That’s an incorrect analysis.

An extraordinary claim has a prior odds of 1/L, therefore extraordinary evidence must have en evidential odds of L/1 or greater (at exactly L/1 you get the claim’s probability all the way up to 50%; and any amount more it goes higher). They are thus the same threshold. They are defined the same way. Thus extraordinary claims (1/L claims) require extraordinary evidence (L/1 evidence).

QED

Let us then say that an event x is extraordinary if its probability is less than some value L, assumed to be sufficiently low. In order to ease notation I will avoid writing dependence on background information and assume that it is understood. So, we write the statement that x is extraordinary as P(x) < L. Let h denote a hypothesis, i.e. a claim, and e the evidence for it. Saying that h is extraordinary then means that P(h) < L. If, given the evidence, the hypothesis is more likely than not this means that P(h|e) > 1/2. We want to see if this implies that P(e) < L, that is, that the evidence is extraordinary. Bayes theorem tells us that

P(h|e) = P(h) P(e|h) / P(e).

Since, by assumption, P(h|e) > 1/2, we have that

P(e|h) = P(h) P(e|h) / P(e) > 1/2.

Multiplying both sides of the inequality by 2P(e) we obtain that

2 P(h) P(e|h) > P(e).

Moreover, since we are assuming P(h) < L, we use the last inequality to establish

P(e) < 2 P(h) P(e|h) < 2 L P(e|h).

So we see that we can guarantee that e is extraordinary ONLY if P(e|h) < 1/2, that is, if the hypothesis is not a good explanation for the evidence. Therefore, if we agree with the proposed definition of extraordinary the maxim is false as stated, since it is missing the assumption that P(e|h) < 1/2. Assuming that, however, would be strange, since we want h to be a good explanation for e.

Although we cannot conclude that the evidence must be extraordinary, there still something we can say about its probability being low. Regardless of the value of P(e|h) we know it is at most 1, since it is a probability, so we can conclude that

P(e) < 2 L P(e|h) < 2 L,

which means that the evidence is at most twice as likely as an extraordinary event. Since L was assumed to be very low, 2L must continue to be low. However, by definition, an event with probability 2L is strictly more likely that extraordinary. If we were to say that any event with probability less than 2L but larger than L continues to be extraordinary, we can always repeat the above arguments for an alternative h’ with P(h’) < 2L and conclude that whatever evidence e’ we have for it must have probability P(e’) < 4L, which, again, does not guarantee that it is extraordinary. If we continue pushing the threshold of extraordinarity in this way we will get to a point where any event will be considered extraordinary, which is absurd. So either the reasoning I present here is incorrect (and thus someone should be able to point out where I have made a mistake), you disagree with this definition of extraordinary, or the maxim is false as stated. If you continue to hold that the maxim is correct, the question would then be what do YOU mean by “extraordinary”?

That’s standard. All delusional people avoid anything that poses a serious critical threat to their beliefs. It’s called “avoidance behavior.” They will only interact with poor or weak critiques, and ignore or lie about good or strong ones.

The good news is that though most delusional people behave that way, there is always a sizable percentage that is too ashamed to, and they eventually escape the delusion. I have already met hundreds of people for whom this is true even from having read just my work; and I’m just one significant critic among thousands, and the deconverts I actually meet are just a small percentage of those there will have been.

In essence, strong critics always divide the target audience between avoiders-and-deniers and worriers-and-checkers. The worriers, those who are bothered by the strength of a critique and can’t let it go and thus try to fact-check or debunk it rather than ignore or dismiss it, are the ones who eventually argue themselves out of the delusion.

As an ex-Christian, I am sharing your arguments regularly with the old crowd. They won’t even engage with your arguments. Now there are many reasons for this, but everyone of them highlights the narrow-mindedness of the ‘narrow path’ to Christ. It’s quite worrying.

Both are correct. If you google around you’ll find many uses of both in professional periodicals, history books, etc.

Yes, I was assuming that “how many people say that” was very low. (For me that number is literally zero, so if someone said that it would be a first of many.) I agree it’s also useful to distinguish between low probability events (lottery, unlikely to happen in my lifetime, willing to bet my life on it, i.e. “extremely” low in your parlance), and even lower probability events (rising from the dead, i.e. “extraordinary” low in your parlance). I do think someone telling you “I will win” could increase P(e|game is rigged), because it’s a sort of “suspicious” behavior. Thanks for the exhaustive analysis!

For me, E.T. Jaynes’s works were transformative, and I was pleased to see you cite “Probability Theory: The Logic of Science” in PH. In many ways, it seems even more suited to “History” than to “Science” (traditionally defined), precisely because assumptions can play such a significant role, approximate results have to be aggregated from multiple studies, and because experiments can’t be replicated in a way that makes frequentist approaches easier to apply.

Craig’s basic misunderstanding is a prime example why your work needs to be amplified.

Don’t underestimate the power of the delusion. In Craig’s universe, God sent his son to die, and come back to life. The truth must point to that, somehow. Logic (and awareness of logic) is something we’ve comprehended recently; but beliefs, I think, are things that have defined reality for us since before we were swinging in trees. I haven’t personally investigated many of Craig’s claims, so you could be right that he really is a conscious liar…. but personally I doubt it (at least, I suspect cognitive dissonance is working its charms on him well enough to prevent him from realising he’s a con). It’s not the amount of evidence that would convince me, but the quality of evidence. Do we have emails of him confiding in someone that he’s fooling everyone? Taped conversations? Mind viruses are the worst….

After reading Hume over 50 years ago at the University of Michigan and re-reading his maxim many times subsequently, I remained perplexed by his phraseology. Aron Lucas facilitated a breakthrough in my understanding of Hume’s maxim in the great article on the Secular Web. I clearly understand Hume’s maxim for the first time, by substituting “improbable” for “miraculous” in the maxim. Many thanks!