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Friday, March 30, 2012

Prior Probability of the Resurrection

People have been arguing for centuries over the evidence for, or against, the resurrection of Jesus Christ. 

What is often forgotten, though, is that the probability of an event requires relating two things: (1) the evidence for that event (which is what we usually think of), but also (2) the prior probability that the event would occur. 


For example, suppose I make the claim, "I just flipped an unweighted coin twenty times, and got all heads!"

What is the likelihood that I am telling the truth?  Some combination of two factors: (a) the odds of achieving this result, which (on exactly 20 flips) would be 2^20, or about one in a million, and (b) the odds that I am lying or mistaken.  (Maybe I wasn't paying close attention, or maybe both the coin and my eyes are old, and the two sides of the coin look the same to me.  Or maybe the coin is, after all, weighted, or perhaps I am suffering a drug-addled hallucination from years of playing Russian Roulette in a smoke-filled bar in Saigon.) 

If I say, "I flipped a coin three times in a row, and got all heads," you would likely nod your head and go back to your Kindle.  If I claim to have done so 300 times in a row, you would have every right to be extremely skeptical.  At some point, the improbability of the event will tip the balance against the improbability (I hope most readers will agree) of various perhaps attenuated "liar" or "lunatic" hypothesis.  In the technical, arcane language of legal philosophy, you "smell a rat."   

Prior probability is often overlooked, or under-emphasized, in popular discussion of the Resurrection of Jesus.  Someone like William Lane Craig, Gary Habermas, or Josh McDowell (for the old school) will argue that the historical evidence for the resurrecion is actually very good.  Then skeptics will try to undermine that claim.

The Christian will generally win that argument, because as historical evidence goes, the evidence for the Resurrection of Jesus IS remarkably good.  The main challenge may be to get the audience to grasp the value of historical evidence, with our bias for forensic or other "scientific" evidence.  But given a fair appreciation of historical method, and a competent description of the evidence, this case can easily be made.  If all that were being asserted were that St. Peter caught a very big fish that day, the event would be universally admitted as having been proved beyond any reasonable doubt. 

But then the skeptic, and maybe even the Christian (even Alvin Plantinga, in some moods), will go home and ask himself, "Yeah, but can you really prove something so contrary to human experience as a resurrection from the dead, from any amount of historical evidence?"

Indeed, read the Book of Acts closely, and it's clear that this objection goes back at least to St. Paul's sermon before the Committee for Skeptical Inquiry that met on Mars Hill in Athens.  "God did what?  Go on!  Get out of here!"  But up to that point, they were listening carefully to Paul's more philosophical and moral arguments for theism. 

A frequent visitor to this site, the Irish skeptic Brian Barrington, once posted a tongue-in-cheek  argument against the Resurrection based on a "prior probability" against it of 100 billion to one.

He didn't explain how he got that figure.  But I think what he based it on, was the number of people who have lived on Planet Earth over the past two thousand years. The assumption, following Hume, seems to be that experience of an event should be deemed proportional to the frequency of similiar events.  If 100 billion people have died without rising, there should be a presumption against any one person rising from the dead, of 100 billion to one.

I pointed out that in fact, no one has any historical evidence about the post mortem fate of the vast majority of human beings since Jesus. Some other resurrections HAVE been alleged -- even in the pages of the New Testament. So even supposing that resurrections strike people as randomly as lightening or meteorites, still the ratio of people whom we know in a strong historical sense have not have risen from the dead,  to those about whom we might have some evidence to the contrary (say Lazarus), is much less than 100 billion to one. 

But of course no one claims Jesus' resurrection was just a matter of luck.  Christians, and even the occasional Jewish scholar who buys into the story, never suppose Jesus won the resurrection ticket in a random cosmic lottery.  On any reasonable account, if God exists and acts in the world, Jesus was far more likely to rise from the dead than, say, Al Capone, or even Joe the Plumber.

"Prior to what?" The prior probability of the resurrection is the probability that Jesus would rise, considered prior to the historical evidence for or against his resurrection. My claim is that there are other facts -- including some which no reasonable person denies -- that vastly increase the probability that Jesus might be resurrected, as opposed to one of the billions of other people who have lived in historical times.

The prior probability of Jesus' being raised would seem to be a function of three issues: (1) Does God exist? (2) How likely would  he be, even while keeping the laws of Nature generally in effect, as He obviously does -- including entropy in general and human death in particular -- raise one person dramatically from the dead? (3) How likely would it be that that person be Jesus?

(1) The probability that God exists will approach one for many people, zero for others, with other people falling at all spots in between.   But the arguments are complicated, and it would be foolish to begin arguing about the resurrection by first trying to solve so involved an issue, that is never solved to everyone's satisfaction.  Theists have arguments for the existence of God that are independent of the resurrection of Jesus. If we begin with belief in God, or the strong probability that God exists, BEFORE discussing the resurrection, we will not be begging the question for ourselves, but might be begging it for non-theist readers.  And we will be leaving out one plausible reason for believing in God.  So what should we do? 

One might deal with the "God question" in one of two ways.  First, we might "compromise" by taking the existence of God at 50-50, just to see how the rest of the argument works.  Or second, we might make the existence of God a variable in the equation, and use the resurrection of Jesus as an argument to faith in God.  Both solutions are ways of bracketting this issue, and perhaps coming back to it later, better-informed. 

Let's go with the first option, here.  I think 50-50 is a fair number to assign, for the following reason.  People are intelligent creatures, but also prone to fooling themselves at times.  (Including, of course, myself, and the reader.)  We may have both worked over the evidence for faith many times, and come to different conclusions.  So who is really fooling himself, or herself?  We can work over the evidence for decades, and both walk away thinking the other person is missing the point. 

In this case, I think it is valid to tentatively make a sort Ad Populum argument, as a short-cut.  Let us appeal to what society at large, our friends and neighbors and humanity in general, believe, as a temporary control over our own opinions. 

Most people believe in God.  Far fewer extremely well-educated people claim to believe in God.  Maybe that's because they're more intelligent, better education, and know more about, say, evolution or the Big Bang.  On the other hand, the Bible predicts that "the wise" are liable to becoming proud and self-dependent, and fail to recognize their need for God.  The Lord "opposes the proud, but gives grace to the humble."  Though of course not all believers LOOK gracious, and some intelligent skeptics seem highly sincere.  Furthermore, miracles and other signs of God's reality might be more likely to occur among those who are needy, if the Bible is any guide, not among the comfortable. 

So there are competing explanations for atheism (I give several others in The Truth Behind the New Atheism), and in some societies, Christians seem more heavily represented among the highly educated than among lower classes. 

So I think a fair shortcut to reworking all the arguments, which we can't do for a premise here, is to compromise between classes and cultures, our own intuitions and arguments and Vox Populi, and call it (for the sake of the argument) a 50-50 shot. 

Again, if you don't like this figure, don't worry -- the evidence of the resurrection may cause it to increase. 

(2) The  likelihood that God, should he exist, would act in this one particular way, by raising someone from the dead, is also hard to be sure about, since we are unable to read the mind of God.  (Of course, it does not follow that He cannot tell us His own mind, which Christians claim He has, in the Bible, for instance, and perhaps in other ways.)  That God would do such a thing, does not seem wildly unlikely, though, given: (a) God is, by hypothesis, good, and would want us to have hope; (b) God created Nature, and therefore presumably has good reason to allow the regularities that we call "natural law" to be manifest most of the time; (c) Much of the offense skeptics take at miracles seems to be aesthetic; it seems inartistic, crude, etc, for the laws to be set aside too easily; (d) Yet one can see intuitive sense in God offering some dramatic sign (as it is called throughout the Bible), some "eucatastrophe" as J. R. R. Tolkien put it, some "good disaster" by which (to cite his friend C. S. Lewis) the "laws of nature would begin to work backwards," so that Entropy would not have the last word, and the "sting of death" be drawn, may seem like amazing good news, presumptuous to assume, but hardly out of character for a good God.  Raising someone from the dead, at a key juncture in history, might well be part of His plan.

It seems presumptuous (aside from His revelation, which is in dispute) to put a number on "what God would do."  But by the same token, it would also be presumptuous to assume dogmatically that He would not do such a thing. 

This question, too, is difficult to solve decisively in a short discussion.  But it does not seem an overwhelmingly unlikely idea a priori.  Only a little evidence that this might, in fact, be God's will, would seem enough to make it sufficiently credible. 

In the same way, since a late spring storm just brought new snow to the Cascade, and I love to ski, even though I have work to do, and should probably stay home and do it, it is not overwhelmingly improbable that I will drive up to Snoqualmie Pass today and do a bit of skiing.  One can see why this might happen.  If you think you see me there in a few hours, it will be very possible that you do. 

Perhaps one could put the possibility that God would do such a thing, a priori, at one in 10.  Given hints in Isaiah that after dying for the sins of the people, the Suffering Servant would "see the light of life," (and given the existence of God already accounted for in [1]), we might fairly put that number higher, say 1 in 4. 

(3) To evaluate how likely a given person might prove to be the one through whom God reveals his power, let's begin with Martin Luther King. Suppose God wanted to dramatically not just want to show that Entropy would not have the last word, and that there was hope for the human race, by raising a prominent person from the dead.  Suppose he also (again, following his character as developed in the OT prophetic works) also wanted to reprimand oppressors by his choice of whom to resurrect.  King might be a good person to pick. His resurrection would not only give people hope for life after death, but also demonstrate that God was on the side of non-violence and human rights.

On the other hand, King's resurrection might also send mixed messages about how to treat women, for example, or force God to "pick sides" in American politics.  And there are other heroic figures who might do just as well -- Gandhi, say, or Socrates. So given that God exists, and He wanted to prove his power and character and the hope of eternal life by raising one man from the dead, one might suppose the chance that he would pick Martin Luther King might be, say, one, in a one thousand -- much higher than someone picked at random, say Otzi the Iceman, one might reasonably suppose.

In this scenario, would anyone be more likely to be raised than Jesus?

Consider the following facts, none of which depend on the historical accounts of Jesus' final days in the NT:

(a) As mentioned above, Isaiah spoke of a Suffering Servant dying, yet then "seeing the Light of Life." Might not this and other passages in the OT be a signal pointing to God's intention not just in general, but specifically related to Jesus?   Christians, including so perceptive a man as Blaise Pascal, have traditionally found many messianic expectations in the Old Testament that seemed to come true in Jesus, more than in the life of any other man or woman.  All in all, the diffuse and complex web of Messianic expectations, that do seem to focus on Jesus (Pascal explains some of the reasons why), do seem to make it far more likely that he would be the one whom God raises. 

(b) The ancient Hindus wrote of God (Prajapati) sacrificing himself for the world. There are parallels in China and in other cultures (see my previous post), and mythological "dying and rising gods."

Again, some of these show remarkable parallels to the person and story of Jesus. If God were to intend his act as a sign not just to Israel, but to the whole world -- and the entire Old Testament underlines the universal character of God's redemptive plans -- then does not the fulfillment of such types in the life of Jesus greatly increase the chance that He would be the One prepared for all mankind -- and that God might (going back to [2]) give humanity a sign of hope and redemption in one such fell, miraculous deed? 

(c) Lin Yutang, the great Chinese philosopher and man of letters, who compiled an anthology of Chinese and Indian literature, said that "no man has taught as Jesus taught." Many others on a similar intellectual plane concur. Is it not more likely that God would choose arguably the world's greatest moral teacher to make His point?

(d) Jesus was, as I show in The Truth Behind the New Atheism and elsewhere, at the center of many of the greatest reforms in history -- inspiring them, setting an example, more so than anyone. Is it not likely God would choose to raise such a person, to set an example for the human raise, and thus endorse his example?  (Leading to such reforms as have in fact occurred?) 

(e) Jesus was murdered by tyrants, backed by the Roman Empire, in a particularly savage way. If God is (as Lao Zi said of the Dao) on the side of the weak against powerful oppressors, would not raising him from the dead be a particularly good way of showing that? 

One could go into a great detail on each of these points, and I have done some of that elsewhere.  But let me now begin to get to the point. 

I haven't introduced any specificially Christian theology into this argument. Even so, these factors seem to show that, if there is a God, and if He intends our redemption through some such act, then the resurrection of Jesus is most to be looked for.  If God exists, and if He wanted to do something dramatic in human history, that would change the world, give us hope, and show that he stands on the side of the righteous and oppressed, Jesus would seem to be the best possible tool through whom to express that. (Whether or not he was, himself, divine.) 

In fact, one might say that not only is Jesus the most likely candidate, he is the only really likely candidate.  Buddha and Lao Zi are almost lost in the historical mists.  Mohammed was a bloody conqueror.  Confucius was a cautious gentleman, whose teachings were useful, but which could not have dramatically changed the world for the better, as the Gospel has.  Gandhi and King were great men, but their movements were inextricably political, and their personal lives not always as inspiring as their ideals. 

So it does not seem wildly unreasonable to suppose, prior to looking at any historical facts at all, that there is at least a one in ten chance that God (assuming Him to exist, and to be of such a mind) would have chosen to raise Jesus, of all people from the dead.

One might even say rhetorically (in reading the Gospels, or in frustration at the cruelty and injustice of life) that it would be a miracle if God did NOT raise Jesus from the dead.

Given all that, the prior probability of the Resurrection of Jesus might be as great as, say, 1 in 80 (2 X 4 X 10), and is certainly far greater than one in 100 billion. 

If there are independent reasons for believing (1) and (2) that make those odds better than even, as I think there are, then the prior probability of the resurrection, before we even begin to examine the actual historical evidence, might rise to 1 in 10, or so. 

If, then, the purely historical evidence for the Resurrection is strong -- and I think Christian historians have shown that it is tremendously strong (see for instance this wonderful piece by Tim & Lydia McGrew)-- then the combined evidence that in fact Jesus DID rise from the dead, as reported, may become quite strong, even prohibitively strong.  In that case, one can "flip" the resurrection equation around, and solve it for "Does God exist?"  It then becomes reasonable to argue from the Resurrection to the existence of God. 

Of course, this conclusion depends on the plausibility of each step of this argument. 

11 comments:

Brian Barrington said...

FIRST, the probability that God exists. This depends on what is meant by "God". For example. The probability that the God I believe in exists is 100%, since I regard God as Total Existence, so, since something exists, it follows that God also exists. It is an absolute certainty.

But you are talking here about the traditional theistic God, who is omnipotent, omniscient and omnibenevolent. There a good reasons to put the probability of such a God existing at a bit less than 50%. First, omnipotence may be a logical impossibility, which would mean the probability of an an omnipotent being existing is zero. Second, if an omnipotent and omnibenevolent God existed there would not be so much unnecessary suffering and pain in the world, therefore this type of God is very unlikely to exist. So being generous, I would put the probability of such a God existing at about one in a 100 million, or even less. Regarding the probability that a specifically Christian God exists (I.e. a triune God consisting of a Father, a Son and a Holy Ghost) - there is no decent evidence at all that such a God exists. I would put the probability of that at about one in a thousand trillion or possibly even less. But for the purposes of this argument we will just stick to our more generic omnipotent, omnibenevolent God, and estimate, very generously, a one in a hundred million chance that such a God exists.

Brian Barrington said...

SECOND, if an omnipotent, omnibenevolent God existed it would likely want everyone to know about the afterlife, and, being omnipotent and all, it would be able to find a way of letting everyone know about it, in the same way that everyone knows about the  existence of the natural world, for example. Anyway, for the purposes of this argument, I'll go some way along with David and halve his probability I.e. I'll put the probability that God would raise one person from the dead at a very generous one in twenty.

THIRD, what is the probability that he would choose Jesus specifically to rise from the dead out of all the billions of people available? Well, God could choose anyone at all for the job, so you could reasonably put the probability at one in tens of billions. But since jesus was a nice guy in some ways, and therefore arguably more suited to the job, I'm going to put the probability of this at a very generous one in fifty. God, being omnibenevolent, would not want hell to exist or anyone to believe in hell. So I'm reducing the probability that he would choose Jesus specifically, because jesus's teaching on hell would be contrary to that of an omnibenevolent God. So one in fifty it is ... Again, being very generous.

If you multiply these three figures together you get a prior probability of Jesus rising from the dead of 1 in a hundred billion.

David B Marshall said...

Brian: Oh, come on, you can do better than the empty blather in your first post. Of course we all know that the "Problem of Pain" is the main weapon in the atheist arsenal -- do you think I wasn't assuming that?

In essence, you're saying there is less than a one in 100 million chance that someone like Alvin Plantinga, who has undermined the value of the Problem of Pain for disproving God in the eyes of many philosophers, sees as clearly as you do. Or, say, Pascal? Isn't that a wee bit arrogant? Isn't there, say, at least a one in ten chance that one of these men sees more clearly than you? Or that Christians who have experienced miracles are not just hallucinating?

You can maintain your inflated numbers, or self-confidence, or bombast, as the case may be: but it seems like paddling out to a little island in the middle of the lake, and then sinking the boat. What can anyone who's not on the island say in response to such silliness? Watch out for mosquitoes.

Doc Johnny said...

I think the three variables that you mention are impossible to quantify. And I think theologically you agree that they are impossible to quantify. Also, what you end up theorizing about is the probability that a being that who may or may not exist may or may not act in a particular way.

Actual probability can be measured and depends on data and analysis. In this case, how many people die? How many people come back from the dead?

Now given this data, you could obtain the probability that a person could come back from the dead. You could then start to isolate based on types of injury, amount of time "dead", etcetera. All these variables are measurable and can be analyzed.

Now you can object that this does not account for the uniqueness of the case surrounding Jesus, and you would be right. Applying probability to specific cases is always extrapolative. I can accurately tell you that a particular disease has a 90% 1 year mortality rate. I cannot accurately tell you that you are 90% likely to die in 1 year.

Probability can only be assessed if you have data points to measure. We can obtain data about wounds, deaths, and people who seemingly survive being labeled as dead. We cannot obtain data about the likelihood of god, the likelihood of his acting in specific ways, or who he might act on.

My overall point is that trying to apply probability to this question in the manner that you are doing is pseudoscientific. Probabilities are not guesses. Probabilities depend on actual numbers. When I tell someone that a disease has a 90% 1 year mortality, we counted the people with the disease, and we counted the people still alive after 1 year.


Multiplying opinion by opinion by opinion is still opinion.

It ends up being a Rube-Goldberg machine for saying "I believe the resurrection is likely."

My overall point is not disagreeing with you about the likelihood of the resurrection of Jesus. My point is that any actual analysis of probability would have to be based on things we can actually measure, not unknowable, unquantifiable opinions about god and how god would or would not act.

David B Marshall said...

Brian: "SECOND, if an omnipotent, omnibenevolent God existed it would likely want everyone to know about the afterlife and, being omnipotent and all, it would be able to find a way of letting everyone know about it, in the same way that everyone knows about the existence of the natural world, for example."

If God existed, Heaven would be visible, audible, smellable, tastable, and tactual?

Come to think of it, maybe Heaven IS all those things. I swear I've heard the testimonials myself on pop music stations.

"God, being omnibenevolent, would not want hell to exist or anyone to believe in hell."

I don't see the logic of this step. It could be, for instance, that God knows twenty people would have become mass murdering tyrants, but for the fear of hell. It could be that Jesus never said anything about hell, at least in the sense that you understand it.

"If you multiply these three figures together you get a prior probability of Jesus rising from the dead of 1 in a hundred billion".

All the "force" of your argument, considered as an argument not as a joke, which of course it is, comes from arbitrarily assigning a huge improbability to the existence of God. Again, on the face of it, it would seem less than astronomically improbable that people like Plato, Confucius, Lao Zi, Zhuang Zi, Jesus, Augustine, Aquinas, Bacon, Kepler, Newton, Pascal, Sun Yat-sen, Tolstoy, Gandhi and Mother Theresa got this right, and you and Richard Dawkins got it wrong. Call me crazy, but I think more even odds would be more reasonable, on skeptical premises.

David B Marshall said...

Doc Johnny: "Probability can only be assessed if you have data points to measure. We can obtain data about wounds, deaths, and people who seemingly survive being labeled as dead. We cannot obtain data about the likelihood of god, the likelihood of his acting in specific ways, or who he might act on."

"My overall point is that trying to apply probability to this question in the manner that you are doing is pseudoscientific. Probabilities are not guesses. Probabilities depend on actual numbers."

Probabilities can, in fact, depend on a range of numbers, or on numbers that are somewhat speculative. "Given A, then there's a 50% chance of B" is a common procedure, and quite legitimate, I think. It is also legitimate to try to specify the probability of A, or to attempt to characterize that probability in some way. Thus we use non-quantitative evaluations like "probably," "it seems pretty likely," and so forth, which I do above for my first two variables. No one is claiming that the probability of all-embracing theories of life can be easily quantified, but there's nothing wrong with such thought experiments.

"Multiplying opinion by opinion by opinion is still opinion."

Yes, but some opinions are better than others. I've explained the heuristic basis for my opinions.

"It ends up being a Rube-Goldberg machine for saying 'I believe the resurrection is likely.'"

On the contrary. Rube Goldberg's machines have, maybe, 40 moving parts, each of which has to work to get the coffee to pour into the cup, or whatever.

Here the situation is quite different. We have millions of reports of miracles. If ANY ONE of them is true, then strict materialism cannot be.

So materialists are running the ultimate Rube Goldberg machine.

The resurrection of Jesus is believed on the account of several independent reports. Only one has to be accurate, for the thing to have actually happened.

Assessing prior probability is evaluating atheists' metaphysical arguments AGAINST those reports. So it is atheism, not theism, that requires all the pieces to work just right here, for the theory to be saved.

"My overall point is not disagreeing with you about the likelihood of the resurrection of Jesus. My point is that any actual analysis of probability would have to be based on things we can actually measure, not unknowable, unquantifiable opinions about god and how god would or would not act."

Well, I appreciate the challenge. I think your comments help shed new light on the issue.

But I don't think there's anything wildly improper about how I try to heuristically quantify, or at least bracket, these two questions. My arguments are more a defeater for skeptical responses to the story that Jesus rose from the dead, than definite attempts to pin God down. Reports that He planned to raise Jesus from the dead, given above, do not seem so improbable that they need to be dismissed, for the reasons given.

Also, I wasn't trying to be "scientific," so my blog can't have been "pseudo-scientific." Science does not corner the market in reason.

Doc Johnny said...

"Here the situation is quite different. We have millions of reports of miracles. If ANY ONE of them is true, then strict materialism cannot be."

Fair enough, but proof of the supernatural does not prove God. There are an infinite number of possible supernatural explanations.

The resurrection of Jesus is believed on the account of several independent reports. Only one has to be accurate, for the thing to have actually happened.

And there are several independent reports of Elvis sightings since his death. These are far more recent. If reports were all that was necessary to substantiate truth, then Elvis is alive.

But I don't think there's anything wildly improper about how I try to heuristically quantify, or at least bracket, these two questions. My arguments are more a defeater for skeptical responses to the story that Jesus rose from the dead, than definite attempts to pin God down. Reports that He planned to raise Jesus from the dead, given above, do not seem so improbable that they need to be dismissed, for the reasons given.

I don't think it is improper, just not useful as proof of anything. Keep in mind I think the same thing when atheists use the same shenanigans to "disprove" God.

Also, I wasn't trying to be "scientific," so my blog can't have been "pseudo-scientific." Science does not corner the market in reason.

Fair enough. Let me state my objections. Probability is a science. I feel that the use of probabilities implies predictive value. To have predictive value, actual data points need to be obtained.

While examining your opinions using the language of probability seems useful, it actually just lends a false sense of objectivity and certainty to the process since the numbers assigned are to a great degree arbitrary.

David B Marshall said...

"Fair enough, but proof of the supernatural does not prove God. There are an infinite number of possible supernatural explanations."

True. But Occam is working for is, here. One God is easier to buy than the whole crowd on Olympus, as the Greek philosophers eventually discovered.

"And there are several independent reports of Elvis sightings since his death. These are far more recent. If reports were all that was necessary to substantiate truth, then Elvis is alive."

I've actually never run across a believable report of someone who knew Elvis after his death. But if there are such reports, why don't you believe them? (That's a real, not rhetorical, question.)

"Fair enough. Let me state my objections. Probability is a science. I feel that the use of probabilities implies predictive value. To have predictive value, actual data points need to be obtained.

"While examining your opinions using the language of probability seems useful, it actually just lends a false sense of objectivity and certainty to the process since the numbers assigned are to a great degree arbitrary."

The main purpose of this blog is to defeat Brian's claim that the Prior Probability of the Resurrection of Jesus is very low, because it is just artitrary who is raised from the dead. (That's the error you're also making with Elvis, BTW.) I think the blog succeeds in doing that. I think my method for obtaining a heuristical probability for God's existence is reasonable, though of course it's not meant to be completely satisfying. (Brian falls back on silly odds against, and of course he can do that, if he wants.) That leaves "what God would do" as the main unknown variable -- and if we have any reason to think he might do this, all in all, I think this argument does render the PP of the Rez pretty high. No one is claiming these numbers have to be treated as anything but a convenience for a useful experiment in thought -- this is not a mathematics journal,it's a blog.

But your criticisms are helpful and, in their way, appropriate.

noiln said...

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dovetheology.com said...

Very good article, David. I think it would be helpful to include links to the issues you say you devote time to elsewhere - would be a good read!

David B Marshall said...

Thanks! Of course, I especially hope people will read my books, linked to off to the right. But also, see my recent post on "my fourteen best posts of the year," of which this was one.