Posted by: Calum Miller | February 18, 2012

Why the resurrection data can be evidence for theism, and why arguments for resurrection need not presuppose theism – formal demonstrations

It is often assumed, among Christians and non-Christians alike, that one can only make an argument for the resurrection after showing theism to be probably true already. Or, perhaps, that the resurrection evidence cannot support theism, but can only support the resurrection given theism. Here, I aim to show, briefly, the formal underpinnings (in terms of probability theory) of how the evidence for the resurrection might support theism. I will show how this evidence can work in two ways: to support theism, and also to support the resurrection independently of whether theism is considered to be priorly unlikely or not. NOTE: I AM NOT ADVOCATING THESE PROBABILITIES AS TRUE. I AM SIMPLY SHOWING HOW, IF PEOPLE DO COME TO AGREE ON SOME PROBABILITIES, THE RESURRECTION DATA CAN BE EVIDENCE FOR THEISM, AND HOW ARGUMENTS FOR THE RESURRECTION CAN GIVE AN OVERALL HIGH PROBABILITY EVEN WITH A LOW PRIOR PROBABILITY FOR THEISM.

I am not here aiming to show that the resurrection data show theism to be true or that they show the resurrection to have occurred, only how it would do so. So, let us begin with defining some conditional probabilities:

Let T = theism, R = the resurrection of Jesus, and D = the specific historical data pertinent to Jesus’ resurrection (e.g. empty tomb, appearances, etc)

P(T) includes natural theology, and is 0.01 (that is, the prior probability of theism – I think it is higher than this, but this is only a demonstration)

P(~T) = 0.99 (the prior probability of atheism = 1 – P(T))

P(R|T) is around 0.0001 (that is, the probability of Jesus being resurrected on theism – unlikely, but not inconceivably unlikely)

P(R|~T) is 0.00000001 (that is, the probability of Jesus being resurrected on atheism – very unlikely, probably much lower than this)

P(D|R) = 0.75 (that is, the probability of empty tomb, appearances, etc, given Jesus’ resurrection. This is, again, an artificially precise number, but the point is that it is relatively high)

P(D|~R) = 0.00000001 ( that is, the probability of empty tomb, appearances, etc, given that Jesus was not resurrected. Again, artificially precise, but very unlikely)

Let us work out P(D|~T) to begin with, the probability of  the data on atheism. This is equal to sum of the individual probabilities of all the different kinds of ways the data could obtain on atheism – in our case, the probability of the data on R and on ~R.  So, P(D|~T) = P(D|~T & R)•P(R|~T) + P(D|~T & ~R)•P(~R|~T). With our given probabilities, this is equal to 0.75 x 0.00000001 + 0.00000001 x 0.99999999, which in total is roughly 0.0000000175

Now, we can do the same for P(D|T), which will be equal to P(D|T & R)•P(R|T) + P(D|T & ~R)•P(~R|T). Plugging in our probabilities, this will be equal to 0.75 x 0.0001 + 0.00000001 x 0.9999, which in total is roughly 0.00007501

Now, it follows from Bayes’ Theorem that  P(T|D) = [P(T)•P(D|T)] / [P(T)•P(D|T) + P(~T)•P(D|~T)] (see the derivation of Bayes’ Theorem here). Plugging in our additional beginning probabilities, we note that this is equal to (0.01 x 0.00007501) / (0.01 x 0.00007501 + 0.99 x 0.0000000175) = 0.0000007501 / (0.0000007501 + roughly 0.0000000175), all of which is roughly 0.977 And so the probability of theism given the empty tomb, etc, would be 0.977, whereas the prior probability of theism was only 0.01. Thus, the data, through looking at the different ways it might obtain (that is, under R and ~R), supports theism significantly, and this support does not presuppose a high prior probability of theism.

Now, when looking at the probability of the resurrection, we can come up with a prior probability of the resurrection based on the prior probabilities of theism and atheism, and the likelihood of the resurrection on each of those alternatives. Again, we do not need to give theism a high prior probability to conclude posteriorly that the resurrection probably happened. P(R) is equal to P(R|T)•P(T) + P(R|~T)•P(~T), and our values for these yield a prior probability of R equal to 0.0001 x 0.01 + 0.00000001 x 0.99, which is roughly 0.00000101

Our posterior probability P(R|D) will then be equal to [P(R)•P(D|R)] / [P(R)•P(D|R) + P(~R)•P(D|~R)] (again, from Bayes’ Theorem). and plugging in our values gives us (0.00000101 x 0.75) / (0.00000101 x 0.75 + 0.99999899 x 0.00000001) = 0.0000007575 / (0.0000007575 + roughly 0.00000001) = 0.987. Thus, the prior probability of the resurrection and of theism might be very low (0.00000101 and 0.01, respectively), and yet the historical data might well be strong enough to overcome this, allowing us to conclude that the resurrection happened.

This, then, serves as a rejoinder to those philosophers of religion who do not see the argument from miracles as supporting theism at all, or who think that one need to presuppose theism to argue for a miracle. Neither of these are true, and I hope to have shown this decisively through formal use of the probability calculus. As noted at the start, I am not intending to argue that the probabilities given at the start are correct, only that it is in principle very much possible for one to argue to theism from miracle evidence, and that it is also possible to argue for a particular miracle without presupposing theism.



  1. “(the prior probability of atheism = P(T) – 1)”

    Surely you mean:

    “(the prior probability of atheism = 1 – P(T))”, otherwise you get a negative result.

  2. True; now corrected. You can tell it’s nearly 4am here!

  3. This post is fantastic! I’ve written some similar posts on using Bayes to combat Richard Dawkins’s stance, and also to support the resurrection as well. I love it!

  4. The problem is this would work for any miracle event because miracles are far more likely given theism. You have to come up with a way to disclude all other possible gods and religions, otherwise all that you have shown is that miracles do indeed occur given theism being not very a-priori likely.

    Indeed, one could declare as a “miracle” any currenlty unknown event or process, then claim god did it on this reasoning. Bayes is supposed to be used as a means of gathering more accurate TEST data. You can’t update on historical events. But that doesn’t mean bayes isn’t valid.

    -Matthew Fuller

  5. Thanks very much, Greg!

    Matthew, that is true. My point here wasn’t to argue for the resurrection, however, only to show that one does not need to presuppose theism in order to argue for it, and that it can be an argument for theism in itself. Your concerns would need to be addressed in a future post on a slightly different topic, of course – I will hopefully get to that soon! If you’re interested in those kinds of considerations, you might want to have a look at Swinburne’s “The Resurrection of God Incarnate” or Tim and Lydia McGrew’s “The Argument From Miracles” in the Blackwell Companion to Natural theology.

    In case you were wondering the *kind* of thing I’d say in response (optimistic of me, I know!) I think Swinburne, the McGrews and I would all agree that, in most other cases of miracle claims, P(Miracle|T) is not quite as high as P(R|T) and that P(D|~Miracle) is not quire as low as P(D|~R). Thanks for taking the time to comment!

  6. “P(T) includes natural theology, and is 0.01…”

    “P(R|T) is around 0.0001…”

    Maybe this is missing your point, but perhaps some philosophers of religion “who do not see the argument from miracles as supporting theism at all, or who think that one need to presuppose theism to argue for a miracle” actually object to assigning probabilities to P(T) or P(R|T).

    For example, if a philosopher thinks that one needs to presuppose theism to argue for a miracle, perhaps they’re saying in terms of Bayes Theorem that one needs to presuppose “a type of theism to which probability values can be assigned” to argue for a miracle.

    Why do you think it’s possible to assign probability values to P(T) and P(R|T)?

  7. Thanks for your response, Geoffrey. I don’t see why it would be impossible to give theism a prior probability and yet give other explanatory hypotheses prior probabilities. The argument I’m making here doesn’t strictly require us to give theism a prior probability, nor does it require a well-defined probability for P(R|T). The idea is simply to show that it is at least possible in principle for the resurrection to *raise* the probability of theism from its previous (perhaps inscrutable) value. I don’t think P(T) or P(R|T) can be given particularly precise values, particularly the latter. And, of course, people’s value for P(T) will differ enormously depending on whether they are convinced by arguments from natural theology. Though I think P(T) is actually a lot higher than given here, I was mainly aiming simply to show how even a low value for P(T) can be overcome by evidence for the resurrection.

    P(R|T) is even harder – William Lane Craig says it is inscrutable, and I think many would agree with him. I have sympathies towards this and would definitely not want to give it a precise value (or even a precise order). But I think we only need to show that it is not negligible, or that it is a fair bit greater than P(R|~T), to show that the argument might at least have some promise, in principle (though, of course, if one gives it a very small value, one will need to argue that P(D|~R) is accordingly exceedingly small for it to have force in practise). One might then give some factors which would ‘boost’ P(R|T) – for example, arguments that Jesus claimed to be God, and predicted his resurrection, arguments that God, if he existed, would likely become incarnate, etc (Swinburne has done a lot on this). One would then not necessarily need to give a particularly high probability to P(R|T) – it may even be as low as something like 10^-5 or whatever, and yet could still potentially be made posteriorly probable by the evidence for the resurrection.

  8. (I haven’t actually worked out the maths of what kind of range particular probabilities would need to be in for this kind of argument to work, but I will hopefully do so at some point in the future!)

  9. (I should also note that proponents of this kind of argument would almost always themselves be very tentative in assigning probabilities. In my experience, the idea is not to give realistic values for these probabilities, but to deliberately underestimate one’s own case (and so assign probabilities which the sceptic would favour) and see whether one can still come up with a reasonably high posterior probability for the resurrection and theism.)

  10. It’s my pleasure, kind Sir.

    I must admit, I’m no expert in Bayes Theorem, probability, or math in general.

    “I don’t see why it would be impossible to give theism a prior probability and yet give other explanatory hypotheses prior probabilities.”

    I don’t know that it’s possible because probability is, as far as I know, only of use regarding natural phenomena.

    To illustrate, a phenomena could have many possible explanations. The natural explanations can have probabilities assigned or estimated based on our understanding of the natural world. Our understanding of the natural world, however, doesn’t necessarily give us any basis to assign or estimate, even in principle, probabilities to the supernatural explanations.

    Where am I going wrong here?

  11. Well, how would you say the probabilities are assigned in the natural world? We don’t always simply have the frequency data to be able to do so, so I think we really have to look at other factors which are key in determining which, of different theories giving the same empirical predictions, might be more intrinsically probable. It seems to me that science favours those which are simpler, which involves postulating simple entities, few entities, few kinds of entities, few properties, few kinds of properties, and so on. Theism, in my opinion, does reasonably well on these.

    I don’t know if it’s enough to say that theism is more probable than not without any evidence at all – I’d be wary of saying it was *that* simple, but remember the prior probability in question here is not simply the intrinsic probability of theism, it is the probability of theism once we have considered all the data of natural theology, including the fine tuning of the universe. Again, what I would suggest here is that some of the evidence is so strong as to overcome even very small prior probabilities (e.g. fine tuning of the universe) – at least, I think, there is enough evidence to say that the probability of theism given natural theology (represented here by P(T) though, of course, it should really say P(T|K) where K = background knowledge including natural theology) is 0.01 – which I think can be in principle overcome by considering the resurrection data (though not necessarily in practice).

  12. Just a few points:
    1) “…and yet the historical data might well be strong enough to overcome this, allowing us to conclude that the resurrection happened.” is essentially saying that the evidence is “extraordinary enough” to make our claim believable. That is the beauty of Bayes Theorem: it solves the Threshold Problem as to how extraordinary evidence must be in order to be “good enough”. So, in other words, this piece you wrote is actually a refutation of your prior post titled “Do ‘extraordinary claims require extraordinary evidence’? The answer is “yes” and Bayes Theorem tells us just how extraordinary the evidence must be.

    2) ” I am not intending to argue that the probabilities given at the start are correct, only that it is in principle very much possible for one to argue to theism from miracle evidence, and that it is also possible to argue for a particular miracle without presupposing theism.” This cuts both ways. The skeptic can also argue against theism without presupposing miracles are impossible. In your specific example the debate would center around the probabilities of P(D|R) and P(D|~R) and what really constitutes the makeup of “D” (i.e. do we really have evidence of an empty tomb or resurrection appearances, or do we just have stories of an empty tomb and resurrection appearances, with the hypothesis that those events actually happened being one of many alternative hypotheses, with each claim being subject to its own Bayesian analysis).

    So, it seems to me that you have accomplished two things, maybe unintended:

    1) You have shown why extraordinary claims (claims with extremely low prior probabilities) require extraordinary evidence is true, and,

    2) You have shown how skeptics can argue against miracles without presupposing theism is false.

  13. Yes – I have always held that propositions with an extremely low prior probability need greater confirmation to be probably true. The reason I might not agree with Sagan’s claim, then, is not because I disagree with this Bayesian principle – but rather simply because of the ambiguity of “extraordinary”. If all that is meant by “extraordinary” is “having a low prior probability” or “providing significant probabilistic support”, then I would agree – but even then the word is being used in two completely different ways. So it’s really a question of semantics.

    And yes, I agree that one can argue against miracles without presupposing theism to be probably false.

  14. This is a very interesting article. Out of curiosity, in the context of this article, how do you define “the resurrection of Jesus”?

  15. Thanks for commenting, Jeffery – always good to have another Bayesian around! I’ve been meaning to respond to your “20 questions” post for a while, but haven’t got round to it yet. For this article, I suppose I would definite it as something like, “the bodily raising of Jesus from the dead”.

  16. Thank you for writing this, it strikes at the heart of what’s wrong with apologetics. Christians do not need to argue for some generic theism or even creationism, we need to be arguing for the resurrection!
    I was an agnostic once…I know what it’s like to be someone who doesn’t know if “God” exists or even if people know what they mean by the word God. But if God came to earth, announced himself, and proved who he was… then none of that matters. God is then who God knows himself to be. The gospel deals with God as a reality, not a theory. We worship the I AM not the “I might be”.

  17. Thanks for your comment, Joshua. I actually think arguments for generic theism can be useful: they do, after all, give arguments for a necessary (though not sufficient) part of the Christian faith. Indeed, even in this article, you’ll see that one of the terms is P(T) which includes natural theology (all the terms should really include conditioning on N as well) – and is therefore affected by arguments in natural theology!

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