The Resurrection Probability Calculus

Dr. William Lane Craig has spoken about the probability of the resurrection, arguing that it is considerably high, and has championed an argument for God’s existence from the resurrection. According to him, in answer to the question of what the probability of the resurrection will be given our background knowledge:

Well, that probability will be determined by the probability of God’s existence and the probability that God would raise Jesus from the dead:

P (R|B) = P (R|G) × P (G|B)

where R = God raised Jesus from the dead; G = God exists; and B = background information. So the person who thinks that the probability of the resurrection hypothesis is low has to show that either the probability of God’s existence is low or that the probability that God would raise Jesus is low.

As an amateur who doesn’t understand probability theory nearly as well as I hope to in the future, I’d like to make what seem to me to be two obvious objections to this. First, if Craig is right about the P(R|B) being equivalent to P(G|B)xP(R|G) then it seems wrong to suggest that the non-Christian (or at any rate the person who denies that the probabilistic argument for the resurrection of Jesus is a success), will not have to argue that the probability that God exists is low, or even that the probability that he would raise Jesus from the dead if he did exist is low. Suppose that the probability of God’s existence given our background knowledge is 0.5, for the sake of argument. Suppose then that the conditional probability that God raised Jesus from the dead given that God exists is 0.5 again for the sake of argument. Then the probability of the resurrection given our background knowledge will be 0.25! Suppose that the probability that God exists (on our background knowledge) is 0.75, and suppose the resurrection (on God’s existence) is 0.75, then the probability of the resurrection on our background knowledge will turn out to be 0.5625. Suppose, in fact, that either one of these probabilities were 0.5, and the other was set to probability n, where o.5<n≤1. That will mean that the probability of the resurrection given our background knowledge will be less than or equal to 0.5, won’t it?

Suppose P(R|G)=0.8 and P(G|B)=0.8, then P(R|B)= 0.64. The trouble here is that in order for somebody to think that the probability of the resurrection given our background knowledge is high, they already have to think that (i) P(G|B)>0.5, and (ii) P(R|G)>0.5. The argument should only appeal, then, to those who already believe P(R|B)>0.5.

So, in the first place, the skeptic need not assign a low probability to either P(G|B) or P(R|G) in order to argue that P(R|B)<<0.5. In the second place, it seems to me that the only way to make the argument work is to establish  (i) P(G|B)>0.5, and (ii) P(R|G)>0.5.

In fact even this isn’t enough though. Suppose that P(G|B)=P(R|G)=0.6; then P(R|B)=0.36!

Thus, to establish the argument one needs one of the following:

1. (i*) P(G|B)>>0.5, and (ii*) P(R|G)>>0.5
2. (i**) P(G|B)>0.5, and (ii*) P(R|G)>>0.5
3. (i*) P(G|B)>>0.5, and (ii**) P(R|G)>0.5

In fact, if P(R|G)=P(G|B) then each of the probabilities have to reach just over 0.7 (say about 0.7089), in order for the P(R|B) to climb over the 0.5 probability marker (which gives us about 0.502). Ideally, then, what we want is:

4. (iii) P(G|B) ≥ 0.7089 ≤ P(R|G)

I am an aspiring Catholic theologian and philosopher, and I have a keen interest in apologetics. I am creating this blog both in order to practice and improve my writing and memory retention as I publish my thoughts, and in order to give evidence of my ability to understand and communicate thoughts on topics pertinent to Theology, Philosophy, philosophical theology, Catholic (Christian) Apologetics, philosophy of religion and textual criticism.
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11 Responses to The Resurrection Probability Calculus

1. lotharson says:

Hello.

All this (kind of) calculus ASSUMES that degrees of belief in a theory are probabilities and can thus be computed as such (Bayesianism).
I find this pretty dubious http://lotharlorraine.wordpress.com/category/probabilities/ on philosophical grounds.
As I am going to explain in future posts, Bayes theorem has been proven for FREQUENTIAL probabilities but never for the intensity of beliefs in our brain.

By the way, though it is utterly unrelated, you might also be interested in my ongoing series on Calvinism.

Cheers from Europe.

• Thanks for the comment, maybe I’ll check that out.

• Why can’t we talk about conditional probabilities of proposition P on proposition Q by appealing to nearest possible worlds (i.e., that the frequency of worlds where Q and P is greater than the frequency of worlds where Q and not-P, or at least that the frequency of the former are all closer’ possible worlds, mutatis mutandis)?

2. Iron Man says:

HON HON HON HON HON HON

LOL MATH SUCKS ROFL

3. William says:

My view is that this isn’t really a matter of evidence or probabilities.

The value of a given piece of evidence is normally assessed by comparing the evidence we have with the evidence that would make the claim certain. For example, if we know that someone had the means and opportunity to commit a murder, but we don’t know what motive they would have had, we can compare this with the accepted standard of motive, means and opportunity and conclude that we have some evidence for the person’s committing the murder. Evidence is partial proof.

However, if there can be miracles, then there is no certainty. We could have motive, means and opportunity for someone to commit a murder, but if we accept that miracles are possible, then we cannot be certain that they committed the murder based on this evidence, because God could have intervened to manufacture all of our evidence, or given the person motive, means and opportunity to commit the murder and then miraculously intervened to prevent the person from carrying out the murder, or any number of other things. There is no standard to assess what counts as evidence objectively on the view that there are miracles.

I think naturalism is an axiom like the law of identity, and indeed a corollary of the law of identity. Things have determinate natures, as the law of identity says, and every action they take is expressive of and consistent with these natures. If we reject inductive skepticism, it follows that we can know what these natures are with certainty by the scientific method. Therefore, a violation of a scientific law that has been established conclusively, within the context in which the evidence justifies the law, is impossible. Examples of proven scientific theories would be Newton’s laws (in an everyday, macroscopic context) or the theory of evolution. If we deny this, then we cannot know or have evidence for anything.

• Thanks for the thought(s) William. Here’s my take on what you propose. First, it seems to me that I could affirm naturalism as you have articulated it, at least provisionally, so long as you accepted that God has a nature. On that view, if miracles are antithetical to naturalism, then God could not even possibly do anything miraculous – of course, I think the words are being bastardized here, but if you suggest that naturalism follows by entailment from the law of identity then the root of the semantic shift is at precisely that point, which I am granting for the sake of argument.

Second, I think you overestimate the scientific method, which is to say, here, that you underestimate the epistemological problem of uncertainty. I hope you would agree that the scientific method is unable to prove a variety of things which we are all justified in believing, such as in the reality of the past. The scientific method cannot give us certainty about such things, even though it often presupposes those things. Moreover, science presupposes certain metaphysical theses, and these include, among others, that inductive reasoning works precisely because there are consistent regularities in the world, that science can be correct precisely because (i) the world is rationally intelligible, and (ii) the human mind is rational in such a way that it has the ability to produce models of the world which are not merely useful but accurate. However, these two assumptions are most consonant with the assumption that God exists, has created the world according to a rational order, and has created mankind in his image in such a way that man is ‘rational’. These, far from being ad hoc stipulations, are actually the very assumptions which historically led to the scientific method in the first place. Even today the greater faction of scientific anti-realists are naturalists, and the overwhelming majority of theistic philosophers and scientists are scientific realists. That, I submit to you, is not coincidental. The point being that science seems to presuppose theism tacitly, and even if it doesn’t the original point is that science does nothing to eradicate the problem of skepticism. If you think we have ‘certainty’ in a completely indubitable and incorrigible sense via the scientific method, then I would respond that we clearly do not – Science doesn’t even purchase certainty about the scientific method itself.

However, if you mean something more modest by certainty, such as having a justified true belief, it seems to me that there are logically possible worlds in which God both exists and acts miraculously, and in which people also have justified beliefs. For instance, one might believe that God exists, on the basis of good reasons, and that belief would be true, thus they would have a justified true belief (i.e., certainty that God exists). Moreover, consider theology to be merely a model of God and his relationship to the world, not unlike scientific models. If one had such a model, corroborated by the empirical evidence, then couldn’t one possibly have a justified belief in God’s acting miraculously along with a justified belief in Science’s ability to latch on to regularities which obtain in the world?

You mention that if God exists then he could clearly have set the world up such that it looks like somebody, call him Dexter, killed somebody else, call her Susie. If God exists and can act miraculously, you suggest, then we cannot have certainty, regardless of the balance of the evidence, that Dexter killed Susie. However, that problem doesn’t disappear by eliminating God. Suppose that I suggest that Aliens made it look like Dexter killed Susie, with all their advanced equipment and motives beyond our comprehension – is that really a good argument to think that therefore no Alien life forms exist? Better yet, what if I said that the government conspired to make it look like Dexter killed Susie? Don’t we, there, have the very same problem? Wouldn’t it be better to say that although it is logically possible that God or Aliens or the Government simply made it look like Dexter killed Susie, we can still have a justified belief that, based upon the weight of the evidence, Dexter did in fact kill Susie? The possibility that God would act to deceive us (ignoring for the moment the inherent contradiction that would yield theologically), although it could in principle be admitted by the Theist, would no more undermine her certainty that Dexter killed Susie than your belief in the Government’s ability to manufacture evidence would undermine your certainty that Dexter killed Susie (assuming the evidence for that is very strong). All this is just to illustrate the point that simply dropping the word ‘God’ from one’s vocabulary does nothing to make the epistemic problem(s) disappear. Adding God to the equation does just as little to undermine our certainty about well established empirical facts. I, for my part, cannot see any reason to think that God’s acting miraculously would be a game changer for either science or epistemology.

• William says:

You argue that a naturalism which asserts that things act according to their natures is trivial, because you can just say that God is natural in the sense that he has a nature. This only works if you assume that we cannot know what the natures of things actually are with certainty. But we can, through observation and experiment, and God contradicts a number of generalizations that a reasonable person would draw from observation, like the generalization that minds have brains. So, I would say that God does not have a coherent nature.

You argue that there are a number of theses that science depends upon but cannot justify with certainty. Let’s go through these one at a time.

1. “The reality of the past.” There is an excellent reason why science cannot justify this belief, and that is that the belief is false. The past does not exist except as a memory in the mind.

2. “That inductive reasoning works precisely because there are consistent regularities in the world.” The existence of regularities is guaranteed by the law of identity, which says that a thing cannot act contrary to its nature. Inductive reasoning is just our way of arriving at those regularities through observation and experiment.

3. “The world is rationally intelligible.” This is also guaranteed by the law of identity. We should be able to measure anything that exists, as well as compare and contrast it with other things, draw generalizations about it, and so forth. Mathematicians can even tell us what the properties of infinity quantities are. The only reason why something would be permanently unintelligible is that it does not exist.

4. “The human mind is rational in such a way that it has the ability to produce models of the world which are not merely useful but accurate.” I’m not sure why observation wouldn’t give someone ample justification for believing this.

Your claim about where science came from is historically incorrect. Christianity was around for thousands of years before science arose. In the late Middle Ages, philosophers began to realize that supernatural explanations yielded less fruitful scientific theories than natural explanations and began insisting that we not appeal to God to explain anything except the beginning of the universe. Early modern philosophers like Bacon built on that principle by maintaining a separation of scientific and religious knowledge, encouraging scientists not to defer to authority, insisting that people actually go out into the world and do experiments rather than sit in a chair and think, and advocating that scientists share their data with each other rather than keep the results of their experiments to themselves. The idea of experimentation in particular was crucial – the ancient Greeks and Medieval philosophers had a more contemplative notion of knowledge and did not really realize that they needed to actively test their ideas. None of these points require Christianity as a foundation, and arguably Christianity would be detrimental to some of them (the idea that we can gain knowledge through revelation certainly seems authoritarian and anti-experimental).

Finally, you argue that introducing God is no more skeptical than believing in aliens or government deception. The problem with the aliens hypothesis is that it is arbitrary, and a claim to certainty cannot be undermined by an arbitrary hypothesis. Just claiming that there might be aliens with no evidence does not undermine a claim to certainty – you have to enter the context of evidence and explain exactly why we should doubt that Dexter murdered Susie. The government example is not an arbitrary hypothesis, because we know that the government sometimes fakes murders, but we have to ask whether the evidence in the specific case of Susie’s murder warrants considering the possibility of a government coverup. If it does, then we should take the possibility seriously – but this does not generate crippling skepticism, because we have remained within the context of evidence, and we know roughly what powers the government has. On the other hand, if the evidence does not warrant considering the possibility of a government coverup, then it is an arbitrary hypothesis and cannot undermine a claim to certainty.

4. Fascinating. So, you’re an ontological presentist, presumably, meaning that you believe that only that which presently exists exists. That means you have to reject a correspondence theory of truth, or else you have to reject that there are propositions about the past with a truth value (whether true or false, unless you think the falsemaker would just be that the propositions do not refer to anything, in which case at least you don’t believe that there are any true propositions about the past). So, to your mind, is it logically possible to travel back in time? If it isn’t logically possible then you must be an A-theorist about time (in fact, to be a presentist means to be a certain sort of A-theorist, unless one construes presentism in a very odd manner). I have to ask, do you really believe that there are not true propositions about the past, or that they don’t have truth-makers, or that their truth-makers are something other than past events?

The generalization that minds have brains does nothing to defeat the idea that there could be a mind without a brain; what we need to do is ask whether we have reasons to think that there are special cases of minds without brains. Even before doing that, though, I think good philosophy of the mind reveals that it is logically possible that the minds we observe be without brains. There is no conceptual necessity for a mind to be co-extended with a brain. I’ve used this example before, but consider the fact that we’ve only ever observed green grass (or brown, or whatever). However, our minds can conceive with ease what it would be like to observe purple grass. We recognize that our observing grass as green does nothing to make purple grass unintelligible. Similarly anyone should be able to, with relative ease, imagine a mind without a brain, if only they have an appropriate understanding of what kind of thing the mind is, and why we distinguish it from the brain in the first place. The distinction isn’t vacuous.

I am not sure what you think the law of identity is. You seem to be suggesting that the law of identity is something real in itself, and this thing insures that there be regularities. I presume that you don’t mean anything as eclectic or platonic. However, you do apparently believe that the world has something like an Aristotelian essence, which is the best way I can make sense of your statement that nature itself is identical with its own nature. I presume you mean that the natural world has a nature, with which the world is identical, and because a thing, pace the law of identity, cannot act contrary to its nature, the world cannot act contrary to its nature. It’s nature, then, is such that there are these regularities, and we can latch on to them. Let me know if that’s approximately what you mean.

You make the same point about the intelligibility of the world, which provides, if not confirmation, at least corroboration for my interpretation above. I presume you also reject Quine’s dispensability thesis, the idea that there is no belief in principle which we cannot dispense with. If you do reject Quine’s thesis, then presumably you accept that there are certain laws, like the law of identity, the law of non-contradiction, the law of excluded middle and so on which provide, as it were, the parameters of intelligibility. However, what say you to the mathematical intuitionists or the quantum physicists who argue that we must dispense with the law of excluded middle? Perhaps you don’t accept the LEM as one of the parameters of intelligibility. I can come at the issue another way though; suppose I say that there are logically possible worlds very much like ours, but where our brains evolved so as to ensure a high probability of our survival and reproduction (or at least survival until reproduction), but where those brains were maladapted to the world such that doing science properly leads straight to paradoxes which the mind can make no good sense of. Perhaps the non-epistemic Copenhagen interpretation is an example of this, but even if it isn’t, surely it gives us some idea of what an ‘unintelligible’ scientific theory would look like. In this logically possible world we’re considering, let’s imagine that the scientific enterprise had run into unintelligible paradox in every direction, in every major field. Wouldn’t the world then be unintelligible to us, even if we could continue to develop ‘models’ of some kind for the purposes of prediction (note that they would no longer be ‘explanatory’ for us). Thus, granting that the world is identical to itself, that it has a nature contrary to which it cannot behave, thus has regularities objectively (mind independently), still doesn’t purchase the conclusion that it must be and remain intelligible to us. There is a logically possible world where the world has regularities contrary to which it cannot act, and in which the nature of that world is unintelligible to us. Do you think this is wrong, and if so why?

Just a historical quibble: “Christianity was around for thousands of years before science arose.” That cannot be correct, Christianity is not even a full 2000 years old, unless you think Christianity began at the birth of Christ instead of at the beginning of his preaching ministry. Christianity was around for ‘at least one thousand’ years before science arose – that is probably what you mean. I go back and forth on this point. It seems to me that science in some significant ways predates Christianity, but it also seems clear that science only reached its modern maturity thanks to the developments of medieval Christian theology.

“if the evidence does not warrant considering the possibility of a government coverup, then it is an arbitrary hypothesis and cannot undermine a claim to certainty.” – This is interesting. Provide for me, if you could, your precise definition of certainty.

X is certain of P if and only if …

I suspect that the way in which you use the word needs to be pinned down precisely before I can make the point forcefully that we can know with certainty that God exists.

That’s all for now, sorry for the late response, and thanks for your patience.

5. Gary says:

I have come to the conclusion that the reason why Christians see the evidence for the Resurrection as “strong” and skeptics such as myself see it as “very weak” is due to our very different views regarding probability. Christians view the Christian supernatural resurrection explanation as the most probable explanation for the early Christian resurrection belief because they presume the existence of their ancient Judeo-Christian god, Yahweh. Since skeptics do not believe that Yahweh exists, they view the Christian supernatural explanation as very improbable; we believe that there are many, much more probable, naturalistic explanations for each piece of evidence that does exist surrounding the death of Jesus.

Many of us skeptics acknowledge the possible existence of evidence for a generic Creator. But we do not believe that this evidence automatically translates to evidence for Yahweh. To us, if there is a Creator, he/she/they/it have determined that the universe is to operate by inviolable laws that are NEVER violated. Yahweh claims to have repeatedly violated these laws. Since science has found no evidence of these violations, we believe that this is but one piece of evidence of Yahweh’s non-existence. And there is more evidence of his non-existence: The consensus of geologists is that Noah’s Flood is a myth. The consensus of archeologists is that the Exodus is a myth. The consensus of biologists is that the Creation story is a myth. The consensus of cosmologists is that the age of the earth as determined by the genealogies in the OT is a myth. In addition, Yahweh believes that a “firmament”, a dome or shell, exists above the earth. Scientists say no such entity exists. There is just too much evidence to believe that Yahweh exists. It is much more probable that he is the figment of the imagination of an ancient, scientifically ignorant people. And without presuming the existence of Yahweh, there are just too many much more probable explanations for the early Christian resurrection belief than a never heard of before or since reanimation of a three day brain dead corpse.

• Note that I don’t run this blog anymore. My new blog is at http://www.tylerjourneauxgraham.wordpress.com

However, very briefly: I see no reason why a naturalist couldn’t concede that the resurrection occurred but was accomplished or caused by aliens using some advanced technology, or that it was some extremely improbable event for which we may, in the future, come up with the correct naturalistic explanation.

So let’s be careful to distinguish the case for the resurrection of Jesus from the case for the Christian worldview wholesale.

Furthermore, I recognized that miracles are popularly thought to be violations of nomological laws, but this may just be confused. Any scientific law you give me, I find I can only make sense of it when I express it as a counterfactual conditional given some implicit set of conditions which ensure that all things are equal. In other words, they are all stipulated for closed systems. For me to stop an apple falling from the ground by catching it is not to violate the law of gravity. The law states simply what ‘would’ happen in a closed system (perhaps where we talk only about the apple and the earth, or something like that). Similarly if God exists then the universe is not a closed system, and for him to operate immediately in the world does nothing to violate any scientific law because the antecendent conditions in the conditional statement expressing the nomological law do not obtain.