Hume and the confirmability of miracles

While discussing the McGrews' Bayesian analysis of the Resurrection (CCRJ), I frequently mentioned that the McGrews and myself were not attempting to derive odds on the Resurrection. Rather, we were focusing on the Bayes factor - aka the likelihood ratio - which if you recall, is the number by which the prior odds ratio p(`thing')/p(`other thing') is multiplied to yield the posterior odds ratio q(`thing')/q(`other thing'). So at the very minimum, one needs to estimate what the prior odds should be in order to derive the final odds. Since neither of our approaches were sufficiently general to capture a truly cumulative Bayes factor, even this may be inadequate, but since the factor I derived - 10⁶ - was calibrated given generous textual assumptions in favor of the Resurrection, we may be able to tentatively estimate an upper bound on reasonable posterior odds using that factor if we have an upper bound on reasonable priors. If that upper bound is less than 10^-6, we may conclude that the Resurrection probably did not occur, i.e. 0.5>q(R).

I opine that barring other arguments in the context of a natural theology, such an upper bound exists. That is, the Resurrection can not reasonably be confirmed from the textual record alone with respect to convincing background knowledge which is shared by skeptics and Christians alike. But I am interested in Hume's more general thesis, which is that miracles by their nature can not reasonably be confirmed. The details of an analysis of the Resurrection are merely an instance of a more general unconfirmability argument.

I pause to obviate a potential objection: I am fully aware that the proper interpretation of Hume's Enquiry concerning Human Understanding, especially Of Miracles, is a hotly disputed topic. I am attributing an unconfirmability argument to Hume; I doubt he can be convincingly interpreted as not making such an argument. But I am not here interested in any historical exoneration or conviction of Hume of philosophical crimes. Rather, I want to work with his apparent argument by recasting it in formal terms, propose that it is inadequate, and attempt to shape a more satisfactory unconfirmability argument.

I am neither discussing nor proposing a definition-dependent impossibility argument, e.g., "A miracle is the violation of mathematical, divine, immutable, eternal laws. By the very exposition itself, a miracle is a contradiction in terms: a law cannot at the same time be immutable and violated." Rather, I am interested in miracles loosely defined as particular exceptions to otherwise exceptionless laws or regularities of nature. In this context, the regularities of importance are putative laws, and their importance is epistemic, not ontological. Whether or not a putative law is true is largely irrelevant: what matters is how well-supported it is.

The proper definition of a miracle is also in dispute. As regards this item, I follow Tim and Lydia McGrew in treating the Resurrection as a paradigm (CCRJ, p.4). As I will explain in the course of this discussion, working with paradigm cases like the Resurrection is all that should be required. Failures of consensus on miracles are relevant to Hume's argument, but they will not prove necessary to reject his conclusion. Which is roughly as follows:

Hume's Argument: There cannot exist evidence E for a miracle M such that

.

With conditionalization, this is equivalent to stating that the posterior odds on a miracle can never be greater than 1/2.¹

The initial prospects of this statement are rather dim. As is commonly pointed out, there are no a priori boundaries on the size of Bayes factors: no matter how small the prior odds on a miracle, there exist finite Bayes factors which can overcome them. Similarly, disputes concerning the definition of a miracle make it impossible to have confidence in such a general statement.

I would also add that we should not be overly interested in such an argument as employed to justify ignoring any potential evidence for miracles. The debate is worthwhile. As I have noted in a slightly different context:

It is very often said, by e.g. PZ Myers and Massimo Pigliucci, that one cannot evidence Christianity or gods because they are not coherent hypotheses. More needs to be said about this, but I would at least suggest the following: if the evidence for the Resurrection really is extremely convincing to reasonable people on the assumption of coherency, we should take an attitude similar to that which I think we take to science: some conceptual fuzziness is to be tolerated, barring flat contradiction, where overwhelming evidence for an aspect of a theory is available. Were I to find the evidence for the Resurrection convincing, I know I would be working very diligently to craft a coherent Christian philosophy to accommodate it. So to me, the coherency difficulty is in many ways secondary, unless that difficulty is so severe that one cannot even begin to discuss relevant evidence. I think we usually manage to do so. Wouldn't you agree?

The argument also conflicts with the empirical, tentative nature of skeptical inquiry. I think we should wish to avoid such categorical statements.

The temptation to fashion such an argument is understandable. But it should be resisted. Any epistemology that does not allow for the possibility that evidence, whether from eyewitness testimony or from some other source, can establish the credibility of a UFO landing, a walking on water, or a resurrection is inadequate. (Earman, p.4)

As Earman also notes, there are events which, were they to occur, surely amount to convincing evidence for a miracle claim:

Suppose, for the sake of illustration, that there is a well developed theology based on the existence of a god called Emuh. who promises an afterlife in return for certain religious observances in this life. Suppose that this theology predicts that on such-and-such a day Emuh will send a sign in the sky. And suppose that on the appointed day, the clouds over America clearly spell out in English the words “Believe in Emuh and you will have everlasting life,” while the same message is spelled out in French over France, in Deutsch over Germany, etc. Then even though these cloud formations may not contravene any of the general principles taken at the time in question to be laws of nature and, indeed, may be explicable in terms of those principles, it would not be untoward to take these extraordinary occurrences to be support for Emuh theology. (p.11)

So Hume's argument, were it valid and coherent, would prove too much. It also ignores the effect of evidence for a theology and its implications for the proposed miracle claim. Were the gloating fiction of LaHaye's Left Behind series to be actualized, it would confirm the Resurrection. I am unsure as to how or why someone would seriously argue otherwise, even if the various details of Christian theology are unclear.

I will not second another common objection: I do not think that Hume's argument, interpreted as I have interpreted it, would destroy the possibility of overturning laws in science. His "straight rule of induction" is problematic in this context (Earman, pp.31-2), but laws are not to my knowledge overturned in the way that a putative miracle claim would overturn them.

Take the Conservation of Mass. Did measurements of nuclear reactions overturn a uniform experience? No, because what changed was not the article where uniform experience applied, but where a novelty was being analyzed. If for example I react 50 grams of sodium with 70 grams of chlorine gas to form salt (NaCl) at approximately standard temperature and pressure and measure a net change in mass of 20 grams, I or my instruments screwed up. Mass is still conserved within significant margins of instrumental error for `ordinary' chemical reactions. The implications of such experience have not been contradicted, but superseded. With miracles, where the putative law needs to otherwise be intact for theological reasons, no such consideration applies. We are not talking supercessions or the overturning of laws as done in the sciences; we are talking about flat-out, singular violations of an otherwise sound natural order. We are talking about an experiment incapable of replication. Were mass conservation to always hold, and the only exception were to occur in one apparently sound experiment, we should have discounted the experiment as flawed if replication failed.

I judge Hume's argument to be a failure and its conclusion to be unsound. But this need not be the end of the story. We may yet build a better monument by clearing away the noisy rubble of Hume's rhetoric and picking out the useful pieces. That will be the subject of my next post.

---

1. This interpretation, though disputed, has a lot of support, perhaps apart from the target threshold 0.5>q(M). Were Hume making an impossibility argument, it is odd that he should emphasize the relative strength of evidences (p.169) and the uncertainty of the relevant propositions (pp.169-70); that he discusses evidence at all would also be strange. In addition to his use of probabilistic terminology, he also casts his argument in terms of degrees-of-confidence: "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." And he continues famously: "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" (p.170).

With terms like `infallible' and `proof' in play, I think that Hume may be interpreted as arguing for a prior probability of zero for miracles, or perhaps an infinitesimal probability (p.171). But I do not think that such a prior is convincing to all - or many - concerned, and it is therefore useless. Many Bayesians accept - here the terminology is unfortunate - the principle of regularity, which states that all possibilities have probability greater than 0, assuming that those possibilities are uncertain and assigned any probability whatever. In any case, we are presumably inviting Christians to the discussion, so we must at least assume that non-zero priors are in play.

There are other ambiguities, and I am lead to second Earman's hostile conclusion (p.20):

I defy the reader to give a short, simple, and accurate summary of the argumentation in "Of Miracles." What on first reading appears to be a seamless argument is actually a collection of considerations that sometimes mesh and sometimes don't. It will take much work to tease out the components of Hume's argument and to evaluate the soundness of individual components and the effectiveness of the entire package.

Immediately after bringing up `proofs' of experience, Hume dives right back into emphasizing the fallibility of evidence, particularly witness testimony (EPHU, p.171). There are other deficiencies in his presentation. After defining miracles as putative exceptions to uniformly evidenced laws, he states the following:

There must, therefore, be a uniform experience against every miraculous event, otherwise the event would not merit that appelation. 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. (p.173)

With this stringency, the mere proposal of evidence for an event disqualifies that event's being a miracle, as experience is no longer uniformly against it. And then, the presentation of additional evidence should leave an opening. I'll stick with the Bayesian interpretation because it is the only plausible interpretation to be found.

Hello lurkers!

I hope that there are lurkers, as there are currently no commenters. That's alright; I did not start this blog in order to accrue a following. But if readers would introduce themselves and their interests, I would be quite happy! I could turn my attention to your interests. Hopefully, I will manage to deter you.

I've been away for several days to help a friend move. My apologies. Thanks to a slight injury, I have further perfected the very-much-not-recommended art of self-surgery. I'm pretty proud of myself at the moment.

On the way to the new place, I decided to put my rationality skills to the test. Boredly approaching the Atlantic shoreline in the Carolinas, I made an estimate about the frequency of pine trees as one approaches the beach. I came up with the following estimate as derived from highway observations:

- Approx or above 50% pines, less than 50% deciduous within 150 miles
- Approx or about 75% pines, less than 25% deciduous within 50 miles

where this estimate completely fails close to riversides.

Now here is the question I sought to intuitively answer (and mostly failed to correctly intuit): Why is it that pines/needle-trees become increasingly common once one approaches the Southeast US coast? I recall seeing this before across the Carolinas, Florida, and etc.

Any ideas?

Belief/Lack of Belief

While thinking about that hoary topic again, I have decided to write a post on the `belief/lack of belief' distinction which is the subject of so much fire in the skeptical and atheist communities.

I opine that atheism is, properly speaking, best described as a belief in nearly all circumstances where it occurs. I am familiar with the objections to this, which are most eloquently and illustratively summarized by the following video:



The goal of describing oneself as an atheist is presumably to give an accurate impression of ones stances. Since the term `atheist' has been used as a curse word - and really originated as such - and false equivalences are so prevalent, I always assume that I will have to clearly explain my stance in detail. For this reason, the distinction between lack of belief and belief, up to the aforementioned goal, is next to useless.

This is why, though I usually disagree with those who insist on the `belief/lack of belief' distinction, I nevertheless agree that it is understandable to make that distinction and insist upon it. If you are an atheist and have ever engaged in any argumentation, I would be surprised if you had not encountered false equivalences like those described in the video. Believers, by using `atheism' in a way syntactically similar to `religion', attempt to draw false equivalences. But I do not think that making the `belief/lack of belief' distinction will help to avoid the boring process of correction. We have to do this anyways.

Whether or not `lack of belief' is better than `belief' depends especially on one crucial fact: the definition or idea of gods in play. This semantic dependency is not obviated by insisting on the `atheism=a lack of belief in gods' equation; rather, `atheism' always carries with it this problem. So why do we identify as atheists at all?

The answer is very simple, and it hinges on how words are normally, conventionally employed. We are atheists with respect to gods as usually defined and as normally present in religion. To call ourselves atheists is to feel that this usage sense has some use, even if we have learned the ready lesson that one must always in advance insist on a relatively precise definition when arguing about gods.

Think about it in an indirect sense: is it really the case that we have no opinions concerning even hitherto undefined gods? Do we for example make statements about the nature of god-belief? Try asserting the following at the same time:

1. The gods of religions exist only in the minds of believers.
2. Some god exists.

Even if a non-descript god is presented for which there is no evidence either way, it is not as though we therefore lack any opinion. We have notions of parsimony - `Occam's razor' - and other intellectual tools that very much apply to such hypothetical entities. We do not think that they are as likely to exist as not, or any other such thing. To see where you stand on such gods, play along with a little thought experiment of mine.

Suppose I have a magic box - do not question the physics of the magic box - that can tell us with 100% accuracy whether or not an otherwise strictly non-predictive entity exists. If that entity exists, the box will output `yes' when you press the button. If not, it will output `no' when you press the button. How much are you willing to stake, in dollars or whatever else you value, that the box outputs `yes'? If you are anything like me, you would stake a considerable amount on `no'. The point is not that we lack a magic box; the point is that this experiment can reveal to us how confident we are about the existence or non-existence of deities which are incapable of being evidenced. Had we such a box, I would be happy to take $1000 off of anyone unwise enough to bet that sum on a `yes' answer. Would you?

I make a similar point in the context of the OTF. `Skepticism', or `atheism', are not free-tickets to being the `default' position. I add, and emphatically, that noise about the `burden of proof' is completely pointless. There are arguments. Talk about them. Once you get a theistic counterpart to carefully present what he or she believes, you should not have any trouble finding the right place to continue. The case is not, as Qualiasoup suggests, that atheism is the natural `default'. Rather, the `default' is whatever the person in the discussion believes at that moment. If anything, they will usually be supernaturalists. There is no one `default'. There is no hypothetical beginning of the discussion at which an absolute burden of proof is in play. This is pure distraction.

I am not thinking how Qualiasoup says that people like me are thinking. I am a probabilist, after all. I am also of the opinion that the `believe/believe that not' distinction is unnecessary in light of probabilism. It's all about whether or not any confidence is assigned to a proposition, and if so, how much. To me, `belief' as normally employed is a secondary, unnecessary term. But I still use it whenever I am talking to people who are not probabilists. So `belief' still has utility, as does `atheist', but as far as I can tell, it only has utility when it is used approximately how I use it.

Now, "I lack belief in your god" and "I believe that your god does not exist" are not contradictory statements. Which is more appropriate may depend on the specifics of the situation, especially whether or not a coherent, understandable concept of god is present. But that is not the point, because this applies to using `atheist' in any sense of the term.

If you want to engage in the arguments, do so. But the common purpose of atheists in insisting on the `lack of belief' label as a means to justify being in a `default' is deeply suspect, and furthermore, it is pointless. You are after all engaging in the arguments anyways. You are not a dictionary entry. Act like it.

This is all very straightforward.

Step 1: You have to ask for a definition.
Step 2: Evaluate that definition.

Is it contradictory? Then you can be a certain atheist. Is it incoherent or nonsensical? Then you can say why the definition is inadequate. Is it capable of being evidenced, but fails to have convincing evidence? That is a claim about the state of the evidence; defend it. Is it coherent and consistent, but incapable of being evidenced? Then demonstrate that this is the case, and if you maintain non-belief, explicate the principles of reasoning that justify your doing so. Whatever you need should be clear in the context of discussion, up to and including any subtleties in definition that are useful.

And whatever you do, do not make a gigantic big deal out of this.

Edit: And there are good reasons to not feel strongly about your self-labelings.

Cosmic Savagery

Via PZ Myers, I came across this video:





There is much more to qualify about the comments in this video, but I think it important to discuss one of the recurring themes: the desire for cosmic justice. I think it important to discuss for a very simple reason, that it is no exaggeration or hyperbole to say that the underlying mentality constitutes a threat to the continued survival of the species in this modern age. The reasons are obvious.

Bertrand Russell summarizes it aptly in his Sceptical Essays:

When we think of mankind, we think primarily of ourself as its representative; we therefore think well of mankind, and consider its preservation important. Mr Jones, the Nonconformist grocer, is sure that he deserves eternal life, and that a universe which refused it to him would be intolerably bad. But when he thinks of Mr Robinson, his Anglican competitor, who mixes sand with his sugar and is lax about Sunday, he feels that the universe might well carry charity too far. To complete his happiness, there is need of hell-fire for Mr Robinson; in this way, the cosmic importance of man is preserved, but the vital distinction between friends and enemies is not obliterated by a weak universal benevolence. Mr Robinson holds the same view with the parts inverted, and general happiness results.

Less frivolously, many of us feel that the universe is not a particularly just place and desire that somehow the good should be rewarded and the evil punished. This retributive mentality does great harm, and I strongly object to it.

Orwell talks about it well:

It is absurd to blame any German or Austrian Jew for getting his own back
on the Nazis. Heaven knows what scores this particular man may have had
to wipe out; very likely his whole family had been murdered; and after
all, even a wanton kick to a prisoner is a very tiny thing compared with
the outrages committed by the Hitler régime. But what this scene, and
much else that I saw in Germany, brought home to me was that the whole
idea of revenge and punishment is a childish daydream. Properly speaking,
there is no such thing as revenge. Revenge is an act which you want to
commit when you are powerless and because you are powerless: as soon as
the sense of impotence is removed, the desire evaporates also.

The desire for retribution, cosmic or otherwise, is one of our worst features. Compare the following statements:

1. Hitler, responsible for the horrible deaths of millions, dies and goes to Hell.
2. Hitler, responsible for the horrible deaths of millions, dies and does not go to Hell.

Who out there is absurd enough to demand the former and claim it a better state of affairs than the latter? What does it do to `offset', by divine justice, karma, `the balance of the universe', or whatever other theory the horrors it contains?

No, I do not want a Hell, not even for Hitler. If there is a Heaven, I even say we should want him there, assuming he would be harmless. What point is there in kicking a caged animal, no matter how vicious? To desire otherwise is to want suffering for its own sake; I cannot abide it, and I stare in dull horror at my neighbors. And yes, sometimes myself.

I stress that this mentality, though usually supported with a theological assertion if any theoretical justification is in offering, is a secular one. It is a very big part of why so many support the death penalty here in America. I wonder why it satisfies them that a murderer, however horrid, should be subjected to ritualistic slaughter after he has been made harmless. The kicking of toothless creatures is a big part of our tribalistic make-up, and its consequences are terrible.

That anyone should want to elevate this reactionary savagery into a governing principle of the cosmos is beyond me. I suspect and hope that those who do have not seriously analyzed their desires.

I leave you with this.

Resurrection round-up

Alright, I've finished my discussion of the McGrews' paper! Eight posts should be a good enough start, I hope. Anyways, start here.

As I say throughout, I think that their paper is an excellent starting point for further discussion. So I will use this post to collect future arguments about the Resurrection which fit into the discussed frameworks. I'll keep this in the sidebar. Anyways, these posts constitute 11,500 words of writing, done over the course of only a few hours (I had been sitting on these thoughts for a while). Doubtless there are errors, typographical and otherwise. If you find any, let me know.

Update (8/6/2011): Here are two posts on Hume's argument and a revision of the same.

Update (9/3/2011): I'll be regularly listing related links here.

1. Reppert on estimating priors.

Tim and Lydia McGrew on the Resurrection, part 8 (conclusion/outline)

[Continued from Part 7.]

I have tentatively argued for a Bayes factor of 10⁶ given the textual assumptions of the McGrews. I have further proposed a way to expand the discussion to include skeptics of these assumptions. In these conclusions, I have agreed with the McGrews that their assumptions are plausible, and that on these assumptions the Resurrection is the best explanation. However, I think that they overreach when they claim, or seem to claim, that their methodology - which they agree is very partial - approximates a more general Bayes factor (p.39).

The rest of their paper (pp.46-68) discusses Hume and the objections of other apologists. As I have said, I am leaving that topic for later. Just as they have not claimed to have derived odds on the Resurrection, I do not claim to have done so either.

As I stated at the beginning, the McGrews have started an excellent conversation. I would recommend anyone who wishes to pursue a rigorous analysis of the evidence for the Resurrection to read their paper. And just as they have qualified, I must qualify that my approach has not been complete.

In particular, I have not gathered all of the relevant evidence in deriving this likelihood ratio. Many apologists insist on Paul's claim that there are 500 witnesses. Skeptics focus on potential `gaps' in the text, which become quite plausible on the (quite plausible) assumption that the relevant texts are polemical and designed to serve the Christian community. In this sense, my Bayes factor is not cumulative. My Bayes factor is also not cumulative with respect to textual criticisms. Again, I've outlined how that may be pursued. A fully rigorous analysis of all of the details will probably lead to a figure quite different from the one which I have suggested. But I think, quite plausibly, that the figure I have suggested is not a gross underestimate. And I think it quite low enough to leave anyone with a reasonable prior ratio unconvinced. If Christians think that the prior odds on the Resurrection are above something like 10^-6, I hope they will tell me why this is. If they have such a value, I assume it will be within the context of a well-formed natural theology. Contrary to the McGrews (p.49), I think this will prove necessary.

So there are my efforts. I hope you will read them all before making critical comments, just as I hope you will read the McGrews' paper. If you spot a technical error or misspelling, I would of course appreciate immediate correction. Otherwise, I ask that you take time to absorb the details, and inquire for yourself how knowledge you have may further inform the discussion.

I also hope that readers can at least retrospectively appreciate the utility of Bayesian techniques in analyzing such arguments. The importance of this or that number is minimal: what is important is that they can be used to capture the arguments, show how they relate to each other, and help us to focus on the important details. They also help to understand how I, a skeptic of the Resurrection, can be quite satisfied with a statement like "the Resurrection is the best explanation of the salient facts" - a statement which is very difficult to refute, I think. Bayesian insights allow me to maintain my skepticism without undertaking hopeless endeavors, for example defending or asserting statements like "the Bible does not constitute any evidence for the Resurrection." In arguments over the Resurrection, we cannot eliminate entirely our differing private intuitions. But we can at least see what is required of those intuitions and reduce their impact when possible.

And I love this stuff for its own sake. I'm soggy like that.

Suggestions for the future: separate the witness of James the Just from that of the other apostles, subsume the testimony of the women within that of the disciples generally, and largely omit any discussion of Paul's conversion as itself evidential in favor of other details. Paul is doubtlessly important, but his importance is in his teaching and his early date. And keep it general, yo.

Tim and Lydia McGrew on the Resurrection, part 7

[Continued from Part 6.]

Hallo and good day, weary traveler! We may now discuss in passing one final detail, i.e. the impact of the testimony of the women (W) on our final Bayes factor, which has currently been reduced from the McGrews' estimate of 10⁴⁴ to 10⁷. If I can convince you that the womens' testimony is not significant, we will arrive at a final replacement Bayes factor of 10⁵. Then, I may finally lay my weary head to rest and fly elsewhere. Especially, I hope to soon discuss Hume's argument and attempt to estimate prior odds on the Resurrection.

I think this may be done as follows: take the alternative hypotheses to the Resurrection which I have proposed to explain the disciples' witness and subsume the womens' testimony within them. This may be done without loss of prior plausibility, especially since initial rumors already play a role. This undermines the `essential independence' of the women from the other disciples proposed by the McGrews (p.41), which I do not think defends probabilistic independence in any case. As I have discussed, the emergence of rumors would very likely have affected the reaction of the disciples. Indeed, since the witness of women would have been regarded as less credible (p.28), we may plausibly expect the disciples to have corroborated their belief - which perhaps began with the women's testimony - with witness of their own. This, I think, tentatively answers the McGrews' challenge: "why [should we not] judge p(W|R) to be at least several orders of magnitude greater than p(W|~R)?" (p.30)

To argue directly that W should have less effect than 10² on the final Bayes factor requires, I think, a questioning of the McGrews' salient facts. If we separate it out from the other details, I think that the McGrews were sufficiently generous in leaving it at 10². In assenting to this, all I am saying is the following: that the women would report an empty tomb, or seeing Jesus, had he not Resurrected has probability 0.01, assuming that they would report it had they really seen it and that the facts are as described by the McGrews. I do not think that this is too challenging an admission for a skeptic. Instead, we need not recognize this contribution as multiplicative, and regard it as diminished by other considerations.

I think this, along with any potential minor significance of Paul in light of the other reductions, can be fairly accounted for by leaving a final estimate of the Bayes factor at 10⁶, and plausibly lower.

Now, we may finally arrive at our destination.