# Prayer and Probability

The reader might wonder about the relation between the previous post and my discussion of Arman Razaali. If I could say it is more likely that he was lying than that the thing happened as stated, why shouldn’t they believe the same about my personal account?

In the first place there is a question of context. I deliberately took Razaali’s account randomly from the internet without knowing anything about him. Similarly if someone randomly passes through and reads the previous post without having ready anything else on this blog, it would not be unreasonable for them to think I might have just made it up. But if someone has read more here they probably have a better estimate of my character. (If you have read more and still think I made it up, well, you are a very poor judge of character and there is not much I can do about that.)

Second, I did not say he was lying. I said it was more likely than the extreme alternative hypothesis that the thing happened exactly as stated and that it happened purely by chance. And given later events (namely his comment here), I do not think he was lying at all.

Third, the probabilities are very different.

## “Calculating” the probability

What is the probability of the events I described happening purely by chance? The first thing to determine is what we are counting when we say that something has a chance of 1/X, whatever X is. Out of X cases, the thing should happen about once. In the Razaali case, ‘X’ would be something like “shuffling a deck of cards for 30 minutes and ending up with the deck in the original order.” That should happen about once, if you shuffle and check your deck of cards about 10^67 times.

It is not so easy to say what you are counting if you are trying to determine the probability of a coincidence. And one factor that makes this feel weirder and less probable is that since a coincidence involves several different things happening, you tend to think about it as though there were an extra difficulty in each and every one of the things needing to happen. But in reality you should take one of them as a fixed fact and simply ask about the probability of the other given the fixed thing. To illustrate this, consider the “birthday problem“: in a group of 23 people, the chance that two of them will have the same birthday is over 50%. This “feels” too high; most people would guess that the chance would be lower. But even without doing the math, one can begin to see why this is so by thinking through a few steps of the problem. 22 days is about 6% of the days in a year; so if we take one person, who has a birthday on some day or other, there will be about a 6% chance that one of the other people have the same birthday. If none of them do, take the second person; the chance one of the remaining 21 people will have the same birthday as them will still be pretty close to 6%, which gets us up to almost 12% (it doesn’t quite add up in exactly this way, but it’s close). And we still have a lot more combinations to check. So you can already start to see that how easy it will turn out to be to get up to 50%. In any case, the basic point is that the “coincidence” is not important; each person has a birthday, and we can treat that day as fixed while we compare it to all the others.

In the same way, if you are asking about the probability that someone prays for a thing, and then that thing happens (by chance), you don’t need to consider the prayer as some extra factor — it is enough to ask how often the thing in question happens, and that will tell you your chance. If someone is looking for a job and prays a novena for this intention, and receives a job offer immediately afterwards, the chance will be something like “how often a person looking for a job receives a job offer.” For example, if it takes five months on average to get a job when you are looking, the probability of receiving an offer on a random day should be about 1/150; so out 150 people praying novenas for a job while engaged in a job search, about 1 of them should get an offer immediately afterwards.

What would have counted as “the thing happening” in the personal situation described in the last post? There are a number of subjective factors here, and depending on how one looks at it, especially depending on the detail with which the situation is described. For example, as I said in the last post, it is normal to think of the “answer” to novena on the last day or the day after — so if a person praying for a job receives an offer on either of those days, they will likely consider it just as much of an answer. This means the estimate of 1/150 is really too low; it should really be 1/75. And given that many people would stretch out the period (in which they would count the result as an answer) to as much as a week, we could make the odds as high as 1/21. Looking loosely at other details could similarly improve the odds; e.g. if receiving an interview invitation that later leads to a job is included, the odds would be even higher.

But since we are considering whether the odds might be as bad as 1/10^67, let’s assume we include a fair amount of detail. What are the odds that on a specific day a stranger tells someone that “Our Lady wants you to become a religious and she is afraid that you are going astray,” or words to that effect?

The odds here should be just as objective as the odds with the cards — there should be a real number here — for reasons explained elsewhere, but unfortunately unlike the cards, we have nowhere near enough experience to get a precise number. Nonetheless it is easy to see that various details about the situation made it actually more likely than it would be for a perfectly random person. Since I had a certain opinion of my friend’s situation, that makes it far more likely than chance that other people aware of the situation would have a similar opinion. And although we are talking about a “stranger” here, that stranger was known to a third party that knew my friend, and we have no way of knowing what, if anything, might have passed through that channel.

If we arbitrarily assume that one in a million people in similar situations (i.e. where other people have similar opinions about them) hear such a thing at some point in their lives, and assume that we need to hit one particular day out of 50 years here, then we can “calculate” the chance: 1 / (365 * 50 * 1,000,000), or about 1 in 18 billion. To put it in counting terms, 1 in 18 billion novenas like this will result in the thing happening by chance.

Now it may be that one in a million persons is too high (although if anything it may also be too low; the true value may be more like 1 / 100,000, making the overall probability 1 / 180 million). But it is easy to see that there is no reasonable way that you can say this is as unlikely as shuffling a deck of cards and getting it in the original order.

## The Alternative Hypothesis

A thing that happens once in 18 billion person days is not so rare that you would expect such things to never occur (although you would expect them to most likely not happen to you). Nonetheless, you might want to consider whether there is some better explanation than chance.

But a problem arises immediately: it is not clear that the alternative makes it much more likely. After all, I was very surprised by these events when they happened, even though at the time I did attribute an explicitly religious explanation. Indeed, Fr. Joseph Bolin argues that you should not expect prayer to increase the chances of any event. But if this is the case, then the odds of it happening will be the same given the religious explanation as given the chance explanation. Which means the event would not even be evidence for the religious explanation.

In actual fact, it is evidence for the religious explanation, but only because Fr. Joseph’s account is not necessarily true. It could be true that when one prays for something sufficiently rare, the chance of it happening increases by a factor of 1,000; the cases would still be so rare that people would not be likely to discover this fact.

Nonetheless, the evidence is much weaker than a probability of 1 in 18 billion would suggest, namely because the alternative hypothesis does not prevent the events from remaining very unlikely. This is an application of the discussion here, where I argued that “anomalous” evidence should not change your opinion much about anything. This is actually something the debunkers get right, even if they are mistaken about other things.

# Fairies, Unicorns, Werewolves, and Certain Theories of Richard Dawkins

In A Devil’s Chaplain, Richard Dawkins explains his opposition to religion:

To describe religions as mind viruses is sometimes interpreted as contemptuous or even hostile. It is both. I am often asked why I am so hostile to ‘organized religion’. My first response is that I am not exactly friendly towards disorganized religion either. As a lover of truth, I am suspicious of strongly held beliefs that are unsupported by evidence: fairies, unicorns, werewolves, any of the infinite set of conceivable and unfalsifiable beliefs epitomized by Bertrand Russell’s hypothetical china teapot orbiting the Sun. The reason organized religion merits outright hostility is that, unlike belief in Russell’s teapot, religion is powerful, influential, tax-exempt and systematically passed on to children too young to defend themselves. Children are not compelled to spend their formative years memorizing loony books about teapots. Government-subsidized schools don’t exclude children whose parents prefer the wrong shape of teapot. Teapot-believers don’t stone teapot-unbelievers, teapot-apostates, teapot-heretics and teapot-blasphemers to death. Mothers don’t warn their sons off marrying teapot-shiksas whose parents believe in three teapots rather than one. People who put the milk in first don’t kneecap those who put the tea in first.

We have previously discussed the error of supposing that other people’s beliefs are “unsupported by evidence” in the way that the hypothetical china teapot is unsupported. But the curious thing about this passage is that it carries its own refutation. As Dawkins says, the place of religion in the world is very different from the place of belief in fairies, unicorns, and werewolves. These differences are empirical differences in the real world: it is in the real world that people teach their children about religion, but not about orbiting teapots, or in general even about fairies, unicorns, and werewolves.

The conclusion for Dawkins ought not to be hostility towards religion, then, but rather the conclusion, “These appear to me to be beliefs unsupported by evidence, but this must be a mistaken appearance, since obviously humans relate to these beliefs in very different ways than they do to beliefs unsupported by evidence.”

I would suggest that what is actually happening is that Dawkins is making an abstract argument about what the world should look like given that religions are false, much in the way that P. Edmund Waldstein’s argument for integralism is an abstract argument about what the world should look like given that God has revealed a supernatural end. Both theories simply pay no attention to the real world: in the real world, human beings do not in general know a supernatural end (at least not in the detailed way required by P. Edmund’s theory), and in the real world, human beings do not treat religious beliefs as beliefs unsupported by evidence.

The argument by Dawkins would proceed like this: religions are false. Therefore they are just sets of beliefs that posit numerous concrete claims, like assumptions into heaven, virgin births, and so on, which simply do not correspond to anything at all in the real world. Therefore beliefs in these things should be just like beliefs in other such non-existent things, like fairies, unicorns, and werewolves.

The basic conclusion is false, and Dawkins points out its falsity himself in the above quotation.

Nonetheless, people do not tend to be so wrong that there is nothing right about what they say, and there is some truth in what Dawkins is saying, namely that many religious beliefs do make claims which are wildly far from reality. Rod Dreher hovers around this point:

A Facebook friend posted to his page:

“Shut up! No way – you’re too smart! I’m sorry, that came out wrong…”

The reaction a good friend and Evangelical Christian colleague had when she found out I’m a Catholic.

Priceless.

I had to laugh at that, because it recalled conversations I’ve been part of (alas) back in the 1990s, as a fresh Catholic convert, in which we Catholics wondered among ourselves why any smart people would be Evangelical. After I told a Catholic intellectual friend back in 2006 that I was becoming Orthodox, he said something to the effect of, “You’re too smart for that.”

It’s interesting to contemplate why we religious people who believe things that are rather implausible from a relatively neutral point of view can’t understand how intelligent religious people who believe very different things can possibly hold those opinions. I kept getting into this argument with other conservative Christians when Mitt Romney was running for president. They couldn’t bring themselves to vote for him because he’s a Mormon, and Mormons believe “crazy” things. Well, yes, from an orthodox Christian point of view, their beliefs are outlandish, but come on, we believe, as they do, that the God of all Creation, infinite and beyond time, took the form of a mortal man, suffered, died, arose again, and ascended into heaven — and that our lives on this earth and our lives in eternity depend on uniting ourselves to Him. And we believe that that same God established a sacred covenant with a Semitic desert tribe, and made Himself known to mankind through His words to them. And so forth. And these are only the basic “crazy things” that we believe! Judge Mormons to be incorrect in their theology, fine, but if you think they are somehow intellectually defective for believing the things they do that diverge from Christian orthodoxy, then it is you who are suffering from a defect of the intellectual imagination.

My point is not to say all religious belief is equally irrational, or that it is irrational at all. I don’t believe that. A very great deal depends on the premises from which you begin. Catholics and Orthodox, for example, find it strange that so many Evangelicals believe that holding to the Christian faith requires believing that the Genesis story of a seven-day creation must be taken literally, such that the world is only 7,000 years old, and so forth. But then, we don’t read the Bible as they do. I find it wildly implausible that they believe these things, but I personally know people who are much more intelligent than I am who strongly believe them. I wouldn’t want these folks teaching geology or biology to my kids, but to deny their intelligence would be, well, stupid.

I suspect that Dreher has not completely thought through the consequences of these things, and most likely he would not want to. For example, he presumably thinks that his own Christian beliefs are not irrational at all. So are the Mormon beliefs slightly irrational, or also not irrational at all? If Mormon beliefs are false, they are wildly far off from reality. Surely there is something wrong with beliefs that are wildly far off from reality, even if you do not want to use the particular term “irrational.” And presumably claims that are very distant from reality should not be supported by vast amounts of strong evidence, even if unlike Dawkins you admit that some evidence will support them.

# Crisis of Faith

In the last post, I linked to Fr. Joseph Bolin’s post on the commitment of faith. He says there:

Since faith by definition is about things that we do not see to be true, there is no inherent contradiction in faith as such being contradicted by things we do see to be true, such an absolute assent of faith seems to imply an assent to the content of faith so strong that one would desire to hold to it as true, “even if it (the content of faith) were to be false”. Can such faith be justified?

Consider the following situation: a woman has grounds to suspect her husband is cheating on her; there is a lot of evidence that he is; even when she asks him and he tells her that he is not, she must admit that the sum of evidence including his testimony is against him, and he probably is cheating. Still, she decides to believe him. I argue that the very act of believing him entails a commitment to him such that once she has given faith to his word, while it is still in fact possible that she is believing him though he is actually lying, this possibility is less relevant for her than it was prior to her giving faith. In this sense, after faith, the “if it were to be false” becomes less of a consideration for the believer, and to this degree she wills faith “even were it to be false”.

A more detailed analysis of the situation: various persons present her with claims or evidence that her husband is cheating on her. Before confronting him or asking him if he is, she collects various evidence for and against it. She decides that since believing him if he is dishonest is not without its own evils, if the evidence that he is cheating (after taking into account the evidence constituted by his statement on the matter) constitutes a near certainty that he is cheating — let’s say, over 95% probability that he is cheating — that she shouldn’t believe him if he says he is not, but must either suspend judgment or maintain that he is cheating. Now, suppose the man says that he is not cheating, and the evidence is not quite that much against him, let’s say, the evidence indicates a 90% probability that he is cheating, and a 10% probability that he is not. She makes the decision to believe him. Since she would not decide to do so unless she believed that it were good to so, she is giving an implicit negative value to “believing him, if he is in fact lying”, a much greater positive value to “believing him, if he is speaking the truth”, and consequently an implicit positive value to “believing him,” (even though he is probably lying).

Going forward, she is presented with an easy opportunity to gather further evidence about whether he is in fact cheating. She must make a decision whether to do so. If she is always going to make the same decision at this point that she would have made if she had not yet decided to believe him, it seems that her “faith” she gives him and his word is rather empty. A given decision to pursue further evidence, while not incompatible with faith, is a blow against it — to the extent that, out of fidelity to him, she accepts his claim as sure, she must operate either on the assumption that further evidence will vindicate him, or that he is innocent despite the evidence. But to the extent she operates on one of these assumptions, there is no need to pursue further evidence. Pursuing evidence, therefore, implies abstracting from her faith in him. To pursue evidence because it is possible that further evidence will be even more against him and provide her with enough grounds to withdraw her assent to his claim of innocence means giving that faith a lesser role in her life and relationship with him, and is thereby a weakening of the exercise of that faith. Consequently, if that faith is a good thing, then, having given such faith, she must be more reluctant to seek a greater intellectual resolution of the case by greater evidence than she was before she had given it.

All of this is true in substance, although one could argue with various details. For example, Fr. Joseph seems to be presuming for the sake of discussion that a person’s subjective assessment is at all times in conformity with the evidence, so that if more evidence is found, one must change one’s subjective assessment to that degree. But this is clearly not the case in general in regard to religious opinions. As we noted in the previous post, that assessment does not follow a random walk, and this proves that it is not simply a rational assessment of the evidence. And it is the random walk, rather than anything that happens with actual religious people, that would represent the real situation of someone with an “empty” faith, that is, of someone without any commitment of faith.

Teenagers will sometimes say to themselves, “My parents told me all these things about God and religion, but actually there are other families and other children who believe totally different things. I don’t have any real reason to think my family is right rather than some other. So God probably doesn’t exist.”

They might very well follow this up with, “You know, I said God doesn’t exist, but that was just because I was trying to reject my unreasonable opinions. I don’t actually know whether God exists or not.”

This is an example of the random walk, and represents a more or less rational assessment of the evidence available to teenagers. But what it most certainly does not represent, is commitment of any kind. And to the degree that we think that such a commitment is good, it is reasonable to disapprove of such behavior, and this is why there does seem something wrong there, even if in fact the teenager’s religious opinions were not true in the first place.

Fr. Joseph’s original question was this: “Can (religious) faith entail an absolute commitment to the one in whom we place faith and his word, such that one should hold that “no circumstances could arise in which I would cease to believe?” He correctly notes that this “seems to imply an assent to the content of faith so strong that one would desire to hold to it as true, ‘even if it (the content of faith) were to be false'”. For this reason, his post never actually answers the question. For although he right to say that the commitment of faith implies giving preferential treatment to the claim that the content of one’s faith is true, it will not follow that this preferential treatment should be absolute, unless it is true that it is better to believe even if that content is false. And it would be extremely difficult to prove that, even if it were the case.

My own view is that one should be extremely hesitant to accept such an assessment, even of some particular claim, such as the one in the post linked above, that “God will always bring good out of evil.” And if one should be hesitant to make such an assertion about a particular claim, much more should one doubt that this claim is true in regard to the entire contents of a religious faith, which involves making many assertions. Some of the reasons for what I am saying here are much like some of the reasons for preserving the mean of virtue. What exactly will happen if I eat too much? I’m not sure, but I know it’s likely to be something bad. I might feel sick afterwards, but I also might not. Or I might keep eating too much, become very overweight, and have a heart attack at some point. Or I might, in the very process of eating too much, say at a restaurant, spend money that I needed for something else. Vicious behaviors are extreme insofar as they lack the mean of virtue, and insofar as they are extreme, they are likely to have extreme consequences of one kind or another. So we can know in advance that our vicious behaviors are likely to have bad consequences, without necessarily being able to point out the exact consequences in advance.

Something very similar applies to telling lies, and in fact telling lies is a case of vicious behavior, at least in general. It often seems like a lie is harmless, but then it turns out later that the lie caused substantial harm.

And if this is true about telling lies, it is also true about making false statements, even when those false statements are not lies. So we can easily assert that the woman in Eric Reitan’s story is better off believing that God will somehow redeem the evil of the death of her children, simply looking at the particular situation. But if this turned out to be false, we have no way to know what harms might follow from her holding a false belief, and there would be a greater possibility of harm to the degree that she made that conviction more permanent. It would be easy enough to create stories to illustrate this, but I will not do that here. Just as eating too much, or talking too much, or moving about too much, can create any number of harms by multiple circuitous routes, so can believing in things that are false. One particularly manifest way this can happen is insofar as one false belief can lead to another, and although the original belief might seem harmless, the second belief might be very harmful indeed.

In general, Fr. Joseph seems to be asserting that the commitment of faith should lead a person not to pursue additional evidence relative to the truth of their faith, and apparently especially in situations where one already knows that there is a significant chance that the evidence will continue to be against it. This is true to some extent, but the right action in a concrete case will differ according to circumstances, especially, as argued here, if it is not better to believe in the situation where the content of the faith is false. Additionally, it will frequently not be a question of deciding to pursue evidence or not, but of deciding whether to think clearly about evidence or arguments that have entered one’s life even without any decision at all.

Consider the case of St. Therese, discussed in the previous post. Someone might argue thus: “Surely St. Therese’s commitment was absolute. You cannot conceive of circumstances in which she would have abandoned her faith. So if St. Therese was virtuous, it must be virtuous to have such an absolute commitment.” And it would follow that it is better to believe even if your faith is false, and that one should imitate her in having such an absolute commitment. Likewise, it would follow with probability, although not conclusively, that Shulem Deen should also have had such an absolute commitment to his Jewish faith, and should have kept believing in it no matter what happened. Of course, an additional consequence, unwelcome to many, would be that he should also have had an absolute refusal to convert to Christianity that could not be changed under any circumstances.

It is quite certain that St. Therese was virtuous. However, if you cannot conceive of any circumstances in which she would have abandoned her faith, that is more likely to be a lack in your imagination than in the possibility. Theoretically there could have been many circumstances in which it would have been quite possible. It is true that in the concrete circumstances in which she was living, such an abandonment would have been extremely unlikely, and likely not virtuous if it happened. But those are concrete circumstances, not abstractly conceivable circumstances. As noted in the previous post, the evidence that she had against her faith was very vague and general, and it is not clear that it could ever have become anything other than that without a substantially different life situation. And since it is true that the commitment of faith is a good reason to give preferential treatment to the truth of your faith, such vague and general evidence could not have been a good reason for her to abandon her faith. This is the real motivation for the above argument. It is clear enough that in her life as it was actually lived, there was not and could not be a good reason for her to leave her faith. But this is a question of the details of her life.

Shulem Deen, of course, lived in very different circumstances, and his religious faith itself differed greatly from that of St. Therese. Since I have already recommended his book, I will not attempt to tell his story for him, but it can be seen from the above reasoning that the answer to the question raised at the end of the last post might very well be, “They both did the right thing.”

Christian philosopher William Lane Craig writes somewhere about what he calls the “ministerial” and the “magisterial” use of reason. (It’s a traditional view — he’s merely citing Martin Luther — and one that Craig endorses.) On this view, the task of reason is to find arguments in support of the faith and to counter any arguments against it. Reason is not, however, the basis of the Christian’s faith. The basis of the Christian’s faith is (what she takes to be) the “internal testimony of the Holy Spirit” in her heart. Nor can rational reflection can be permitted to undermine that faith. The commitment of faith is irrevocable; to fall away from it is sinful, indeed the greatest of sins.

It follows that while the arguments put forward by many Christian philosophers are serious arguments, there is something less than serious about the spirit in which they are being offered. There is a direction in which those arguments will not be permitted to go. Arguments that support the faith will be seriously entertained; those that apparently undermine the faith must be countered, at any cost. Philosophy, to use the traditional phrase, is merely a “handmaid” of theology.

There is, to my mind, something frivolous about a philosophy of this sort. My feeling is that if we do philosophy, it ought to be because we take arguments seriously. This means following them wherever they lead.

There is more than one way to read this. When he says, “this means following them wherever they lead,” one could take that to imply a purely rational assessment of evidence, and no hesitancy whatsoever to consider any possible line of argument. This would be a substantial disagreement with Fr. Joseph’s position, and would in fact be mistaken. Fr. Joseph is quite right that the commitment of faith has implications for one’s behavior, and that it implies giving a preferential treatment to the claims of one’s faith. But this is probably not the best way to read Dawes, who seems to be objecting more to the absoluteness of the claim: “The commitment of faith is irrevocable,” and arguments “that apparently undermine the faith must be countered, at any cost.” And Dawes is quite right that such absolute claims go too far. Virtue is a mean and depends on circumstances, and there is enough space in the world for both Shulem Deen and St. Therese.

The reader might be wondering about the title to this post. Besides being a play on words, perhaps spoiled by mentioning it, it is a reference to the fact that Fr. Joseph is basically painting a very clear picture of the situation where a Catholic has a crisis of faith and overcomes it. This is only slightly distorted by the idealization of assuming that the person evaluates the evidence available to him in a perfectly rational way. But he points out, just as I did in the previous post, that such a crisis is mainly overcome by choosing not to consider any more evidence, or not to think about it anymore, and similar things. He describes this as choosing “not to pursue evidence” because of the idealization, but in real life this can also mean ceasing to pay attention to evidence that one already has, choosing to pay more attention to other motives that one has to believe that are independent of evidence, and the like.

# Why They Don’t Return

As a framework for continuing the present discussion, we can consider a person’s religious opinions as though they had a numerical probability. Of course, as was said earlier, probability is a formalization of degrees of belief, and as a formalization, it can only be an approximate representation of people’s real behavior. Evidently people are not in fact typically assigning such numbers. Nonetheless, the very “rigidity” of such numerical assignments can help us to understand the present issue.

In some cases, then, a child will effectively take the probability of his religious opinions to be 100%. As said in the linked post, the meaning of this is that, to the degree that 100% is the correct approximation, it is approximately impossible for him to change his mind, or even to become less sure of himself. P. Edmund Waldstein might be understood as claiming to be such a person, although in practice this may be more a matter of a mistaken epistemology which is corrigible, and consequently the approximation fails to this extent.

In the previous post, one of my conditions on the process was “given that he is capable of looking at the world honestly.” This condition basically does not apply to the person assigning the probability of 100%. In effect, he is unable to see any evidence against his position.

But suppose our approximate probability is very high, but not 100%, as for example 99.99%. This is not a balanced assessment of the real probability of any set of religious claims, but is likely a good approximation of the assessment made by a child raised very devoutly in a religion. So if the person correctly assesses the evidence that arrives throughout his life, that probability must diminish, as described in the previous post. There will of course be individual cases where a person does not have the 100% assignment, but cannot or will not correctly assess the evidence that arrives, and will either continually increase his assignment, or leave it unchanged throughout his life. The constant increase is more likely in the case of converts, as in the linked post, but this also implies that one did not start with such a high assignment. The person who permanently leaves it unchanged might be more correctly described as not paying attention or not being interested in the evidence one way or another, rather than as assessing that evidence.

But let us consider persons in whom that probability diminishes, as in the cases of Shulem Deen and of St. Therese discussed in the previous post. Of course, since evidence is not one sided, the probability will not only diminish, but also occasionally increase. But as long as the person has an unbalanced assessment of the evidence, or at least as long as it seems to them unbalanced compared to the evidence that they see, the general tendency will be in one direction. It can be argued that this should never happen with any opinion; thus Robin Hanson says here, “Rational learning of any expected value via a stream of info should produce a random walk in those expectations, not a steady trend.” But the idea here is that if you have a probability assignment of 99% and it is starting to decrease, then you should jump to an assignment of 50% or so, or even lower, guessing where the trend will end. And from that point you might well have to go back up, rather than down. But for obvious reasons people’s religious opinions do not in fact change in this way, at least most of the time, regardless of whether it would be reasonable or not, and consequently there are in fact steady trends.

So where does this end? The process causing the assessment to diminish can come to an end in one way if a person simply comes to the assessment that seems to him a balanced assessment of the evidence. At this point, there may be minor fluctuations in both directions, but the person’s assessment will henceforth stay relatively constant. And this actually happens in the case of some people.

In other persons, the process ends for reasons that have nothing to do with assessing evidence. St. Therese is certainly an example of this, insofar as she died at the age of 24. But this does not necessarily mean that her assessment would have continued to diminish if she had continued to live, for two reasons. First, the isolated character of her life, meant that she would receive less relevant evidence in the first place. So it might well be that by the time of her death she had already learned everything she could on the matter. In that sense she would be an example of the above situation where a person’s assessment simply arrives at some balance, and then stays there.

Second, a person can prevent this process from continuing, more or less simply by choosing to do so, and it is likely enough that St. Therese would have done this. Fr. Joseph Bolin seems to advocate this approach here, although perhaps not without reservation. In practice, this means that one who previously was attending to the relevant evidence, chooses to cease paying attention, or at least to cease evaluating the evidence, much like in our description of people whose assessment never changes in the first place.

Finally, there are persons in whom the process continues apparently without any limit. In this case, there are two possibilities. Either the person comes to the conclusion that their religious opinions were not true, as in my own case and as in the case of Shulem Deen, or the person decides that evidence is irrelevant, as in the case of Kurt Wise. The latter choice is effectively to give up on the love of truth, and to seek other things in the place of truth.

As an aside, the fact that this process seems almost inevitably to end either in abandoning religious claims, or in choosing to cease evaluating evidence, and only very rarely in apparently arriving at a balance, is an argument that religious claims are not actually true, although not a conclusive one. We earlier quoted Newman as saying:

I have no intention at all of denying, that truth is the real object of our reason, and that, if it does not attain to truth, either the premiss or the process is in fault; but I am not speaking here of right reason, but of reason as it acts in fact and concretely in fallen man. I know that even the unaided reason, when correctly exercised, leads to a belief in God, in the immortality of the soul, and in a future retribution; but I am considering the faculty of reason actually and historically; and in this point of view, I do not think I am wrong in saying that its tendency is towards a simple unbelief in matters of religion. No truth, however sacred, can stand against it, in the long run; and hence it is that in the pagan world, when our Lord came, the last traces of the religious knowledge of former times were all but disappearing from those portions of the world in which the intellect had been active and had had a career.

Newman explains this fact by original sin. But a more plausible explanation is that religious claims are simply not true. This is especially the case if one considers this fact in relation to the normal mode of progress in truth in individuals and in societies.

But getting back to the main point, this explains why they “do not return,” as Shulem Deen says. Such a return would not simply require reversing a particular decision or a particular argument. It would require either abandoning the love of truth, like Kurt Wise, or reversing the entire process of considering evidence that went on throughout the whole of one’s life. Suppose we saw off a branch, and then at the last moment break off the last little string of wood. How do we unbreak it? It was just a little piece of wood that broke… but it is not enough to fix that little piece, with glue or whatever. We would have to undo all of the sawing, and that cannot be done.

While there is much in this post and in the last which is interesting in itself, and thus entirely useless, all of this evidently has some bearing on my own case, and I had a personal motive in writing it, namely to explain to various people what expectations they should or should not have.

However, there is another issue that will be raised by all of this in the minds of many people, which is that of moral assessment. Regardless of who found the truth about the world, who did the right thing? Shulem Deen or St. Therese?

# All Who Go Do Not Return

Shulem Deen begins his memoir of the above name:

I wasn’t the first to be expelled from our village, though I’d never known any of the others. I’d only heard talk of them, hushed reminiscences of ancient episodes in the history of our half-century-old village, tales of various subversives who sought to destroy our fragile unity. The group of Belzers who tried to form their own prayer group, the young man rumored to have studied the books of the Breslovers, even the rebbe’s own brother-in-law, accused of fomenting sedition against the rebbe.

But I was the first to be expelled for heresy.

The call came on a Sunday evening, while Gitty and I were having dinner with our children.

“Shulem, this is Yechiel Spitzer,” a deep male voice said, and then paused. “Can you be at the dayan’s office for a meeting at ten?”

I wasn’t entirely surprised by the call. I had heard from friends that word was getting around the village: Shulem Deen has become a heretic.

If heresy was a sin in our all-Hasidic village in Rockland County, New York, it was not an ordinary one. Unlike the yeshiva student who ordered a taxicab each night to get away for an hour of karate lessons, or the girl spotted wearing a skirt that didn’t fully cover her knees, or the schoolteacher who complained of the rebbe’s lengthy Sabbath noon prayers, heresy was a sin our people were unaccustomed to. Heresy was a sin that baffled them. In fact, real heresy, the people in our village believed, did not happen in our time, and certainly not in our village, and so when they heard there was a heretic in their midst, they were not sure what to make of it.

The meeting itself is not the most pleasant:

“We have heard rumors,” Mendel began. “We have heard rumors and we don’t know if they’re true, but you understand, rumors alone are bad.”

He paused and looked at me, as if expecting me to show agreement of some sort.

“People say you’re an apikorus. People say you don’t believe in God.” He raised his shoulders to his ears, spread his palms, and opened his eyes wide. “How does one not believe in God? I don’t know.” He said this as if he were genuinely curious. Mendel was an intelligent man, and here was a question that, given the time and inclination, one might seek to discuss. But now was not the time, and so he went on to tell me more about what people were saying.

I was speaking ill of the rebbe.

I was no longer praying.

I disparaged the Torah and the teachings of our sages.

I was corrupting other people. Young people. Innocent people.

In fact, people were saying that I had corrupted a yeshiva boy just last week. Corrupted him so badly that the boy left his parents’ home, and— Mendel didn’t know if this was true, but so people were saying— went to live with goyim in Brooklyn. It was rumored that the boy planned to attend college.

People were saying, Mendel further informed me, that something must be done. People were very concerned, and people were saying that the bezdin must act.

“If people are saying that the bezdin must act, you understand, we can’t very well do nothing.”

Yechiel Spitzer, sitting at the very end of the table, twirled a few hairs beneath his lower lip and absentmindedly placed one hair between his front teeth. The three rabbis sat with their eyes downcast.

“You understand,” Mendel went on, “that this is not about causing pain to you or your family.”

Here he paused and looked at the dayan, before putting his palms flat on the table and looking at me directly.

“We have come to the conclusion that you must leave the village.”

I was being expelled, though in those moments, I wasn’t sure how to feel about it. My initial thought was to defend myself, to declare it all lies, hateful gossipmongering. But the truth was, I no longer belonged here. This was a community of the faithful, and I was no longer one of them.

How did things come to this point? To those who are curious, I recommend the book. However, Shulem remarks on this matter:

“What made you change?” people would ask in later years, and the question would frustrate me because the things that made me change were so many and varied that they felt simply as life feels: not a single moment of transformation but a process, a journey of inquiry and discovery, of beliefs and challenges to those beliefs, of uncomfortable questions and attempts to do away with them, by brute force if necessary, only to find that that was not possible, that the search was too urgent and necessary and giving up was not an option. Yet I found no neat answers but only muddled and contradictory ones, until hope gave way to disillusion, which would in turn give way to hope once again, but dimmer and weaker each time, until I would swing back to confusion and disillusion in an endlessly maddening cycle.

Overall, the process is so gradual that he does not even remember a definite moment of changing his mind, but only remembers the realization that he has already done so:

I remember that I was in the dining room, and through the thin walls I could hear Gitty busy in the kitchen: “Akiva, finish your toast,” “Freidy, stop bothering the baby and get dressed,” “Tziri, brush your hair and get your backpack.” The sounds all blended together. One by one, each of the children recited the morning blessings, groaned about unfinished homework assignments, lost shoes, misplaced hair accessories. I swung my prayer shawl over my shoulders, whipped up my sleeve, and wrapped the leather straps of my tefillin around my arm. And as I stood there, the black leather cube on my left arm bulging against the sleeve of my starched white shirt, my body enveloped in the large, white, black-striped shawl, the thought came to me:

I no longer believe in any of this.

I am a heretic. An apikorus.

For a long time, I had tried to deny it. A mere sinner has hope: An Israelite, although he has sinned, is still an Israelite, the Talmud says. But a heretic is lost forever. All who go do not return. The Torah scroll he writes is to be burned. He is no longer counted in a prayer quorum, his food is not considered kosher, his lost objects are not returned to him, he is unfit to testify in court. An outcast, he wanders alone forever, belonging neither to his own people nor to any other.

It was at that moment, sometime between fastening the knot of my tefillin against my occipital bone and racing through whatever chapters of prayer I still chose to recite, that I realized that my heresy was simply a fact about myself, no different from my brown eyes or my pale skin.

Earlier we discussed the possibility of changing your mind without realizing that you have done so, and there is no reason that this cannot apply even to things as important as religion. This seems to have happened in Shulem’s case, although of course he was aware of the process in a general way, as can be seen in the previous quotation.

If a person is raised in a religion from childhood, and taught to adhere very strongly to that religion, then given that he is capable of looking at the world honestly, he will almost inevitably go through a process much like the one described by Shulem Deen, even if it may have a different ending.

This will happen almost always, without regard to how much truth there is in the religion, or how much truth is lacking there. For at least in the devout cases of which we speak, the parents will teach the child that their religion is very certainly true, and it is unlikely that they will go out of their way to present arguments and evidence against it. And if they do present such arguments and evidence, they are likely to present them in the least favorable way, rather than in the most favorable way. None of this is very surprising, nor does it have much to do with religion in particular, but is simply the way that people generally present their opinions to others. But it follows from all this that the child is not given a balanced view of the evidence relative to his religion, but one which makes the evidence seem more favorable than it actually is. As was said, this would be likely to happen even if the religion in question were entirely true.

The consequence is that once the person begins to get a more balanced grasp on the evidence, which will be the natural result of living in the world, they will  begin to see that their religion was less certain than they supposed it to be.

This will happen even in the case of very devout people who are entirely enveloped, as it were, in the belief and life of their own religious community, and who seem to have virtually no contact with unbelievers or their reasons. Thus for example St. Therese, in her autobiography, speaks of the conflict between her belief in heaven and her doubts about it:

I was saying that the certainty of going away one day far from the sad and dark country had been given me from the day of my childhood. I did not believe this only because I heard it from persons much more knowledgeable than I, but I felt in the bottom of my heart real longings for this most beautiful country. Just as the genius of Christopher Columbus gave him a presentiment of a new world when nobody had even thought of such a thing; so also I felt that another land would one day serve me as a permanent dwelling place. Then suddenly the fog that surrounds me becomes more dense; it penetrates my soul and envelops it in such a way that it is impossible to discover within it the sweet image of my Fatherland; everything has disappeared! When I want to rest my heart fatigued by the darkness that surrounds it by the memory of the luminous country after which I aspire, my torment redoubles; it seems to me that the darkness, borrowing the voice of sinners, says mockingly to me: “You are dreaming about the light, about a fatherland embalmed in the sweetest perfumes; you are dreaming about the eternal possession of the Creator of all these marvels; you believe that one day you will walk out of this fog that surrounds you! Advance, advance; rejoice in death which will give you not what you hope for but a night still more profound, the night of nothingness.”

St. Therese mentions two things favorable to the existence of heaven: her own desire to go there, and the fact that “I heard it from persons much more knowledgeable than I.” No particular reason is given supporting the denial of heaven, except the “voice of sinners,” which corresponds in a certain way to one of her positive reasons, since St. Therese is aware that many of the sinners in question are also much more knowledgeable than she is. While not explained clearly in her autobiography, she stated verbally a few months before her death:

“If you only knew what frightful thoughts obsess me! Pray very much for me in order that I do not listen to the devil who wants to persuade me about so many lies. It’s the reasoning of the worst materialists which is imposed upon my mind: Later, unceasingly making new advances, science will explain everything naturally; we shall have the absolute reason for everything that exists and that still remains a problem, because there remain very many things to be discovered, etc., etc.”

Why is it such a bad thing to explain everything? Her fear, of course, is that the explanation will imply that there is no life after death.

Of course the arguments implied here, on one side and the other, are not very complicated or technical, because St. Therese is not a scholar. And although she may be able to see some reasons supporting their position to some extent, as in this case the progress of science, to a large extent her doubts are simply caused by opposing authorities: people more intelligent than she who do not believe what she believes. In this sense, St. Therese is a clear example of the point under discussion, where someone raised to hold to their religion with great certainty, comes to be less certain of it when they realize that the evidence is not all on one side.

Nonetheless, St. Therese is evidently not attempting to weigh the evidence on one side and the other, in order to decide which side is more likely to be true. She writes, again in her autobiography:

My dear Mother, I may perhaps appear to you to be exaggerating my trial. In fact, if you are judging according to the sentiments I express in my little poems composed this year, I must appear to you as a soul filled with consolations and one for whom the veil of faith is almost torn aside; and yet it is no longer a veil for me, it is a wall which reaches right up to the heavens and covers the starry firmament. When I sing of the happiness of heaven and of the eternal possession of God, I feel no joy in this, for I sing simply what I WANT TO BELIEVE. It is true that at times a very small ray of the sun comes to illumine my darkness, and then the trial ceases for an instant, but afterward the memory of this ray, instead of causing me joy, makes my darkness even more dense.

This should be understood in the sense of voluntary belief. While St. Therese feels the weight of opposing reasons, she chooses to accept one side regardless. In that sense, she does not need to find an exact measure of the weight of the reasons for one side or the other, because her mind is already made up: regardless of how things stand exactly, she will still choose to believe.

One difference between St. Therese and Shulem Deen, then, is that while St. Therese had an experience somewhat similar to his, she chooses to prevent this process from leading to unbelief. In his case, in contrast, there may not have been any particular distinct moment of choice one way or another, according to his description.

If we return to our child, however, there are a number of other possible results. We will discuss these in a future post.

# Settled Issues

In chapter 5 of his book Probability Theory: The Logic of Science, E. T. Jaynes discusses ESP:

I. J. Good (1950) has shown how we can use probability theory backwards to measure our own strengths of belief about propositions. For example, how strongly do you believe in extrasensory perception?

What probability would you assign to the hypothesis that Mr Smith has perfect extrasensory perception? More specifically, that he can guess right every time which number you have written down. To say zero is too dogmatic. According to our theory, this means that we are never going to allow the robot’s mind to be changed by any amount of evidence, and we don’t really want that. But where is our strength of belief in a proposition like this?

Our brains work pretty much the way this robot works, but we have an intuitive feeling for plausibility only when it’s not too far from 0 db. We get fairly definite feelings that something is more than likely to be so or less than likely to be so. So the trick is to imagine an experiment. How much evidence would it take to bring your state of belief up to the place where you felt very perplexed and unsure about it? Not to the place where you believed it – that would overshoot the mark, and again we’d lose our resolving power. How much evidence would it take to bring you just up to the point where you were beginning to consider the possibility seriously?

So, we consider Mr Smith, who says he has extrasensory perception (ESP), and we will write down some numbers from one to ten on a piece of paper and ask him to guess which numbers we’ve written down. We’ll take the usual precautions to make sure against other ways of finding out. If he guesses the first number correctly, of course we will all say ‘you’re a very lucky person, but I don’t believe you have ESP’. And if he guesses two numbers correctly, we’ll still say ‘you’re a very lucky person, but I still don’t believe you have ESP’. By the time he’s guessed four numbers correctly – well, I still wouldn’t believe it. So my state of belief is certainly lower than −40 db.

How many numbers would he have to guess correctly before you would really seriously consider the hypothesis that he has extrasensory perception? In my own case, I think somewhere around ten. My personal state of belief is, therefore, about −100 db. You could talk me into a ±10 db change, and perhaps as much as ±30 db, but not much more than that.

The idea is that after Mr. Smith guesses 7 to 13 numbers correctly (when by chance he should have a probability of 10% of guessing each one correctly), Jaynes will begin to think it reasonably likely that he has ESP. He notes that this is his subjective opinion, saying, “In my own case,” and “My personal state of belief.”

However, Jaynes follows this up by stating that if this happened in real life, he would not be convinced:

After further thought, we see that, although this result is correct, it is far from the whole story. In fact, if he guessed 1000 numbers correctly, I still would not believe that he has ESP, for an extension of the same reason that we noted in Chapter 4 when we first encountered the phenomenon of resurrection of dead hypotheses. An hypothesis A that starts out down at −100 db can hardly ever come to be believed, whatever the data, because there are almost sure to be alternative hypotheses (B1, B2,…) above it, perhaps down at −60 db. Then, when we obtain astonishing data that might have resurrected A, the alternatives will be resurrected instead.

In other words, Jaynes is saying, “This happened by chance,” and “Mr. Smith has ESP” are not the only possibilities. For example, it is possible that Mr. Smith has invented a remote MRI device, which he has trained to distinguish people’s thoughts about numbers, and he is receiving data on the numbers picked by means of an earbud. If the prior probability of this is higher than the prior probability that Mr. Smith has ESP, then Jaynes will begin to think this is a reasonable hypothesis, rather than coming to accept ESP.

This does not imply that Jaynes is infinitely confident that Mr. Smith does not have ESP, and in fact it does not invalidate his original estimate:

Now let us return to that original device of I. J. Good, which started this train of thought. After all this analysis, why do we still hold that naive first answer of −100 db for my prior probability for ESP, as recorded above, to be correct? Because Jack Good’s imaginary device can be applied to whatever state of knowledge we choose to imagine; it need not be the real one. If I knew that true ESP and pure chance were the only possibilities, then the device would apply and my assignment of −100 db would hold. But, knowing that there are other possibilities in the real world does not change my state of belief about ESP; so the figure of −100 db still holds.

He would begin to be convinced after about 10 numbers if he knew for a fact that chance and ESP were the only possibilities, and thus this is a good representation of how certain he is subjectively.

The fact of other possibilities also does not mean that it is impossible for Jaynes to be convinced, even in the real world, that some individual has ESP. But it does mean that this can happen only with great difficulty: essentially, he must be convinced that the other possibilities are even less likely than ESP. As Jaynes says,

Indeed, the very evidence which the ESP’ers throw at us to convince us, has the opposite effect on our state of belief; issuing reports of sensational data defeats its own purpose. For if the prior probability for deception is greater than that of ESP, then the more improbable the alleged data are on the null hypothesis of no deception and no ESP, the more strongly we are led to believe, not in ESP, but in deception. For this reason, the advocates of ESP (or any other marvel) will never succeed in persuading scientists that their phenomenon is real, until they learn how to eliminate the possibility of deception in the mind of the reader. As (5.15) shows, the reader’s total prior probability for deception by all mechanisms must be pushed down below that of ESP.

This is related to the grain of truth in Hume’s account of miracles. Hume’s basic point, that an account of a miracle could never be credible, is mistaken. But he is correct to say that the account would not be credible unless “these witnesses are mistaken or lying” has a lower prior probability than the prior probability of the miracle actually happening. His mistake is to suppose that this cannot happen in principle.

Something like this also happens with ordinary things that we are extremely sure about. For example, take your belief that the American War of Independence happened before the Civil War. You can imagine coming upon evidence that the Civil War happened first. Thus for example suppose you found a book by a historian arguing for this thesis. This would be evidence that the Civil War came first. But it would be very unpersuasive, and would change your mind little if at all, because the prior probability of “this is a work of fiction,” or indeed of “this a silly book arguing a silly thesis for personal reasons” is higher.

We could call this a “settled issue,” at least from your point of view (and in this case from the point of view of pretty much everyone). Not only do you believe that the War of Independence came first; it would be very difficult to persuade you otherwise, even if there were real evidence against your position, and this is not because you are being unreasonable. In fact, it would be unreasonable to be moved significantly by the evidence of that book arguing the priority of the Civil War.

Is it possible in principle to persuade you to change your mind? Yes. In principle this could happen bit by bit, by an accumulation of small pieces of evidence. You might read that book, and then learn that the author is a famous historian, and that he is completely serious (presumably he became famous before writing the book; otherwise he would instead be infamous.) And then you might find other items in favor of this theory, and find refutations of the apparently more likely explanations.

But in practice such a process is extremely unlikely. The most likely way you could change your mind about this would be by way of one large change. For example, you might wake up in a hospital tomorrow and be told that you had been suffering from a rare form of amnesia which does not remove a person’s past memories, but changes them into something different. You ask about the Civil War, and are told that everyone agrees that it happened before the War of Independence. People can easily give you dozens of books on the topic; you search online on the matter, and everything on the internet takes for granted that the Civil War came first. Likewise, everyone you talk to simply takes this for granted.

The reason that the “one big change” process is more likely than the “accumulation of small evidences” process is this: if we want to know what should persuade you that the Civil War came first, we are basically asking what the world would have to be like in order for it to be actually true that the Civil War came first. In such a world, your current belief is false. And in such a world it is simply much more likely that you have made one big mistake which resulted in your false belief about the Civil War, than that you have made lots of little mistakes which led to it.

# Everything Proves It

G. K. Chesterton, in his book Orthodoxy, discusses the meaning of being “entirely convinced” of something:

It is very hard for a man to defend anything of which he is entirely convinced. It is comparatively easy when he is only partially convinced. He is partially convinced because he has found this or that proof of the thing, and he can expound it. But a man is not really convinced of a philosophic theory when he finds that something proves it. He is only really convinced when he finds that everything proves it. And the more converging reasons he finds pointing to this conviction, the more bewildered he is if asked suddenly to sum them up. Thus, if one asked an ordinary intelligent man, on the spur of the moment, “Why do you prefer civilisation to savagery?” he would look wildly round at object after object, and would only be able to answer vaguely, “Why, there is that bookcase … and the coals in the coal-scuttle … and pianos … and policemen.” The whole case for civilisation is that the case for it is complex. It has done so many things. But that very multiplicity of proof which ought to make reply overwhelming makes reply impossible.

We could think about this in terms of probability. The person who is “entirely convinced” would be like the person who assigns a probability of 100%, while someone who is “partially convinced” might assign a somewhat lower probability.

As Chesterton says, the person who assigns the lower probability has no difficulty defending his position. He can point to various things which he has found, arguments and evidence, that support his position.

But what about the person who assigns the probability of 100%? According to Chesterton, he is in difficulty because he finds that everything supports his position. And indeed, this is reasonable. For if some things support your position and some things do not, how could you suppose that there is no chance that you are mistaken? On the other hand, if you think that literally everything supports your position, you might well suppose that you cannot be mistaken about it.

Of course, as we have said many times on this blog, it is unreasonable in fact to claim such certainty, and it is unreasonable in fact to claim that everything supports your position. So being “entirely convinced” in Chesterton’s sense here is a bad thing, not a good thing.

Chesterton goes on to apply this to his belief in Catholicism:

There is, therefore, about all complete conviction a kind of huge helplessness. The belief is so big that it takes a long time to get it into action. And this hesitation chiefly arises, oddly enough, from an indifference about where one should begin. All roads lead to Rome; which is one reason why many people never get there. In the case of this defence of the Christian conviction I confess that I would as soon begin the argument with one thing as another; I would begin it with a turnip or a taximeter cab.

This is not a good thing, for the above reasons. We could describe this situation in another way. We saw in the previous post that consistent testimony where there should be inconsistent testimony leads to a weakening of the evidence. If a dozen eyewitnesses agree in every respect, this is not good evidence for their claim, but good evidence that they are collaborating. In a similar way, if it seems to you that “everything proves it,” this is very good evidence that you are incapable of distinguishing between things that support your position and things that do not.

This also provides a fuller explanation for the fact that the person who is entirely convinced in Chesterton’s sense finds it difficult to argue for his position. Chesterton’s point that when there are too many possibilities, it is difficult to choose one of them, has some validity. But more fundamentally, the person who is entirely convinced in this way is not even engaging in reasonable argument in the first place; while the person who is partially convinced at least has the possibility of engaging in this kind of argument.

# Too Much Evidence

Lachlan J. Gunn et alia argue in a paper, “Too good to be true: when overwhelming evidence fails to convince,” that in some situations, as you apparently gather evidence in favor of a theory, it becomes less and less likely to be true:

In this paper, for the first time, we perform a Bayesian mathematical analysis to explore the question of multiple confirmatory measurements or observations for showing when they can—surprisingly—disimprove confidence in the final outcome. We choose the striking example that increasing confirmatory identifications in a police line-up or identity parade can, under certain conditions, reduce our confidence that a perpetrator has been correctly identified.

Imagine that as a court case drags on, witness after witness is called. Let us suppose thirteen witnesses have testified to having seen the defendant commit the crime. Witnesses may be notoriously unreliable, but the sheer magnitude of the testimony is apparently overwhelming. Anyone can make a misidentification but intuition tells us that, with each additional witness in agreement, the chance of them all being incorrect will approach zero. Thus one might naïvely believe that the weight of as many as thirteen unanimous confirmations leaves us beyond reasonable doubt.

However, this is not necessarily the case and more confirmations can surprisingly disimprove our confidence that the defendant has been correctly identified as the perpetrator. This type of possibility was recognised intuitively in ancient times. Under ancient Jewish law, one could not be unanimously convicted of a capital crime—it was held that the absence of even one dissenting opinion among the judges indicated that there must remain some form of undiscovered exculpatory evidence.

This is an interesting situation, and someone might suppose that it is one where the evidence changes sides.

But this does not follow. The reality is a bit different. Suppose you flip a coin thirty times, and get heads every time. Each time you get heads, you receive evidence in favor of the hypothesis, “I am having a really lucky streak.” But each time you also receive evidence, and stronger evidence, in favor of the hypothesis, “This coin is biased.” After flipping it thirty times, you thus are likely to become very convinced of the latter hypothesis, and thus convinced that the former is mistaken. But this did not happen because at some point the evidence went from one side to the other, but because the evidence supported two different theories, and one more than the other.

However, regardless of exactly how we describe the situation here, this does have important consequences for the evaluation of multiple instances of testing the same thing. In the case of the police line-up discussed in the article, we know from experience that people are far from infallible in their identification. So if you have a large number of people who identify the same person, without any exception, that is strong evidence that the process is biased; perhaps the police encouraged the people to identify a particular person, for example. Likewise, eyewitness testimony tends not to be perfectly accurate. Consequently, if we take the testimony of many eyewitnesses to the same complex events, and there is not the slightest discrepancy, this is good evidence that their testimony is biased. Perhaps someone created a story and instructed them not to deviate from it, for example. And on the other hand, minor discrepancies in such accounts do not weaken their testimony, but strengthen it (although not to the discrepant point itself), by making it less likely that they are biased in such a way.

# Hume’s Error on Miracles

Though experience be our only guide in reasoning concerning matters of fact; it must be acknowledged, that this guide is not altogether infallible, but in some cases is apt to lead us into errors. One, who in our climate, should expect better weather in any week of June than in one of December, would reason justly, and conformably to experience; but it is certain, that he may happen, in the event, to find himself mistaken. However, we may observe, that, in such a case, he would have no cause to complain of experience; because it commonly informs us beforehand of the uncertainty, by that contrariety of events, which we may learn from a diligent observation. All effects follow not with like certainty from their supposed causes. Some events are found, in all countries and all ages, to have been constantly conjoined together: Others are found to have been more variable, and sometimes to disappoint our expectations; so that, in our reasonings concerning matter of fact, there are all imaginable degrees of assurance, from the highest certainty to the lowest species of moral evidence.
A wise man, therefore, proportions his belief to the evidence. In such conclusions as are founded on an infallible experience, he expects the event with the last degree of assurance, and regards his past experience as a full proof of the future existence of that event. In other cases, he proceeds with more caution: he weighs the opposite experiments: he considers which side is supported by the greater number of experiments: to that side he inclines, with doubt and hesitation; and when at last he fixes his judgement, the evidence exceeds not what we properly call probability. All probability, then, supposes an opposition of experiments and observations, where the one side is found to overbalance the other, and to produce a degree of evidence, proportioned to the superiority. A hundred instances or experiments on one side, and fifty on another, afford a doubtful expectation of any event; though a hundred uniform experiments, with only one that is contradictory, reasonably beget a pretty strong degree of assurance. In all cases, we must balance the opposite experiments, where they are opposite, and deduct the smaller number from the greater, in order to know the exact force of the superior evidence.

While Hume is right to say that convictions should in some way be proportionate to the evidence for them, we can already see here the cause of a serious error. As I have pointed out elsewhere, Hume does not have a developed mathematical theory of probability. Hence his talk of how one should “deduct the smaller number from the greater, in order to know the exact force of the superior evidence.” If one takes this literally, this would suggest that something with no experiments supporting it has a force of 0; something with 35 experiments supporting it and nothing against has a force of 35; something with 40 experiments supporting it and 5 against has the same force; and so on. All of this, of course, is evidently absurd.

He then brings up the example of the testimony of witnesses:

To apply these principles to a particular instance; we may observe, that there is no species of reasoning more common, more useful, and even necessary to human life, than that which is derived from the testimony of men, and the reports of eye-witnesses and spectators. This species of reasoning, perhaps, one may deny to be founded on the relation of cause and effect. I shall not dispute about a word. It will be sufficient to observe that our assurance in any argument of this kind is derived from no other principle than our observation of the veracity of human testimony, and of the usual conformity of facts to the reports of witnesses. It being a general maxim, that no objects have any discoverable connexion together, and that all the inferences, which we can draw from one to another, are founded merely on our experience of their constant and regular conjunction; it is evident, that we ought not to make an exception to this maxim in favour of human testimony, whose connexion with any event seems, in itself, as little necessary as any other. Were not the memory tenacious to a certain degree; had not men commonly an inclination to truth and a principle of probity; were they not sensible to shame, when detected in a falsehood: were not these, I say, discovered by experience to be qualities, inherent in human nature, we should never repose the least confidence in human testimony. A man delirious, or noted for falsehood and villainy, has no manner of authority with us.
And as the evidence, derived from witnesses and human testimony, is founded on past experience, so it varies with the experience, and is regarded either as a proof or a probability, according as the conjunction between any particular kind of report and any kind of object has been found to be constant or variable. There are a number of circumstances to be taken into consideration in all judgements of this kind; and the ultimate standard, by which we determine all disputes, that may arise concerning them, is always derived from experience and observation. Where this experience is not entirely uniform on any side, it is attended with an unavoidable contrariety in our judgements, and with the same opposition and mutual destruction of argument as in every other kind of evidence. We frequently hesitate concerning the reports of others. We balance the opposite circumstances, which cause any doubt or uncertainty; and when we discover a superiority on any side, we incline to it; but still with a diminution of assurance, in proportion to the force of its antagonist.
This contrariety of evidence, in the present case, may be derived from several different causes; from the opposition of contrary testimony; from the character or number of the witnesses; from the manner of their delivering their testimony; or from the union of all these circumstances. We entertain a suspicion concerning any matter of fact, when the witnesses contradict each other; when they are but few, or of a doubtful character; when they have an interest in what they affirm; when they deliver their testimony with hesitation, or on the contrary, with too violent asseverations. There are many other particulars of the same kind, which may diminish or destroy the force of any argument, derived from human testimony.
Suppose, for instance, that the fact, which the testimony endeavours to establish, partakes of the extraordinary and the marvellous; in that case, the evidence, resulting from the testimony, admits of a diminution, greater or less, in proportion as the fact is more or less unusual. The reason why we place any credit in witnesses and historians, is not derived from any connexion, which we perceive à priori, between testimony and reality, but because we are accustomed to find a conformity between them. But when the fact attested is such a one as has seldom fallen under our observation, here is a contest of two opposite experiences; of which the one destroys the other, as far as its force goes, and the superior can only operate on the mind by the force, which remains. The very same principle of experience, which gives us a certain degree of assurance in the testimony of witnesses, gives us also, in this case, another degree of assurance against the fact, which they endeavour to establish; from which contradiction there necessarily arises a counterpoize, and mutual destruction of belief and authority.
I should not believe such a story were it told me by Cato, was a proverbial saying in Rome, even during the lifetime of that philosophical patriot. The incredibility of a fact, it was allowed, might invalidate so great an authority.
The Indian prince, who refused to believe the first relations concerning the effects of frost, reasoned justly; and it naturally required very strong testimony to engage his assent to facts, that arose from a state of nature, with which he was unacquainted, and which bore so little analogy to those events, of which he had had constant and uniform experience. Though they were not contrary to his experience, they were not conformable to it.

While we might disagree that someone would be reasonable in refusing to accept testimony concerning the effects of frost, Hume’s general points here are fairly reasonable.

But when he attempts to apply to this miracles, he basically attempts to reason from the invalid mathematical points in the previous text:

But in order to increase the probability against the testimony of witnesses, let us suppose, that the fact, which they affirm, instead of being only marvellous, is really miraculous; and suppose also, that the testimony considered apart and in itself, amounts to an entire proof; in that case, there is proof against proof, of which the strongest must prevail, but still with a diminution of its force, in proportion to that of its antagonist.
A miracle is a violation of the laws of nature; and as a firm and unalterable experience has established these laws, the proof against a miracle, from the very nature of the fact, is as entire as any argument from experience can possibly be imagined. Why is it more than probable, that all men must die; that lead cannot, of itself, remain suspended in the air; that fire consumes wood, and is extinguished by water; unless it be, that these events are found agreeable to the laws of nature, and there is required a violation of these laws, or in other words, a miracle to prevent them? Nothing is esteemed a miracle, if it ever happen in the common course of nature. It is no miracle that a man, seemingly in good health, should die on a sudden: because such a kind of death, though more unusual than any other, has yet been frequently observed to happen. But it is a miracle, that a dead man should come to life; because that has never been observed in any age or country. There must, therefore, be a uniform experience against every miraculous event, otherwise the event would not merit that appellation. And as a uniform experience amounts to a proof, there is here a direct and full proof, from the nature of the fact, against the existence of any miracle; nor can such a proof be destroyed, or the miracle rendered credible, but by an opposite proof, which is superior.
The plain consequence is (and it is a general maxim worthy of our attention), ‘that no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavors to establish; and even in that case there is a mutual destruction of arguments, and the superior only gives us an assurance suitable to that degree of force, which remains, after deducting the inferior.’ When anyone tells me, that he saw a dead man restored to life, I immediately consider with myself, whether it be more probable, that this person should either deceive or be deceived, or that the fact, which he relates, should really have happened. I weigh the one miracle against the other; and according to the superiority, which I discover, I pronounce my decision, and always reject the greater miracle. If the falsehood of his testimony would be more miraculous, than the event which he relates; then, and not till then, can he pretend to command my belief or opinion.

There are various ways to read this, but each way leads to problems. Hume has told himself that he believes that he has found a conclusive proof that accounts of miracles should never be accepted; and this implies that he must be saying that his condition, that the testimony should “be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavors to establish,” can never be satisfied.

But there is no reasonable understanding where this condition can never be satisfied. Hume seems to be equating “more miraculous” with “less probable,” but there is no degree of probability that could not be established by witnesses in principle. Even if each witness has only a small chance of telling the truth, multiple independent witnesses could in principle establish any degree of probability whatsoever.

The basic problem here seems to be Hume’s mathematically incorrect understanding of probability. If something has never been seen to happen, he says, this is a full proof that it cannot happen. Thus he seems to imply that there is a 0% chance of it happening. But this is evidently unreasonable. In reality, of course, we often see particular things happen which never happened before. And similarly, it is simply not true that a miracle is only called a miracle because it “has never been observed in any age or country.” There have been many reports, in many ages and many countries, of dead people coming to life again. So the only way Hume could say that a dead person coming to life is a miracle in this sense, is by assuming that all of these reports are false. This is simply to assume what he is trying to prove, and in any case we think that resurrection is a miracle whether or not these reports are true. In other words, to say that resurrection is a miracle is not to say that these reports are false, but that if they are true, they are reports of miracles.

Taken in another way, Hume seems to be saying, “A miracle requires a suspension of natural laws. But false testimony does not. Therefore if we have the choice of believing that there was a miracle or of believing that there was false testimony, we should always choose to believe that there was false testimony.” The problem is that, again, if you evaluate this in terms of probabilities, the suspension of natural laws might well be more probable than a particular possibility which does not suspend natural laws. If someone predicts the result of a coin flip 100,000 times in a row, it does not violate natural laws to think that this happened by chance, with a fair coin. But it is much more probable that natural laws were violated, than that this happened by chance with a fair coin.

While it would only relate to Hume personally, we might also note that according to Hume, induction cannot even establish a probability, let alone a necessity.  So according to his position, the experience of dead people remaining dead does not make it improbable that one would rise, let alone excluding it as impossible.

Hume himself seems to sense that there is something wrong with his position, even if he cannot quite work out what it is, again probably on account of the lack of a mathematical theory of probability. Consequently he adds a number of arguments:

For first, there is not to be found, in all history, any miracle attested by a sufficient number of men, of such unquestioned good sense, education, and learning, as to secure us against all delusion in themselves; of such undoubted integrity, as to place them beyond all suspicion of any design to deceive others; of such credit and reputation in the eyes of mankind, as to have a great deal to lose in case of their being detected in any falsehood; and at the same time, attesting facts performed in such a public manner and in so celebrated a part of the world, as to render the detection unavoidable: all which circumstances are requisite to give us a full assurance in the testimony of men.
Secondly. We may observe in human nature a principle which, if strictly examined, will be found to diminish extremely the assurance, which we might, from human testimony, have in any kind of prodigy. The maxim, by which we commonly conduct ourselves in our reasonings, is, that the objects, of which we have no experience, resembles those, of which we have; that what we have found to be most usual is always most probable; and that where there is an opposition of arguments, we ought to give the preference to such as are founded on the greatest number of past observations. But though, in proceeding by this rule, we readily reject any fact which is unusual and incredible in an ordinary degree; yet in advancing farther, the mind observes not always the same rule; but when anything is affirmed utterly absurd and miraculous, it rather the more readily admits of such a fact, upon account of that very circumstance, which ought to destroy all its authority. The passion of surprise and wonder, arising from miracles, being an agreeable emotion, gives a sensible tendency towards the belief of those events, from which it is derived. And this goes so far, that even those who cannot enjoy this pleasure immediately, nor can believe those miraculous events, of which they are informed, yet love to partake of the satisfaction at second-hand or by rebound, and place a pride and delight in exciting the admiration of others.
With what greediness are the miraculous accounts of travellers received, their descriptions of sea and land monsters, their relations of wonderful adventures, strange men, and uncouth manners? But if the spirit of religion join itself to the love of wonder, there is an end of common sense; and human testimony, in these circumstances, loses all pretensions to authority. A religionist may be an enthusiast, and imagine he sees what has no reality: he may know his narrative to be false, and yet persevere in it, with the best intentions in the world, for the sake of promoting so holy a cause: or even where this delusion has not place, vanity, excited by so strong a temptation, operates on him more powerfully than on the rest of mankind in any other circumstances; and self-interest with equal force. His auditors may not have, and commonly have not, sufficient judgement to canvass his evidence: what judgement they have, they renounce by principle, in these sublime and mysterious subjects: or if they were ever so willing to employ it, passion and a heated imagination disturb the regularity of its operations, their credulity increases his impudence: and his impudence overpowers their credulity.
Eloquence, when at its highest pitch, leaves little room for reason or reflection; but addressing itself entirely to the fancy or the affections, captivates the willing hearers, and subdues their understanding. Happily, this pitch is seldom attains. But what a Tully or a Demosthenes could scarcely effect over a Roman or Athenian audience, every Capuchin, every itinerant or stationary teacher can perform over the generality of mankind, and in a higher degree, by touching such gross and vulgar passions.
The many instances of forged miracles, and prophecies, and supernatural events, which, in all ages, have either been detected by contrary evidence, or which detect themselves by their absurdity, prove sufficiently the strong propensity of mankind to the extraordinary and the marvellous, and ought reasonably to beget a suspicion against all relations of this kind. This is our natural way of thinking, even with regard to the most common and most credible events. For instance: There is no kind of report which rises so easily, and spreads so quickly, especially in country places and provincial towns, as those concerning marriages; insomuch that two young persons of equal condition never see each other twice, but the whole neighbourhood immediately join them together. The pleasure of telling a piece of news so interesting, of propagating it, and of being the first reporters of it, spreads the intelligence. And this is so well known, that no man of sense gives attention to these reports, till he find them confirmed by some greater evidence. Do not the same passions, and others still stronger, incline the generality of mankind to believe and report, with the greatest vehemence and assurance, all religious miracles?
Thirdly. It forms a strong presumption against all supernatural and miraculous relations, that they are observed chiefly to abound among ignorant and barbarous nations; or if a civilized people has ever given admission to any of them, that people will be found to have received them from ignorant and barbarous ancestors, who transmitted them with that inviolable sanction and authority, which always attend received opinions. When we peruse the first histories of all nations, we are apt to imagine ourselves transported into some new world; where the whole frame of nature is disjointed, and every element performs its operations in a different manner, from what it does at present. Battles, revolutions, pestilence, famine and death, are never the effect of those natural causes, which we experience. Prodigies, omens, oracles, judgements, quite obscure the few natural events, that are intermingled with them. But as the former grow thinner every page, in proportion as we advance nearer the enlightened ages, we soon learn, that there is nothing mysterious or supernatural in the case, but that all proceeds from the usual propensity of mankind towards the marvellous, and that, though this inclination may at intervals receive a check from sense and learning, it can never be thoroughly extirpated from human nature.
It is strange, a judicious reader is apt to say, upon the perusal of these wonderful historians, that such prodigious events never happen in our days. But it is nothing strange, I hope, that men should lie in all ages. You must surely have seen instances enough of that frailty. You have yourself heard many such marvellous relations started, which, being treated with scorn by all the wise and judicious, have at last been abandoned even by the vulgar. Be assured, that those renowned lies, which have spread and flourished to such a monstrous height, arose from like beginnings; but being sown in a more proper soil, shot up at last into prodigies almost equal to those which they relate.

Hume is making some reasonable points here. But note that all of these things are contingent. They could have been otherwise in general, and they might well be otherwise in particular cases, even actual ones. Consequently they cannot possibly amount to a full proof that miraculous accounts should not be accepted. The fact that Hume feels the need to point to these contingent facts shows that at some level he is aware of the fact that his argument is not conclusive, although he wishes it to be.

In the end, Hume’s argument does not establish anything, but only expresses his own incredulity, as in this example:

There is also a memorable story related by Cardinal de Retz, which may well deserve our consideration. When that intriguing politician fled into Spain, to avoid the persecution of his enemies, he passed through Saragossa, the capital of Arragon, where he was shewn, in the cathedral, a man, who had served seven years as a doorkeeper, and was well known to every body in town, that had ever paid his devotions at that church. He had been seen, for so long a time, wanting a leg; but recovered that limb by the rubbing of holy oil upon the stump; and the cardinal assures us that he saw him with two legs. This miracle was vouched by all the canons of the church; and the whole company in town were appealed to for a confirmation of the fact; whom the cardinal found, by their zealous devotion, to be thorough believers of the miracle. Here the relater was also contemporary to the supposed prodigy, of an incredulous and libertine character, as well as of great genius; the miracle of so singular a nature as could scarcely admit of a counterfeit, and the witnesses very numerous, and all of them, in a manner, spectators of the fact, to which they gave their testimony. And what adds mightily to the force of the evidence, and may double our surprise on this occasion, is, that the cardinal himself, who relates the story, seems not to give any credit to it, and consequently cannot be suspected of any concurrence in the holy fraud. He considered justly, that it was not requisite, in order to reject a fact of this nature, to be able accurately to disprove the testimony, and to trace its falsehood, through all the circumstances of knavery and credulity which produced it. He knew, that, as this was commonly altogether impossible at any small distance of time and place; so was it extremely difficult, even where one was immediately present, by reason of the bigotry, ignorance, cunning, and roguery of a great part of mankind. He therefore concluded, like a just reasoner, that such an evidence carried falsehood upon the very face of it, and that a miracle, supported by any human testimony, was more properly a subject of derision than of argument.

This is a distorted account of the miracle of Calanda. Ironically, Hume’s position is actually supported to some extent by the errors contained in his own account of the miracle: we cannot “trace its falsehood,” in the sense that we cannot determine whether the account has been distorted by Hume himself, by the Cardinal, by the residents of Zaragoza, or by others, or some combination of these, but it is easy enough to determine the fact that it has been so distorted. Nevertheless, Hume is not proving anything here, but simply asserting that he would not believe in a miracle no matter how good the testimony brought in its favor.

This is not a reasonable attitude, but sheer stubbornness.

# The Second Mistake

The second mistake that we mentioned at the end of this post was that given the thesis that God is hidden, arguments against a religion become arguments in favor of it. Pascal suggests such a position when he says, “that darkness in which they are, and with which they upbraid the Church, establishes only one of the things which she affirms, without touching the other, and, very far from destroying, proves her doctrine?”

This is not completely wrong, but there is less truth than error in it. If you explain a difficulty by adding something to your account, as the Mormon does in this example, then technically the difficulty does support the new account. The problem is, as was said there, that the new account overall remains less probable than the original account was without the addition. And the difficulty remains evidence that the original account was simply wrong; it does not “change sides” to support only the new account.

There are several things that need to be considered in the present case. The first is in what sense it is an additional explanation when one says that the explanation for the existence of difficulties is the hiddenness of God. It is somewhat different from the example of the Mormon, and in a way that favors Christianity. The Mormon may have been aware of all widely known facts about his religion, without being aware of the problem regarding the Book of Abraham. But this could not be true of the Christian in the case under discussion. It may be a Christian does not realize that his religion implies that God is hidden; but this is on account of a lack of consideration. The very things that he already knows and believes imply that this must be true in order for his religion to be true. In the Mormon case, on the contrary, nothing about the Mormon religion implies there needs to be some special explanation regarding the Book of Abraham. So when the Mormon adds a special explanation, this is a real addition that must reduce the probability of his general claim. But the Christian in the case under discussion is simply explaining what was implicit in his claim in the first place. In this sense it does not reduce the probability of his claim, but leaves it as it is.

On the other hand, if one has not yet considered the fact that Christianity requires the thesis of hiddenness, it may be that the prior probability for Christianity should be less than what one supposed, when one was judging it without this consideration. In this sense, it may reduce the probability for a particular individual who has not yet fully considered the situation.

Finally, this does not in fact imply that concrete difficulties with Christianity are not evidence against it, because even if the difficulties fit with the claim that Christianity is true but hidden, they may sometimes fit even better with the claim that Christianity is false. For example, suppose there were no historical evidence for the Virgin Birth (this is a counterfactual, since in reality there is historical evidence for it, even if weak relative to the strength of the claim, namely the assertions of the Gospels); such an absence of evidence would fit with the idea that the doctrine is true but hidden, but it would fit even better with the idea that the doctrine is false. Consequently, the evidence would make “the doctrine is true but hidden” more probable relative to “the doctrine is true and not hidden,” but overall the evidence would be likely to make “the doctrine is false” more probable than it was initially before checking the evidence.