our scenario, no such background evidence establishes the existence of sorcery. Now, it could have. It just hasn’t turned out that way. And this is a crucial point. This would all be different for Harry Potter, who in his universe is presented vast background evidence establishing magic is real; so for him, the prior probability of a sorcerer Klaatu using magic would even be higher than of his “secretly” using technology to “pretend” to be a sorcerer. Hence this all depends on background evidence. But notice, there has to be some finite, definable prior probability of magic for Potter to have been convinced of it with enough evidence. And the same evidence would convince you.

So far I have given examples where an eyewitness has to come to an epistemic conclusion. Importantly, Hume never argued that an eyewitness could not be persuaded a miracle happened (a point often missed by apologists attacking Hume). Hume’s point was that miracles could not be believed on a basis of testimony. His point being that the probability the testifier is lying or in error (even if themselves an eyewitness; even more so if they are passing on a story many times removed, though Hume did not depend on the latter to make his point) is always going to be higher than the probability a miracle really occurred (the more so when we include a third alternative cause, that of literary contrivance: miracle stories never intended to be taken as actual historical events, on which see Evan Fales’s thorough epistemic treatment in Reading Sacred Texts, in connection with my article on Establishing the Biblical Literalism of Early Christians).

As Hume put it, the testimony would have to be at least as miraculous as the event testified to. And he’s right. Likewise, Hume argues, no human testimony is that reliable; as proved by its embarrassingly high rate of unreliability across history, as anyone can observe. But this simply shifts the question to how improbable would a testimony have to be (on any other explanation of it than that it was correct) for us to believe a miracle testified to really occurred. Multiple independent testimony is especially powerful in this respect (Aviezer Tucker, “The Generation of Knowledge from Multiple Testimonies,” Social Epistemology 2015). This is not unlike the same question with respect to whether UFOs are alien spaceships, and we actually know the prior probability of that is far higher than for “miracles” in the proper sense, since the components are again established (Earth is already a spacefaring civilization, others are calculably certain to exist, and visiting other planets is again just a matter of scaling existing technologies), whereas this is still not the case for “magic.” There is a reason apologists cannot pass the Outsider Test for Faith and thus fail to grasp that their Formalized Gullibility would warrant believing all manner of embarrassing things, like that real magic-wielding witches were executed at Salem in 1692 (see Matt McCormick’s Atheism and the Case against Christ).

Apologists will chafe at my calling miracles magic, but that’s just their delusional aversion to admitting what things are because it makes them look silly. There is a developed apologetic based on inventing bogus definitions of magic in order to escape this cognitive dissonance, but that is all concertedly bullshit. If I can just think things into existence, that’s magic. It doesn’t become less magic when the person doing that is themselves even more magical, miraculously lacking even a body or limitations of any kind beyond logical necessity. A god is more magical than a sorcerer; not less so. But we can account it a miracle even if Jesus was raised from the dead by an ordinary sorcerer—even his own sorcery (just as Harry Potter resurrected himself, or could have done). This is of course a problem for the apologist: it is far more probable Jesus used magic to resurrect himself, than that a God did it. Because the “God” hypothesis requires absurdly more numerous and complex suppositions (hence my point in The Argument from Specified Complexity against Supernaturalism); a Christian God, vastly more so (hence Loftus’s point in The End of Christianity). It’s much simpler to posit far less ambitious explanations of the same effect, especially when they have considerable epistemic advantages. For instance, that sorcery exists but gods do not is a better explanation of the vast diversity of miracle claims across religious traditions, because it makes that outcome more probable, whereas a god-hypothesis predicts a far more consistent tradition—without gerrymandering, which only makes the hypothesis less probable, not more so. Even more uncomfortably for Christians, that aliens resurrected Jesus is necessarily more probable than that sorcery did! And since “that sorcery did” is necessarily more probable than that a god did, if Christians really believed their own arguments, they should believe in alien visitors, not gods.

But let’s put all that aside and simply ask what is the prior probability of the most minimal (the least ambitious) theory of miracles operating in the world. This would be something closer to mere sorcery or minor theism (fallible, limited gods; who possibly have even died out since). Unlike alien visitors and the like, we have no background evidence that such things as sorcery or gods even could exist, much less do. Therefore, epistemically, the probability they do has to be less than the probability of alien visitors. But since the idea of alien visitors having resurrected Jesus is already wildly remarkable, and would be a profound discovery if any evidence were sufficient to lend it credence, we can say, a fortiori, the highest possible prior probability of a miracle is effectively the same as “aliens did it.” This is not saying they are the same. Necessarily, the prior probability of real miracles has to be lower. But any evidence that was good enough to convince me aliens did it, would certainly lead me to take more seriously the proposal that gods did. It would not alone be enough to sway me to that conclusion; but it would put swaying me to that conclusion within reach. It would just take a certain amount of differential evidence favoring gods over aliens. Which would have to be no less comparable in scale.

So, I can honestly say, evidence for the resurrection of Jesus that were really only a million times more likely on it actually having happened than on any other explanation of the tales and beliefs of it that arose (such as trickery or error or lying), would be enough to persuade me at least the resurrection occurred, thereby making it, at the very least, evidence aliens have visited Earth. It follows, therefore, that further evidence that was a million times more likely on it being caused by gods than aliens would persuade me of that in turn (and here I mean minor gods; evidence of specifically the Christian God would have to be vastly better still, because that claim is vastly more complex and ambitious in its details). Which gets me to a 1 in 10^12 epistemic probability a miracle has occurred. Because any evidence that was 10^12 times more likely on that explanation than any other would persuade me—and should. There isn’t really any good case to make that we should reject even such evidence as that. You’re just being irrationally resistant to evidence at that point.

Of course, getting to that ratio is a lot harder than apologists think. They will fabricate evidence-claims of that kind, but all their math is bogus. For example, “the Gospels” are not independent or eyewitness testimonies, and the most reliable evidence we have is not of what the Gospels attest, but of far more mundane—and thus not miraculous—events: see Resurrection: Faith or Fact and Then He Appeared to Over Five Hundred Brethren at Once! And the evidence that does exist doesn’t rule out other explanations of it (like trickery) at all—much less by a factor of a trillion. But Hume was at least correct that the priors are never their friend.

As I represent in the above table, that miracle claims are baloney (lies, distortions, exaggerations, mistakes, or not even originally intended to be taken as fact) is the most common cause found for them by far, and thus starts with a near 100% prior probability. That some known natural phenomenon really is the cause (and thus witnesses simply in their ignorance mistook something as miraculous), is the next most common cause found, and thus the next most likely. And that’s by thousands of times; because it is extremely rare that it has turned out to be some unknown natural phenomenon later discovered to be natural (like ball lightning or neurotypical hallucination). And that’s surely hundreds of times more likely than “aliens did it” or that some technological Cartesian demon is responsible (which to date has never been found as the actual cause of any reported miracle). And that’s surely more likely than any kind of sorcery. Which has to be more likely than “gods did it” (since “gods” entails more, and far more fabulous, assumptions and thus must occupy a much smaller piece of the “magic” probability space). And that there are various minor fallible gods is more likely than that there is one even more fabulous God (per the specified complexity required); and that there is only one peculiarly specific God is even less probable still (per the conjunction fallacy). Epistemically, Christians are pretty well screwed here. Which is why they need to lie about all this.

And that’s just prior odds. It’s all the worse when you reintroduce evidence, which actually lowers the probability further. For example, the final odds that a good god exists has to be many trillions to one against. A callous or indifferent god is vastly more likely, because it agrees so very much better with observations; but such a god is unlikely to waste time on any miracles, other than to dink with people and deceive them—ergo, one should sooner conclude that if a miracle did raise Jesus from the dead, it was by a wicked agent deceiving us for a nefarious purpose and not the reasons Christianity avers. Because that’s what all the world’s other evidence indicates is the trend. But even that’s implausible; ergo, that a good God exists, even more so. Since “a good and all-wise-and-powerful god” by its own definition is an ever-present cause of unlimited capability, its effects should nearly everywhere be observed, nearly all the time; that they are not, rules that explanation out. And “making excuses” to avoid this consequence still just reduces the probability; it does not increase it as pretended.

Back to TMM on Craig on Hume

That’s why Hume’s point holds not on his own stated terms but in proper Bayesian terms: to get a miracle to be a probable fact (to warrant believing a miracle has occurred), the probability of the testimony to that miracle on all possible alternative causes (e.g. lies, errors, or fictions) has to otherwise be lower than the prior probability of the miracle itself. Here, on a hypothesis of any mere physics-defying magic (gods involved or not), that probability is at least one in a trillion (1 in 10^12). Which means we need testimony that is a trillion times more likely to exist if true than if a product of lies, errors, or fictions. Hume asserted this could never happen, which was his only mistake. One does not need to claim that. Because even though it can happen (we can have testimony that’s as good as that), it just has not yet happened for any miracle to date. Hence my conclusion in Resurrection: Faith or Fact? That we could have had such evidence is still of no help to Christians. Because the material point is that we don’t. Ergo, we can never conclude any miracle really happened. This is not in the sense Hume intended (which was that we “could never” have such evidence), but in the corrected sense Earman updated Hume to (which is that, “so far,” we simply “don’t” have such evidence), as illustrated by my example in Proving History of a hypothetical “miraculous darkening of the sun” in 1983. I even give examples of how we could have had a sufficient amount of such evidence from 31 A.D. We just don’t. And that’s the insurmountable problem for the apologist.

So the problem is not, as TMM first imagines, that the prior probability of miracles is inscrutable and therefore insurmountable by any evidence. We can all easily imagine sufficient evidence that would persuade us (we could have found ourselves in the Harry Potter universe, for example). It logically necessarily follows that the prior probability of miracles (of “magic” in any sense) is both scrutable and surmountable—even on a sole basis of testimony. Hume was wrong to say otherwise. But he was right to say that we have as yet never encountered a testimony so reliable as to be sufficient to warrant believing any miracle claim. And that’s an empirical, not a logical problem. Yes, the Christian apologist faces an even more difficult hurdle than just that, because they are not actually arguing for “just any” magical cause, but a very specific one, which entails a great deal more in terms of observations, yet actual observations run entirely counter to their thesis. So even if we had confirmed many miracles, all else being equal, Christianity would still be the least likely explanation of them. But the point here is that we don’t even have evidence enough to believe any magic exists at all—Yahweh’s or otherwise. Therefore, a fortiori, the probability Christianity is true can be no better than trillions to one against—even before we add up all the evidence against its being true, which reduces its probability trillions of times more.

But if all you want to prove is something less ambitious—like, say, Jesus really rose from the dead owing to his secret discovery of sorcery, in an effort to trick the world into obeying his desired moral reforms, then hid away until his natural death years later, a theory that actually better fits all the evidence than the Christian’s far more outlandish theory does—well, you still can’t even do that. Not because it’s impossible (it’s not “can’t” in that sense), but because we have yet to uncover any such evidence as would do it—and what you “can’t” do is argue it’s reasonable to believe something so remarkable on insufficient evidence. Commonplace things, maybe. Like winning lotteries. That happens all the time. But this is an unprecedented thing. You therefore need way better evidence for it than for any ordinary claim imaginable. And where Hume’s argument does successfully hit the road is precisely there: no testimony to any miracle has been sufficiently miraculous in itself to believe it. Maybe that will change some day. But we can only work with what we have now. “Hypothetical” evidence is still not evidence.

Imagine, instead, your theory was that Jesus used magic to rise from the dead and then fly everywhere in the whole world on a single day demanding obedience or destruction with a booming voice across the skies, and then at a word immediately disintegrated into magma all people and nations that refused. Imagine, too, that you actually had sufficient evidence that this really did all happen. What would that evidence consist of? We’d have archaeological confirmation—cooled magmites originating and situated exactly in accord with the worldwide testimony of thousands of witnesses, whose reports we would have directly from them, in inscriptions, preserved letters on papyrus, and the like. And we’d have otherwise “impossible” (as in: extraordinarily improbable) coordinated testimony from nations and peoples across the whole Earth from the very same day corroborating numerous essential details: not just Roman records across a dozen provinces, but Chinese records would uniformly show this, as would Mesopotamian, Ethiopian, Indian, Persian, Mesoamerican, even tribal lore in the Americas and Australia. That by itself would confirm the events happened, sufficient to believe they did, but we’d still be more warranted in suspecting a secret advanced civilization was responsible (whether Lemurian or Alien).

To get us to “sorcery” we would need at least as much evidence sorcery is an available mechanism as we have now that advanced technologies are. A single one-off event in history cannot suffice there, as that leaves the preponderance of evidence in favor of what we do have vast evidence of: technology. That’s why Picard was right to doubt Ardra was really a supernatural agent. So we’d need quite a lot of evidence that sorcery was a thing first. If we had sciences proving it, to the point that we had sorcery wings in hospitals and sorcery units in armies, for example, well, that would do. And it would do because at some point in this accumulation of evidence, the probability all of that was faked or secretly still somehow just tech would drop below the prior probability against it—just as has already happened the other way around: the accumulated evidence of “miracles by technology” is so vast now, that the probability it has all been faked or secretly really sorcery has dropped below the prior probability that it could all just be mere tech. Remember, before evidence exists, the prior probability between magic and no magic is 50/50; it only isn’t that now because of a vast accumulation of evidence. People tend to forget all that evidence when deciding what’s more likely. We are not just suddenly-materialized intellects possessed of no background knowledge of the past and present world.

Then We Get to Bayesian Fraud

TMM is correct say there is something shady in Craig’s chosen analogy, the question whether “a husband murdered his wife” as presented in court. But Craig is not wrong in his logic: he is correct to say that the prior probability the husband is guilty is low, but evidence could substantially raise its posterior probability; and if so for a murder verdict, so also for a miracle verdict. Of course what Craig elides here are two things TMM is pouncing on with his point. First, we actually know husbands murder their wives; in fact we have a known and measured base rate of it and thus can actually identify its prior probability to within fairly reasonable margins of error: it is, horrifyingly, around 1 in 4. That’s right. In the United States, if a married woman is murdered, there’s a 25% chance her husband did it. Before you look at any evidence either way. We have nothing like this for miracles. The number of miracle claims that have been confirmed to be genuine (and not just lies, errors, or edifying fictions), is exactly zero. After thousands of years now. This is more comparable to asking whether “an angel from heaven flew down and murdered his wife.” We have zero cases of that. So that’s starting at quite a low prior, to say the least.

Second, what would it take to prove that in court? Craig opens with a court analogy but then quickly drops that analogy when getting to claims like the resurrection—for which no evidence exists that could even be submitted in court (all that exists is unauthenticated documents; not even any hearsay survives, which requires a living witness to or authenticated record of something an actual witness said; but even that would be rarely admissible). What evidence of any legal quality is there for the resurrection? Craig mentions for murder we have “DNA, fingerprint evidence, the motive, the lack of an alibi, the eyewitness testimony of someone who saw” it (which can only mean an actual living eyewitness); but no evidence anywhere like this exists for the resurrection. So you couldn’t even prove a murder on its evidence, much less a miracle! Imagine trying to prove in court that, “No, your honor, I didn’t kill my wife, an angel from God did it, and as evidence I have a bunch of anonymous unauthenticated yarns about it and the assurance of it in an unauthenticated putative letter written by a guy who admits he wasn’t there but was told of it by that damned murdering angel in a dream!” Oh, and, “No, the prosecution can’t cross-examine him, or anyone, regarding this claim; in fact, sir, neither can I, as I have never met any of these people, nor anyone else who has.” Needless to say, the prior probability that “an angel did it” isn’t going to go up on that evidence. That evidence wouldn’t even pass the lowest bar of being admitted in court!

So there is definitely a problem here. It’s just not the one TMM keeps harping on (that there is no surmountable prior probability of it even in theory). We could have enough evidence to prove it in court—we just don’t. Because what we need would be vastly more than anything we have. And contrary to TMM, it is perfectly clear how much evidence we could have had that would convince us. Just put yourself in Helen’s position. Or Harry Potter’s. But that’s just the first three minutes or so of the video. After that point TMM is right to call Craig out for a different mistake: Craig’s false claim that “Hume only considers the prior probability.” This is basically lying. The prior evidence Craig is a liar generally is vast; and it’s very improbable Craig is this ignorant of Hume’s actual argument, and even if he is, he is then lying about his being familiar with it, and thus either way is a liar. Hume properly takes into account both the prior probability and the likelihood ratio (the probability of the evidence, in the case “testimonies,” on either a claim being true or its being false). Hume’s argument is that the latter can never be large enough in favor of a testimony’s reliability to overcome the prior probability against it. Earman’s corrected Humean argument is that the latter is never large enough in favor of a testimony’s reliability to overcome the prior probability against it. The difference is that, per Earman, some evidence someday could produce a large enough likelihood ratio; so we cannot say that will never happen. We still have to examine testimony to ascertain if it is “that reliable” or not. So far, it never has been. Which itself is evidence against the reliability of any such testimony (that miracle claims have such a peculiarly low rate of reliability compared to mundane claims evinces that miracle claims in themselves have a particular reliability problem: Proving History, pp. 114-17). But even that can in principle be overcome. Just ask Helen or Harry Potter.

This relates to a second point TMM gets right against Craig: that Craig appears to be scamming his audience by claiming that “Bayes’ Theorem” tells us that anything you deem evidence for a conclusion increases its probability. That’s false. And it’s so egregiously and obviously false that I can only conclude that Craig is, again, lying. It is simply not credible that Craig somehow “forgot” the entire content of Bayes’ Theorem and how it works: for any item of evidence the pertinent number is not how probable that evidence is on the conclusion you want to reach, but how probable it also is on alternative explanations of the same evidence—exactly as TMM correctly explains. And quite often this means what you thought was “evidence for” a conclusion is actually not evidence for it at all or is even evidence against it. In the one case it does not increase the probability at all; in the other, it actually decreases it.

For example, in Craig’s “murder husband” analogy, a witness who saw the husband go into the house shortly before the murder ceases to be evidence for his having murdered his wife if the evidence altogether establishes that he was there when his wife was murdered by someone else, and there is nothing differentially improbable about his being home with her at the time. In such a case, the evidence is equally probable on both theories, and therefore increases the probability of neither. Or if the evidence altogether instead establishes that the husband was in another city at the time (say, by several independent unimpeachable public video records), and the “witness” did not know that at the time they testified, then the impeachment of their testimony does not increase the probability the husband did it, but actually, rather, increases the probability that the witness murdered his wife (as otherwise why did they lie about who they saw there?); yes, perhaps not enough on its own to convict them, but the material point here is that what you think is evidence “for” a conclusion, might actually be evidence “against” it: it all depends on the ratio of probabilities between the competing hypotheses explaining that testimony.

In that latter scenario, “the witness did it” makes the evidence of their testimony more probable than “the husband did it” does. Ergo that testimony decreases the probability that the husband did it, by increasing the probability someone else did (in case it’s not obvious, within the assumption that she was murdered at all, the probability that “the husband did it” and that “someone else did it” must sum to 100%, as that exhausts all logical possibilities—so any increase in the one necessarily decreases the other). This is why Christian apologetics always operates by leaving evidence out; because once you put back in what was left out, everything Christians claim is evidence for their religion being true, actually, like this impeached witness, turns out to be evidence against it being true (see Bayesian Counter-Apologetics for a detailed survey of this fact).

Craig is thus, in my opinion, simply lying when he says, and I quote, “And so, Hume’s argument is mathematically, demonstrably fallacious, because it only considers the prior probability” and “not the probability of the evidence.” This is TMM’s big gotcha, and he’s right. TMM more charitably calls this Craig’s “misunderstanding” Hume, but Craig is not some two-bit YouTube amateur. He literally studied this subject for his Ph.D. and has written on it extensively. It is effectively impossible for him to “misunderstand” Hume in this way. He is simply deliberately not telling the audience the truth about Hume’s argument. Which is especially weird because he didn’t have to. He could have stuck to the more honest apologetic line that Hume did consider the effect of evidence on a miracle’s probability but undercounted how strong it could be. That argument would also be false, but is more easily a product of incompetence or error than this boner of a lie Craig chose to tell instead.

Keeping the Actual Distinct from the Possible

That covers everything in TMM’s video up to around timestamp 7:45. From there, TMM makes the point that even though, yes, “lots of mutually corroborating testimony” will (usually) increase the probability of an event, even a resurrection, but “if there is a competing explanation for those testimonies whose probability is higher still,” then in fact the evidence increases the probability of that competing explanation, and not of “resurrection.” This may sound like a contradiction but we can charitably read TMM here as saying all else being equal such a testimony will increase the probability of a resurrection but if instead all that testimony is more probable on another explanation, the reverse is the case: it will decrease the probability of a resurrection, by increasing the alternative. TMM could also have said here that even when that isn’t the case—for example, even if, somehow, we have testimony that is in fact more probable on a real resurrection than on any other explanation of that testimony—it is still possible for testimony to be insufficient as evidence. For example, if testimony increases the probability of a resurrection tenfold, but its probability started at 1%, we still end up with only a 10% chance the resurrection occurred, which is actually the conclusion it probably didn’t. We actually have no such testimony in this case—to have such, we’d need something far more potent, like several trusted scientists alive today attesting under oath to having witnessed and verified a resurrection, which is far less probable on alternative explanations than the anonymous, ambiguous, propagandistic, methodologically naive, and frequently edited and forged texts comprising the New Testament. But I mention this only to be theoretically complete as to the possible outcomes “a testimony” can have on the credibility of a miracle claim.

I should also point out that this is a distinct and different point from one TMM also makes (but that viewers might accidentally conflate): that the implausibility of a testimony actually itself reduces the probability of a testimony’s reliability. I mentioned this above already, but in Bayesian terms this works out as follows: when we say e (a claim someone made being offered as evidence) is inherently unbelievable because it makes claims wildly contrary to what we know of reality, we mean, in b (all our background knowledge) we have no confirmed examples of anyone making such a claim telling the truth but countless confirmed examples of their not telling the truth; accordingly, e is typically going to be evidence against a story being true rather than the other way around (see Craig vs. Law on the Argument from Contamination and Is the Principle of Contamination Invalid?).

Hume’s mistake was to declare this an inviolable universal rule. But of course it can’t be. If we could only ever believe what we had seen ourselves, nearly all human knowledge would be impossible. We actually epistemically depend on testimony to inform us of things we are directly unfamiliar with; our entire civilization is built on this (see A Vital Primer on Media Literacy). In fact all scientific progress, especially of a paradigmatic nature, likewise depends on this (the rise of Relativity Theory and Quantum Mechanics, for example, as well as germ theory, plate tectonics, viral ulcers, and so on), as it is simply not possible for even every scientist much less every human being to replicate every paradigm-changing scientific experiment before believing it has occurred. Hume himself faced the conundrum of why someone in the South of India should believe on testimony that water solidifies into white dust and even rock. Hume himself offered as an example of the impossible that fire cannot burn under water; so what do I do with all the third-party evidence I have that it does? (There are several kinds of fire that burn under water; none known to Hume.)

So what is it that makes the difference between my reading about magnesium flares or undersea acetylene welding or the Olympic Torch carried under lakes and seas, and my reading about fire-breathing dragons and mass resurrections of the ancient dead? The difference is that the former kinds of accounts (based on markers such as their express methodology and how they are composed and sourced, who the sources are, what evidence accompanies them, and more particularly their content’s inherent plausibility on known background facts) tend, on examination, to be true (usually when I fact-check such stories, they almost always turn out to be directly, verifiably true, or attested by such a great number of independent witnesses whose accounts, or accounts I can confirm they endorse, I can access directly, as to make their inaccuracy itself miraculously improbable). Thus the prior probability that they are true, i.e. P(h|b) for any h = “testimony is true,” is on a well-established base-rate high. And that is information in b (background knowledge). It thus must influence all my probability estimates.

By contrast, the other kinds of accounts (again based on markers such as their express methodology—or usually, lack of one—and how they are composed and how they are—or usually, aren’t—sourced, who the sources are, what evidence accompanies them—or usually, doesn’t—and more particularly their content’s inherent lack of plausibility on known background facts) tend, on examination, to be false, or suspiciously incapable of verifying. This did not have to be the case. It just happens to be the case. Which is itself evidence against the supernatural in general because this split is, conspicuously, always along exactly those lines: as soon as the supernatural is claimed in a story, it falls into the “usually falsified; never verified” reference set. That is a very improbable coincidence if the supernatural is as real as any other contents of the world. This is not as strictly the case for non-supernatural claims—those have plenty of falsehoods among them, too—but the ratio of true-to-false (which means, of well-verified-to-well-falsified) is far higher in their case, and particularly from sources with an established past record of reliability, and on subjects very-well-attested by reported evidence.

This was the fact Hume was noticing and attempting to codify into a law. His mistake was in thinking that zero confirmed reliability in one class of story permitted the assumption that “stories” can never be sufficiently reliable. But Laplace’s Law of Succession, and examples of our trusting testimony to physics-changing discoveries, prove that it can. It’s just that what it takes to recover the reliability of such an account far exceeds what has ever been accomplished in any case of reputed miracle. Indeed the Scientific Revolution was very much about admitting this fact, and thus setting a very high bar of evidence before believing things are other than we think we know. No longer did Gilbert believe any of the tall tales about magnets in his medieval textbooks; if he couldn’t prove a claim with evidence to such a replicable degree as to render too improbable any other explanation but that it was true, it was simply not to be credited. The modern science of magnetism is the result. Medieval intellectuals, by contrast, pretty much just believed whatever trash they were told. Primitive ignorance and rampant folly was the result. All of which now is what has resulted in the conclusion that we simply aren’t likely in a world that has any magic in it. Had we been, the results of history and the Scientific Revolution would have turned out differently. It’s still not impossible that we are in a world with magic somewhere in it; nor impossible that we could be convinced we are by enough evidence. It’s just that that hasn’t happened. So we remain where the evidence has left us. And that’s that.

This difference in prior probabilities of a claim’s truth by reference class is obvious to anyone who thinks about it. Compare the claims “I own a car,” “I own a nuclear missile,” “I own an interstellar spaceship,” and “I own a chalice that grants immortality.” You are not suspicious of the first of these; it’s so typical for people to own cars, that the prior probability it’s true I own one is amply high, but even if owning a car were uncommon, you still would likely not doubt it, because it’s so unusual for people to lie or be mistaken about that sort of thing. But as soon as I say “I own a nuclear missile,” now you won’t believe me, without some pretty good evidence of the fact (for examples of which see The Manhattan Project). Why? Because you know the frequency with which people like me “own nuclear missiles” is lower than the frequency with which people might lie or be mistaken about that.

You’d be even more suspicious of my claim to own “an interstellar spaceship,” because you have vast evidence that nuclear missiles at least exist and it’s conceivable, if highly improbable, for a private person to get a hold of one, whereas you have no evidence to believe there even are interstellar spaceships on Earth for me to get a hold of. So you would immediately assign this claim a very low prior, and require even more exceptional evidence it was true before believing it, even than you’d require for the “nuclear missile” claim. It follows, necessarily, that the prior probability you’d assign to “I own a chalice that grants immortality” being true will be far lower than even for “I own an interstellar spaceship,” because you at least know from vast background evidence that interstellar spaceships are a definite technical possibility, in precisely the way you know “magical chalices” are not. But now I’m still nowhere near to “there is probably a God who raises people from the dead.” Magical chalices could exist without gods; in fact, there are more imaginable universes with magic and not gods, than with gods. So to believe a god is doing magic requires far more evidence than you’d require to even just believe magic exists. But you can still conceive of what evidence I could present to convince you I do indeed have “a chalice that grants immortality.” It’s just not a scale of evidence I can ever meet—unless I actually had one. Which is how you know I don’t.

The same holds for the resurrection of Jesus, even if by petty sorcery or magical chalice. It’s just all the worse that not only do we lack the required scale of evidence to believe it, we possess a massive scale of evidence warranting our disbelieving it (see, for example, The Rain Miracle of Marcus Aurelius: A Case Study in Christian Lies and How To Fabricate History: The Example of Eusebius on Alexandrian Christianity). Thus it is important to notice that TMM concludes “the idea that the authors of the New Testament were mistaken or lying” seems “like a far more probable explanation of their testimony than the occurrence of a resurrection” (emphasis mine). He starts by talking about “mutually corroborating testimony” but switches to “the authors of the New Testament.” You might mistake this for saying these are the same thing. They are not. There is no mutually “corroborating” testimony in the New Testament. There is a single witness (Paul) to a purely psychological event (Christ appearing “in” him, by “revelation,” and not meeting anyone in the “flesh”). Everything else is either derivative of that (made-up tales in the Gospels decades later) or corroborative of nothing miraculous (Paul attests others had his experience; but there is nothing miraculous in his experience—so far as we can tell from what Paul writes, that Jesus really rose from the dead is a fallacious inference he and they are drawing from their inner imagination). So, indeed, it is far more likely these stories are unremarkable (in Paul) and fabricated (in the Gospels) than actual, because that’s how such things almost always turn out in every other case the world over and throughout history.

Conclusion

Hume was not wrong about anything Craig dishonestly claimed he was. Hume’s only error was in confusing an actual absence of data for an impossibility of it. But once you correct that error, a revised Humean case survives all scrutiny: yes, it is possible to have enough evidence to believe in magic; but no, no one has presented any such evidence of any yet. To the contrary, what has panned out is that, whenever we get to tell for sure, claims to magic tend uniformly to be false (either lies, errors, or fictions); and evidentially are never impressive even by ordinary, much less extraordinary standards. Craig’s “evidence for the resurrection” is so absurdly poor it wouldn’t even be permitted in a court of law; and it certainly could never win a conviction there, not even by a preponderance of evidence standard. Miracles do have upper bound epistemic prior probabilities that we can discern; and there can in principle be enough evidence to convince us of them. But no evidence ever offered for any miracle claim has ever in history come anywhere even near to meeting that standard. Which fact in itself has now become evidence that miracles simply aren’t a thing that happens in this universe—as opposed to tons of highly unusual and even surprising things that do happen, yet have entirely natural causes, which we have, strangely, found vast quantities of evidence for—the kind and scale of evidence apologists can conspicuously never present for magic.

In the end, I disagree with TMM’s excess skepticism about the potential impact of “hundreds of independent testimonies” to a thing. He seems to think this impossible; whereas I only conclude that it is not actual. The difference is crucial, as I explain in Then He Appeared to Over Five Hundred Brethren at Once! That “five hundred men” would independently err or conspire to lie about meeting and dining with Jesus for a month after his death could be less probable than a resurrection (it would depend on all the concurring circumstances of and within their testimonies). For example, if we had the sworn independent testimonies of five hundred scientists today who both witnessed this and verified by every possible test that a man was indeed dead and even decaying for days and indeed fully healed and well afterward, I’d be inclined to believe it, and seek only to find out how he accomplished the feat. I’d even believe just one person’s testimony, in the right circumstances—for example, I’d not doubt Helen’s testimony to the technological resurrection of Klaatu if I had, indeed, all the same background evidence of a recent emissary from an ultra-advanced extraterrestrial civilization, as I would indeed have personally witnessed in that case. Likewise, I’d even believe her testimony that it was accomplished by magic, if Klaatu were the leader of an army of magic-using faeries that recently stormed D.C. from their kingdom near Hogwarts. Because then I’d have all that background evidence. But that isn’t what we have. We have no record of five hundred men “meeting and dining with Jesus for a month after his death,” much less from those actual men, much less independently of each other, much less a reliable body of men who verified by every possible test they weren’t mistaken or being duped. What we have is a guy who admits he met Jesus only in his imagination, telling us he knew hundreds of others who likewise claimed to have met Jesus in their imaginations, all of whom actually only “inferring” by religious, not scientific, logic that Jesus must therefore have risen from the dead. That doesn’t cut it.

“No, your honor, I didn’t kill my wife, an angel from God did it, and as evidence I have a bunch of anonymous unauthenticated yarns about it and the assurance of it in an unauthenticated putative letter written by a guy who admits he wasn’t there but was told of it by that damned murdering angel in a dream!” And, “Oh, that guy’s letter also says he knows hundreds of other dudes who swear they saw that angel maybe in a glimmer of sunlight!”

That’s not even ordinary evidence, much less the extraordinary required. It is, in fact, as evidence, total crap.