## Sunday, July 31, 2011

### 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.

## Friday, July 29, 2011

### 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.

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.

## Thursday, July 28, 2011

### 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.

### Tim and Lydia McGrew on the Resurrection, part 6

[Continued from Part 5.]

We are now ready to focus on D, the facts about the witness of the disciples, and the independence assumptions which the McGrews make in arguing that this multiplicatively contributes 1039 to a final Bayes factor. We have already reduced the cumulative Bayes factor from 1044 to 1041 by arguing that Paul's conversion does not significantly contribute to the likelihood of the Resurrection. As noted in the last past, we are also accepting the characterization of the factual record which the McGrews make, leaving dispute of that question to a more general analysis, the outlines of which are to be found in part 2. The relevant facts, and some of the argument, which I assume in this post are to be found in part 5. You will not understand this post if you have not closely followed the previous discussion.

For simplification, we may assume - and safely - that p(D|R)=1. If you think that this compromises accuracy or is unfair, try ascribing to it a value of 1/2 or 1/10 or even 1/100, and see how much it affects my conclusions. It won't prove to be crucial. There may be room to dispute it, but that discussion falls outside of the matter at hand, which is to question the conclusions of the McGrews while sharing their premises. This done, we may focus entirely on p(D|~R). This single value is the most important that occurs in their analysis, so it merits the closest focus.

Recall that there are 13 disciples in D; I denote them as D1,...,D13. If we assume conditional independence under ~R, then

$p(D|\sim R)=p(D_1|\sim R)\times \cdots\times p(D_{13}|\sim R)$.

Recall that the McGrews do not argue that this assumption is plausible. Instead, they feel that the failures of this assumption tend to favor their argument, that is,

$p(D_1|\sim R)\times \cdots\times p(D_{13}|\sim R)\leq p(D|\sim R)$.

Since the quantity on the left hand side represents the denominator of relevant Bayes factor - where the numerator is 1 by assumption - increasing it decreases the Bayes factor. Given the general likelihood formula (p.26), the McGrews argue in my notation and with our shared assumptions the following (p.42):

$[p(D_1|\sim R)]^{13}\leq p(D_1|\sim R)\times p(D_2|\sim R\cap D_1)\times p(D_3|\sim R\cap D_1\cap D_2)\times\cdots\times p(D_{13}|\sim R\cap D_1\cap\cdots D_{12})$

where p(D1|~R) is an approximate average of the 13 p(Di|~R). From the messiness of this formula, one can see the allure of independence. But remember the crucial detail is the inequality, not the equality, of the left hand side and right hand side in the above formula. And note that regardless of independence, one must explain at least one disciple's testimony without relying on the others.

Here is where we turn to their defense of this thesis (pp.40-46), which, just as in our previous discussion of the strength of a disciple, goes wrong for the following reason: the McGrews assume that the only significant, explanatory alternatives to the Resurrection are those which claim that the disciples were frauds, pranksters, cynical, or the victims of hallucination or delusion. As I mentioned, I am quite willing to agree that the disciples were devout, sincere, and dedicated followers of Jesus, and that they were not likely the victims of elaborate pranks, and that they were not likely to forge conspiracies without good reason. But just as in the discussion of Paul's conversion, the McGrews expend little effort imagining how it is that Jesus' followers may have realistically reacted to Jesus' death. They never say, or do not closely pursue, something like, "given that the Resurrection did not occur, here are several likely ways that the disciples may have reacted..." They instead assume that no such plausible reaction would at all explain their later behavior.

I think it plausible that the Crucifixion occurred and that the Crucifixion surprised the disciples. I doubt they would expect their Messiah to die the humiliating death of a criminal. But I think this for a good reason: I do not think that anyone there would have expected that, except for those who already condemned him as a heretic. Why is this important? Because this is exactly how skeptics argue. They do not need to invent vivid hallucinations to explain the disciples' commitment. Instead, they ask how well-intentioned followers would likely react and argue from there.

For example, suppose that you are one of Jesus' disciples. He has been dead for some time now, and you are frustrated. You are frustrated because you still believe that Jesus was the Messiah, and because you believe that the Messiah has appointed you to spread his message. The end of the world is coming soon, and Jesus will be the judge. It is vitally important that others in these troubled times understand and believe. But unfortunately, the audience is not receptive. What good, after all, is a dead Messiah for liberating the Jews from the Romans and restoring the Law? There are rumors among followers and admirers of Jesus, especially in distant regions, that he could not have died as he did. Why shouldn't there be? Nobody expected this, of all horrors, to happen. But you did not waste your life by abandoning your family to follow Jesus. His moral message is still sound, and the brotherhood of your Christians is still precious to you. The communalistic message of Christianity can provide for the poor and the beggars and others who live as you have lived.

You are faced with a choice: (1) lie, exaggerate, fail to discredit rumors, or otherwise insist that Jesus better fulfilled the expectations of doubters, or (2) quash the rumers, alienate the faithful, and suffer virtually no success against the skepticism of your peers. Pace the McGrews, you have every incentive to choose (1), as do the other Appointed. You are already willing to risk persecution and death for Jesus' teachings. Why risk dying without successfully guiding others to Salvation? The Messiah did not do as you expected, but he will one day return to rule the world, and he had always explained why your mortal expectations were presumptuous before. Yes, Jesus is currently dead, but why should his teachings die before he returns? You've been tested before; now you are ready to pass the test.

Even this scenario, which I do not think too implausible, is too simplistic. The dilemma may have been the same, but the psychology leading to the preaching of the Resurrection was probably nowhere near this crude. If you like, you can imagine other, similar scenarios which you find more plausible. The question you must address is this: were Jesus to die and remain dead, how likely is it that the disciples would have gone quiet, abandoned Jesus' teachings, and renounced their appointed status? To answer this is to partly answer the important question: what do you expect to have happened if the Resurrection did not occur? Do you expect that the disciples might have rationalized the Crucifixion? If so, how? Is a rationalization like the Resurrection story deeply unlikely, say p(rationalization'|~R)=10-10?

It is hypotheses like these, unconsidered by the McGrews, that make the difference and undermine their defense of independence, because we need not dispute that the disciples were courageous, committed men in order to maintain it. In fact, these hypotheses are the product of their dedication. To ask how strongly this affects the Bayes factor is to ask how likely it is that the disciples would not have agreed to lie in the service of the Truth' or would not have been able to rationalize this testimony in some other manner, or at the very least, would not have initially struck down the rumors among the faithful, and eventually come to believe them, or believe that their essence' was true, or some other such thing. I do not think I need to list precedents of such behavior.

I think that this consideration alone, pursued in detail, is enough to push the Bayes factor way down; I think that others cleverer than myself can explore other plausible alternatives as well. But if we assume that this hypothesis does not share the bulk of p(~R), it is worth asking what the rest of the elements in ~R might look like. The McGrews appear to assume that it is dominated by completely un-explanatory hypotheses, i.e. H:p(H|~R)=0, but I do not think that this is the case. At the very least, I think that this is a very poor approximation whenever we start tossing around numbers like 1039.

How exactly should this affect p(D|~R)? Suppose that the hypothesis I outlined with respect to a single disciple gives us something like p(D1|~R)=0.01, an estimate which I think rather conservative. Now we ask how this should affect the other disciples: here, look at the same dilemma again, adding that other disciples are already claiming R. By the time we reach 7 or 8 disciples, the others follow very naturally. In this way, we can be sure that the independence assumption is overly hard on skeptical alternatives. More formally, we can give a tentative estimate of p(D|~R)=10-4 or p(D|~R)=10-5, and plausibly something higher.

So a tentative readjustment of the multiplicative contribution of the disciples to the cumulative Bayes factor is 105. I think that this is generous, and I think that skeptics can ground this or a lower value. But I am still admitting the following: given the characterization of the texts and the history of the McGrews, the testimony of the disciples is best explained by the Resurrection. What I dispute is how significant this detail must be.

Taking stock, our reduction of the Bayes factor is from 1044 to 107. The last fact which the McGrews discuss is the testimony of the women (W). That will be the topic of my next post. After that detail - perhaps unimportant relative to the other aspects of the argument - I can round all of this noise up and see where it leaves us.

### Tim and Lydia McGrew on the Resurrection, part 5

[Continued from Part 4.]

In addition to the items summarized in the last post, we have now seen that there is little reason to regard Paul's conversion as itself making significant contribution to a cumulative Bayes factor, even when taking the relevant salient facts of the McGrews at face value - where they are consistent and actually salient. We have therefore reduced the McGrews' final Bayes factor from 1044 to 1041. Admittedly, this change is very slight. As previously discussed, the most important features of CCRJ are as follows: the independence assumptions and the effect of the disciples, particularly in conjunction. These ideas alone contribute a multiplicative factor 1039 to the McGrews' final likelihood ratio. Before discussing the details surrounding the disciples, their independence should be treated first.

The McGrews do not argue that their conditional independence assumptions are plausible. On the contrary: "the invocation of independence assumptions at several points is contestable; in fact, we believe that the case of the calculation for D the independence assumption almost certainly breaks down." So why do they invoke it? Because "this fact does not necessarily lessen the strength of the argument. Everything depends on the balance of considerations regarding the direction and extend of the breakdown of independence under R and ~R" (p.40). This approach is quite sensible: start out with an admitted simplification, critique it, and argue that the critiques generally favor the proponent of the argument.1 I think that the only other even approximately accurate response available on the web - at least that I know of - fails to adequately address this.

Allow me to motivate the discussion by being self-referential. If one grants a significant number, say n, of events Ei where i is a positive integer less than or equal to n, and one can assume that the Ei are conditionally independent under a hypothesis H and its negation ~H, then even a very small individual factor

$\frac{p(E_i|H)}{p(E_i|\sim H)}$

is a big deal. With only 13 witnesses, even assigning a Bayes factor of 10 to each individual - even this seems uncharitable - still gives you a cumulative factor of

$\left[\frac{p(E_i|H)}{p(E_i|\sim H)}\right]^n=10^{13}$,

where the notation is exchanged as needed. This is a big number, even if its nowhere near that proposed by the McGrews.2

As we noticed, the McGrews will not be surprised if we directly challenge independence. Instead, we have to focus on pages 40-46 of the CCRJ. In order to do this, we have to start discussing and disputing facts about the disciples. This is because of the straightforward fact I noted in the other post in the context of evidential variety: the invocation of independence across a significant number of events filters out' hypotheses which do not convincingly allow these events to be correlated. Less abstractly, the fact that the disciples not only witnessed as they did individually but did so in agreement must be plausibly explained by some element in ~R, if we grant that agreement.3 Bad' hypotheses, for our purposes, are those which are intrinsically implausible and those which are only plausible with respect to one or a few disciples. Even if a hallucination happened to plausibly explain the witness of one disciple, it does not without lots of dubious conjecture help to explain the correlation of the witnesses.

The key facts about the disciples4 are as follows:

1. They were not expecting the Crucifixion or Resurrection in advance.
2. They were willing to die - or at least take great risk - in order to spread Jesus' message. (pp.17, 20-24, 30-37.)
3. There was little to no material or social advantage in preaching this message.
4. They attested to a physical resurrection. They claimed to have seen him and spoken with him for some lengthy period of time, e.g. 40 days. (pp.17-8)
5. They did not present a self-flattering account of the Resurrection. For example, they claimed to not readily understand, to be fearful, to have doubts, and to be chided and corrected by Jesus. (pp.17-8)

A hefty confidence in these is required to stick to the McGrews' analysis. For example, the importance of consensus amongst the disciples, and that there were no omitted complaints from purported witnesses, is of central importance (p.32). While I agree with the McGrews that it is very difficult to seriously attribute most or all of the disciples' testimony to cynicism or fabrication, my confidence in the uniformity of their testimony with respect to these key facts is much less. There is little reason, had there been substantive dissent, to expect that it would have made it into the gospels. Still, my purposes at this time are not to dispute these items. However, such discussion will be an important part of a general analysis like that presented in part 2.

Now, I agree with the McGrews that external' theories, e.g. the stolen body hypothesis', do very little to explain the facts in question (p.32). The point is to account for these facts in a convincing manner, and their origin is not at all explained by people stealing Jesus' body to corroborate the claims of early Christians. What we are looking for are internal theories', i.e., those which convincingly explain the actions and statements of the disciples. It is here that I think the McGrews grossly underestimate the strength of potential naturalistic alternatives.

Before moving on, I should say something about the significance of these facts, especially (5). The idea is that were the disciples to be manufacturing stories or exaggerating in order to advance their cause, we might reasonably expect them to not be so hard on themselves. I do not think, especially in the context of Christianity, that this is a significant consideration. I think we are all familiar with examples of false humility, or we understand at the very least that confessions of guilt and personal inadequacy are very often encouraged in Christian communities, even admired. Perhaps my personal experience is overshadowing the general picture, but I understood early as a child that confession of guilt in church and begging for redemption was admired and expected of me. That the disciples do not flatter themselves is insignificant in the context of Jesus' teachings, quite apart from the Resurrection, "for all have sinned and fallen short of the glory of God." There are similar considerations for (1). Evidently, and I think everyone involved agrees, Jews did not expect that the prophesied Messiah would be anything like Jesus. That the disciples understood, empathized, or even shared in part this lack of expectation is not overly surprising. What matters is how we would expect the disciples to "normally behave" after Jesus' death.

To address naturalistic alternatives to the Resurrection which explain (1)-(5), the McGrews rework in probabilistic terms the good ol' "nobody would die for a lie" claim. This argument has its merits, as far as it goes, but it does not go as far as the McGrews think.
It is sometimes urged that kamikaze pilots, suicide bombers, and Nazis were willing to give their lives for what they believed was true. The objection may be put more broadly. Virtually every religion has its zealous adherents who have been willing to die for what they believe; why, then, should the willingness of the apostles to die as martyrs be of special epistemic interest? The answer is that this description blurs the distinction between the willingness to die for an ideology and the willingness to die in attestation of an empirical fact. (p.32) [citations omitted.]
It is clear that neither kamikazes, Nazis, nor suicide bombers died to affirm the reality of something that they had seen with their eyes and their hands had handled. Thus, their deaths and the falsehood of some of their beliefs tell us nothing about the probability that a man will die to make an affirmation like that of the apostles when it is in fact false. The educational resources of an entire nation, applied over the course of a decade or more to minds at their most impressionable stage, may be sufficient to induce in the young the general belief that their country or their religion is worth dying for. But what would induce grown men to break with the religious community in which they had been raised and confess with their blood that they had seen with their own eyes and handled with their own hands their dead rabbi raised again to life? (p.33) [Emphasis in original.]
The McGrews partially answer their own question or are really asking it to themselves. The disciples had already broken with their religious background to follow Jesus - with the possible exception of James. They were already committed to the heretical philosophy of Jesus. They were not merely dying to attest the Resurrection; they were dying for the belief that the Messiah had arrived, that only His followers would find salvation, and that the end was near. The distinction which the McGrews make is valid, but it is extremely narrow in this context.5 To divorce the Resurrection from the other claims, moral and otherwise, of Christianity is misleading.

The previous considerations apply to the "disciples witnessed but did not really believe" section of alternatives. Now we can look at the "disciples witnessed and really believed" section of alternatives. Here is where we find claims about delusion and hallucination:
Suppose, on the other hand, that the witnesses did have good reasons for their belief in the resurrection but were nevertheless mistaken. How is this supposed to come about? The hallucination theory has at least this advantage over both external naturalistic explanations and the appeal to enthusiasm: the supposition that the disciples suffered from sufficiently vivid and persistent hallucinations provides the resources to explain why they firmly believed they had seen Jesus risen. But this gain in explanatory power comes at a prohibitive cost in prior probability... (p.33) [Citations omitted]
Instead of looking at how the McGrews corroborate this last statement, try imagining for yourself why this should be. Is it really at all likely that these committed disciples should individually experience vivid hallucinations or powerful delusions on the event of their Messiah's death, and that these delusions, if that is what they were, should have the staying power that they did? How common are events like this at all? If we want to substantially reduce the Bayes factor proposed by the McGrews in a way that should be at all convincing to both skeptics and believers, we need to look elsewhere for alternatives. As the McGrews note, "the problem with the hallucination theory is that it has a vanishingly small probability conditional on ~R. The sort of complex, repeated, integrated hallucination that would be required to maintain even one disciple's testimony and willingness to die for it would represent a serious mental illness" (p.35). There are qualifications to the McGrews' argument that I would add, in particular that extreme distress can very plausibly result in a resilient, complex delusion, and need not require the positing of serious mental illness, but I will not dwell on these details. The fact remains that there is little reason to think such a thing happened, or would likely happen, and ultimately result in the texts as we find them.

Assuming plausibly that p(D|R) is quite high - say 1 for purposes of simplification - our attention focuses entirely on p(D|~R). We want to know whether or not p(D|~R)=10-39 is at all realistic. Although I agree with the McGrews that hallucination, delusion, and cynical fabrication are very unlikely, we still need to know whether their disjunction is well below 10-39. Unlikely as I think these are, I think that they are at least slightly more credible than that. However, if we are to propose a substantial increase in p(D|~R), we are going to have to look elsewhere. Namely, we want to look for alternative hypotheses H such that p(H|~R) and p(D|H) are significant. In the case that H is fully explanatory' of ~R, the former quantity is p(H)/p(~R). Assuming that our prior confidence in ~R is high, this approximates the absolute probability p(H). The moral of the story? If you are proposing alternatives to the Resurrection which are designed to completely explain it, you have to be careful to avoid ad hoc thinking: you have to argue that all concerned should find that alternative explanation plausible for reasons other than that the Resurrection has low prior plausibility.

Failing to find any explanatory hypothesis of significant probability, the McGrews assign p(a disciple'|~R)=0.001, which with independence gives a Bayes factor of 1039.

In order to discuss significant alternatives, question their defense of independence, and propose an alternative Bayes factor, we need to compare the independence case to the more general likelihood formula, which they provide in a footnote (p.26). This work will be formal, abstract, and address the most important part of their argument, so I give this work its own post.

1. To avoid confusion, the vernacular sense of independent witnesses' is completely opposite to that of the probabilistic sense of independent witnesses'. What we expect witnesses to do, if none influence the other and the circumstances are clear, is to produce correlating accounts. That is, what one witness says usually strongly affects what we expect other witnesses to say, which is the very opposite of conditional independence.

2. We'll see that the disciples should contribute heavily to the Bayes factor if one is willing to grant near-certainty to all of the important premises entertained by the McGrews. I do not think that skeptics should want (or need) to conclude something other than the following: if the gospel accounts are accurate as regards the relevant, secular evidence for R, then they strongly evidence R. I accept what probability tells me I must, if I assent to the following:
In other words, is there anything in the reports coming out of the first-century church that is more like what you should expect if Jesus was raised than if Jesus was not raised. If the answer to that question is yes, then the evidence confirms the resurrection, but it might still be rejected by reasonable people on the grounds that a Resurrection would commit you to the existence of God, or other features of Christianity that you consider to be improbable. Fine, but you can at least say, in response to the evidence, that the evidence directly bearing on the resurrection of Jesus is easier to explain if the Resurrection occurred than if it didn't. In other words, we can isolate one particular piece of evidence from the total evidence we have that bears on the issue and ask whether this piece of evidence supports the Christian claim that Jesus was resurrected or not.
Again, I think that the most important part of rejecting confirmation of R as derived from the Bayes factor is in rigorously defending a very small prior odds and presenting at least a very minimal case for skepticism as regards characterizations of the text like those employed by the McGrews. Both of these feats are easily done, I think, but I leave them to my betters for now.

3. There's much more to be said about this, but that discussion falls well outside of the salient facts as characterized by the McGrews.

4. Recall that there are 13, counting the original disciples minus Judas, Mathias, and James the Just (called Brother of Jesus'). The details about James are sufficiently different that I would in future analyses recommend separating him from the rest of the disciples. In particular, he is argued to not have been a believer before Jesus died (p.22), and quite plausibly was not present to witness the events attested by the others (p.34).

5. I add also that we have many cases where people are willing to die for false witness. Does anybody here watch House? Have there ever been false confessions to murders? I'll leave this tangent to others.

## Wednesday, July 27, 2011

### Tim and Lydia McGrew on the Resurrection, part 4

[Continued from Part 3.]

I hope that by now readers - you better have read the previous parts if you are reading this! - understand that the McGrews' CCRJ cannot be dismissed in an ordinary blog comment which spells out things that are intuitive and obvious to a layman. I hope that I've convinced you already not to invest too heavily in highly dubious alternative explanations to the Resurrection. (I would hope that I wouldn't have to say that.) And I hope you have some idea of what to expect now that we are now actually calculating a Bayes factor to replace that proposed by the McGrews on their assumptions, roughly how we can expect to do that, and the limitations on our conclusions which that method imposes.

As I mentioned in Part 3, I for some time avoided mentioning explicitly the set of salient facts. As one reason, I gave that I do not accept the relevance of that set of salient facts. In particular, I do not think that the conversion of Paul should significantly affect a cumulative Bayes factor. Now that we've checked the depth of the waters so as to avoid high-diving into a kiddie-pool, we may now safely plunge into the subject. Recall that the McGrews place a value of 103 on the Bayes factor derived from conditioning on Paul's Damascene moment (See CCRJ pp.37-9 for their argument):

$\beta_P=\frac{p(P|R)}{p(P|\sim R)}=10^3$

More specifically (p.39):

$p(P|R)=10^{-1};\text{ and }p(P|\sim R)=10^{-4}$.

As against this, I propose that

$\beta_P\approx1\longleftrightarrow p(P|R)\approx p(P|\sim R)$,

where the bi-directional arrow denotes the equivalence of these claims. That equivalence is mathematically trivial, but conceptually challenging: why should I expect Paul's conversion to be equally likely whether or not the Resurrection occurred? I give a hint - also mentioned by the McGrews but not adequately pursued - that there is a qualitative difference between P and the other facts - recall that namely these are the women's testimony (W) and the witness of the disciples (D). I'll give another hint: how significant would Paul's conversion be in contrast to the range of reasonable argument over the contribution of the disciples - which alone, to the McGrews, is 1039 if you recall - if the factor produced by Paul's conversion can only be convincingly argued to be something as low as 5 or 10? In answering these, understand why I am not requiring that all involved recognize an exact equality. Given this sense of approximation and that I will be successful, reintroducing Paul's conversion as a significant fact should only be done after accepting that the more direct witness testimony is far less powerful and that the important range of priors includes some relatively large odds on the Resurrection.

The McGrews expend most of their brief effort on p(P|~R) and may not give any argument for p(P|R) at all. They say that "on the assumption of R there is no difficulty whatever in accounting for P" (p.39), making no other argument, unless their emphasizing that Paul would later insist on a physical resurrection is significant (p.38). On the contrary, I do not think that the facts around Paul's conversion support this no difficulty' assertion at all. To see how this is the case, let's think about how their comments at the expense of a significant p(P|~R) reflect on p(P|R):
Delusions that change the minds of vicious persecutors and transform them into faithful martyrs are unfortunately quite rare; one looks in vain for comparable conversions among the notorious murdering zealots of the ages. (p.38)
While I think we probably can find cases with important parallels, I'm not sure that the McGrews want to insist on this statement. If the Resurrection had occurred and is supposed to make at all probable that someone like Paul would be transformed through a vision into an apostle, why then are Pauls so rare? Do we lack similar circumstances where a persecuted Church is suffering under a hostile regime? We can ask other uncomfortable questions as well. Why would we expect Jesus to wait for martyrs like Stephen to become such before making Paul a martyr? Is that how we should expect Jesus to send messages? Why should skeptics accept that, and further, if that is the case, why are Pauls so rare? And why Paul? If he was, as the McGrews insist, an unrepentant, guiltless zealot up to the Damascene moment, why not appear to more powerful persecutors? Why wait until the 4th century to send the Emperor a sign? Nothing about the Resurrection makes Paul's conversion likely, and if it did, we may generate a powerful argument against the Resurrection from the rarity of such conversions.

Before going further, something needs to be said about the inconsistencies in the scriptures describing Paul's conversion. I agree with the McGrews that one should not make too much of the minor inconsistencies in scripture, which they admit while pointing out that this is what we expect of any historical text (p.6). I'm not interested in inviting the hyper-literalists to the discussion. But I think that they are incautious wherever those non-fatal inconsistencies emerge. As I will discuss later, this is also important when they are describing some (relatively minor) secular facts surrounding the witness of the disciples. I think, in the argumentation for Paul's contribution especially, that the McGrews are sometimes uncharacteristically lax, uncharitable to skeptics, and preemptively dismissive. But then, this essay is part of an anthology, and there may have been space constraints. This would not excuse the casualness with which they approach these details, but it is worth keeping that eventuality in mind, along with the tentative and very restrictive nature of their conclusions.

To support these strong statements:

1. They claim that Jesus appeared to Paul in the flesh (p.38). To cut a quote from Loftus out of context and into context:
...Paul [claimed] to have experienced the resurrected Jesus in what is surely a visionary experience (so we read in Acts 26:19, cf. II Cor. 12:1-6; Rev. 1:10-3:21--although he didn't actually see Jesus, Acts 9:4-8; 22:7-11; 26:13-14)...
I have not seen McGrews cite Revelations, but Acts and II Corinthians are in necessary, regular appearance, and it is in Acts that it is not said that Paul actually saw Jesus, but only heard a voice. For example:
And he fell to the earth, and heard a voice saying unto him, Saul, Saul, why persecutest thou me? And he said, Who art thou, Lord? And the Lord said, I am Jesus whom thou persecutest: it is hard for thee to kick against the pricks. And he trembling and astonished said, Lord, what wilt thou have me to do? And the Lord said unto him, Arise, and go into the city, and it shall be told thee what thou must do. And the men which journeyed with him stood speechless, hearing a voice, but seeing no man. And Saul arose from the earth; and when his eyes were opened, he saw no man: but they led him by the hand, and brought him into Damascus. (KJV Acts 9:4-9)
This passage also says that Paul's companions heard a voice. That is explicitly contradicted elsewhere. This the McGrews recognize and accordingly claim to place no weight on the experience of Paul's companions earlier in CCRJ (p.24). They do appear to insist on the visual hallucination (p.25), however, but I'm unsure as to whether the difference between a visual, as opposed to merely auditory, vision is supposed to have a great effect. They rely heavily on the complexity of the vision' - I wonder where that is - in dismissing delusion and hallucination as explanations of Paul's conversion. (Complexity-of-recall concerning hallucinations would validate the seeings of many a schizophrenic.) This does not change that Paul's conversion is almost surely sincere and almost surely involved a dramatic moment - more on this next - but I do not think that the McGrews were sufficiently diligent in explaining what exactly they reject, what they allow others to reasonably reject, and how those details should or should not be significant in making their case. And they did not make clear at the outset that the account of Paul's subsequent blindness would be involved, as it turned out to be, just as the supposed reaction of his companions on the road to Damascus would later be employed (p.38).

2. More culpable is their treatment of a hypothesis proposed by another scholar:
Perhaps aware of just how feeble [explanations like Paul not really converting or that he was the victim of a clever prank] might be, Strauss suggests delicately that Paul might have been overcome by feelings of doubt and guild during a thunderstorm [citation omitted]. This remarkable conjecture might be worth discussing were it not for the fact that the doubt, the guilt, and the thunderstorm are all invented out of whole cloth. Having made the insinuation, Strauss wisely drops this hypothesis and takes refuge behind the claim that the book of Acts cannot possibly be historical. (p.38)
Let's stop and ask: is the suggestion that Paul - evidently a man capable of great feats of conscience, as zealots often are - might have felt guilt over his actions ridiculously implausible? Even SS Concentration Camp guards occasionally felt guilt and uncertainty. This is dismissed out of hand, but the previously cited passage (Acts 9:4-9) makes the accusation of whole-cloth fabrication that much weaker: "it is hard for thee to kick against the pricks" naturally translates as "you know what you're doing is wrong, and it bothers your conscience." It appears elsewhere in the KJV, and does not appear in the NIV in Acts 9, so I'll leave whether or not this passage is crucial to those with more experience in textual criticism. It's just not obvious prima facie that Strauss was wise to drop that very very silly and unserious hypothesis of his. No effort was made here to state the case charitably. No mention of the fact that Strauss' proposed external stressor, a thunderstorm, was not crucial to his idea. Any strong stimulus or external stressor may have triggered a dramatic event, assuming Paul was predisposed to conversion. And though the McGrews say that it is "an odd sort of hallucination that is followed by three days of blindness," they give no reason why that blindness - if at all considered a common fact - should be expected (by skeptics) had Paul received a genuine vision (p.38). Nor do they mention that temporary blindness is known to be associated with severe emotional trauma and stress.

I agree with the McGrews that Paul can not credibly be thought of as a conscious fraud. I think that his conversion was sincere, and it is very well evidenced, as is Paul's historical personage. But they fail to imagine why Paul, a sincere convert, might manufacture a vision story because he's a sincere convert. Imagine that you are Paul: you have been overcome with horror at the guilt of your actions, and you wish to make amends. As a new-found Christian, you care deeply about Christians and the fate of the persecuted churches. There is, however, a small problem: you do not expect these beleaguered communities to accept a killer with open arms and smiles. You especially do not expect this beleaguered communities to recognize your education and expertise as conferring status upon you. What would you do to convince them? Claim a vision, and proudly, and take the very real risks of doing so.

This emotive-incentive hypothesis', along with a serious look at Strauss' explanation, allows us to safely assert that p(P|~R) is not significantly less than the value of p(P|R), which has no convincing support.

Before moving on to the disciples and the women, a post on the assumptions of conditional independence, its importance, and how the McGrews actually employ it is in order, especially for the 13 disciples. Once this is addressed, I think that the tentative finish line will be in clear view.

### Tim and Lydia McGrew on the Resurrection, part 3

[Continued from Part 2.]

I think that Part 2 alone accomplishes the goals which I set out in Part 1, most importantly, that the Bayes factor proposed by the McGrews, 1044, is a gross overestimate of the strength of the textual record. I also provided Bayesian formulas which are sufficiently comprehensive to be acceptable to all concerned and that provide a framework in which layman and experts can further pursue the various details and arguments. Albeit in passing, I further concluded that a layman, so long as he treats the disagreement in the expert community as even minimally legitimate, may be quite confident in the following statement: a cumulative Bayes factor produced by the textual record should not confirm the Resurrection against any reasonable prior odds on the same, assuming that a well-formed natural theology which confers a quite substantial prior is not in play.

So why pursue the topic further? As I mentioned, it is possible that natural theology can do the prerequisite work; and it may one day be the case that any legitimate disagreement whatever about the salient facts vanishes, unlikely as the latter prospect seems. There are other reasons which I also think are important.

Careful readers might have noticed that I have regularly referred to a set F of salient facts discussed by the McGrews without ever mentioning what these are. This was deliberate: I wanted to avoid thinking that detail overly important in the previous discussion, as I have explained. Another reason is that I do not wholly accept the McGrews' F as being relevant. This is obviously important when analyzing their argument, but it is especially important since the method rigorously implemented by the McGrews takes a fairly common starting point by arguing from salient facts'. And of these, the trouble I mention is quite common, and I think it can only be clearly explained in the Bayesian framework employed by the McGrews.

The McGrews include the following as elements of F: the eyewitness testimony of 13 disciples (The original ones minus Judas, plus Mathias and James the Just) (D), the testimony of the women (W), and the conversion of Paul (P). The McGrews posit the conditional independence of these facts modulo R and ~R. (They do not do this casually. I'll explain what this means and discuss it later.) So their final Bayes factor is calculated as follows:

$\beta=\beta_P\times\beta_W\times\beta_D$

where

$\beta_P=\frac{p(P|R)}{p(P|\sim R)};\ \beta_W=\frac{p(W|R)}{p(W|\sim R)};\ \beta_D=\frac{p(D|R)}{p(D|\sim R)}$.

As I have mentioned, their final Bayes factor is 1044. The contributions of each term are, respectively, 103, 102, and 1039. So obviously, the really big factors are the independence assumptions - these are internal to D as well, each disciple multiplicatively contributes 103 to the total - and the strength of D. Minor variations in that specific estimate will swamp the other contributing factors. That said, I still want to argue the following: the conversion of Paul contributes nothing.

This will be our first voyage on the sea of calibrating a conditional probability, so some things need to be said before we undertake this adventure. In fact, there is so much to say in advance that Paul will have to wait for my next post.

Apologists often claim - and I think need claim to be successful - that the circumstances surrounding the Resurrection are in many important senses unique. And I agree. What this means in practice is that the most important means of calibrating a probability, frequency data, are in many decisive ways unavailable by virtue of there being no suitable reference class. In even more jargony terms, a frequentist, empirical probability p(A) is the number of occurrences of A over the number of trials, N. The set that N counts is called a reference class, and the indeterminacy or disputability of such classes is a powerful objection to objective interpretations of probability like frequency. Even if in a very limited sense some data are available, I do not think they capture realistic' probabilities, as cases are so limited.

So how can we sensibly calibrate such conditional probabilities? I think that there are boundaries on the reasonable values which we might maintain; if you argue for a Bayes factor as small as 2, you are at one or several points uninviting Christians from the conversation, and your pretension to argument is merely an exercise in self-gratification. Yes, I think that cumulatively, the textual record should be recognized as providing an increased likelihood for the Resurrection in any sensible discussion. But do not worry about it too much - the question is the strength of that likelihood with respect to priors. Still, we can at least stop saying that there is no evidence for the Resurrection or Christianity.1 However, if we are faced with numbers like 1044, we should be immediately suspicious: how can you assert such a bold number with any confidence when we lack frequency data and are discussing unfamiliar events?

I think that the McGrews attempt this in a sensible, plausible way, though it only constitutes a rough beginning. To put it loosely: they look at the relevant conditional probabilities concerning a fact. They claim, usually plausibly, that the numerator p(fact|R) is high. Then they attempt to show that outstanding alternative explanations affecting the denominator p(fact|~R) are not plausible. But they are not pretending that this is comprehensive. They clearly illustrate and discuss why many of the important alternatives to the Resurrection are not at all explanatory of the fact in question. I will quote from this explanation, since it helped me to correct a common error in probabilistic thinking, namely the over-estimation of the importance of unlikely alternative hypotheses:
...we must be on guard against a plausible error. It might seem that our analysis of cumulative case arguments in terms of Bayes factors puts the emphasis on likelihoods in such a way that finding any sub-hypothesis under ~R that gives a high probability to some piece of evidence always represents a significant gain for the proponent of ~R. But when an auxiliary hypothesis Ha is very improbability given ~R, its contribution to the explanation of a fact F is negligible even when it has high likelihood [...] It is easy, also, to slip into a different false assumption -- that in making a probabilistic argument of the sort in question for R, we are obliged to restrict ourselves to those sub-hypotheses under ~R which make some attempt to explain the facts in question. (CCRJ, p.27)
I omit a formal illustration in the ellipses, but I think this sufficiently important that I have talked about it elsewhere already. The moral of the story here is that by focusing on hugely improbable theses like mass hallucination', skeptics are needlessly driving themselves to distraction and discredit.2 A very important question is as follows: what sub-hypotheses actually dominate ~R in the sense that p(sub-hypothesis|~R) is significant? The McGrews answer correctly as follows: "The answer is that most of it is going to the generic hypothesis that Jesus died and that all went on as usual thereafter," but they continue with a more difficult assertion: "which provides no explanation, not even an attempted explanation, of the evidential facts in question" (p.28).

As I will elaborate when discussing the particular facts, the important question is as follows: what exactly does normal mean when we've accepted the background assumptions of the McGrews? Would we expect as normal' in the circumstances that the disciples would abandon the teachings of Jesus altogether and fail to evangelize?

As we lack suitable frequency data or other accepted calibrating methods, we have to rely heavily on intuitions - intuitions about expectations of the psychology, interests, and behavior of 1st-century followers under duress in exceptional circumstances. One may think of the method as a psychological analogue to propensity: one attempts to empathize with the people in question and attempt to imagine what they might do. A very limited knowledge of their circumstances, cultural differences, and a layman's knowledge of the topic means that we have to be very cautious. It also means we should be suspicious of Bayes factors which are supposed to be large enough to convince us that a well-evidenced, putative law of nature has been violated.

1. 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?

2. You'll notice that this is a common theme of mine. I agree with the McGrews (p.11, elsewhere) that critics of Christianity have often been unreasonably prejudiced against reasonable claims which seem to credit Christianity - and apparently for that reason. I think that this prejudice can be remedied by having skeptics understand that they do not need to invest heavily in this-or-that dubious alternative to the Resurrection by understanding the significance of likelihoods, properly understood. In failing to understand, they send the (perhaps accurate) impression that their views are prejudicial, and where observers are led to believe that skeptics have to rely on an absurd confidence in a mass hallucination to argue against the Resurrection, they will understandably feel that the argument favors the Christians, or at least feel safe in equivocating between skeptics and Christians on the matter. And they'll tend to make familiar noises about the true ulterior motive of atheists, that though they deep down believe in God, they hate Him and want to live in sodomy, adultery, abortiony, or other sinnery. If you ask me what exactly I think happened in 1st century Palestine, I would say very little to satisfy you. What probability teaches us is that we must learn to be comfortable with generalized alternatives instead of specific, highly detailed explanations. Those are by nature hard to come by with any confidence on any issue, much less issues of distant historical inference. Proving a particular negative is very difficult, but a generalized negative like ~R is often manageable. As we will see, it requires work and some cleverness, but it can be done.

### Tim and Lydia McGrew on the Resurrection, part 2

[Continued from Part 1].

Recall the odds form of Bayes' theorem:

$\frac{p(R|F)}{p(\sim R|F)}=\frac{p(F|R)}{p(F|\sim R)}\times\frac{p(R)}{p(\sim R)}$

where R denotes the Resurrection and F the set of salient facts. The left hand side of this equation is the posterior odds on R, and the RHS is the Bayes factor (or likelihood ratio) multiplied by the prior odds on R. Recall from Part 1 that the Bayes factor is the primary topic of discussion in the McGrews' article (CCRJ).

The first question to ask is whether or not this is a sufficiently general formula, and I do not think that it is. I think that there is room for reasonable doubt of F as characterized by the McGrews. Further, I think that this reasonable doubt, if established, should limit the cumulative Bayes factor. To see why, treat F as an element of a more general set of characterizations of the texts which we all agree exist; call it History' (H). Then trivially, F and its complement in H partition H.

Before going further, I should defend what I am about to do - you may see it coming already if you are a practiced Bayesian - since the McGrews provide plausible reasons for the approach that they take. They recognize the divisions amongst serious historians (p.4), asking in response to those divisions concerning source hypotheses the following question: "Faced with such a Babel of conflicting voices, what should the interested layman do?" (p.9). Their response is to sidestep inessential(?) questions about sources and propose a well-evidenced set F from which to argue. On the contrary, I think that layman should accept a consequence of a lack of expertise where expertise exists: namely, to be more general, and to account for that in argument. As Bertrand Russell proposes in his Sceptical Essays, a layman should regard no matter as certain where experts are divided. I think that the tools of Bayesianism allow us to formally obey this principle so as to allow experts to continue the argument.

How important is this? It allows us to be more cumulative, though if we pursue the details we are forced to do more work. But as it is, a truly comprehensive case is beyond the capabilities of any single person, as far as I am aware. I do not think we should shy away, and I think I can argue for the all-importance of this work as follows: consider the following generalization:

$\pi_q=\beta_{comp}\times\pi_p$

where this equation is the analogue of the previous. The crucial difference is that we are conditioning on H, not F:

$\beta=\frac{p(H|R)}{p(H|\sim R)}$.

Since {F} is a subset of H - simply F for clarity - it and its complement partition H. So

$\beta=\frac{p(H|R)}{p(H|\sim R)}=\frac{p(F\cup (H\setminus F)|R)}{p(F\cup (H\setminus F)|\sim R)}$

$\beta=\frac{p(F|R)+p(H\setminus F|R)}{p(F|\sim R)+p(H\setminus F|\sim R)}$

Here we can notice something important before attempting to calculate the details. Suppose that given the salient facts, the Resurrection is certain, i.e. p(F|R)=1 and p(F|~R)=0. Then this becomes

$\frac{1+p(H\setminus F|R)}{p(H\setminus F|\sim R)}$

As we will see, the McGrews argue from F for a Bayes factor of 1044. Let's see what this requires of the more general formula in order to be even nearly as large as that even on the generous assumptions made so far: given that

$\frac{2}{p(H\setminus F|\sim R)}\geq\frac{1+p(H\setminus F|R)}{p(H\setminus F|\sim R)}\approx 10^{44}$,

we have that

$\frac{1}{p(H\setminus F|\sim R)}\geq \frac{10^{44}}{2}=5\times 10^{43}$.

So for the Bayes factor found by the McGrews to be realistic, $p(H\setminus F|\sim R)$ must be very, very tiny indeed.

The significance of this should be obvious, but allow me to elaborate further: you can accept that the salient facts of the McGrews are plausible - the most probable characterization of the record, even - and still reject their conclusions as outlandishly overstated without calculating an alternative Bayes factor yielded by conditioning on those facts. The reality of proposing cumulative Bayes factors as extreme as 1044 is this: you gotta be damn certain of your assumptions, not merely reasonable in accepting them.

For this reason, I do not have to step into unfamiliar territory and survey the conflicts in scholarship in order to reject such a large Bayes factor. I leave the details to my betters, and I propose a way of pursuing them now.

The partition of H as I have presented it may be refined to make explicit the various hypotheses of interest. Write it out as follows:

$H=\{H_1,...,H_n\}$

Reserve the first blocks for the most accepted hypotheses, which are probably too general to be wholly satisfactory for analysis, and reserve later blocks for outstanding refinements of those hypotheses. To ensure that you are being comprehensive, reserve the nth block for a non-descript other' category. Next, you can apply probability kinematics to this partition:

$q(R)=p(R|H_1)q(H_1)+\cdots+p(R|H_n)q(H_n)$

where q is your posterior probability and p is your prior probability. Note that in this case the debate over the record is taking place in q, and the debate over how that record influences probability(R) is taking place in p. Alternatively, one can approach it as I did above using normal conditionalization as follows:

$\frac{q(R)}{q(\sim R)}=\frac{p(H_1|R)+\cdots+p(H_n|R)}{p(H_1|\sim R)+\cdots+ p(H_n|\sim R)}\times \frac{p(R)}{p(\sim R)}$

I did warn you that it would be complicated. To keep it simplish, one can focus on the most important hypotheses and keep the other' block stocked with the rest.

If we accept that a layman may reasonably ascribe odds to characterizations in light of which the evidence for the Resurrection is at all distant from one, we are stuck with far humbler conclusions about acceptable cumulative Bayes factors. I'm not going to suggest a number at the moment, but I think it safe to say this: whatever you think of Hume's argument, the Bayes factor for the Resurrection derived from the textual record is insufficient to overcome a reasonable ascription of prior odds, barring some natural theology which substantially increases the odds from those suggested by treating the Resurrection as a potential exception to a regularity. Up to this specific miracle, I contradict the conclusions of the McGrews otherwise. (If you are feeling ambitious, you can suggest a number and attempt to make this statement general by limiting the evidential strength of historical investigation. I have ideas in this direction, but I leave that for a discussion of Hume's argument.)

I still think it's worth going into the details. After all, such a natural theology might be found. So in the next post, I will begin to analyze the Bayes factor as analyzed in light of the textual assumptions made by the McGrews. I also add that the McGrews mention their dependence on F (p.39) and are not blind to the possibility of saying more about the issue. But I am not sure that they anticipated the extent to which that assumption prevents the strong conclusions which they make.

### Tim and Lydia McGrew on the Resurrection, part 1 (introduction)

When I was beginning to do serious reading on Bayesian philosophy a few months ago, the first thing I wanted to do was this: find examples in action, then figure out what I needed to understand them. In my experience, this is a good way to start studying abstruse topics like philosophy and mathematics. As a beginning, you cannot expect to read a chapter of a technical text and be able to recall the theorems and important argumentative techniques. You certainly cannot expect to re-derive' the chapter you just read. Instead, you absorb the techniques through practice. Like puns, nerds love a good continuity, so I sought applications relevant to my previous interests. What better topic for me could there have been than those same arguments about gods which had served me so well as a springboard into philosophy? Naturally, the first topic that came to mind was one which had always bothered me: Hume's argument against the possibility of confirming a miracle.1 By obvious tangents, I soon discovered Tim and Lydia McGrews' The Argument from Miracles: A Cumulative Case for the Resurrection of Jesus of Nazareth (CCRJ).

I have been writing down my thoughts about this for a while, but I had only been doing so for auto-didactic purposes related to Hume's argument. Since I've only found one (inadequate) rebuttal, I feel that I should share my thoughts.

Here I'll pull a Vonnegut: I will say that I disagree with the conclusions of the paper, and then I will try to realize that in an interesting way. I'm not here to surprise the readers. But I do not want these posts to be thought of as a debunking'. If you come away from these posts feeling that reading the McGrews' CCRJ has been rendered unimportant by my awesomely insightful analysis, then I have failed you miserably. If you come away feeling that the McGrews are lazy, unscrupulous apologists, then I have failed you super miserably.2 They sought to start an analytical conversation and send the topic our way. Stop, read the paper carefully, and please, please try to understand it before breaking the internet. I heartily recommend it as an application of Bayesian principles, and I have learned much from it.

My response will have limitations. First, I'm new to this stuff and they are not. Second, I am not an expert on Biblical criticism and historical method. Third, and partly as a function of the first two, my discussion will necessarily be incomplete. As I hope you will see, I am going to try to build on what the McGrews started. I will attempt to suggest a better formula to analyze so that those with more expertise than myself might converse in a common framework. Since the CCRJ is the best relevant article that I have managed to find, I will use it to open a more general discussion. As I have yet to find any other Bayesian analysis on the same level, I think this the best place to start and I plan on referencing it heavily in the future. If there are other, better presentations, I would appreciate a reference.

As I interpolate it, the core of the CCRJ is as follows3:

1. There are salient, secular facts surrounding the Resurrection (R, which Christians and skeptics can reasonably accept.
2. These facts (F) constitute powerful evidence for the Resurrection.

These are the two items which I wish to discuss. As against these, I propose two counter-statements:

1. The set F proposed in the CCRJ may also be reasonably disputed, and this observation severely constrains the evidential strength of any argument from them.
2. The McGrews grossly overestimate the strength of F as evidence for the Resurrection.

The first Big Issue is the most obvious one: what exactly is this textual record we have, and how should we deal with it? That will be the topic of my next post.

Outline and summary:

1. Part 1: You just read it, I hope.
2. Part 2: I prove that even minor disputes about textual assumptions would undermine the McGrews' Bayes factor as an estimate of a genuinely cumulative Bayes factor. I propose a general analysis to avoid this trap, and one which allows experts to continue the discussion.
3. Part 3: I introduce the McGrews' argument, discuss how one can question it and why, and what one should expect in the details.
4. Part 4: I argue that Paul's conversion does not significantly contribute to the argument.
5. Part 5: I introduce the disciples, discuss their importance, and the effects of independence assumptions. I introduce their arguments and begin presenting my own.
6. Part 6: I present an alternative hypothesis to explain the disciples.
7. Part 7: I discuss the testimony of the women and arrive at a final replacement Bayes factor.
8. Part 8: I make some concluding notes and comments.

1. Yes, pace Keith Parsons, I think that Hume argues that in some circumstances it is impossible to confirm a miracle. No, I'm going to leave that topic for a different series of posts. No, I don't think that his unconfirmability argument is valid, for reasons discussed by Earman in Hume's Abject Failure. I think I have a way of updating the argument, with restrictions, such that it covers the Resurrection, but I leave that for later. For this topic and generally, one can start by reading the (and recently updated) SEP article on miracles, which was itself written by Tim McGrew. In particular, I would like to cite part of the conclusion of that article:
For the evidence for a miracle claim, being public and empirical, is never strictly demonstrative, either as to the fact of the event or as to the supernatural cause of the event. It remains possible, though the facts in the case may in principle render it wildly improbable, that the testifier is either a deceiver or himself deceived; and so long as those possibilities exist, there will be logical space for other forms of evidence to bear on the conclusion. Arguments about miracles therefore take their place as one piece—a fascinating piece—in a larger and more important puzzle.
2. Partly, I feel a need to say this because apologist' is very often used as a curse word. So if I call the McGrews or others apologists', I want to renounce the bad connotations. I am often guilty of using the term pejoratively, so I think it important to give this disclaimer. In other contexts, I have used (and will use) it pejoratively without apology. I think this is legitimate, but generalization as mixed with emotive judgment should be done cautiously.

3. The McGrews are clear about what they are doing and how. They are not here arguing for theism or Christianity. Rather, they are focusing on the Resurrection. They are also not focusing on rebutting Hume's argument or determining what the prior odds on the Resurrection should be. They are arguing for a very large cumulative Bayes factor in its favor. Nor are they claiming that their case is cumulative in the broadest sense; they are restricting the discussion to the relevant textual evidence and its characterization. There are also issues which are inessential to this discussion, such as the proper definition of miracle', which they do not concern themselves with here (CCRJ, pp.2-4).

## Monday, July 25, 2011

### Secular morality and metaphysics

Over at Dangerous Idea, Victor Reppert asks how secular ethics must relate to a metaphysic, suggesting the following: "Even if a personal God isn't required for ethics, doesn't it seem plausible that at the very least some sort of metaphysics is required that most naturalists today would have a hard time accepting."

If I catch Reppert's sentiment correctly, he appears to accept - or at least allow - what atheists call the autonomy of ethics'.1 (David O. Brink has an essay of that name in The Cambridge Companion to Atheism, ed. Michael Martin.) What he is asking is a little broader: what (uncomfortable) metaphysical commitments must naturalists/secularists make in formulating a satisfactory account?

In a trivial sense, some sort of metaphysical statement must be made whenever giving a descriptive account of ethics. But I do not think that secular thinkers find or need find such statements to be problematic. Ethical naturalism, for example, may be defined in the broadest sense as figuring out what we want in fact, and then how to get it in fact. To me, questions of the former type are, strictly speaking, subjective, even if many opinions are widely held. The latter type, the goal-oriented' question, is often capable of uncontroversial, empirical answers.

For example:

1. I want people to live satisfying, intellectually fulfilled lives.
2. Poverty and constraints on liberty prevent the actualization of (1).
3. I should therefore work against poverty and tyranny given (1).

There are of course objections to this sort of ethical naturalism; I think that it is woefully incomplete. But it is a good start, and I see no difficult metaphysical entanglements in proposing such reasoning.

I have a more or less Humean stance on the nature of morality; I think it is rooted in sentiment. But is it not true that abstract notions - principles - appear to affect our moral thinking? I think so, but I do not think that this presents any fatal difficulty. This is because we are capable of sentimental attachment to principles. This is why I, while I would be considered a non-realist in many important respects, am nevertheless a moral universalist, in the sense that I think we can discover approximating categorical statements which usually capture our desires and interests and provide something I think we all should want: a clear basis for law, moral argumentation, and the communicating of judgments. So while "it is not contrary to reason to prefer the destruction of the world to the scratching of my finger", I am still a big proponent of the Universal Declaration of Human Rights. (Were you to ask me why I think it is that we can find approximating principles, I would start to babble about evolution and culture and other things.)

But this can only ever be an approximation: there are psychopaths, after all. So why do we get to imprison them? Because we can, based on stuff we want. (For the Aristotelians, I include widely-accepted virtues' in that category.) I understand that this may feel deeply unsatisfactory, but I do not think that wrong actions' are in the same category as wrong statements'. Is it unreasonable for us to imprison the psychopath? In the sense that the root of our action is not in reason, yes, but it's equally unreasonable to not imprison him. The decision is ultimately sentimental and may even be crudely emotive at the fundamental level.

I am not and cannot be a Kantian. I think that the much-touted categorical imperative is vacuous and that trying to deny the circumstantial nature of moral judgments is flatly absurd. I am not a utilitarian: though a rough idea of general happiness' is part of our moral thinking, I do not think that all the various virtues and desires are subsumed by a one-dimensional measure. I think most people would have a problem with organ lotteries.

I think that people want a different account because they want to still their uneasy consciences about tossing others in prison. They want to be as certain of their moral judgments and prejudices as they are of facts like gravity. They want to be correct, not merely right, about the objects of their strongest passions. In practice, there is little difference, except that those who feel self-righteous tend to be crueler.

But passion is passion, which is not a truth-finding mechanism. I prefer to take a more authentically scientific attitude to morality, which is piecemeal and allows for goal-indifference. Like the piecemeal social engineering' advocated by Popper in The Open Society and Its Enemies, I want to ask questions like the following: "how do we arrange our institutions so as to prevent murder, supposing we should want to prevent it or not?" Properly considered, the moral demands of the skeptical ethicist are more stringent than those of the metaphysical ethicist. The former is forced to discover more and to think of morality feature of her surroundings as created by herself, and in application she will probably need to take far courageous stands and make great sacrifices to fulfill her desires. As Russell commented in his Sceptical Essays, utilitarians, in contrast to traditional moralists, must live a far more onerous life. I think that this counter-intuitive fact holds for active moral thinkers generally, even that of we skeptical types.

So that, roughly, is what I take to be a roughly correct, if overly broad-brushed, picture of morality, which is a mixture of moral skepticism, Humean considerations, and pragmatism. Nothing about this says anything about grounding' any particular value or norm. This project is unnecessary and possibly even misguided.

What I'm not saying: I'm not a moral nihilist. We make moral distinctions, but they are sentimental (See Hume). I'm not a psychological egoist (see Hume again). I think that normal people are altruistic in some degree, although self-interest' is a big part of our makeup. Norms and values are not arbitrary'. Norms are like satisfactory definitions: they capture widespread intuitions and ideas; values are not switches which we flick on and off in our heads.

For pragmatic reasons, I speak in ordinary moral language. I quite regularly say things like "...so we should...", and I often refer to abstract concepts like human flourishing' or even purpose'. But when saying such things, it is important to remember their nature. I may say that humans should read philosophy in order to flourish intellectually - where intellectual flourishing is desirable - but I do not think that this statement is `true' in any normal sense of the term. It is not a fact about the universe or a fact about the mind of God. Rather, it is a sentiment-ridden assertion made by a natural being, and those who cannot empathize with my sentiments may think it curious, or white noise, and would not risk failing to understand some truth about the universe in ignoring it.

I think I'm in the minority with respect to the philosophical community on these stances, but that appears to apply to everyone. Apart from the linked SEP articles (and some of the citations in those articles), those interested may want to follow Russell Blackford, or more famously, Simon Blackburn. The papers on Blackburn's site are excellent starters; Hume's Enquiry Concerning the Principles of Morals is a quick, enjoyable read; Blackford's monstrous review of Harris' Moral Landscape is a good example of moral skepticism in practice. (I haven't read The Moral Landscape, but from this and other reviews, I can't pretend that I'm interested.)

1. This is supported in the comments:
Look, I made a very specific claim. I maintained that the modern secularist typically does not only reject theistic morality, but also rejects morality based on transcendent forms that we can know, and an inherent purpose for human existence. So what I had in mind was leaving theism out of it and trying to see that there have to be some "metaphysical" commitments made in order to have anything that looks like an adequate moral theory. For the purposes of this discussion it would be better if everyone forgot about the fact that I happen to be a Christian theist and addressed the problem I actually posed.

Edit 1 (7/27/2011): I altered some of the wording, but not, I think, in a way that affects the message.