Loke: Investigating the Resurrection

A post was split to a new topic: Vincent Torley’s Response to Loke’s Argument for the Resurrection

Announcement here. @Andrew_Loke will be taking questions on this livestream about the book tomorrow night.

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I’d really prefer not to. (i) At first glance, it seems to be a bad argument (even if McGrew’s attempted rebuttal was worse). (ii) It is irrelevant to the point I was trying to make. & (iii) Loke seems to mangle it sufficiently badly that we’d probably need to find another source to base any discussion on.

I’d be more interested in contrasting Loke’s and Alter’s (as reported by VJ Torley) very different takes on 1 Corinthians 15:3–11. (When I first read that section of Loke, it provoked me to look up the route from Corinth to Jerusalem on Google Maps.)

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Ruling stuff out with non-zero probabilities is irrational.

*If I were to accept this statement I cannot conclude that you said this, because according to this statement I cannot rule out the possibility that an alien wrote this instead of you (this possibility has non-zero probability). Since if I were to accept this statement I cannot conclude that you say this, I shall ignore it for the purpose of communicating with you in what follows, and I shall continue to use ‘rule out/exclude’ for hypotheses which have very low (not non-zero) probability.

You then go on and compare Jesus’ resurrection with thought experiment of documents where one or more people claim to have seen unicorns in the sky etc. I replied that you ignored my point that one should ‘distinguish between the probability of a miracle claim considered apart from the evidence and the probability of the claim given that evidence’ (McGrew 2013), i.e. while there’s no good evidence for the unicorn case there’s good evidence for Jesus’ resurrection, and that we can arrive at a high posterior probability for Jesus’ resurrection using argument by exclusion without having to first assign a number for the prior probability of Jesus’ resurrection

You replied ‘I ignored nothing and accused me of just

(1) handwaving in the direction of “when free agents are involved all bets are off and anything can happen”, and (2) by a rather poor treatment of the disjunction of all naturalistic variants and combination hypotheses together, attacking a particular versions of it (one with some legendary development and numerous identical halluscinations, which nobody here actually advocates), and (3) by completely ignoring all other supernatural hypothesis, apparently in the belief that the “God did the resurrection which actually happened” is the only alternative to all possible naturalistic ones.’

Concerning (1), what you wrote is a MISREPRESENTATION; that was not what I said. Rather what I said was ‘in the absence of evidence the claim is indeed unlikely, not because we can read the mind of God to know that the prior probability that He would not make him tall is low, but because we know that there have been many mistaken claims of miracles. In other words, there are common naturalistic alternatives which—in the absence of evidence to rule them out—are very likely, and given that the probabilities must all add up to 1 this would imply that the miraculous claim is unlikely.’

Concerning (2), you claim that my treatment was ‘poor’ by claiming (2.1) ‘we just have some claims in some documents, aka “for the Bible tells me so” and (2.2) citing the Paulogia video.

Concerning (2.1), you fail to understand the historical-critical method of reading the Bible whereby historians do not simply accept or reject a claim because ‘the Bible tells me so’, rather historians evaluate a claim in a text by taking into considerations the religious, social, and cultural background of the author and readers of the text, their understanding of other texts, their interactions with surrounding cultures, the challenges that they faced etc,’ which is precisely what I did in Chapters 2 to 7 of my book.

Concerning (2.2), the hypothesis in the Paulogia video is similar to Ehrman’s combination hypothesis which I discussed in Chapter 7 of my book and it is likewise contradicted by multiple evidences uncovered by the historical-critical method:

Paulogia claims Jesus placed in mass grave—this ignored the evidence for guards at the tomb (see Chapter 6 of my book)

Paulogia claims Peter has bereavement hallucination…as you summarized: ’two people could have had experiences of seeing Jesus alive for their own reasons, and they do not have to be identical at all. All they need to claim is to have seen him after his death, and legendary developments can do the rest.’—this ignored the evidence that many people ‘saw’ Jesus at once (Chapters 2 to 4 and Chapter 7).

As for legendary development, I did not ‘subjectively declare aspects of it implausible’, rather I note in Chapter 8 that (as you said) ‘there is overwhelming historical evidence, up to and including the present day, of legendary development of easily falsifiable facts ’ (see pages 192-196 of this evidence and also the discussion of Schneersohn pages 162-163), and explained the reasons and evidences for thinking that the case for Jesus’ resurrection is different in Chapters 2 and 3.

You claim that I ‘dismiss the idea that Jesus was left to decay without a burial, or being buried in an anonymous mass-grave, because you say it’s not unlikely Jesus would have been allowed to get a proper burial because this is known to have happened. But possibility doesn’t get you probability . In other words, a crucial aspect of a particular variant of naturalistic hypotheses have here been summarily dismissed by appeal to a mere possibility .’

However, what you wrote above is a MISREPRESENTATION. You fail to note that I offered other arguments in Chapter 6 for rejecting those naturalistic hypotheses, such as the argument for guards at the tomb. This shows that you have not read my book carefully.

Concerning (3), you claim ‘you’re treating the resurrection as the only supernaturalistic hypothesis, as in you think the remainder left over when subtracting the probability of the disjunction of all naturalistic hypotheses, must be the probability of the Christian God resurrecting Jesus.’

But this is yet another MISREPRESENTATION. Having arrived at the probability of (at least) 99.4% for Jesus’ resurrection by subtraction on pages 184-186, I did go on and consider whether this may be explained away by other “supernatural” possibilities on pages 189-192, and explain why they are less likely than God raised Jesus from the dead.

When I said previously that the prior probability of the resurrection is indeterminate, what I meant was we cannot determine that it is low (or high) based on methodological naturalistic considerations alone, but that does not mean we cannot argue for the resurrection by exclusion (without having to first assign a prior probability), and neither does it mean that, having established the resurrection and considering various SUPERnaturalistic explanations for or against it, we cannot take into account other considerations (such as other arguments from natural theology) for preferring ‘God raised Jesus from the dead’ over other supernaturalistic hypothesis (as I explained on pages 189-192).

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Whenever we encounter a claim that something supernatural occur, following Ockham’s razor the first step is to consider the naturalistic alternatives to the claim rather than another supernaturalistic alternative—there is nothing arbitrary and self-serving about this!

Thus, when we look into the historical records and encounter the claim that the earliest Christians claim that God raised Jesus from the dead, following Ockham’s razor the first step is to consider the naturalistic alternatives to the claim (e.g. could it be naturalistic mass hallucination) rather than another supernaturalistic alternative (e.g. could it be supernaturalistic mass hallucination, which btw no one in first century was claiming). This same first step is likewise used for evaluating other miraculous claims (e.g. the claim that Bodhidharma rose from the dead, see Chapter 8 of my book), so there is nothing arbitrary or self-serving about it.

And naturalistic alternatives need to be evaluated by methodological naturalistic method—there is nothing controversial about this!—Whereas methodological naturalistic method should not be used to exclude supernatural hypotheses because a supernatural hypothesis ex hypothesi is not supposed to be a hypothesis about how the natural world when left on its own operates. Nevertheless, if we have a good naturalistic explanation as shown by the application of the methodological naturalistic method to demonstrate the likelihood of the explanation thru the rigorous application of history, psychology, science and other such disciplines.—as in most claims of supposed resurrection— then following Ockham’s razor the naturalistic explanation wins! And naturalistic explanations win most of the time e.g. with respect to the claims of Bodhidharma’ s resurrection, Schneersohn’s resurrection etc (see chapter 8).

However, the case of Jesus’ resurrection is unique in that all the naturalistic alternatives can be shown to fail, and this is the evidence that it did happen (one way to show that something could happen is to show that it did happen!). It is only then that we need to consider whether the evidence can be explained away by other SUPERnatural hypothesis (e.g. supernatural mass hallucination), and these can be evaluated on the basis of other considerations as Daniel noted.

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*Thank you Tim for engaging with the arguments on this thread. The probability that I am talking about is not physical probability, but epistemic probability, which I argue to be very low for the naturalistic alternatives on the basis of evidences presented in my book. I explained on pages 184-186 that the margin of error of these probability estimates is 0.1%. I noted on pages 202-203 that, despite the incomplete historical record we possess (and despite your claims about ‘the length of times, the thinness of source data, the tangential nature of much of the evidence, and the inevitable caveats-and-disagreements-between-experts on the details of what they mean), the following considerations are well established as shown by the evidences presented in chapters 2 to 7 (which you would need to engage rather than simply dismiss):

1 Jesus’ crucifixion in around AD 30.

2 Individuals (including former unbelievers Paul and James) and groups of

people (including Jesus’ apostles who were familiar with him) claimed

to have seen the resurrected Jesus shortly after that.

3 The threat of persecutions against them and their willingness to die for

their religion.

4 The importance of the claim of Jesus’ transphysical bodily resurrection

for the earliest Christians.

5 The reverent fear of being judged by YHWH for being false witnesses

were present among them.

6 The scepticism of bodily resurrection were present among them and

their audiences.

7 The commonsensical idea of ‘checking out’ the ‘eyewitnesses’ were pre-

sent among them.

8 The mobility and ‘networking’ among the earliest Christians.

9 The author of Matthew’s account of the guard of Jesus’ tomb would not

have committed a ‘credibility suicide’ by inventing an easily falsifiable

story for his apologetic purpose.

General considerations:

10 No group of people would be willing to sacrifice everything for what

they do not believe to be true and be condemned by God after death for

being false witnesses.

11 Without a corresponding external stimulation of the relevant sensory

organ, the mental states internal to each person within a group of peo-

ple would not agree on various details concerning their experience of

the external world.

12 Many cases of hallucination do subsequently achieve insight that their

experience is hallucinatory after the experience has ended.

13 No mere human being would have been able to naturalistically cause his

own body to manifest ‘transphysicality.’

14 A half-dead person still suffering from the wounds of the crucifixion

would not be able to convince people that he was the risen Lord of life.

These well-established considerations indicate that the epistemically “small probabilities” are as small as I thinks they are. My calculation that the probabilities sum to 1 has already taken into account the margin of estimation errors of 0.1%, as explained in Chapter 8: ‘For example, it has been argued in Chapter 4 that mistaken reports due to hallucinations, i.e. the intramental hypothesis—widely regarded by sceptical scholars as the most plausible naturalistic alternative to Jesus’ resurrection–would not work because of multiple considerations against it… indeed, collective hallucinations are not found in peer-reviewed medical literature (Bergeron and Habermas 2015). Thus the intramental hypothesis has negligible probability or ‘vanishingly small probability’ to use McGrew’s term… even if one assigns to each of the six naturalistic alternatives a probability of one in a thousand (which is very generous and much higher than the vanishingly small probability calculated by McGrew), that still leaves the resurrection with a probability of 99.4%.’

You raised the concern that, if we reject methodological naturalism, the probabilities of all outcomes (natural or supernatural) are incalculable, as they rely on the whim of whichever supernatural entity (or entities) control the outcome (whether that entity is Yahweh, Ra, Odin, or whoever, and no you don’t get to cherry pick which). We can estimate, given methodological naturalism, and a bunch of date, what the probability will be of rain tomorrow. Given the existence of the Goddess Demeter (for example), we can’t (it all comes down to whether she’s pleased with us or not).’

However, as I said before, I do not reject methodological naturalism totally. What I said was methodological naturalism should be used to evaluate naturalistic hypothesis, and they should not be used to exclude supernaturalistic hypothesis. When considering an event, given Ockham’s razor we should first consider what probability it would occur naturalistically using methodological naturalism (and not assume whether Goddess Demeter exist and would intervene). It is only after we have been able to exclude the naturalistic alternatives (as in the case of Jesus’ resurrection) that we then consider supernaturalistic hypothesis, which can be evaluated on the basis of other considerations as Daniel noted.

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Welcome to the discussion and thank you for your comments and question. The definition of resurrection is discussed on pages 19-22, 48, where I note that understanding the Jewish eschatology portrayed in Daniel 12 is important for understand resurrection as a concept when applied to Jesus.

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But that’s not what you are doing. You are not just considering naturalistic alternatives to a claim that “something supernatural” happened. You are narrowing all the possible “supernatural” options to just the one that you favour, without any rhyme or reason.

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@Andrew_Loke

The probability that I am talking about is not physical probability, but epistemic probability, which I argue to be very low for the naturalistic alternatives on the basis of evidences presented in my book.
My background is in Statistics, not Epistemology, so I hope you will understand why I used the jargon of the former not the latter. However, I think the logic of my argument still applies.

I have seen epistemic probabilities defined as degrees of rational belief (or in the case of naturalistic explanations, under your argument, disbelief). My point is that each time you make an assumption, supposition, and/or what Vincent called “hypothetical reasoning”, the rational solidity of the belief (or disbelief) derived from that line of argument becomes additionally fuzzier.The more assumptions, and the more heroic or disputed the assumptions, the fuzzier the disbelief, until you might get to a point where a disinterested observer might well simply squint and shrug, as all there is is fuzz.

the following considerations are well established as shown by the evidences presented in chapters 2 to 7
  1. What you presented were arguments, not evidence. Take one example, that caught my eye, that of questioning the ‘500 witnesses’:

a) You must assume that the journey would be sufficiently practical (in spite of the long distance from Corinth to Jerusalem, hazards, disruption to livelihood, etc) and proportionate (that even the most fanatical of skeptic would judge such a journey worth while, just to test a single claim in a letter) to have a reasonable likelihood of occurring.

b) You explicitly assume that a list of the 500 names might have existed, and implicitly further that it was available (e.g. because a copy had been distributed to each Christian outpost), but that no record of this list has survived.

c) In doing this, you must assume that the list had been created in the first place. This would have taken quite a lot of effort. Look at how much difficulty police routinely have tracking down far smaller groups of eye witnesses, even with modern communications and legal authority.

d) You must assume that the intrepid skeptic was able to track down eye witnesses, who (even if still alive) may well no longer be in Jerusalem, be misidentified on the list, or merely simply have too common a name to be practical to track down.

I’m sorry, but based on this, I have little epistemological certainty that the claim of the 500 eye witnesses was well-tested, and can be accepted as prima facie factual.

  1. On the subject of “evidence”, evidence can be misrepresented. I would be far more credulous as to the ‘superiority’ of “our present critical Greek text of the New Testament”, if I did not already know that the majority of 5,839 manuscripts date from the 12th Century and later, more than a millennium after the events they are meant to be attesting to. Likewise I believe I have read that the “dozen or so of these manuscripts have been dated to have been written within 150 years of the originals” are fragments.
which you would need to engage rather than simply dismiss
Actually, it is very easy to dismiss your list: in legal jargon, the phrase would be "objection you honor, assumes facts not in evidence." (And no, your arguments are not evidence.)

I would describe your list as “widely asserted” (among Christian apologists) rather than “well established”.

Your asserted “margin of estimation errors of 0.1%” would be quite low, even under many modern laboratory conditions, to assert them for a lengthy argument, based on a large number of disputed interpretations, of an event 2000 years ago is utterly implausible.

However, as I said before, I do not reject methodological naturalism totally.
This is rather like being "a little bit pregnant". It's an all or nothing thing. Either you accept MN and are willing to apply it to all the hypotheses under consideration, or you forgo its norms and regularities for all of them. To ask for an exception solely for your preferred hypothesis is the very definition of a Special Pleading.

In any modelling, you simply cannot compare results obtained under two different sets of assumptions, and expect them to be directly comparable.

Ockham’s Razor would suggest that it is more likely that you overestimated the certainties of your argument, not that a supernatural event occurred.

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This is the equivalent of how you arrived at that 99.4% number:

We have a bag full of 1000 marbles.

2 are red.

2 are blue.

2 are black.

2 are orange.

The rest are white.

We calculate: 6 out of the 1000, or 0.6% are either red, blue of black.

That means, if we pull a marble at random out of the bag, the odds are 99.4% that it will be orange.

Do you agree, or has an error been made somewhere?

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I see your point @Faizal_Ali, but that seems like it is resolvable by making this a two part argument.

  1. Can naturalistic explanations explain the evidence concerning the Resurrection?

We can use methodological naturalism to approach this question. If the answer is “No”, this would be exceedingly unlikely, so conclude that it is likely or at least suggestive something supernatural happened, even if we do not know what yet.

  1. What was the supernatural event that happened?

Now we can’t use methodological naturalism to approach the question. From here, other approaches noted by @Andrew_Loke and @dga471 are important. It seems you are camped out on this second question. I think it is worth engaging in more depth, but probably only after we settle the first one.

@Faizal_Ali, do you agree with his answer to the first question? Let’s concede that this would not mean, necessarily, that Jesus rose from the dead. It would only be suggestive that something supernatural happened.

IF that doesn’t work for you, how about concluding it seems like something very unusual or very peculiar happened?

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I don’t think that is defensible, which is the point I try to illustrate in my most recent comment here.

The problem is that “Supernatural” occurences remain vanishingly rare. So much so that there is widespread opinion that they never, ever happen. So to claim that the naturalistic events that could account for what is claimed to have happened are highly unlikely does not lead to the conclusion that a supernatural event is therefore likely. It also remains highly unlikely. We then have a seven possible explanations (by Loke’s count), all of which are highly unlikely.

It is not justifiable to then arbitrarily assert that one of them DID happen just because it has been deemed “supernatural.” Just as in my example above, the orange marbles don’t become any more common just because the red, black and blue marbles are rare.

No. Religious zealots believing something seemingly impossible to have happened, without any good reason to support their belief, is not at all unusual or peculiar. It happens all the time.

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I would suggest that MN can never rule out all naturalistic alternatives. At best it can tell us that either (i) we have missed or misjudged some factors in one or more of the naturalistic explanations, or (ii) only if we are already willing to accept the supernatural, that a supernatural explanation is a possibility (but not, due to (i), a certainty).

As such this line of argument can never yield a compelling argument for the supernatural for somebody that is not already predisposed to it.

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I don’t think this works out. You’re basically saying that an explanation which is normally unlikely can never become likely because of new evidence and the insufficiency of alternative explanations.

Let’s compare this to the Alvarez hypothesis. Alvarez and Alvarez hypothesized in the 1980s that the mass extinction of the dinosaurs was caused by the impact of a large asteroid hitting Earth. However, large asteroid impacts on Earth are very rare - no more than one every 100 million years or so. It is a very, very unlikely event. Thus, no matter how much evidence Alvarez and others amass in favor of their hypothesis and against other explanations, according to your reasoning their theory is still unlikely.

Surprisingly, and contrary to your methodology, this theory is widely accepted nowadays: it is now called the Chicxulub impact and the asteroid impact which made a crater in Mexico is thought to be the most likely candidate.

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No, I’m saying an unlikely explanation does not magically become likely because there are lots of other unlikely explanations. That should be self-evident.

And that is now accepted because large amounts of naturalistic evidence was provided by to support it. Not by mentioning a lot of other unlikely scenarios, saying they are all unlikely, and therefore their unlikely scenario is now 99.4% certain.

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Not much different from the Loke case. He argues that all other explanations for the facts are unlikely, then he offers some other independent arguments (in other books) that supernatural entities likely exist. These supernatural entities have the capability of explaining the facts, unlike the other alternatives, so this raises up the probability of the supernatural explanation.

Now, I can see your point that merely proving other explanations are unlikely doesn’t automatically warrant a 99.4% probability for the remaining supernatural explanation(s). I’m not convinced of that part of his argument, nor of him assigning 0.1% probabilities. But Loke’s argument is more persuasive in conjunction with other positive arguments for the supernatural.

Let’s imagine a different scenario. Say we discover independent evidence that there are super-advanced aliens from a different galaxy who are able to steal bodies from graves, teleport, and masquerade as humans, and that they are also interested in human affairs and have the capability to operate on earth. Would you think that would then be a likely explanation for the facts surrounding the resurrection?

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To expand a little bit more on what I think @Andrew_Loke should have done, in light of criticisms such as yours. This is tentative, and I might need to reread Chapter 8. This is from page 186:

For example, even if one assigns to each of the six naturalistic alternatives a probability of one in a thousand (which is very generous and much higher than the vanishingly small probability calculated by McGrew), that still leaves the resurrection with a probability of 99.4%.

I think this procedure of assigning a high probability to the supernatural purely by exclusion of naturalistic ones is not rigorous. As seen in this thread, it is vulnerable to criticisms that Loke is unfairly assessing the naturalistic hypotheses first. If he had assessed the supernatural first and deemed it to be low probability (for example, due to supernatural events being rare in general), then one of the naturalistic hypotheses could have ended up as the last hypothesis assessed and thus being non-excluded. Loke replies to similar criticisms as these (from Shapiro et al.) on pp. 182-183, but the following is what I think is a more rigorous way to frame the evidential situation.

What Loke should have done, is to independently assess the probability that supernatural powers exist that are capable of resurrecting Jesus. This might include arguing that the Christian God exists and that the religious context of Jesus and Christianity makes it likely that he could be resurrected. (For example, if it’s Thor who exists, he might not care about resurrecting Jesus, so his existence alone wouldn’t make Jesus’ resurrection much more likely). This is analogous to his independent assessment of all the alternative naturalistic hypotheses.

Here, Plantinga’s Argument from Dwindling Probabilities (PAFDP) might come to haunt us: namely, that you need a conjunction of several background propositions to be true, such as:

  1. A personal God exists.
  2. God cares about human affairs such that he might want to reveal himself through a person in history like Jesus.
  3. Jesus is the individual that God wants to reveal himself through.

But this is OK, even if (for example) we conclude from our independent arguments that the odds of each proposition being true is for each figure which is not too great (for example, to be conservative, P(R) = P(1) \times P(2) \times P(3) \approx 0.5^3 = 0.125), as we will shortly see.[1]

Next, what Loke should do is what we call renormalization. The probabilities must add up to 1, yet we have considered all the logically possible alternatives, including supernatural hypotheses:
(6 \times 0.001 + 0.125) R = 1.
From that we get the renormalization factor R is 7.63. Thus, the renormalized probability for the alternative naturalistic hypotheses is 0.00763, while the probability of the Resurrection hypothesis is 0.95 - still large, though not as large as the initial guess of 0.994.

Note that the above renormalization procedure is (I think) mathematically similar (or even equivalent) to McGrew’s claim (in reply to PAFDP) that we have to update the priors for the relevant propositions underpinning the resurrection hypothesis in light of the evidence for the resurrection. The subtle difference might be that in the above calculation, I’m not only increasing the probabilities of the Resurrection hypothesis, but also the naturalistic ones, so as not to be accused of unfairly favoring the supernatural.

Now, this might not seem super-rigorous, as what if there are supernatural entities other than the Christian God or unknown naturalistic entities that might be able to resurrect Jesus, such as the super-advanced aliens I mentioned? Why aren’t we putting these probabilities into our list of hypotheses? However, one could cogently argue from independent reasons that the odds of these entities existing is negligibly small that they will not significantly affect our calculation.

Conclusion

First, I think that Faizal is right that it’s unfair to independently assess the probabilities of the naturalistic hypotheses as low, and then not do the same independent assessment of the supernatural hypothesis. One has to argue independently that the supernatural hypothesis also does not have a low probability.

Second, the overall success of the resurrection-by-God hypothesis does depend on the success of other arguments for the truth of Christianity. Some people might find these arguments more or less persuasive. I don’t think there is a single universally “correct” assignment of each of the probabilities in question. It depends on the background beliefs of each individual. For a Muslim, for example, the probability of statement 1 (from my list of 3 statements) is 1, but the probability of Jesus being the right individual (statement 3) might be very low, so that might be the area where the Christian apologist has to spend the most time arguing.

Third, despite what I view to be a gap in Loke’s reasoning in chapter 8, the possibility of doing a probability renormalization above shows the usefulness of his new systematic and logical framework for assessing the different hypotheses. This is why as I said, that framework is probably going to be his biggest contribution to Resurrection studies (regardless of how persuasive his individual arguments are against these alternate hypotheses), similar to Herman Gunkel’s classification of psalm types or the Aarne–Thompson–Uther Index for folk tales.

Notes:
[1] Furthermore, if we properly consider the holistic religious context of Jesus (p. 183), the probabilities of propositions 2) and 3) might be even higher anyway, so PAFDP might have even more limited effect. We can also consider other factors such as Wright’s argument for the uniqueness of the Christian movement.

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Thank you for your response and welcoming me to the discussion Dr. Loke!

I think your book is worthwhile reading for anyone interested in the Resurrection debate. For those who want a briefer 1 hour introduction, I recommend the discussion with Dr. Sean McDowell linked above by Dr. Swamidass. And for anyone who has 10 -15 minutes, the original article would I think provide the key framework: Resurrection of the Son of God: A reduction of the naturalistic alternatives

Finally, I think a visual illustration of the framework and mapping out of the naturalistic options like a branching algorithm would intuitively reinforce one of your key contributions, if this hasn’t already been done.

Thanks again for advancing the discussion!

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I think a good next step, if anyone has enough time, would also be a wiki where each of these naturalistic alternatives are numbered and listed according to Loke’s new scheme, and the argumentation for each listed and considered, so that the results of future debates and new contributions could be updated.

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Thank you for your suggestions! I think somebody has done a visual illustration before, but sadly I have lost it. It would be good if someone can do it and post it on wikipedia as Daniel suggested!

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