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.