An unconfirmability argument

[A caution: This post is long, verbose, and overly technical. Read the dialogue I posted at the end. If it is comprehensible to you, you should have an adequate grasp of the contents of this and the previous post.]

In my last post, I interpreted Hume's argument against the confirmability of miracles and found it to be unsound. Now I want to affirmatively answer another question: given that Hume's argument is unconvincing, is there another unconfirmability argument which applies to paradigm miracles like the Resurrection?

If so, we are not doomed to Earman's (apparent) conclusion, that we must analyze the details of every miracle claim if we are to safely reject miracles (p.3). Rather, we can specify categories of claims, narrower than `miracle', that cannot be confirmed by certain categories of evidence. Though we might not safely assert something like "no evidence you have should convince me of a miracle," we may be able to say something like "by itself, the evidence you present is by nature incapable of overcoming the prior probability of the type of miracle you assert." I will leave open the possibility that miracles like the Resurrection (R) can be confirmed; I am only closing a class of potential means of doing so. I do not think that I am providing or can provide "an everlasting check to all kinds of superstitious delusion" which "will be useful as long as the world endures" (EPHU, p.169); rather, I am giving something which, if successful, would obviate any non-pedagogical, rational need for any detailed Bayesian analysis of the Resurrection like this one, so long as we lack other significant arguments in favor of Christianity. One notices the number of qualifiers required to invest in such an argument, and there will be more. This form of argument does not constitute a license for ignorance, and it will require creative adaption to specific miracle claims.

A lot hinges on the problem of determining appropriate priors in subjective Bayesianism. In order to provide some convincing estimate of the prior odds on R, a trick - I think novel to me - must be applied.

To motivate this trick, I'll cite its precedent and inspiration: the method of reparation, originally due to Richard Jeffrey. This tool was invented for a specific problem, the problem of old evidence. Roughly, the problem is as follows: whenever a new theory is crafted, its ability to explain known phenomenon is considered to be of epistemic significance. For example, relativity theory's ability to `predict' the perihelion precession of Mercury - a phenomenon which had long defied explanation in classical physics - is considered powerful evidence for that theory. But the hypothetico-deductive principle, which states that verification of an uncertain prediction of an uncertain hypothesis increases the probability of that hypothesis, is incapable of yielding this well-founded intuition; the `prediction' is already known, i.e. not uncertain.

Reparation solves this problem by positing a hypothetical prior probability, an ur-distribution or ur-prior, in which the theoretical prediction itself in addition to the corresponding observation is treated as uncertain. The `discovery' of that implication and the evidence then raises the probability of the hypothesis in question in a straightforward way. I want to extend this to a problem which plagues estimating the prior probability of miracles, i.e. the problem of old evidence and explanation, or if you prefer, the old everything problem. When priors are not in dispute, this method is unnecessary; this is not an actual case of confirmation. But on the very safe assumption that the prior probability of R, in the absence of other argument, is calibrated with respect to confirmation of natural principles, it is quite useful. We `know' that the prior ratio is small, but we do not know how small. This is important when disputes depend on whether or not that prior is greater than 1/1000 or less than 10^-44. We need something better than the `vague intuitions' rightly deplored by the McGrews (CCRJ, p.50).

The idea is to posit an ur-prior u which is (partially) devoid of current background knowledge. The voided background knowledge is treated as uncertain in u, the recapturing of which yields through conditionalization a suitable prior p. In this context, the `recaptured' knowledge is the bulk of the `uniform evidence of sense' discussed by Hume, which is assumed to be wholly confirmatory of the law L. As I am working with the Resurrection, L is the principle that dead people remain dead. By definition and subset rule, we have for any probability pr that

.

By quick algebra, we yield the following inequality:

.

So by calculating our confidence in L, we set a maximum confidence in R.

As I mentioned, we want to `recapture' p using u by conditioning, that is,

,

where O is the set of observations confirming the putative law. But here I seem to have merely shifted the problem elsewhere, since we now need the u-odds on ~L. This regress is halted by the following abstract consideration: at some theoretical point of time, a hypothetical rational agent should have crossed the `more likely than not' threshold in favor of the law. For this reason, we may assume that the u-odds on ~L is 1. From there we must estimate the raw confirmatory force of the remaining experience as captured by the u-Bayes factor. Cleaning up the previous mess, we want to analyze

.

Now here is the crucial question: what does ~L look like after this partial recapitulation of background knowledge? An `anti-law' which states that all dead people resurrect will have probability zero, but this is an extreme; the preexisting information will select for those non-laws which ascribe higher probability to individual non-resurrections. These include, amongst other things, statements like "everybody who dies remains dead except under very rare conditions." However, even this will be made less probable by O so long as the further prediction of its elements are uncertain. Unless one claims to be able to specify in advance who should be exempted from death based on limited confirmation, the effect remains strong as regards statements like "everybody stays dead except for Jesus." (Lazarus and other proposed resurrections are not part of the background knowledge being recaptured here.)

To go further, I assume that O contains N instances of confirmation beyond those which lead our hypothetical agent to equivocate, each of which are of equal weight¹, i.e.



where I use the combinatorial notation [N]={1,...,N} for the sake of cleanliness. Supposing that each element of O is independent modulo ~O, we have that

.

So if N=10⁶, even a quite large value for u(O₁|~L) yields an absurdly small upper bound on the prior odds on R. If this assumption of independence is to fail in favor of the theists, there must be in the domain of u some significant set of propositions in ~L that correlate the elements of O. In other words, there must be plausible causal processes not implying L which probabilistically tend to keep dead people dead.

And here is where the remainder of our scientific knowledge and experience come into play. We do not consider L to be a fundamental feature of the universe but a consequence of other properties. Were a technology developed that allowed for the reanimation of corpses, we need not say that a putative law of nature had been overturned; instead, we would say that the `ordinary course' had been altered by the introduction of a new yet naturalistic element. Without such technology, we expect other principles to produce L. These include thermodynamic principles. When Jesus died, there are three sorts of miraculous possibilities: either his vital tissues never fatally decayed by supernatural sustenance, his vital tissues decayed but were reconstituted gradually or suddenly prior to his reanimation, or he reanimated without the function of his vital organs. Each of these possibilities runs against other established regularities. Properly speaking, then, L is not merely the generalized rule that dead people remain dead but also the set of combinations of rules which produce that outcome. For the Resurrection to occur, every one of those combinations must fail to hold.

So a theory which advantageously counters the above independence assumption must probabilistically correlate the elements of O, have significant plausibility in its own right, and fail to assume or imply any element in L, be it the putative law itself or any combination of principles which produce that law. In this case and in the case of other miracles which violate mass/energy conservation and/or thermodynamic principles, we must toss out modern science before examining any evidence proposed against its principles. This is an absurd task, but that absurdity is consequent to proposing a suspension of the natural order.

In such a manner I tentatively maintain that the independent assumption as employed is valid in establishing an estimated upper bound on the prior odds on R. So long as we agree that it is disgustingly low, a precise value is unimportant for the strengthening of the argument. Since as you may have noticed this method of unconfirmability argument requires some detailed information concerning the relevant miracle, specifying any particular value is unimportant. For an approximate value's import to materialize, we must also delve into the category of evidence in play.

Hume appears to recognize the need for categorical bounds on the strength of evidence in his essay:

When any one 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. (EPHU, p.174)

I think that whenever Hume attempts to place such a bound on `testimonial evidence', he overreaches by virtue of that category's broadness. "I would not believe such a story were it told to me by Cato" (p.172) does not a sufficiently general argument make, nor are vague allusions to particular failures of human reporting which remain in the background, as these do not necessarily apply to all claims that humans make. Instead, it is better to focus on more task-specific categories. It might be possible to do so, but there is no reason to argue for more than is required.

Now I must specify what I mean by a categorical bound in formal terms: roughly, a categorical bound B is a real number such whenever evidence E for an uncertain hypothesis H lies in a set C, any Bayes factor produced by E is less than or equal to B.² That is,

.

If H is a miracle claim violating a putative law L supported by recaptured observations O, then tautologically the argument is successful if

.

What remains is to show that the evidence proposed by the miracle-claimant lies in such a set C.

What would C look like? If we want to be trivial, we can use uncontroversial examples. For example, two gamblers who have vastly differing prior odds on the fairness of a coin should not be able to resolve their dispute by tossing the coin a small number of times to check the outcomes. With precise values available, one could derive the minimum number of tosses which could possibly result in agreement.

One may extend this notion by thinking of the epistemic limitations on experiments like "tossing a coin a certain number times" as instances of methodological bounds. In the medical sciences, anecdotal and testimonial evidence as applied to broad categories of claims is almost completely useless for confirmation; careful study is required. What this reflects is a well-founded confidence in systematic errors frequently present in anecdotal reports which cannot be sufficiently discounted in the absence of controlled study. Anecdotal evidence for a novel, popular medical hypothesis cannot by nature discount those established theories and credences, a task which is necessary in order to confirm the new remedy against a significant prior implausibility.

Similar considerations apply to many classes of potential miracles and the evidence most commonly presented for them. With the Resurrection, we need not attempt the task Hume appears to set for himself in arguing that any set of witness testimony is incapable of producing convincingly calibrated Bayes factors above something like 10¹⁰⁰⁰. In this case, we may instead suggest that the methods of historians in establishing the historicity of those reports and their background, while possibly very good, are not so good as to allow a confident assertion that the historical record produces a Bayes factor above 10¹⁰⁰⁰. Myth and utter fabrication, even if wildly improbable in this case, have precedents, and I do not think they are capable of being discarded with absolute or nearly absolute confidence given the difficulties in method.

Hoping my proposals have been convincing thus far, I conclude by giving the sort of conversation which two Bayesians could have.

Christian: Ah, skeptic! Just the person I was wanting to see! I have crafted a convincing case that the Resurrection did in fact occur, and I was wanting your feedback.

Skeptic: That sounds very interesting, but before we go into the details, would you consider the Resurrection, had it occurred, to have constituted a suspension of the natural order?

Christian: Certainly, as you well know. Otherwise, the Resurrection would be meaningless. If Jesus' Resurrection were, say, a mere product of absurdly unlikely but possible quantum fluctuations, then any argument to theism or Christianity from that event would be undermined. It would be an isolated physical anomaly; nothing more, nothing less, and surely not a communicative sign of divine endorsement of the validity of Jesus' teachings.

Skeptic: I'm glad that you and I agree. And within this possibly suspended natural order, would you admit that local, epistemic generalizations hold and that tremendous confidence in those generalizations is yielded by what Hume would call `uniform sense data'?

Christian: While I obviously do not accept Hume's argument, I agree that incredulity prior to the examination of the evidence is wholly reasonable. That is why I have analyzed the evidence; one cannot say prior to analysis whether or not the evidence is sufficient or insufficient. You and I both know that this is merely a matter of probability.

Skeptic: I do not accept Hume's argument either, but one may with certain conditions be able to obviate the need to examine all of the details in advance by grounding a posteriori bounds on Bayes factors produced by certain types of evidence.

Christian: Part of this worries me, as it sounds like an excuse to avoid examining the evidence, which, as a good skeptic and Bayesian, you should be interested in doing. In any case, I suspect that such an approach would undermine important areas of scientific research, were it to be accepted.

Skeptic: I admit that this idea is a time-saver and has its ideological attractions, but allow me to specify some of those conditions. Hopefully, when you are satisfied with their stringency, your worries will vanish. But before I may do so, I nevertheless must know the nature of the argument that you are proposing. As I said before, the bounds to which I refer would be a posteriori, not some analytic consequence of Kolmogorov's axioms, uncontroversial metaphysical theses, or sound subjective Bayesian principles. I could not pretend to Hume's rhetoric and claim "an everlasting check to all kinds of superstitious delusion," or claim to have silenced any potential reasoned argument on your part or the part of your comrades in arms, be they future comrades or present confederates. In order to state exactly what I can say in advance of detailed analysis, I have to consider at least some of the details.

Christian: That at least sounds more interesting than another platitudinous regurgitation of Hume's breathless meanderings. Fine, I will play along. I am arguing, as against many prominent skeptics, to and from the historicity of the texts with respect to several key facts, especially those facts concerning the secular claims of witness testimony.

Skeptic: Have you accorded these facts certainty in your analysis as opposed to a more general analysis, for example using Jeffrey conditioning or classical conditioning on a partition of the historical possibilities?

Christian: For the facts concerning the witnesses, I strengthened the relevant arguments so as to make those facts not only plausible, but so overwhelmingly likely as to ensure that errors of omission do not seriously undermine the strength of the argument.

Skeptic: How overwhelmingly likely?

Christian: I think that I see roughly where you are heading with this. By your earlier hintings, it is clear that you are relying on some estimate of the prior odds of the Resurrection. Riddle me this: How do you propose to estimate prior odds on the Resurrection in any convincing way? You and I are both critics of equivocation and objective Bayesianism. You and I both acknowledge the limitations of current theories of calibration, especially as applied to claims like the Resurrection.

Skeptic: Properly speaking, I do no such thing.

Christian: Help me out here.

Skeptic: I rely on the notion that miracles, to occur, require a suspension of the natural order. As you have probably anticipated, I rely on the epistemic status of that natural order with respect to any potential exemption to gauge a suitable prior on the Resurrection...

Christian: Sorry to interrupt, but clearly you seem to be contradicting yourself.

Skeptic: Only if you assume that I need a specific range of prior odds. Instead, I use the deductive implications of putative laws to straightforwardly derive inequalities via the subset rule which by basic algebra translate into an upper bound on the prior odds of the Resurrection. I only need inequalities and bounds, not specific, well-defined ranges of reasonable discussion.

Christian: Ah, I see. You're assuming that the only relevant calibrating factor is the relation of a potential suspension of the natural order to the epistemic status of the natural order, of course.

Skeptic: That's right; hence why I do not claim that my approach, even if valid, would constitute the end of the discussion. One may still need to engage the evidence, but only if an adequate, well-established natural theology is formulated so as to calibrate the priors differently.

Christian: Which of course would present a serious difficulty, since the primary and standard means of evidentially filtering Christianity out of the more general category of theism is by arguing for the Resurrection. Now I am curious: supposing you could bound the prior odds on the Resurrection below 10^-1000, what would you say to me if I claimed to have produced a Bayes factor based on a confidence some salient facts which is greater than 10¹⁰⁰⁰?

Skeptic: I would say that you have proposed the a posteriori equivalent of proving the rationality of the square root of two, Euler's number, or pi.

Christian: That's quite a strong statement; how do you mean it?

Skeptic: I might agree that your proposed facts are plausible, even extremely convincing. But I would nevertheless insist that they cannot be sufficiently plausible as to yield such a factor. Formally, I would put evidence like that you have proposed into a set of similar evidences and claim that Bayes factors in favor of the Resurrection produced by an element in that set are below 10¹⁰⁰⁰.

Christian: In which case, your argument would be tautological or trivial unless you can convincingly establish, before engaging all of the details, that my textual evidence cannot be stronger. Again, I do not see how you are avoiding the shortcomings of Hume.

Skeptic: Well, you have surely noted my insistence on your specifying the type of evidence in question. I doubt you fail to imagine how that might be relevant.

Christian: I have a rough idea: are you proposing theses, like those of Hume, against testimonial evidence? Just because testimonial evidence is always subject to some precedented, possible counter-thesis, that does not mean that one can say that testimonial evidence as a category must be at least this or that weak by that virtue. The details decide how significant those considerations need to be.

Skeptic: I agree, which is another reason why I do not claim to be vindicating Hume's essay. `Testimonial evidence' is perhaps too broad a category to be subject to sufficiently small, convincing categorical boundaries. As I said before, some specifics are required. Allow me to motivate those which apply to the textual record on which you plan to rely: you can envision cases where an experiment, by its nature, cannot overcome discrepancies in prior odds so as to yield agreement between two rational agents, correct?

Christian: In highly idealized scenarios like fair dice rolls or well-understood machines and programs, sure, but I do not see the relevance to a scenario so complex and multivariate as eyewitness testimony.

Skeptic: You may at least be able to anticipate a generalization of simple and uncontroversial lessons to broader notions like `historical methodology', correct?

Christian: Not exactly, as I see such a category as too vague to easily bound.

Skeptic: Again, it depends on specifics. What is the method which you used to arrive at your initial, secular factual claims? Presumably, you do not claim to have directly observed the events in question.

Christian: Of course not.

Skeptic: And so there is some significant uncertainty in the indirect inference methods, i.e. historical methods, which you employ?

Christian: At least in a trivial sense, but that need not translate into any boundary.

Skeptic: Actually, it does, unless you claim that there is no minimally significant alternative to your facts which your methods can not diminish to an arbitrary degree. For example, can you rule out as strongly as you like the possibilities of fraud and later myth-making with respect to these secular facts?

Christian: I wouldn't say that, but again, I see no reason why, in advance, I can not devalue such possibilities sufficiently as to overcome the prior implausibility of the Resurrection.

Skeptic: If by `in advance' you mean in advance of all background knowledge, surely you are correct, but I mean the reliability of your methods with respect to our current knowledge about its reliability. If that reliability is such that the probabilities of hypotheses like frauds and myths cannot be convincingly grounded below 10^-1000, I have established my case. For such extreme values, I would say that this can be said further in advance than I need to argue, but to firmly secure your methods into the category which I require, I will need to know a few more specifics.

Christian: I think I understand now, and I think that I see how you will be able to secure your conclusions were I to spell out more details. I suppose that I will have to qualify my paper with a placeholder for the time being and play with the formalisms to double-check your statements.

Skeptic: That sounds fine. In the meantime, I would be happy to read your paper. After all, you might be able to calibrate the relevant priors differently. It is still worth reading, for this and other reasons, if your conclusions are as strongly supported as you have suggested.

Christian: I look forward to your review. However, I hope you only resume technical blogging after all that wine you just drank leaves your system.

Skeptic: You're breaking the fourth wall.

---

1. The simplifying assumption of equal-weightedness is not generally true. If alternatives to the law include something like `dead people remain dead unless you perform a certain magic ritual', then only failures of that ritual will contrast the hypothesis with L. We can recapture the plausibility of the assumption by stipulating that a theoretical agent at this theoretical threshold point has effectively ruled any particular such hypothesis.

2. I've been playing with this notion for some time, and I know of several generalizations if anyone is interested.

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.

The Outsider Test for Faith

I've entered the fray (jparris8) over at Debunking Christianity, and as those familiar with John Loftus and the blog know, there is no avoiding discussion of `The Outsider Test for Faith' (OTF). I had heard of this before while watching the threads at Victor Reppert's blog, and as I will inevitably have to discuss it again and again, I will save myself and others misery by collecting my thoughts on the OTF in this post.

I will begin by discussing what Loftus describes as the most comprehensive explanation of the OTF on the web.

The question I’ll be addressing today is whether we should adopt a believing or a skeptical predisposition prior to examining the evidence for a religious set of beliefs. I’ll argue that a skeptical predisposition is the preferred one to adopt.

He did say `prior', so I have to say something Bayesian: one can leave aside entirely the question of prior absolute probabilities and focus on a cumulative Bayes factor, i.e. the number by which your old odds ratio on Christianity is multiplied to yield your new odds ratio on Christianity. Granted, this is a very big task, but once done, you can say what initial commitment to Christianity would be required to have confirmed it as `more likely true than not' or to whatever value you feel significant. That analysis would necessarily include evidence concerning contradictory religious beliefs.

But I have a nagging feeling that Loftus wants to do more with the OTF than the above generalization suggests, just as he obviously wants it to be more than a handy thought experiment.

There is overwhelming, undeniable and non-controversial evidence for the test itself that can be found in the sociological, anthropological, and psychological data. I’ll start with some of this data that forms the basis for the test.

Here I have an issue: he is claiming `undeniable' evidence for a norm. As good is/ought distinguishing Humeans, we should be looking out for the goals or desiderata - presumably themselves uncontroversial and undeniable to theists - by which this evidence translates into support for the OTF.

What might this be? In comments and posts, Loftus is ever touting this:

The basis for the outsider test has been stated adequately by liberal Christian philosopher John Hick: “It is evident that in some ninety-nine percent of the cases the religion which an individual professes and to which he or she adheres depends upon the accidents of birth.” That is to say, if we were born in Saudi Arabia, we would be Sunni Muslims right now. If we were born in Iran, we’d be Shi’a Muslims. If we were born in India, we’d be a Hindus. If we were born in Japan, we’d be Shintoists. If we were born in Mongolia, we’d be Buddhists. If we were born in the first century BCE in Israel, we’d adhere to the Jewish faith at that time, and if we were born in Europe in 1000 CE, we’d be Roman Catholics. For the first nine hundred years we would’ve believed in the ransom theory of Jesus’ atonement. As Christians during the later Middle Ages, we wouldn’t have seen anything wrong with killing witches, torturing heretics, and conquering Jerusalem from the “infidels” in the Crusades. These things are as close to being undeniable facts as we can get in the sociological world.

Ok, this is all fine and well (call it and other details the fact of context-dependence): but why does this imply the OTF? I suppose it will have to wait:

Michael Shermer, a former Christian turned atheist, has done an extensive study of why people believe in God and in “weird things.” He argues: “Most of us most of the time come to our beliefs for a variety of reasons having little to do with empirical evidence and logical reasoning. Rather, such variables as genetic predispositions, parental predilections, sibling influences, peer pressures, educational experiences, and life impressions all shape the personality preferences and emotional inclinations that, in conjunction with numerous social and cultural influences, lead us to make certain belief choices. Rarely do any of us sit down before a table of facts, weigh them pro and con, and choose the most logical and rational belief, regardless of what we previously believed. Instead, the facts of the world come to us through the colored filters of the theories, hypotheses, hunches, biases, and prejudices we have accumulated through our lifetime. We then sort through the body of data and select those most confirming what we already believe, and ignore or rationalize away those that are disconfirming. All of us do this, of course, but smart people are better at it.”

Ok, so there are reductive accounts of religious tendencies. Ok, why does this imply the OTF? Damn, I still have to wait:

This whole inside/outside perspective is quite a dilemma and prompts me to propose and argue on behalf of the OTF, the result of which makes the presumption of skepticism the preferred stance when approaching any religious faith, especially one’s own. The outsider test is simply a challenge to test one’s own religious faith with the presumption of skepticism, as an outsider. It calls upon believers to "Test or examine your religious beliefs as if you were outsiders with the same presumption of skepticism you use to test or examine other religious beliefs." Its presumption is that when examining any set of religious beliefs skepticism is warranted, since the odds are good that the particular set of religious beliefs you have adopted is wrong.

Ok, now tell me: why do the apparent cultural dependency of religion and the at least partial success of reductive accounts in explaining common tendencies in religion suggest the OTF?

The OTF is no different than the prince in the Cinderella story who must question forty-five thousand girls to see which one lost the glass slipper at the ball last night. They all claim to have done so. Therefore, skepticism is definitely warranted. This is especially the case when an empirical foot match cannot be had.

Ok, warranted. Not a big deal, what's important is the demand that theists apply the OTF.

The amount of skepticism warranted depends on the number of rational people who disagree, whether the people who disagree are separated into distinct geographical locations, the nature of those beliefs, how they originated, how they were personally adopted in the first place, and the kinds of evidence that can possibly be used to decide between them. My claim is that when it comes to religious beliefs a high degree of skepticism is warranted because of these factors.

These can be factors, but they are not decisive. Anyways, TELL ME ALREADY.

If she refuses to [apply the OTF] then she must justify having such a double standard. Why does she test other religious beliefs differently than her own? For someone to object that what I’m asking is unfair, she has the burden of proof to show why her inconsistent approach to religious faith is justified in the first place.

Pardon me, but assessing `other beliefs' as `outsiders' and our own as `insiders' applies to everyone. About anything. Forget religion. That's not `double standards', that's the necessary product of having beliefs. A `double standard' with respect to the evidence and argument, if it should emerge, should emerge in the normal way... in the course of normal argument. The OTF adds nothing to this.

I suspect I'm not going to get anywhere on the is/ought gap today. And I see I'm not going to get anywhere on the `X is warranted/X is binding on all reasonable people' gap, either. Oh well, I can do other things:

Nonetheless, if after having investigated your religious faith with the presumption of skepticism it passes intellectual muster, then you can have your religious faith. It’s that simple. If not, abandon it like I did. I suspect that if someone is willing to take the challenge of the outsider test, then her religious faith will be found defective and she will abandon it along with all other religious faiths, like it has me.

See, I'm not sure that a positive or negative result for the OTF translates into any probability threshold, and by that virtue it need not translate into any final stance on one's beliefs. This ignores completely the problem of the priors. It ignores completely the incompleteness of evidential calibration that we have. It assumes from the outset that believers should adopt for-the-sake-of-examination probabilities about skeptical alternatives - yes, positive claims - and apply them to religious claims, alternatives which are presumably non-binding to adopt in the first place. Else, why insist on the OTF? The OTF would merely be restating an undeniable state of the evidence, not a norm mysteriously implied by the existence of other religions. So then what happens when believers adopt their former commitments, something which is apparently not intrinsically unreasonable to do?

A believer who failed the outsider test would conclude, and need only conclude, that skeptics of her faith can be reasonable. It would not entail that her beliefs are unwarranted and incapable of reasonable commitment. At all. And I'm not sure what the point of the OTF is if not that.

I now move on to some other objections. As far as I can tell, none of his responses to potential objections substantively address of mine, but he does say interesting things in his response to one objection:

After all, someone can be right if for no other reason than that she just got lucky to be born when and where she did. But how do you rationally justify such luck? This is why I’ve developed the challenge of the outsider test in the first place, to test religious faiths against such luck.

Here, Loftus is definitely saying something about prior distributions. Namely, that believers must equivocate across contradictory religions in assigning prior probabilities, or that they should match the likelihood of the truth of their religion to the `chance' of their believing it.

First big problem: the demand is impossible if you believe that the number of possible religions and/or gods is uncountably infinite. No non-zero probability of each particular god/religion is possible if that number is (countably) infinite.

Second big problem: the initial facts Loftus describes may not be neutral with respect to some initial distribution. The fact that a religion has died out can affect its probability. The fact that Christianity is `a religion amongst others' does not entail that there are no important qualitative differences between Christianity and other religions which could lead someone to reasonably assign it a quite significant prior: we lack coherent descriptions of large numbers of extinct religions or sufficient detail to distinguish them from others apart from relabeling of terms. Of extant religions, many lack coherent, unified systematic treatments like that presented by Christian philosophers who have spent centuries dealing with rational argument.

Third big problem: if the initial confidence in a religion is to be determined by the fact that there are many other religions, and we should equivocate across them, we make very big commitments to features shared by most or all of the important contenders. In particular, the OTF would have us practically assume supernaturalism and some supernatural cosmogony, though any particular version may be initially implausible. I don't think Loftus should be so eager to demand the starting point he does. I think he's sneaking in the assumption that materialistic alternatives dominate this space. If that's the case, we are more than past the importance of the number of alternatives. Unless he's asking us to assume supernaturalism, this sure does look, smell, and taste like an unjust demand that believers assume naturalism.

I could say more. Lots more. (Must personal, subjective background experience count for nothing?) But I really don't see the point, since Loftus never presents a sound, valid argument. Since the conclusion of his argument is that supernaturalism should be part of the outside position, I suppose I can be happy about that.

There is a way of preserving the legitimate core of the OTF without lapsing into absurdities and unreasonable dictations. It's called normal argument. Inconsistent treatments of evidence by believers, where important, will show up there. I do not see how the OTF does anything more than to prolong outstanding confusions. I would even say that it does more harm than good to the rational faculties of those who are introduced to it.

Bonus fun-time exercise: What would Calvin say about all this?

Edit 7/25/11: I cleaned up some of the sentences, added a few phrases, and added the last paragraph. I also want to add another observation:

If we should set our prior odds on Christianity to match those of our being Christians - as calibrated by sociological data - we get a very bad result for skeptics of Christianity. What, for example, should be our prior odds on the Resurrection before arguing over the evidence? Normally, and I think correctly, we calibrate this with respect to the well-documented tendency of dead people to remain dead. But what happens if we set our prior odds to the frequency of belief in the Resurrection? Since Christians argue that the Resurrection was an exceptional event, the fact that we normally observe dead people remaining dead would do nothing to reduce these odds before engaging the texts. If we take as a low estimate that Resurrection-believers account for 10% of the population, then our prior odds on the Resurrection will be absurdly high. I think that Christians can very easily overcome such a prior improbability with even a skeptical account of the New Testament. Yes, I think that Christians can easily pass the OTF as presented, at least with respect to the Resurrection.