A popular slogan among many contemporary atheists is that “extraordinary claims require extraordinary evidence”. The slogan itself goes back to the late astronomer Carl Sagan , though similar ideas were expressed by David Hume, who wrote “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.”  Indeed, so confident was David Hume about this principle that he said, humble man that he was, “I flatter myself that I have discovered an argument…which, if just, will, with the wise and learned, be an everlasting check to all kinds of superstitious delusion, and consequently, will be useful as long as the world endures.” 
This principle has led many skeptics to push the bar of demonstration so unreasonably high that it cannot possibly be cleared by any amount of testimonial evidence. And atheists seldom attempt to define what precisely is meant by “extraordinary”, or what sort of evidence would be sufficient to demonstrate an extraordinary event. As such, it has become a lazy excuse of many atheists for not dealing with the evidence for miraculous events such as the resurrection, but instead to dismiss it by appeal to Sagan’s dictum or to David Hume’s treatise against miracles (which the skeptic has seldom read for himself). In this article, I will argue that the dictum that extraordinary claims require extraordinary evidence, and the philosophy of David Hume that it encapsulates, are fundamentally wrong-headed.
Signs and the Order of Nature
What is the stated purpose of miracles in religious contexts? According to Scripture, miracles function as signs that authenticate the message of the person performing them. For example, when Jesus is asked by disciples of John the Baptist whether He is the Messiah, or whether they should be waiting for another, Jesus replies, “Go and tell John what you hear and see: the blind receive their sight and the lame walk, lepers are cleansed and the deaf hear, and the dead are raised up, and the poor have good news preached to them. And blessed is the one who is not offended by me,” (Mt 11:4-6). In other words, Jesus’ miracles were signs that authenticated His message. The gospel of John refers to Jesus’ miracles as “signs.” Towards the conclusion of his gospel, John writes, “Now Jesus did many other signs in the presence of the disciples, which are not written in this book; but these are written so that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name,” (Jn 20:30-31).
If the purpose of miracles in religious contexts, therefore, is to function as signs, then they have to take place against the backdrop of a stable, uniform, natural order, since it is by contrast with a stable, uniform, natural order that miracles are able to serve as signs. Consider, for instance, the relevance of the abnormal character of the resurrection to the epistemic value of Jesus being raised from the dead. If people were routinely rising from the dead then the resurrection of Jesus would lose its epistemic value, since it would not be an event that could be distinguished from the way that nature normally operates. It is precisely because it is unique, and is an interruption of the regularities of nature that we can appeal to the resurrection as God’s vindication of the claims of Jesus of Nazareth to be the Messiah and Savior of the world. Thus, since miraculous signs require that there be a stable natural order, the existence of such a stable natural order cannot be taken as an argument against the occurrence of miracles in religious contexts. This point was first raised in response to David Hume by William Adams, who wrote that “An experienced uniformity in the course of nature hath always thought necessary to the belief and use of miracles. These are indeed relative ideas. There must be an ordinary regular course of nature, before there can be any thing extraordinary. A river must flow, before its stream can be interrupted.”  Philosopher Dr. Tim McGrew, of Western Michigan University, concurs: “The fact that in our ordinary experience dead men stay dead cannot be a significant piece of evidence against the resurrection considered as a miraculous sign — that is, it will not do the work that Hume wants it to do in the very sort of religious context where he is most implacably skeptical.” 
Can Testimony Ever Be Sufficient to Establish a Miracle?
It may be admitted that miraculous events do require more evidence to establish them than do mundane events, since the prior probability (that is, their probability given the background information) is lower than for mundane events. I shall return later to why miraculous claims are rightly treated differently from mundane ones. However, I shall note here that any proposition with a non-zero prior probability can, in principle, be demonstrated with sufficient evidence. The eighteenth century British Bishop Thomas Sherlock wrote concerning the resurrection, “I do allow that this case, and others of like nature, require more evidence to give them credit than ordinary cases do. You may therefore require more evidence in these than in other cases; but it is absurd to say that such cases admit no evidence, when the things in question are quite manifestly objects of sense.” 
In 2000, philosopher John Earman, himself a religious agnostic, published a book by the provocative title Hume’s Abject Failure — The Argument Against Miracles.  Earman cites the nineteenth century philosopher and mathematician Charles Babbage, who wrote that “if independent witnesses can be found, who speak the truth more frequently than falsehood, it is ALWAYS possible to assign a number of independent witnesses, the improbability of the falsehood of whose concurring testimonies shall be greater than that of the improbability of the miracle itself,”  Thus, Hume’s so-called “everlasting check” fails, since a cumulative case can, in principle, be adequate to overcome the intrinsic improbability of a miracle, thereby being sufficient to warrant belief. Typically, when an atheist states that “extraordinary claims require extraordinary evidence”, they have in mind a single spectacular piece of evidence that is sufficient to overcome the intrinsic improbability of the miracle itself. However, what can be accomplished by a single spectacular piece of evidence can, in principle, also be achieved by numerous pieces of less spectacular evidence, perhaps none of which individually is of particularly great weight but collectively is equivalent in weight to a single piece of spectacular evidence.
The Cambridge philosopher Arif Ahmed, himself an outspoken atheist, has sought to meet the challenge of Earman and Babbage to Hume by arguing that the presumption of independence is often false.  For example, in the case of a stage illusion or trick, given that at least one person has been fooled, the probability that many other people will be fooled as well is significantly increased. Thus, successive pieces of evidence fail to add significant force to the case that a true miracle was observed. Tim McGrew responds to this point by noting that “The cumulative testimonial evidence might have a theoretical limit to its force, but it might not. Without further information about the specific case in question, we cannot say anything more. Everything depends on the details,” . McGrew further observes concerning the claimed resurrection of Jesus, “The witnesses are not all confined to one vantage point, as they were in the case of the stage magician. If they agree, it is much more difficult to find a single simple explanation for how they could all have been fooled. Their testimonies are not bare assertions that the event in question happened. They may include details that interlock with details in other testimonies in ways that increase their credibility.” 
The Problem of Defining “Extraordinary”
Tim McGrew has dubbed this slogan that “extraordinary claims require extraordinary evidence” the Argumentum Sagani in honor of Carl Sagan . In his presentations on the subject, he parodies the argument as follows (though he notes that this parody is not original with him):
1. Extraordinary claims require extraordinary evidence.
2. The claim that a miracle has occurred is extraordinary.
3. Any evidence supporting it ought to be extraordinary as well.
4. I am not sure what I mean by “extraordinary.”
5. But whatever you come up with, it’s not going to work.
6. No one is justified in believing any miracle claim.
The problem with the word “extraordinary” here is that it is rarely clearly defined. The mantra that I would adopt instead is that all claims require sufficient evidence. What counts as sufficient evidence will depend upon the relevant prior probability. And, indeed, the only relevance that the fact that a given event is supernatural has epistemically is that it suppresses the prior. However, even prior probabilities that are extremely small (but non-zero) can, in principle, be overcome if adequate evidence is forthcoming. This failure on the part of atheists to define what they mean by “extraordinary” in this context leads to them setting the bar of evidence so unreasonably high that the burden of proof cannot possibly be met.
What is the problem with the word “extraordinary”? If by that word we simply mean an event that is highly improbable or unique, then any event can be defined with sufficient specificity to meet that criteria. For example, consider Joe’s marriage to Sally. Joe being married to someone with the specific traits and characteristics of Sally is enormously improbable — especially when one considers the numerous other couples who had to meet, and the specific sperm cells that had to meet specific egg cells, all the way back to the dawn of humanity, in order for Joe and Sally to both be living at the same time. And yet Joe would be able to offer sufficient evidence that he is in fact married to Sally – adequate evidence to overcome a low prior probability. Is the fact that Joe married Sally an extraordinary event? Well, it depends on what you mean by “extraordinary.”
The point I am trying to make here is that you cannot simply define an extraordinary event as an occurrence that is highly improbable or unique (i.e. that it is something that lies outside of what normally happens), since that takes us into the realm where we can show that lots of events are very improbable or unique, if they are defined with enough specificity. Instead, the argument here is going to need to be more sophisticated. So, let me try to steel man the atheist’s argument and formalize why we tend to treat the resurrection differently from how we would treat the case of Joe marrying Sally.
Why Do We Treat a Miracle Claim as Different from a Mundane Claim?
Why, then, do we treat a miracle claim, like “Jesus rose from the dead”, differently from how we might treat a more mundane claim, like “Joe married Sally”? Clearly, it is not that the former is more improbable than the latter, or that the former is a very unique or an unprecedented event, since mundane claims, like “Joe married Sally”, can be defined with enough specificity to make them highly improbable, unique and unprecedented events as well. Obviously, the very same problem would be encountered if Joe had married any other woman, and so this consideration may be ‘cancelled out’. This is equally true of a lottery. The lottery being won by any given individual is extremely improbable. But since this is equally true of all participants in the lottery, we can ‘cancel’ that consideration. On the other hand, suppose that someone wins the lottery who happens to be the spouse or son or a close friend of the person running the lottery. In that case, we get more suspicious because that can be thought of as more probable on an alternative hypothesis than that of chance coincidence. Notice here that we do not get suspicious simply because something extremely improbable has happened — because this individual winning the lottery is no more improbable than any other individual selected at random winning it. Rather, we get suspicious because we consider that this particular individual winning the lottery is more expected (more probable) if something suspicious has happened than it is on the hypothesis of chance.
This is a good parallel for how the skeptic thinks about a miracle report such as the claim that Jesus was raised from the dead. Just like the hypothesis of fraud picks out the lottery example for special suspicion, so in this case, the skeptic argues, the hypothesis of hallucination or deceit picks out the resurrection reports for special suspicion. But note that this is not based on a purely inductive approach that argues that the event of the resurrection or of the suspicious lottery win is improbable by itself (because of the problem I raised above). Instead, the argument ought to be something to the effect that in the past we have found such claims to be more explicable by way of alternative hypotheses, so most likely this one is too. This then moves us into a discussion of alternative explanations of the evidence under consideration and whether those explanations are reasonable.
Further Problems With Inductive Assessments of the Prior Probability of Miracles
Another point that is worth bearing in mind is that it, if the prior probability of miraculous events is judged in purely inductive terms, Christians and skeptics are likely going to disagree about the priors anyway. The atheist presumably would take every miraculous report from the Old and New Testaments as another example of a failed miracle report, adding it to his case against miracles, whereas the Christian beliefs that those miracles in fact took place. Thus, simply putting the resurrection into the reference class of “alleged miracles” and then asking for a purely inductive prior probability of its occurrence is quite unhelpful. In fact, the atheist’s attribution of those miracle accounts to naturalistic causes is itself in significant measure a consequence of his non-empirical metaphysical judgments.
It is hardly as though there are millions upon millions of other cases where a miracle was reported and the skeptic, having thoroughly investigated each claim, has found compelling empirical evidence for their falsity. This is why the appeal of David Hume to the uniform testimony of mankind against miracles is wrong-headed. Indeed, there are many miracle reports in existence. Craig Keener has documented many examples in his two volume set on miracles . Thus, as far as that is concerned, Hume is simply wrong. Human experience is not at all uniform on this matter. In fact, Hume’s whole argument here is circular, since in order to argue that the uniform experience of man precludes miraculous events, he has to dismiss the numerous reports of such miracles as false – the very point he is endeavoring to establish.
In the case of the resurrection of Jesus, what is the relevant background information, which will inform our assessment of the prior probability? We have strong independent reason to think God exists and that He sometimes performs miracles. However, God doesn’t appear to perform those miracles particularly frequently, so they do fall into a reference class that is rather special on the basis of considerations such as God’s wanting to reserve them for special occasions so that they can be recognized as a sign against the background of a regular natural order. However, this implies a much higher prior than the prior assigned by the skeptic, even before we factor in other considerations that could raise the prior probability in Jesus’ particular case. It is to these other considerations that I now turn.
The Relevance of Religio-Historical Context to Estimating the Prior
For the reasons expressed above, I think that a purely inductive or frequentist approach to estimating the prior probability of the resurrection of Jesus is mistaken. The prior probability, I would suggest, can instead be raised by (a) pointing to the independent evidence from natural theology that God exists (if there is independent demonstration that there is a God who could perform the miracle, then this increases the prior probability of Him actually performing a miracle at least somewhat); and (b) fleshing out, to borrow a phrase from William Lane Craig, the religio-historical context of the resurrection. By elaborating this religio-historical context, one can show that God plausibly might have motivation for raising Jesus of Nazareth specifically from the dead. William Lane Craig notes, “A miracle without a context is inherently ambiguous. But if a purported miracle occurs in a significant religio-historical context, then the chances of its being a genuine miracle are increased. For example, if the miracles occur at a momentous time (say, a man’s leprosy vanishing when Jesus speaks the words, ‘Be clean!’) and do not recur regularly in history, and if the miracles are numerous and various, then the chances of their being the result of some unknown natural causes are reduced. In Jesus’ case, moreover, his miracles and resurrection ostensibly took place in the context of and as the climax to his own unparalleled life and teachings and produced so profound an effect on his followers that they worshiped him as Lord.” 
How can this religio-historical context be elaborated in an objective way? I would point here to instances in the gospels of what I call Messianic convergence. That is, instances in the gospel accounts where an episode in Jesus’ life intersects in some striking way with the Old Testament Scriptures but which also enjoys strong historical support. This is best explained by giving examples, so I will give a few here.
The evidence is compelling that Jesus died on the Day of Passover, the 15th of Nisan. I won’t get into the details here of how we know the gospels are historically reliable on this detail, since I only wish to illustrate the principle. Given the theological theme in the New Testament of Jesus being the fulfillment of the Passover lamb (e.g. 1 Cor 5:7), this is quite striking. This is not by any means a conclusive proof that Christianity is true, but that striking correspondence does seem to be somewhat more probable on the hypothesis that Christianity is true than on the falsehood of that hypothesis, and may therefore be counted as evidence (not proof) that Jesus is indeed the Messiah. I suggest that a Bayes factor of 10 (meaning it is 10 times more likely on the hypothesis of Christianity than on its falsehood), though admittedly somewhat subjective, is a fairly conservative estimate.
This argument may be developed as a cumulative case. In previous articles (e.g. here), I have discussed the undesigned coincidence that corroborates John’s statement that Jesus entered Jerusalem five days before Passover, on the week leading up to His death, which would correspond to the 10th day of the month of Nisan (Jn 12:12). It turns out that the instructions given in Exodus 12 regarding the Passover stipulated that on the 10th of Nisan the Jews were to select their Passover lamb and bring it into their homes (verse 3). Isn’t it striking, then, that Jesus’ death on the Day of Passover just so happened to be the same year that Jesus entered Jerusalem on the 10th day of Nisan?
Now, we need to be careful here since it may be pointed out that this coincidence is not wholly independent of the previous one, since Jesus coming into Jerusalem on Nisan 10 would not matter at all if it weren’t for the fact that he then died subsequently at least around that time. If we take his entry into Jerusalem on Nisan 10 to be significant, we must be assuming that he died right around that time, which means that that one entails the other. The question we can ask, however, is how much additional evidence it provides that Jesus also entered Jerusalem on Nisan 10. Another factor for us to consider is that Passover is a particularly likely time for a Jew to enter Jerusalem and also a time when he could count on ministering to a large crowd of people who had made their pilgrimage to Judea for the feast. Even with a Bayes Factor of 2 (meaning that it is twice as likely on the hypothesis than on its falsehood), however, the overall cumulative Bayes factor doubles.
Another example is the fact that Jesus was born in Bethlehem, fulfilling the Old Testament prophecy of Micah 5:2. I realize that this proposition is controversial. However, I will not argue the case (which I think can be made strongly) for Jesus being born in Bethlehem in this article. Here, I only intend to show how in principle such evidences can be relevant to the prior probability of Jesus’ resurrection. To be conservative, I will assign a Bayes factor for Jesus’ birth in Bethlehem to 1000. This is a generous Bayes factor, since clearly much, much less than 1 out of every 1000 individuals that have been born since the time of Micah has been born in Bethlehem, and the probability of Jesus in fact being born in Bethlehem, on the hypothesis that He is indeed the Messiah (based on the prophecy of Micah 5:2), is very high (approaching 1).
Another example is the fact that Christianity became the dominant international religion that it became. The Old Testament predicted that the Messiah would be the light to the gentiles, that God’s salvation might reach to the ends of the world (Isaiah 42:6; 49:6). Jesus Himself, during His ministry, said, “And this gospel of the kingdom will be proclaimed throughout the whole world as a testimony to all nations, and then the end will come,” (Mt 24:14). This entails that it is quite probable that, on the hypothesis that Jesus really is the Messiah, Christianity would bring people of all nations to a recognition of the God of Israel. However, this seems to be really quite improbable on the falsehood of that hypothesis. Until 313 A.D. (when the Edict of Milan, under the Emperor Constantine, guaranteed religious freedom and made Christianity legal), Christians endured intense persecution under multiple Roman Emperors. Under the circumstances, the odds of Christianity prevailing and becoming an international religion seemed vanishingly small, and yet it did. Again, then, we have a significantly top-heavy likelihood ratio where Christianity’s spread across the world is much, much more expected on the hypothesis of its truth than on its falsehood. I suggest that a Bayes Factor of 1000 is reasonable for this one.
If my assigned estimated Bayes Factors for the above four examples are reasonable, then our cumulative Bayes Factor is already 20,000,000 (meaning, our evidence is 20 million times more likely if Jesus is the Messiah than if He isn’t). There is admittedly a certain degree of subjectivity involved in assigning those Bayes factors, but I have tried to be reasonable and conservative, and I am only attempting to show how the argument may be probabilistically modeled given certain sets of assumptions. Those were only four examples, and there are many more that I could provide. Cumulatively, I would argue, this sort of evidence leads one to think that God may plausibly have motivation for raising Jesus of Nazareth specifically from the dead as God’s vindication of Jesus’ claims, message, and teaching (certainly more so than if He were some obscure miscellaneous Joe Blow). This argument can be developed still further (for more examples, see my earlier article here).
Another consideration here is that the Old Testament predicts that the Messiah would be raised from the dead (Isa 53:10), and Jesus Himself claimed numerous times that His resurrection from the dead would be God’s vindication of His radical Messianic claims. For example, there is strong historical support for the veracity of Jesus’ statement in John 2:19 that He would “Destroy this temple, and in three days I will raise it up” (speaking of His body), though again I will not get into the details of the supporting evidence here. Since that is itself a prediction that He would be raised from the dead, it may be taken as historical that Jesus really did predict ahead of time not only His impending violent death, but also His resurrection. This and other predictions of Jesus’ resurrection in the gospels can also be historically confirmed to be authentic, but we need not get into the details here. If Jesus really did predict ahead of time not only His impending violent death but also His resurrection from the dead, this also gives us reason to suspect that God may plausibly have motivation for raising Jesus of Nazareth specifically from the dead.
Overcoming the Subjectivity of Prior Assignment
One may challenge the appropriateness of a Bayesian approach to miracles on the basis that the assignment of the prior probability of a miraculous event is quite subjective. However, this obstacle can be overcome by back-solving for how low of a prior the pertinent evidence could overcome. For example, in their chapter in the Blackwell Companion to Natural Theology, Tim and Lydia McGrew argue for a Bayes factor of the evidence pertaining to the resurrection of 1044, “a weight of evidence that would be sufficient to overcome a prior probability (or rather improbability) of 10-40 for R and leave us with a posterior probability in excess of 0.9999.”  Likewise, I have similarly argued in a previous article that the cumulative Bayes factor of the evidence for God’s existence, on the most charitable of assumptions, is sufficient to overcome a prior probability of 10-18 and still yield posterior odds of God’s existence of 0.9999.
Thus, one can calculate how small a prior would need to be in order to overcome the positive evidence for a hypothesis. Even if we do not know precisely what the prior is, if we are reasonably confident that the prior is higher than that value, then a more precise estimate of the prior becomes less important.
To conclude, while the dictum that “extraordinary claims require extraordinary evidence” may have rhetorical appeal, a closer inspection reveals that it is fraught with oft-overlooked problems. Sagan’s dictum has regrettably often been used to shut down inquiry and discourse rather than foster it and encourage an open-minded and careful investigation of the public evidence bearing on resurrection of Jesus of Nazareth. My hope and prayer is that this article clears the way for more productive dialogue between Christians and skeptics on the epistemology of testimony and the provability of miracles.
 Carl Sagan, Broca’s Brain: Reflections on the Romance of Science, reprint edition (New York: Ballantine Books, 2011), 78.
 David Hume, An Enquiry Concerning Human Understanding (Rhode Island: Angelnook Publishing, 2014), 143.
 Ibid., 135.
 William Adams, An Essay in Answer to Mr. Hume’s Essay on Miracles, 3rd ed. (London: B. White, 1767), 15.
 IAP, “Timothy McGrew – The Formal Epistemology of Testimony: Beyond Hume and Earman,” October 30, 2019, video, 57:03, https://www.youtube.com/watch?v=GH11Ur8cjwM.
 Thomas Sherlock, The Trial of the Witnesses of the Resurrection (California: CreateSpace Independent Publishing Platform, 2015), 29.
 John Earman, Hume’s Abject Failure: The Argument Against Miracles, 1st edition (Oxford: Oxford University Press), kindle.
 Arif Ahmed, “Hume and the Independent Witness”, Mind 124, no. 496 (August 2015), DOI: 10.1093/mind/fzv076
 IAP, “Timothy McGrew – The Formal Epistemology of Testimony: Beyond Hume and Earman,” October 30, 2019, video, 57:03, https://www.youtube.com/watch?v=GH11Ur8cjwM.
 Craig S. Keener, Miracles: The Credibility of the New Testament Accounts, Volume 1 (Michigan: Baker Academic, 2011).
 J.P. Moreland and William Lane Craig, Philosophical Foundations for a Christian Worldview, 2nd Edition (Illinois: IVP Academic, 2017), 930.
 Tim McGrew and Lydia McGrew, “The Argument from Miracles: A Cumulative Case for the Resurrection of Jesus of Nazareth”, in The Blackwell Companion to Natural Theology, 1st Edition, ed. William Lane Craig and J.P. Moreland (Wiley-Blackwell, 2012), kindle.