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Mutual support and circularity

In the interview with Luke Muehlhauser, linked below, we got briefly into the subject of mutual support vis a vis the existence of God and miracles like the resurrection. The question, related to a debate my husband and I have had with Alvin Plantinga, arose (though not in so many words): Isn't the existence of God a premise for belief in any miracle, in which case isn't it circular to say that a miracle increases the probability of the existence of God?

I feel that I wasn't as clear as I could have been in responding to this question, so I'd like to say a little more about the issue of mutual support. Here is the technical paper that addresses this subject, but here I want to try to do this non-technically as much as possible.

Most of us are inclined to reason about things in the following way: Use only propositions for which you have lots and lots of support. Base your predictions on what these propositions really lead you to believe, and go from there.

So, for example, a man whose wife always makes him a chocolate cake for his birthday is most concerned with the question of whether, this year, his wife will continue the tradition. He takes his wife's existence for granted, because it's overwhelmingly supported. He also doesn't worry about whether he's hallucinating all his past apparent memories of chocolate cakes over the years. His only concern is whether she will be too busy this year, and he reasons forward, using what is strongly supported already, to make a prediction about this year's birthday. He may, for example, justifiably conclude that he will get a chocolate cake this year.

In this context, it seems sort of pointless and trivial to argue after the fact that the smell of chocolate cake he notices when he comes home on his birthday is evidence for his wife's existence. No one is worrying about that, and so no one thinks twice about the fact that the evidence runs in that direction.

But if you stop to think about it, it's true: The direct sensory evidence of the baking chocolate cake is evidence that Someone exists who made the cake, especially since he has been gone all day and knows he didn't do it.

Is there any circularity going on here? Not at all. He expected the cake, because he had other evidence from the past both of his wife's existence and of her intention to make the cake. He now has even more evidence of interaction with his wife in the form of the smell of the baking cake. We have, of course, reached a point of diminishing returns here, since the probability for his wife's existence was close to 1 already, even before he noticed that Someone had been in the house baking the cake. To make the epistemic force of the sensory evidence of the cake more vivid, we would perhaps have to imagine the poor fellow as suffering from some sort of mental illness that makes him temporarily doubt that he has a wife, in which case the direct sensory evidence of the chocolate cake would take on a lot more practical importance for him than it has in the ordinary situation!

Here's another example: Suppose that I have a long-distance friend named Bill Smith whom I have never met. Bill and I have talked a number of times on the phone, and the last time we spoke, he asked for my e-mail address and said that he would like to send me a paper. A week or so later, up pops an e-mail in my in-box that purports to be from Bill Smith and that has the paper attached.

Once again, we have evidence here going in two directions but no circularity. The evidence of my previous phone conversations gave me some reason to believe that Bill Smith exists already and also to expect the e-mail. The direct evidence--the "after-the-fact" evidence--that I've actually received an e-mail from someone by that name gives me more reason to believe in the existence of Bill Smith.

And note something that is perhaps clearer in this example than in the cake-baking example: Most of us receive e-mails occasionally from people whom we previously had no contact with. (I certainly have.) Once we have the e-mail, even though we did not have a high previous probability for the existence of the writer or reason to predict the arrival of the e-mail, we have reason to believe in his existence. In other words, the e-mail is evidence for the existence of its author independent of the lowness or highness of our prior probability for the existence of the author.

It's certainly true that my previous good evidence for the existence of Bill Smith and our conversation gave me a higher prior probability for the arrival of an e-mail than I would have had otherwise. But that is not a necessary condition for my believing, on the basis of my sensory evidence and knowledge of e-mail, etc., alone, that I have indeed received an e-mail from a person named Bill Smith (who therefore exists).

In the same way, our prior reasons for believing that God exists (or for doubting God's existence) will be quite relevant to the prior probability of any given miracle. Just as I will not expect to receive an e-mail from a person of whom I have never heard, so I will not expect a miracle from a God for whom I believe there to be little or no positive evidence. On the other hand, if I believe there to be some independent evidence for God's existence, this will raise the prior probability of a miracle. And if I believe there to be enormous evidence--for example, if I believe that the arguments of natural theology render the existence of an omnipotent, omniscient, and omnibenevolent eternal God a certainty--this will "help" the prior probability for a miracle a lot, though it will still not make me positively expect most specific miracles.

But direct evidence for a miracle--for example, testimonial evidence in a context that makes it extremely hard to explain as a lie, etc.--is evidence for the existence of God as well, and it is evidence for the existence of God either way--that is, regardless of whether my prior probability for the existence of God was high or low. Just as the evidence of an e-mail purporting to be from Bill Smith is evidence for the existence of Bill Smith even if my prior probability for both his existence and the coming of the e-mail was low, so it is in the case of miracles and the existence of God.

What this means is that we can have evidential "lines" going in more than one direction without any circularity.

We might be inclined to say that

A: Bill Smith exists

is evidence for

B: I receive an e-mail from Bill Smith

and also that B is evidence for A when the e-mail pops up in my inbox. Drat! A circle! But not really: To say that A is a premise for B is a convenient shorthand, but it comes at the cost of some loss of clarity, and in this type of situation it can be positively misleading. The clearer way to think of it is to think of a set of other evidence, such as

A1: I seem to remember a conversation on the phone with Bill Smith on June 1

A2: I seem to remember a conversation on the phone with Bill Smith on August 25



A3: Bill Smith asked for my e-mail address.

A1-A3 provide one set of evidence that Bill Smith exists and that he intends to send me an e-mail, which raises the the probability of B.

On the other hand,

B1: I have received an e-mail which says that it is from someone named Bill Smith

is evidence for A, independent of whether I have A1-A3. Thus, A1-A3 and B1 are all evidence pertinent to B.

And that's all there is to it.

Look, Ma, no circles!

Now, let's see if I've succeeded in expressing this clearly. What think you, readers?

Comments (46)

The explanation of bidirectional evidence was very clear to me (I don't have any philosophical training). In fact I think you may have even explained your point a little more than necessary.

The use of the word "miracle" was still iffy, though. That's where the question-begging seems to come in. The chocolate cake would be evidence of the wife's existence only if the mentally deranged husband considered the likelihood (or posterior probability, whichever you prefer) of that hypothesis to be higher than the likelihood that cake spontaneously evolves from flour, sugar, etc., or that cake is created by intelligent extraterrestrials, or baked by helpful neighbors, or whatever. The cake doesn't have "BAKED BY YOUR WIFE" written on it, and exceptional phenomena don't have the word "MIRACLE" stamped on them.

For this non-philosopher at least, it's your use of the word "miracle", or even "putative miracle", instead of a word like "anomaly" that seems responsible for the question-begging. But that's not because "miracle" assumes a god. It's because miracle is an overly-specific hypothesis. Maybe that's what your interlocutors had in mind, I don't know.

The chocolate cake would be evidence of the wife's existence only if the mentally deranged husband considered the likelihood (or posterior probability, whichever you prefer) of that hypothesis to be higher than the likelihood that cake spontaneously evolves from flour, sugar, etc., or that cake is created by intelligent extraterrestrials, or baked by helpful neighbors, or whatever.

I _think_ you are using the phrase "evidence for" here in the sense of "what you should conclude," right?

Philosophers tend to use the phrase in a somewhat different way, so that something can be "evidence for" some hypothesis even if that isn't what I should conclude. Speaking in strict terms, something is "evidence for" an hypothesis H if H is more probable with the evidence than without it. So even if the man is convinced that he has no wife and therefore thinks it more probable in the end that a kindly burglar broke into his home in his absence and baked the cake (knowing it was his birthday), that doesn't mean that the evidence doesn't _raise_ the probability that he has a wife over what it was before.

I think it's clear that someone's rising from the dead--truly being dead for over twenty-four hours, body cold, etc., and then coming back to life and walking around talking--especially taking place within a religious context (where the person claimed to be at least a representative of a particular God) is _evidence for_ the existence of that God in the sense that the existence of that God is more probable than it was before the person got up from the dead and started walking around and talking.

Lots of things don't have "made by so-and-so" written on them, but they still reasonably increase the probability that so-and-so was acting. We see this in all kinds of mundane cases. (For that matter, even if something _does_ have "made by so-and-so" written on it, we can question that label, as in a case of plagiarism.) That raising of the probability can occur even if in the end someone concludes that some other hypothesis is the explanation that comes out on top. In fact, the raising of the probability can occur even if the person is reasonable for concluding something else given all evidence.

Naturally, in this case I think it's not only obvious that a coming-back-to-life in a religious context is _evidence for_ the existence of the God in question; I also think that's the explanation that comes out on top and is the thing to be concluded, taking everything into account. But those two are not quite the same thing.


Thanks for going into this; it’s very helpful. I admit I haven’t read your paper yet, but here are some comments; if they’re addressed in the paper, just let me know and I’ll do the homework.

You seem to assume some principle of sufficient reason, which is not uncontroversial. We may want to understand and explain things, but the universe is not necessarily bound by our wishes! So there’s no a priori guarantee of success, and failure may because (a) we did not manage to hit the ‘right’ explanation yet, or (b) because there’s no explanation. But if we can’t tell whether (a) or (b) is the case, I think it’s fine to keep searching. We can do that without necessarily being committed to some principle of sufficient reason.

It is also controversial that ‘direct evidence for a miracle … is evidence for the existence of God either way--that is, regardless of whether my prior probability for the existence of God was high or low’. Is there some a priori reason why would God opt to show humans that he can perform a miracle rather than that he is omnibenevolent? That it’s problematic to show that God is omnibenevolent lowers the probability that God exists at all; that miracles may not be witnessed doesn’t! So, even if one is a theist, one has no reason to expect a divine miracle, let alone specifically a resurrection, unless one is already committed to the OT as being divine revelation. Here’s where I see the circle.

Since you refer to natural theology, I think Aquinas recognised that he needed to bring revelation in at some point, that natural theology couldn’t do the whole job. But by undermining the rationality of non-Christians, your argument (in the other thread) undermines the project of natural theology itself.

Lydia, yeah, that was sort of my misunderstanding about the word "evidence". Thanks for straightening me out there.

This assertion still seems problematic, though:

But direct evidence for a miracle...is evidence for the existence of God as well...

Is this even true if evidence is defined as something that raises the probability? Here's why it's still unclear to me. Let

G = "God exists"
M = "this was a miracle"
E = the evidence

By your definition of evidence, P(M|E) > P(M). But you seem to be concluding from this, together with "M is a subset of G", that
P(G|E) > P(G).

To see why this doesn't seem to follow, note that the last inequality is equivalent to

P(M|E) + P(G and ~M|E) > P(M) + P(G and ~M)

It's logically possible that the evidence E could lower the probability of "G and ~M" a lot more than it raises the probability of M.

Is there a mistake in my reasoning so far? If so, then thanks for pointing it out! If not, then maybe you meant only that your assertion was true, not mathematically, but for this particular subject matter? But that seems to require a pretty serious argument. I could imagine someone arguing, contrarily, that evidence for some apparent "miracle" is even stronger evidence for some atheistic theory, and arguing from there that it is not evidence for the existence of God. Evidence for a miracle doesn't have to be something dramatic and biblical like a resurrection; it could be a long run of wins at the roulette table - weak evidence, but evidence nonetheless. (If you disagree about that, then we're back to the point about events not being stamped with the word "miracle".)

I think this is sort of what I was getting at in my previous comment, even though I misunderstood the word "evidence". The main point was that alternate explanations seem to be ignored or discounted - explanations that could lower the posterior probability of G and in particular of "G and ~M". You're right that using the word "miracle" is not circular here (it was my misunderstanding of "evidence"), but it's still unclear to me why the overall assertion is true.

Overseas, my comment about evidence for a miracle and the lowness or highness of the prior was merely a statement that the evidential force (upon theism) of direct evidence for a miracle does not depend on having a particularly high prior probability for theism. I'm sorry, but that's pretty much just the probability theory. Some hypothesis H does not have to have a prior above some cut-off k before some evidence E can be evidence for H. If your argument about omnibenevolence is just supposed to be some sort of anti-theistic argument from God's general non-obviousness, you're going to have to throw that into the prior. But that _still_ won't make it impossible for evidence for a miracle to be evidence for theism.

I have no special "down" on the rationality of mankind. As with everything else, your mileage may vary. But this doesn't mean that natural theology arguments can't work for those who can see them. I don't happen to agree with my esteemed colleague Ed Feser about _how much_ we can conclude from natural theology, but an argument is an argument. If it works, it works. There's no point reasoning in some roundabout way about how rational people are in order to try to figure out whether natural theology is gonna work. Just go to the argument and evaluate it as best you can.

Yes, I do think the PSR is knowable a priori, but I'm not sure exactly where it comes in here in any invidious way. If some skeptic is going to admit that Jesus got up and walked around and talked after being dead for three days but appeal to a supposed failure of the PSR to say that "there is no explanation," then he's obviously getting very desperate. And the same mutatis mutandis for the origins of Christianity and of the disciples' belief that Jesus was risen. "Explanation? No, I don't _have_ to explain things. There might not _be_ an explanation, so don't _bother_ me with asking about the best explanation."

If you've never heard the sound of anti-rational foot-stomping before, I'll tell ya: That's a sample.

Aaron, I'm looking at your assertion about the inequalities. I'm not sure which two you mean to say are equivalent. I _think_ you mean to say that

P(M|E) > P(M)

is equivalent to

P(M|E) + P(G and ~M|E) > P(M) + P(G and ~M)

But you may be saying that

P(G|E) > P(G)

is equivalent to

P(M|E) + P(G and ~M|E) > P(M) + P(G and ~M)

I admit to having some difficulty seeing the equivalence in either event. I may just be missing something, but could you provide a derivation for whichever equivalence you mean?


Evidential carry-through is a function of two things: First, it's a function of probabilistic relevance. Second, it's a function of an independence relation called screening-off. It seems to me that your hypothetical scenario in which some E raises the probability that a miracle has occurred but lowers the probability that God exists is a scenario in which we are supposed to envisage a failure of the screening-off relation at some point.

Could you give a concrete example? Do you really mean to acknowledge that E raises the probability of a miracle? And are we defining "miracle" itself here as meaning only a very extraordinary event (for example) or as actually _entailing_ the action of a supernatural agent?

In the main post, I meant "miracle" to entail theism, by analogy with the way that "Bill Smith wrote me an e-mail" entails "Bill Smith exists." This is a convenience that worked well for examples showing the two-way evidence relation. Notice that B1 only asserts that the e-mail _says_ it is from someone named Bill Smith; nonetheless it does seem (under all ordinary evidential circumstances) to be evidence that Bill Smith exists by way of being evidence for B, which entails A. A parallel to B1 in the case of God would be various types of testimony under particular background religious circumstances, etc.

If you would rather work with an intermediate premise that refers to the event (such as Jesus' walking-around-alive-after-being-dead) without calling it a miracle, that would be some sort of B1* that would go in between B1 and B as applied to the miracle case. We could use a word like "resuscitation" or something like that to designate the event without considering it to be an actual miracle.


I asked if there’s an a priori reason to think that God would want to show humans that he can perform a miracle rather than that he is omnibenevolent, given that (i) witnessing evil lowers the probability that God exists, while (ii) not witnessing a miracle does not lower the probability that God exists. Do you have a view on this? As you said, ‘that's pretty much just the probability theory’.

‘But that _still_ won't make it impossible for evidence for a miracle to be evidence for theism.’

You’re right if anything can be evidence for the omnipotent! But how does that help your case about the resurrection?

‘Yes, I do think the PSR is knowable a priori, but I'm not sure exactly where it comes in here in any invidious way.’

I don’t think the PSR is knowable a priori and see no contradiction involved in its negation. It doesn't come in at all really because it’s epistemically inert and makes no difference on the ground. We’re dealing with hypotheses and there’s no guarantee of getting things right whether PSR holds or not. It just sounded like a weird argument from the smell of chocolate to the existence of a wife; as if the wife would have vanished had the neighbour baked the cake! I can see the intended parallel; what I still can’t see is why you’d expect the miracle of resurrection a priori and apart from revelation.

what I still can’t see is why you’d expect the miracle of resurrection a priori and apart from revelation.

I never said you would. In fact, I've expressly talked (surely in this thread, I'll have to check later) about the fact that the event does _not_ have to have a particularly high probability on the theory. What we are talking about is comparative explanatory power. The Bayes factor is the ratio P(E|H)/P(E|~H). This can be any order of magnitude you like even if the individual numbers are both very low in absolute terms.

Whether witnessing any evil whatsoever lowers the probability that God exists depends on whether you can defend the proposition that our witnessing _any evil at all_ is more likely if God does not exist than if he does. Good luck with that. You can post the publication data on your paper here when you get it accepted.

Is there a good primer somewhere for the notations used in this thread?

Don't worry too much about it, DmL. I was careful not to use any of it in the main post. P is usually used in these contexts for "the probability of." The solidus looks like this


and is used to mean "given."

So P(M|E) means "the probability of M given E."

symbols like greater than (>) or less than (

Heh, looks like the tags gotcha, but I think I get it now, thanks.

Some clarifications that were asked for:

Notation: "P(A|B)" means conditional probability: the probability of A given B. "~A" means "not A". Since I'm already using some math notation, in the following I'll also use "A^B" for "A and B" and "A\B" for the set difference "A minus B",i.e., "A and ~B".

I did indeed mean "miracle" to entail the existence of God: a suspension of the natural order by God (who exists). This was basic to my statement.

I do mean to acknowledge that E raises the probability of a miracle. That's exactly what I meant by "P(M|E)>P(M)".

I'd have to think some more to come up with a good nonpathological, concrete, real-world example. For now, I'll just show why I think your statement is not mathematically true. A proof of my claim would of course just be a counter-example with numbers for the various probabilities. Instead of giving that, it would be more interesting to point out the equivalence I stated above, because that explains why your statement was not "mathematically" true.

This is what I was claiming: "P(M|E) > P(M)" does not imply "P(G|E) > P(G)". In order to keep this at an elementary level, I'll stick to the case P(E)>0. What I said is also true for the case P(E)=0, but conditioning on probability-zero events takes us above the level of elementary probability. Now, each of the following statements is equivalent. It looks much, much more complicated than it is: all I'm doing really is giving an elementary proof for this one special case that conditional probability is additive. It would be very clear from a Venn diagram.

P(G|E) > P(G)

P(G^E)/P(E) > P(G) (elementary definition of conditional prob)

P(G^M^E or G\M^E)/P(E) > P(G^M or G\M) (rewriting as disjoint unions)

[P(G^M^E) + P(G\M^E)]/P(E) > P(G^M) + P(G\M) (additivity of P on disjoint events)

P(G^M^E)/P(E) + P(G\M^E)/P(E) > P(G^M) + P(G\M)

P(M^E)/P(E) + P(G\M^E)/P(E) > P(M) + P(G\M) (because G^M = M, since M is a subset of G by definition of "miracle")

P(M|E) + P(G\M|E) > P(M) + P(G\M) (elementary definition of conditional prob)

Now as I said before, I think one can easily construct a numerical counter-example where P(M|E) is just a little bigger than P(M), but P(G\M|E) is a lot smaller than P(G\M), so that P(M|E)>P(M) but P(G|E)

Oops, the last part of my post got cut off. I said,

Now as I said before, I think one can easily construct a numerical counter-example where P(M|E) is just a little bigger than P(M), but P(G\M|E) is a lot smaller than P(G\M), so that P(M|E) is greater than P(M) but P(G|E) is less than P(G). That is, E is evidence for a miracle but evidence against the existence of God.

I'm less confident that I could construct a good "story-problem" to go along with the counter-example. For now at least, I'll leave that as an exercise for the reader.

P.S. Lydia, you asked for a concrete example. I don't have one, but I have a story. A deist has a prior belief (interpretive framework) that God probably exists and that miracles are almost certainly impossible. He sees evidence, E, such that the probability that a given occurrence was a miracle given E is 1/4. But he also reasons, as follows, that the posterior probability there's a God and that there was no miracle given E is small. The occurrence was so extraordinary, so close to a suspension of the natural order, that if it wasn't a miracle then E must be evidence that it was caused by a super-powerful extraterrestrial. (The deist is almost certain it wasn't caused by God, because God, if he exists, almost certainly doesn't intervene.) But the existence of such a super-powerful extraterrestrial throws the whole God hypothesis into doubt: the super-powerful extraterrestrial can now explain everything God can, given not-M. So the probability of God given the evidence is just the probability there was a miracle given the evidence (1/4) plus the probability there's a God but no miracle given the evidence (quite small), so the posterior probability there's a God is less than 1/2. But the prior probability there's a God was high.

To summarize: evidence E raised the probability of a miracle, which was very low a priori. But E also lowered the probability of God's existence, which was high a priori. Therefore, in your technical, philosophical meaning of the word "evidence" - an observation that raises the probability of a hypothesis - E is evidence for a miracle but evidence against God.

I hope people enjoyed my little story. I'm not saying the deist is a good philosopher, only that his reasoning is rational (probably mistaken) and psychologically not too implausible. It's what I was getting at from the very beginning here. As I said, I could turn this into a mathematical counter-example with numbers and all, provided I'm not mistaken about the whole thing. Constructing the counter-example would be easy but tedious.

Aaron, in order for your story to work as an example, it's going to have to be the case that P(E|~G) > P(E|G)--that is, that the average probability of E given the negation of "God exists" is lower than the average probability of the evidence given "God exists."

I think this is very, very questionable. I suspect that our disagreement there will turn on the notion, which I think you are implying, that if this occurrence confirms the existence and activity of the alien it _disconfirms_ the existence of God. I'm going to have to look later at this more, but it seems to me right now that the problem lies somewhere in the statement that P (There is a God and no miracle occurred) must be significantly smaller than it was before _combined with_ your statement that "[T]he existence of such a super-powerful extraterrestrial throws the whole God hypothesis into doubt" because the alien can explain the event without the existence of God. But as you have set it up, there could be a God, no miracle, _and_ the extraterrestrial. That is to say, the ET hypothesis is not at all incompatible with the existence of God. As a matter of fact, especially since your deist is committed to the idea that God doesn't intervene, it's difficult to see how the non-occurrence of a miracle--with or without the performance of extraordinary feats by ETs--_could_ disconfirm the existence of God. After all, non-miracles are exactly what the deist expects given the existence of God anyway, so why should ETs going around performing extraordinary non-miracles disconfirm the existence of God? So the ET hypothesis could be strongly confirmed without this in any way disconfirming theism as the deist conceives it.

The deist has no reason, for example, to believe that God would _prevent_ the existence of ETs who can do such extraordinary things. In fact, to imagine a God who intervenes to "zap" the ETs when they get too uppity with their technology would _itself_ be contrary to his deism! So the existence of ETs doing extraordinary things just seems to come back to prior evidence regarding that, entirely independent of the existence or non-existence of God, which means that "ETs did this" has no negative relevance to "God exists."

Somewhat relatedly, going back to your brief mention of extraordinary results at roulette, above: "Does not confirm theism" of course does not entail "confirms atheism." The fact that we may even quite rationally not treat the occurrence of an extraordinary roulette run as _confirming_ theism does not mean that it _disconfirms_ theism. To get that, you would have to argue that the probability of the event given atheism is actually _higher_ than the probability of the event given theism. But why think that? Chance occurrences are perfectly compatible with a robust theism, Christianity, etc.

Another point is that you are not giving the deist any opportunity or "space" in his probability distribution to reconsider his insistence that God does not intervene. This is often a difficulty that is put to Bayesians: Sometimes evidence needs to make us rethink propositions that we might want to hold constant to provide our "framework" and keep things from being messy. So, for example, the proposition, "God never performs miracles" should almost certainly be updated (depending on the details, but granting your scenario as much as possible), and this would make a difference to the entire thing. In this respect, it's rather like a situation where we are "told" that the only two urns from which a ball can be drawn have some incredible, overwhelming proportion of black balls (billions of black to one white, or something of that kind), and then we are asked to condition on the occurrence of a white ball. In the real epistemic world we need to be permitted to reconsider what we were "told" about the urns in the first place, the probability of which really can't, in the nature of the case, be 1.

Formally speaking, it is true that some evidence E can confirm a sub-hypthesis of H while disconfirming H overall. This is, I find, most easy to illustrate in cases where we switch H and ~H. Those are just the examples I find it easiest to generate, but it makes no difference to the formal point. This could come up with conspiracy theories: The evidence E would somewhat confirm (The event did not happen and there is a massive conspiracy to make it appear that it did happen) while overall confirming (The event happened), thus disconfirming overall "The event did not happen." Since H (the event happened) does not form a partition with (~H and conspiracy), it is possible for both to be confirmed despite the fact that they are incompatible.

I believe that this is your formal point, if I'm understanding you correctly.

But I think you're going to find it very difficult to argue convincingly that any case where evidence is alleged for a miracle constitutes such a case.

In fact, I would not say this even about miracle claims that I disbelieve. For example, I'm a Protestant and don't, in fact, believe that this or that miracle occurred by means of prayers to a particular saint. But I don't take it that the occurrence of the evidence alleged (whatever it is) _disconfirms_ Catholicism. I just think it's weak evidence that either neither confirms nor disconfirms the specifically Catholic hypothesis concerning the event (Bayes factor is a wash) or else that confirms so weakly as to make no serious dent in my Protestantism.

‘depends on whether… our witnessing _any evil at all_ is more likely if God does not exist than if he does.’

Pick your favourite evil then, Lydia! The question stands. Whether you believe that the PSR holds or that God is rational etc etc won’t make any difference: It’s cheap to postulate as many reasons as we like as long as we don’t feel inclined to say what they are. In the other thread I said: ‘The analogy here with the problem of evil, in your own terms, is that no one seems quite sure how to formulate the imperative for a miraculous sign in the first place or that it’s not sufficient to claim that the resurrection is merely ‘a possible sign' from God amongst infinitely many other possibilities: You clearly need something stronger, which I don’t see how you can get to without circularity.’ So I'd be very interested to find out precisely where you see revelation coming in, how and why but I'm afraid I can't tell which ‘theory’ you mean here; I’ve asked you before: ‘What’s the ‘hypothesis’ and what’s the ‘data’?’

A miracle, let alone a miracle report, clearly can’t be ‘raw data’. Miracles and miracle reports are highly theory-laden observations/observation reports. That people may be inclined to evaluate miracle reports and dismiss just some, but not others, as ‘honest mistakes’ only helps to compound the problem of question-begging circularity.

What you say in response to Aaron sounds prima facie sensible to me. But if it’s OK for the Jewish believer to continue to await the Messiah since not having dispatched the Messiah is no evidence against the Jewish God, or for the non-Jewish theist to remain a non-Jewish theist because nothing is evidence against an omnipotent being, what precisely is the claim about how serious a 'dent' a resurrection ought to make?


or for the non-Jewish theist to remain a non-Jewish theist because nothing is evidence against an omnipotent being,

I certainly have never asserted any such thing. What I have implied is that the probabilistic POE is much harder to get a handle on than you seem to think it is--and not for want of trying. I suspect that the probabilistic POE may not actually be the best hypothetical weapon for a possibility of evidence against the existence of God. I myself certainly do not consider that God's omnipotence somehow makes it impossible for there to be evidence against his existence. I'm not sure why this should be thought to be the case.


Thanks for the derivation. I've now worked through it, with a little help from someone quicker with the notation and the algebra than myself. You're a mathematician, I presume, then?

What you are showing is that non-deductive support is not in general transitive and also that even when some proposition M entails some hypothesis H, it does not follow that evidence for M is evidence for H.

This is correct formally, and as I said several comments up, transitivity of confirmation depends on the screening-off condition. Clearly you are formally contemplating situations in which the screening-off condition fails and there is no transitivity of confirmation. As I have said, I find these situations quite easily to construct where we reverse H and ~H--as in the case of a conspiracy theory. The evidence confirms (somewhat) the conspiracy theory. The conspiracy theory _entails_ that the event did not happen. But the evidence actually confirms that the event _did_ happen, hence there is no transitivity of confirmation from the evidence to the conspiracy to the non-occurrence of the event.

So, as you originally asked, I am not saying that it is a _theorem_ for some evidence E and some propositions or other G and M that if E confirms M and M entails G, E confirms G.

In fact, the not-in-general-transitivity of confirmation is something formal epistemologists like to talk a lot about. There is a paper by Tomoji Shogenji (referenced in our Erkenntnis paper) that you might like a lot on this subject that introduces the screening-off condition, and we do a lot with screening-off in the paper as well.

But I think it is going to be extremely difficult for you to argue with any plausibility that the type of evidence typically put forward for a miracle is actually evidence _against_ the existence of God overall. Your deism scenario does not do it, as far as I can tell (see discussion above).

In particular, I think you have given insufficient attention to P(E|~G), which must be higher than P(E|G) in order for E to disconfirm G. The fact that some explanation which is compatible with atheism (e.g., the action of a finite being or of random physical processes) can account for the data, combined with a low probability of the data given theism, does not give us this likelihood inequality by itself.

By the way, you probably know this, and it is a confusing use of terminology, but probability theorists don't use the word "likelihood" to mean "posterior." They use "likelihood" to refer to the conditional probability P(E|H). To make things still more confusing, they usually use "H has high likelihood" to mean "E has high probability given H." I occasionally slip up on this latter usage and refer to "the likelihood of E," because that seems more natural to me. But in any event, a "likelihood inequality" refers to an inequality between P(E|H) and P(E|~H).

1. Wife bakes cake (10:00 am)
2. Wife is replaced by Stepford wife (11:00 am)
3. Husband comes home for lunch (12:00 am)

Can the man reasonably conclude: a)his wife is alive or b)his wife baked the cake just because he smells the cake when he enters the house?

The Chicken

If the word supernatural is inherent in the word miracle, how can there not be circularity? A genuine miracle points to God. The real question is how do we know something is a miracle.

The real question is how do we know something is a miracle.

The same way we know that Abe Lincoln was assassinated, or that O.J. did it: that’s the way all the evidence points.

No one can prove with metaphysical certainty that a miracle took place, or that the ‘miracle’ was caused by God. It’s not a science like mathematics, which can be known with such certainty. As Aristotle said, to require absolute certainty from a science in which it is never found, like history, is the sign of bad education. Moreover, to refuse to accept the preponderance of evidence that points to a miracle would be just as stupid as denying an absolutely certain law of mathematics; for the preponderance of evidence is to law, history, and miracles what absolute proof is to mathematics.

But one might object, “Not so, since what is only probable may be wrong, but what is absolutely certain cannot be.” This is true, but not at all to the point. For in both cases one would be denying that which one is morally bound to accept, to wit, evidence in the former science and proof in the latter.

Faith is an assent to the truth, not the probability of truth, based on the reality and authority of the truth-teller, not the probability of his existence and authority. Thus, faith is not a valid part of history or law.

The Chicken

Faith is an assent to the truth, not the probability of truth, based on the reality and authority of the truth-teller, not the probability of his existence and authority. Thus, faith is not a valid part of history or law.

The science of miracles is not faith; and to suggest that it is is to hand the game right over to the agnostics, and to let them off the hook.

The science of miracles is like the science of history. If the evidence points to a high probability of a certain supernatural event having taken place, one is not bound by faith to believe it with certainty. However, one is morally bound to acknowledge the direction to which the evidence points. This is what the agnostics in their intellectual dishonesy refuse to do. And it does no good, as they do, to claim that one is not so morally bound on account of alleged a priori evidence against the existence of a supernatural order -- for the simple reason that there is no such evidence, and those who claim otherwise are merely inventing an a priori lie in order to provide cover for all the a posteriori lies they employ in order to deny the evidence for miracles. (A similar type of a priori lie was shown in the O.J. Simpson case: Since, as it was presumed by some, white cops are always trying to frame black men, none of the evidence against O.J. had to be accepted.)

This is not to say that the science of miracles is not related to faith, it certainly is; nor that some miracles ought not be believed by the faithful simply because the Church says they are worth of belief, they certainly should. All I’m saying is that at bottom the credibility of miracles depends on empirical evidence, and such evidence itself compels assent.

Chicken, I think you would be interested in my earlier post on so-called "self-committing propositions." Briefly, I think George is right, here. It certainly is a mistake to take that since some proposition has to be assented to "by faith" that means that it cannot be genuinely probabilified by evidence.

It's here:


A distinction must be made between what St. Thomas called the preambles of faith and faith. The existence of God is a preamble of faith and may be probabilized. Faith, itself, cannot. I will agree that direct probabilistic arguments may be made for the existence of God, but only subsidiary arguments for the Trinity, since it is revealed directly by God and depends on his existence.

The Chicken

To clarify my position, in saying that the science of miracles is not faith, I do not mean to suggest that there are not certain miracles that are de fide, the Resurrection being one of them. Therefore, the Resurrection is to be believed primarily because it has been revealed by God, and it is to be believed with certainty. Above, on the other hand, I was mainly referring to those miracles that have occurred after the age of revelation and have left behind plenty of evidence of having taken place.

Lydia, Descartes apparently thought God could perform contradictions! But to get a grip, take your Bill Smith example: ‘Bill and I have talked a number of times on the phone, and … he … said that he would like to send me a paper’. If you accept the OT as the word of God, there’s an analogy between the resurrection and the paper Bill said he’ll send you; I've granted that. But if you don’t want to restrict your result exclusively to those who accept the OT as the word of God, this means that you can’t invoke any prior conversation with Bill in order to explain why you got to receive a paper rather than, say, a bunch of roses or nothing at all: Nobody will be impressed if you just insist that you’re certain a priori that ‘there is a reason’ why you’d be sent a paper, except you have no idea what the reason is! The POE analogy was meant against such a move.

In the other thread you seemed to have no problem dismissing certain miracle claims. Here, you gave a nice example of how you can field or accommodate a miracle report without giving up your particular religious/metaphysical framework, so I wondered if you meant to kick out the likelihoods I was willing to grant: If Bill specified no timeframe for sending you the paper, perhaps what you got was spam or not the paper you were promised at any rate. You still believe Bill will send you the promised paper at some point, that he’s out there and is trustworthy. That he may not have sent you the paper yet is no evidence against that, it’s never been.

So why the Bayesian God would choose to perform a miracle rather than wipe out evil - as if he couldn’t do both - you don’t say. Whether a miracle counts as ‘raw data’ or a ‘hypothesis’ you don’t say either. If you don’t want to engage, Lydia, suit yourself.

Overseas, you've been sort of pushing increasingly for quite a while, and I've been hesitating to say this, but...

I'm sorry, but you're out of your depth. It comes out in the way you fire seemingly at random.

You are so careless that you appear, at least, not even to notice that I specifically discussed a case in which you have not had any prior conversations with Bill Smith but in which you still received an e-mail purporting to be from Bill Smith. (One can specify the description that we want to use to specify who "Bill Smith" is beyond the name if needed to fill out the example.)

Nor is data concerning the OT (which an atheist can examine as well) the only thing that might correspond to _other evidence_ for the existence of God, and hence to the "earlier conversations with Bill Smith." That's meant only to stand in for whatever other evidence someone does have that God exists. It could include natural theology arguments, for example.

Strictly speaking, "raw data" are the data of experience. Honestly, I have no great interest in going into that with you in any great detail, given the carelessness with which you have treated all the other things I've written to you. But data and hypothesis can also be described at various levels of the epistemic structure--for example, sometimes it makes sense and changes nothing in the argument if I speak of "there is a desk in front of me" instead of "I have a set of sensory experiences as-if of a desk in front of me."

I have never said anywhere that there is some conflict between God's "wiping out evil" and God's performing some individual miracle. Indeed, God would have to perform _many_ miracles to "wipe out evil." Whether the fact that he does not do so counts as evidence _against_ his existence, however, is another question. Again, you are too hasty to follow what I've actually said in detail.


Do you have any references on the screening-out process you describe, above? I think I know what the name refers to and if it is, then it could be a stop to transitivity, but I can imagine a "leaky screen," if things are as I think they are, in which case transitivity might squeak by, but I will wait for your references and try to do a little research, myself.

What I am concerned with is the case where a super-intelligent race (not God) is listening in on us and just happens to provide our miracles at random. In other words (warning: bad example), what about the tooth fairy? A boy puts a tooth under a pillow and prays that God transform it into a quarter and the next day, it is. He then prays to God to find a new notebook in his back pack and the next day, he does. He thinks God is performing the "miracles" all the while his mother is listening in on his prayers at night. Some times, "God" does what he asks. Sometimes "he" doesn't. He does not know why, but trusts that "God" knows what he is doing.

Here, the evidence raises the possibility of miracles, but also raises the possibility of a false God, not God.

How does one know that something is a genuine miracle and not the product of another preternatural force or a super -intelligent (but hidden) alien race?

For example: we could fairly easily duplicate the burning bush, in the near future, with a holographic projector. One would feel heat from the light scattering, but the tree would not burn, since we are simply projecting a tape loop.

I am sure that philosophers has dealt with this, but I have no idea where to look to be brought up to speed.

In this context, it seems sort of pointless and trivial to argue after the fact that the smell of chocolate cake he notices when he comes home on his birthday is evidence for his wife's existence. No one is worrying about that, and so no one thinks twice about the fact that the evidence runs in that direction.

What about my Stepford wife proposal? He can have 99% probability that his wife exists and she baked the cake and be wrong on both counts. If you don't like Stepford wives, then, how about a twin sister or (in the near future) a robot maid. The evidence of the smell is circumstantial at best that his wife were the Someone made the cake. One could not convict the wife of cooking the cake based on that evidence. Patterns can be violated unless there is strict historical continuity. For instance, in non-linear mathematics, if a variable is increased from x to x + a, the process might continue, but if increased from x +a to x + a + e, where 0

And note something that is perhaps clearer in this example than in the cake-baking example: Most of us receive e-mails occasionally from people whom we previously had no contact with. (I certainly have.) Once we have the e-mail, even though we did not have a high previous probability for the existence of the writer or reason to predict the arrival of the e-mail, we have reason to believe in his existence. In other words, the e-mail is evidence for the existence of its author independent of the lowness or highness of our prior probability for the existence of the author.

I guess I think like a defense attorney. The e-mail is evidence that Someone sent an e-mail, but I would never, if I were Mr. Spock, jump to the conclusion that Bill sent me the e-mail. For all I know, his roommate killed him, read his e-mail and sent the e-mail to you just to make it look like Bill were still alive.

In other words, probabilities can be short-circuited as in the case of a frame-up. It as recently become possible to fake DNA evidence, which is perhaps the most solid evidence most prosecutors think they have (a complicated issue, to be sure). One can have ten pieces of evidence that point in one direction and still be wrong.

Why this is not the case in the Resurrection is that the pay-off for such a frame-up is so low: misery, suffering, death, but in the case of Bill, the pay-off (utility in Ramsey's decision theory) for the murderer could be very high.

I am not saying that one cannot use Bayesian statistics for argumentative purposes, but one must consider the possibility of the Ludic Fallacy rearing its ugly head.

Since I am merely a learner, I will shut up and listen to your comments. I admit the strong possibility that I am wrong from the start, but hey, if you don't try, you don't learn.

The Chicken


Why does Bill cease to exist if you can’t invoke the content of a prior conversation with him? Never mind.

You seem too preoccupied with atheists rather than with the vast majority of theists who are not Christians. Followers of other religions are certainly not the tabula rasa you make them out to be: They have their own holy books and miracle stories, and therefore the machinery to accommodate or field off 'the data' without a serious dent, like you do all the time. You assume that non-Christians are mere generic theists, a hair’s breath away from converting to Christianity through Bayesian epistemology upon becoming apprised of ‘the data’; I suspect what reasons you may have for making this claim, but I can’t see the grounds you have for making it. Incidentally, by ‘observation report’ I meant things like ‘red now’. Desks and chairs are OK and I accept there’s a spectrum, but ‘miracle’ is nowhere near a chair. I know you think you can get your counterfactual self to track a rational reconstruction of how you’d come to believe what you do even if that’s not how you did come to believe it. I don’t see what the feat would entail, unless it’s that whoever disagrees with Lydia is irrational or thereabouts. If that’s the price for ‘showing Ma’ I don’t know how many philosophers would be prepared to pay it; certainly not Aquinas, who openly pulled out the revelation; so if I’m out of my depth at least I have company.

If miracles cost the Bayesian God nothing, I can’t see why we can’t expect him to perform as many as it takes to fix the priors. That he doesn’t perform them and that we can’t figure out why he doesn’t undermines the claim that God’s a Bayesian, if not that he exists. I can’t see how you expect divergent priors to get to converge rather than become ‘the whole story’ anyway unless you have a long run of events. Unless you think there’s some ‘objective’ way to assign priors, in which case I agree circularity may be the least of your problems!

In the previous thread I suggested that you post something on mutual support and appreciated it when you did, so I’m sorry you felt ‘sort of’ pushed. I guess it’s because I chose to rephrase questions when I repeatedly failed to elicit any straight answers: I assumed I hadn’t made myself clear rather that you were being deliberately obstructive or evasive.


All good questions, but I'll have to take them one at a time and probably respond piecemeal.

Screening-off is an independence relation. The transitivity of non-deductive confirmation is notorious slippery. It's well-known in probability theoretic circles that A can be positively relevant to B, B to C but that A can fail to be positvely relevant to C. Shogenji has shown (reference in the Erkenntnis article I link above) that we can guarantee transitivity of confirmation if a partiion on B "screens off" A from C. This means that

P(A|B & C) = P(A|B)


P(A|~B &C) = P(A|~B)

In the Erkenntnis article we extend this idea to longer chains beginning with foundational evidence. To trace a line of evidence, you start with a piece of foundational evidence and, as it were, check to see how it relates to some higher-level "target" proposition. So, if you have foundation F and "target" proposition D, you check to see if a partition on A screens F from D, if a partition on B screens A from D, and so forth. (I believe, though I'd have to check, because it's been a while, that if a partition on B screens A from D and a partition on A screens F from D, a partition on B screens F from D automatically. If not, one would have to add that in as a condition.) So you have these nested screening-off relations which build up a line of transitivity that shows how the evidential relevance of evidence F "runs" to D. Where no one proposition screens but F clearly seems relevant to D, we have to find some complex Boolean partition on a complex node that uses something like R _or_ S _or_ T, like that.

_All_ that this does (one sometimes gets what seem thin returns for a lot of investment of time in this area) is to show _how_ F supports (or undermines) D. In other words, one learns that F does that "through" its impact on these other propositions.

What this _doesn't_ do is make non-deductive inference any stronger or less uncertain.

If I'm understanding your examples, Chicken, they mainly concern places where we are simply _wrong_ in our non-deductive conclusions. There's no way of getting away from that possibility. It is logically possible that the man's wife has been replaced by a robot in the meanwhile.

It doesn't at all follow, though, that we can't be rationally justified in our inferences. After all, I suppose you would agree that the man who comes home is _reasonable_ to believe that his wife made the cake and to feel an upsurge of gratitude to her as he smells it. It's not like he has to _worry_ about her being replaced by a robot. And the same with my e-mail from Bill Smith.

So "supports" or "confirms" never means "makes certain to the point where we can't dream up alternative scenarios."

That's just the nature of the beast. You, MC, could in logical possibility be the product of a Cartesian deceiver bent on making me converse with someone who doesn't exist.

But I don't lose sleep over the possibility. :-)

As you say, context is everything. Your burning bush example makes a point I have often made--namely, that some miracle reports should be interpreted differently depending on the technology available at the time.

One interesting thing, though: We have no reason to think that the ability to raise dead men (dead for days, not just thirty minutes) is a mere matter of technology. In fact, we have some reason to think it is _not_ merely a matter of technology.

I'm not opposed to the use of science fiction stories as imagination pumps, but I think it's important that we not treat "aliens" as if they are omnipotent.

In other words, MC, when I say that something (the e-mail, for example) is "reason" or "evidence" for the person's existence, I don't mean either a) that it is _overwhelming_ evidence or b) that I couldn't be wrong in the conclusion. I just mean that it raises the probability, and I think myself that a sufficiently detailed e-mail that doesn't sound like spam, etc., that gives a plausible reason for getting in touch with me, is sufficient evidence for me to believe the conclusion that the person exists, though not in an indefeasible way, of course, if other evidence were to come in.

I _answer_ such e-mails as if the person exists, that's for sure. I figure, if all the person is asking for is a philosophy paper, it's hard to see what nefarious purpose he could be up to, even if he's giving a false name.

The context of the Internet and the wider spread of practices like using pseudonyms, sock puppetry, etc., certainly has to be taken into account. I have expressed some suspicion about a pretense of having known me previously from an e-mail correspondent. (And the other day when I did that it turned out that it really was an old electronic friend whose name I'd merely temporarily forgotten, and boy, did I feel stupid and caddish.)

But yeah, the processes of non-deductive inference are all non-certain and messy. No getting around that.


Why does Bill cease to exist if you can’t invoke the content of a prior conversation with him?

See, this is the kind of totally bizarre comment that I just do not have time for. And you do it all the time. Many I let pass without comment, because you seem impervious to correction when I try to correct you. I've never said anything _remotely like this_, and if you think that I have, that reflects badly on your ability to engage in this conversation.

On the Internet, as in the rest of life, if you spend your time attributing to other people silly positions that bear no discernible relation to what they say, acting in a way that shows that you are either incapable of or unwilling to comprehend what they are talking about, this is a major demotivator for conversation with you. What you take for evasiveness has been in a number of cases simply...tact.

I've spent a lot of time trying to answer you, Overseas, and my statements go into your brain and come out unrecognizable. I really can't see why I should bother. (No, I certainly don't think that other people are on the verge of becoming Christians if I just present evidence to them. That's just another weird misinterpretation of your own.)

So, if I more or less ignore your comments from here on out, you will think what you will think, but at least I've given you the opportunity to consider the possibility that you aren't quite as sharp at this kind of thing as you evidently think yourself to be and that you are _extremely annoying_ in the way that you shallowly (mis)represent other people's statements and positions, apparently for the sake of having something to say that you think sounds smart. Save that for the college dorm bull sessions (or whatever corresponds to those in your life), because I'm not going to waste further oodles of my time with it.

Hi, Lydia,

Very nice!

You need not assume theism to argue for the resurrection. Right. For instance a small prior for theism (without considering the direct evidence for the resurrection) may well be good enough with the direct evidence. I have disliked most objections to evidential resurrection apologetics from the supposed circularity – the alleged need to presuppose theism ad definitively or probably true to argue for the resurrection (and from it to theism). But here is a related worry, in the form of a fictional dialogue.

Skeptic: No god, no resurrection.

Lydia: Right. But atheism is not 100% certain even prior to the direct evidence for the resurrection.

Skeptic: Do me a favour and assume atheism is very, very probable prior to the direct evidence for the resurrection.

Lydia: O.K. Even so, and with good enough direct evidence for the resurrection and not 100% prior certainty about atheism, there’s still no clear obstacle.

Skeptic: Yet, the prior probability of atheism is, ex hypothesi, almost 100%. So, we can approximate the right posterior probability of the resurrection by taking the prior probability of atheism to be 100% certain. In fact, you treat similarly the historical facts about the testimonies of the disciples, the women, and St. Paul (= the direct evidence) – those facts are not 100% certain either. Still, you seem to handle them as 100% when approximating the posterior probability of the resurrection on all of them by means of Bayes’ Theorem. Beware double standard! Thus, under the mentioned assumption, you should treat atheism as 100% certain in approximating the posterior probability of the resurrection. But then this posterior probability is 0; for on atheism there’s no resurrection.

Lydia: ________? (Inserting Jeffrey’s Rule, instead of Bayes’ Theorem, for calculation of the probability of the resurrection by means of the testimonies?)

Sorry if this has been obviously replied above!

Vlastimil, here is how I would reply:

An atheist can, if he wishes, regard my argument for R as a kind of exercise. By arguing the enormously high Bayes factor for the evidence adduced, this exercise presses the atheist to do at least one of the following:

a) defend a prior probability so low that it still cannot be overcome even by evidence that strong,

b) challenge the Bayes factors as overly generous or the independence assumption--in other words, challenge the form of the argument as we actually give it,

c) challenge the facts as being so poorly supported that treating them as given even as a calculative convenience is a huge overstatement of the case and thus makes the argument as given totally uninteresting (as if we had shown the impact of conditioning on the existence of leprechauns),

d) reconsider his atheism.

Option b is right in there with the rest. And if the atheist chooses to do that and, say, to make an argument _against_ one of our facts, it would of course be ridiculous for us to say, "But we are treating that fact as if it had probability 1 as an approximation, so we don't need to bother with your argument." We are treating the proposition as having a probability close to one _for purposes of a particular argument_ regarding R. When that proposition itself is what is being called into question, responding to an argument against it obviously requires dropping that approximation and discussing whatever evidence the other person adduces against it, the evidence we have for it, and so forth.

So the true parallel to the atheist's move that you are envisaging is not our treating the facts as given for the sake of the argument we actually give for R. That's merely a convenience that allows us to show clearly and cleanly the evidential force of these facts, taken as facts. (After all, the Bayes factor we estimate is so huge that it can take the "hit" of some sort of recalibration for the mere fact that the facts are less than certain, so long as they are very highly probable, while remaining a very strong argument.) The parallel would be, rather, refusing to treat them as anything other than given when and if they are challenged. In other words, here person A is presenting an argument against an admittedly non-certain proposition held by person B. For person B actually to respond to the argument, B needs to do the standard things--challenging premises, challenging the inference, etc. Simply saying, "But I consider the probability of the proposition you are challenging to be approximately 1, so I don't have to do anything about your argument," is--whatever else--not a response to the argument.

Hi Vlastimil!

Perhaps I’m particularly sensitive to circles, and it's not clear if you see a circle or not but consider this supplement:

A kindly neighbour who wishes to stay anonymous has always acknowledged our guys’ birthday too and left a bunch of roses on the door-step one year, a bottle of wine the next etc. A third-party who catches the birthday-boy with signs of ‘mental illness’ and doesn’t know of the wife or her habits but believes in anonymous kindly neighbours may well conclude that the chocolate cake was the neighbour’s doing and that our birthday boy is a deluded bachelor. The claim is that we have no reason to prefer the hypothesis that it was the wife who baked the cake this time rather than the neighbour, unless we accept the hypothesis of a wife who always and solely produces chocolate cakes for our birthday boy. Can you see the circle now?

Whether the wife has chosen to celebrate her husband’s birthday this year by baking a cake is one question and who it is who’s baked this cake is another; the second question does not necessarily presuppose theism. Whether this is a cake we see before us is, of course, a separate question too!

Vlastimil, I thought of adding this:

One sign to me that a calculation done with our facts taken as given is of interest is this: I've seen it happen that a skeptic will start out talking as though it's quite easy to explain these facts in some naturalistic way: People lie a lot, etc. Standard Humean approach. Then when the details are pressed home and the explanatory difficulties made clearer, we see a retreat back on questioning the facts, trying to throw the entire textual account into question even as a record of what was claimed, etc.

So it is IMO worth doing to show that the facts as given textually _aren't_ actually easy to explain non-miraculously.

And I would also point out that in our argument we don't assume or come anywhere near to saying that the prior probability of theism is 1, which would be another type of parallel move to the atheist's move you wonder about.

The facts we adduce are themselves entirely non-miraculous and secularly describable. They do not assume theism or atheism at all. They are statements about what ordinary human mortals said and did, with the question being how best to explain them.



First, a minor correction of my fictional dialogue above. A sentence of mine in it was put pretty badly. I think I should have rather said: you seem to handle the testimonies as 100% certain when approximating the posterior probability of the resurrection on really given direct evidence for the resurrection by means of Bayes’ Theorem applied to the testimonies as if they were given.

But most importantly, what you're saying in your response does make sense to me. I guess I'm satisfied.

Finally, you note: "the Bayes factor we estimate is so huge that it can take the "hit" of some sort of recalibration for the mere fact that the facts are less than certain, so long as they are very highly probable, while remaining a very strong argument."

I assume you would show this by Jeffrey's Rule, right?




Thank you. Try to translate and express your idea transparently for the case of the resurrection and theism, please. So far I don't get the supposed analogy. Sorry for not being enough attentive.

Bayesian analysis is a means of arriving at the best possible decision given the evidence and assoicated probabilities. As such, it is used for inductive hypothesis testing. It fails, I think, in exactly three cases: 1) if all evidence has exactly 50/50 probability, such as in the case of a true paradox. Bayesian analysis cannot provide reasons to believe on side of the paradox over the other, 2) in the case where the sample space exceeds the size of the state space. this happens in mathematical chaos, where, even though the sample is state space is bounded, so the probability of finding the sample in the space is 100%, the sample space is larger, so the sample probability is greater than 100%. In this case, the sample space bends back in on itself to fit in the state space and probabilities become quasi-random, even though there is ann underlying order to the sample, and 3) if the evidence is truly random. If a serial killer decides to kill people based on the roll of the dice, there is nothing a Bayesian analysis can do.

There is one interesting theorem, however, to consider, called the Ramsey theory (a subcategory of 3, above), that states that for a sufficiently large random sample, pockets of order will occur. The interesting thing is that Bayesian analysis will work (I think) within the pocket of order, so that even if the unniverse truly were random and we happen to be living in a pocket of order, Bayesian analysis should still work, for the most part, provided the outside randomness doesn't leak too much into the ordered subset. Can Bayesian analysis be used to argue that the universe is not random? I don;t know.

The Chicken


Vlastimil, I think that to use the JC expansion of Bayesian updating one would need to plug in a prior probability for R, at least some "dummy" or "illustrative" prior, but this could certainly be done. E.g., If the prior of R were X, and if this fact or set of facts moves to probability Y in the new distribution, what would the new posterior for R be? Right at the moment, I'm not thinking of a way to generate what one might call a "mere Bayes factor" (without using priors--just the Bayes factor part of it) for a rise in intermediate probability of some evidence, but there may be a way to do it.


I don’t think you’ve been inattentive, though you’d better watch it around here!

Consider this: Did the chocolate cake have ‘Happy Birthday’ written on it? The scenario only stipulates the ‘smell’ of baked chocolate. Perhaps there was also the smell of sewage in the house because there was a plumber fixing a leaking pipe. The wife had called him in before leaving for work in the morning. The birthday-boy does not even notice the offensive smell because he does not connect the presence of the plumber with his birthday. He’s looking for the chocolate cake she usually makes. He does not know that she’s making arrangements for a surprise party at home in the evening and that this is the reason she wanted the leak fixed.

What makes the chocolate cake a 'birthday cake'? I doubt the birthday-boy wants to know whether his wife exists, or who it was who baked the cake. Even the deluded bachelor wants to know what his wife is like, if she cares, if she remembered his birthday or maybe forgot. But would it still be a birthday-cake if she'd baked it though she'd forgotten it was his birthday? He would probably take it to be a birthday-cake, but that's because the 'evidence' is theory-laden and context-dependent. Our birthday-boy has certain expectations based on what he takes or hopes his wife’s intentions to be. But we don’t know what his wife’s intentions are, and intentions are certainly not observable ‘data’: They are only guesses, hypotheses made by the husband. He ascribes intentions to the wife and forms certain expectations which constrain the range of facts he’ll pay attention to. So he misses the plumber altogether because he’s only looking for a cake. He cannot see the connection between the plumber and his birthday, because he does not expect there to be a connection; but he can see a connection between his wife and the cake, which may have been baked by the neighbour.

If we make the chocolate cake stand for the miracle of the resurrection, the way I read the analogy is this: The birthday-boy is a Jewish believer at the time. ‘Wife’ stands for God, and the way our Jewish believer comes to form expectations about the wife’s intentions is based on some reading of the Old Testament. But all it takes for the Jewish believer to refuse to convert is to doubt that it’s a ‘special day’ today: No cake can be a ‘birthday cake’ today, that special day is still to come. I think it’s clear that, if a Jewish believer can do it, it’s not necessary for one to be an ‘atheist’ to doubt the significance of the resurrection miracle. Different religions may stipulate different divine intentions. I hope it’s also clear that unless one is committed to some reading of the OT one is unlikely to expect the miracle of the resurrection at all. So one is likely to miss it, like the plumber. But if challenged, one’s reaction will again depend on one’s prior commitments: One can always claim that there’s but a single wife - the one our birthday-boy is married to - who always bakes him a chocolate cake only on his birthday so that if she has baked the cake then it’s his birthday and she’s the only one who knows how to bake a chocolate cake anyway so everyone else who thinks he has a capable wife is a deluded bachelor etc. Such a move only makes the circle more pronounced. Now has ‘transparency’ increased or decreased?!

Overseas, thanks.

Have you read Lydia's cumulative case for the resurrection? Given Lydia's Bayes factor, and the very low prior (which is not focus of the paper), for the resurrection is right, the resurrection is almost 100% probable. How to proceed form it to Judeo-Christian theism? Again, that's not the focus of the paper, but you'll find a brief hint there: the resurrection covers only a tiny part of the probability space under the negation of this theism -- for reasons analogical to the reasons for the claim that the disciples' (or the women's or Paul's) testimony covers only a tiny part of the probability space under the negation of the resurrection: every explanatory hypothesis under ~R (and analogically under ~JCT) has negligible antecedent probability and/or explanatory power. Of yourse, many will deny or be incapable to appreciate this (if it is true), but I do not see why it could not be true and fleshed out in detail.

Above, I noted an interesting comment of yours (which is beyond the point of this thread): "Followers of other religions are certainly not the tabula rasa you make them out to be: They have their own holy books and miracle stories, and therefore the machinery to accommodate or field off 'the data' without a serious dent ..."

I like to ask myself this way: 1. Are the apologetical cumulative packages of different world-views (including naturalism) mutually neutralizing (as Hume suggested at least wrt religious world-views?) 2. Is the right answer to (1), or to the question whether a particular world-view significantly beats the other (incompatible) ones, relative to which package one begins with, or, more generally, to the temporal of gaining the evidences? (If yes, that would seem to be a problem for the probabilistic method of assessment.) 3. If no and a particular world-view significantly beats all the other ones, are believers in some other world-view who are studied in its evidences and not having the evidences of the best supported world-view generally capable to appreciate these evidences latter later (when gaining them) or culpable for not doing so? What do I think? 1 - no. 2 - no. 3 - ? In fact, it seems to me Christianity is the winner. But it's a daunting challenge to show this with a tolerable degree of philosophical rigour. (I think Lydia's view is similar.) It would take mountains of research and time to sift in a Bayesian way, say, the clash of the historical evidences for the resurrection, discussed by Lydia, and the contemporary evidences for reincarnation, discussed by Ian Stevenson and others at http://www.scientificexploration.org/journal/articles.html I admit this.


‘In fact, it seems to me Christianity is the winner. But it's a daunting challenge to show this with a tolerable degree of philosophical rigour. (I think Lydia's view is similar.)’

If that’s the bottom line perhaps there’s little one could disagree with, provided that circularity and philosophical rigour are incompatible. But I took this post to be an explication of the concept of ‘mutual support’ as a way to escape question-begging circularity, which was moreover meant to go wider than just the case of Christianity and the resurrection. This is why I didn’t think you’d been inattentive when you asked for a ‘translation’ of the birthday-cake analogy. So I don’t quite see what papers about e.g. reincarnation may have to do with working our way out of circularity and, no, I haven’t read any.

But I’m not sure what you take this thread to be about if you can't see the connection between my earlier comment which you say lies 'beyond the point of this thread' and the way I've expanded on the birthday-cake example of mutual support to expose the embedded circularity and theory-ladenness of factual reporting. You don’t say if you can read the analogy the way I suggested. If the wife called the plumber to fix the guest-bathroom because she's throwing a surprise-party this doesn't mean she did not also bake the cake. But does the cake count as a ‘birthday cake’ if it was baked for reasons other than it’s our guy’s birthday today? We know he believes it’s his birthday today but we have no independent evidence that it is; perhaps he’s a deluded bachelor who thinks it’s his birthday every time a kindly neighbour leaves him a chocolate cake.

So re your (1),(2)and (3), the interesting question, in my view, is what may count as ‘evidence’ for a religion and why, or if miracles in particular are somehow special. That what might have counted as a miracle 100 years ago may not count as a miracle today does not mean that God was not behind the phenomenon, just because we’ve found a sufficient ‘natural’ explanation for it. And conversely, just because we haven’t found a sufficient explanation so far doesn’t mean that there’s none to be found: We may just not have figured it out yet. So miracles aren’t a priori or unconditionally special, and if you were to say that miracles are special because your favourite religion contains a prediction about a miracle which has already occurred and that this is why it's your favourite religion, then you would be begging too many questions! Jewish people would agree with you over the prediction, though not over whether it’s been fulfilled yet. But they don’t take non-fulfillment of the prediction as disconfirmation of their Jewish belief, any more than Christians take that e.g. there’s been no second coming in the past 2,000 years as disconfirmation of their Christian belief.

So what does it mean precisely for one religion to ‘beat’ another religion? I find it quite remarkable if religions can only be confirmed by evidence but not be disconfirmed. What do you think?


The first par (in my last reply) is pertinent to the focus of this thread - from miracles to (specific) theism, and, I think, to your worries, too. Just reading Lydia's paper on the resurrection may well be the most direct way to concede it.

The 3rd par (about the reincarnation evidences, etc.) is pertinent to the words I cited.

By "beating the other world-views" I've meant being significantly more probable than not (on all evidence).

Sorry, no more time.


Look, this chat started in another thread and I’m importing background; all I cared to see was whether ‘mutual support’ amounts to something more than a euphemism for circularity. So I’m interested now that you mention ‘all the evidence’: What’s that? If 'all the evidence’ is an exercise in censorship to ensure a fit with pre-existing beliefs, it will be no miracle if we end up proving what we’ve assumed. We’re just making a circular argument. Of course it’s OK to seek to establish some internal coherence and consistency; I just wonder how one might hope to establish anything more than that.

There can be nothing wrong with refusing to revise beliefs in the face of the evidence for a particular miracle witnessed by some people, if it’s OK to refuse to revise beliefs in the face of the evidence of suffering constantly experienced and witnessed by all people. If we find alternative naturalistic explanations credible only when faced with a miracle we don’t believe but find shortcomings with each alternative other than that our God did it when it comes to a miracle we do believe, and if we flatly refuse to believe that who did it may also be nasty or clumsy at times, despite the evidence of evil all around, then we’re already committed to some theistic religion and ‘the evidence’ can’t have anything to do with it. So this is the context and what I've expressed a view on. If you can tell me now what you want me to concede, I won't refuse to consider it.

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