Bayesian Analysis of Craig Keener, “Otho: A Targeted Comparison”

[Recently I wrote a critical review of Christian scholar Craig Keener’s new volume Biographies and Jesus: What Does It Mean for the Gospels to Be Biographies?, with emphasis on chapter 6–“Otho: A Targeted Comparison of Suetonius’ Biography and Tacitus’ History, with Implications for the Gospels’ Historical Reliability”–which is written by Keener himself. Ancient historian Richard Carrier sent me some further analysis, which makes both a deductive and inductive critique of Keener’s arguments. Carrier’s feedback can be found below. -MWF]

Applying Bayes’ Theorem to your article’s point:

Keener says we can be sure Suetonius et al. worked from sources, because they say they worked from sources. Then he says we can assume the same of the Gospels, because the Gospels have other similarities to Suetonius et al., except for that one.

This is a straightforward fallacy of false generalization. “All X’s did Y, and all Y’s entail doing Z, therefore all X’s did Z” does not lead by any valid logical inference to “The Gospels are an X,” precisely because the Gospels did not do Y (so the first premise in the argument fails to obtain). So any other similarities there may be are analogically irrelevant to whether the Gospels did Z. Only doing Y can entail Z. He would need to find examples of texts that we can be certain did Z, without doing Y (Y being “naming and discussing sources”). Without arguing in a circle.

So for the deductive logic.

But an apologist will insist it’s inductive. But then Bayes’ Theorem enters.

Keener’s argument that “we can be sure Suetonius et al. worked from sources” has this form:

The probability that they would say Y and not have done Z is low; therefore, given Y, the probability they did Z is high. But Keener has no evidence this relation holds for anything other than Y.

Where Y is in e, as are all other similarities between Suetonius et al. and the Gospels, then:

P(Z|e) = P(Z)P(e|Z) / [ P(Z)P(e|Z) + P(~Z)P(e|~Z) ]

Suppose we break the evidence into just Y, and then X for all the other parallels.

For just Y:

P(Z|Y) = P(Z)P(Y|Z) / [ P(Z)P(Y|Z) + P(~Z)P(Y|~Z) ]

Assume we’re neutral on the prior (we might not be, but that depends on other arguments and we are just analyzing this one), so that P(Z) = P(~Z), then:

P(Z|Y) = P(Y|Z) / [ P(Y|Z) + P(Y|~Z) ]

Which –> 1 as P(Y|~Z) –> 0.

That’s Keener’s argument.

But when he turns to the Gospels, he falsely treats X as if it were Y. But that doesn’t work. His argument from X is:

P(Z|X) = P(Z)P(X|Z) / [ P(Z)P(X|Z) + P(~Z)P(X|~Z) ]

And with a neutral prior that’s:

P(Z|X) = P(X|Z) / [ P(X|Z) + P(X|~Z) ]

Keener presents no evidence that that –> 1 as P(X|~Z) –> 0, nor any evidence that P(X|Z) is even high.

What instead he does is argue:

P(Z|Y&X) = P(Y&X|Z) / [ P(Y&X|Z) + P(Y&X|~Z) ]

Which gets him the Y result, then he uses the same argument for the Gospels, but “forgets” the Gospels don’t have Y. He is thus conflating Y with X. To argue X correlates with Z in the absence of Y requires actual evidence that that is ever the case. He presents none. Presenting examples that correlate Z with Y&X simply does not constitute evidence that Z correlates with Y. That’s the generic Bayesian analysis of the fallacy of false analogy in a nutshell.

Conversely, you point out that the generic similarities in X are actually known or credible properties even of fiction, so that in fact the evidence there is actually argues *against* any distinct correlation between X and Z (it may be at best 0.5, such that P(Y&X|Z) = P(Y&X|~Z)), so it’s even worse than Keener having no evidence that X correlates with Z; the evidence actually is against there being any such correlation, at least in any reliable sense. That’s the “at best” 0.5 correlation; but it’s possibly worse, if the absence of Y is telltale of fiction, another argument of yours. And that need not be a correlation of 1, it could be, say, 0.8, allowing 20% of examples of no-Y still being Z texts. Look what happens when you are even that generous (and still using a neutral prior as if no other considerations mattered, which we know isn’t the case), assuming no correlation exists between no-Z texts containing X:

P(Z|X) = P(X|Z) / [ P(X|Z) + P(X|~Z) ] = 0.2 / (0.2) + (0.5) = 0.29.

If no-Z texts typically contained X, it’s even worse. Only if no-Z texts rarely contain X would it get better; but that would require the very evidence Keener doesn’t present: that X correlates with Z.

Introducing Y also changes the result, of course. But that’s precisely what the Gospels don’t do. Likewise any other generic factor that might up the odds of Z; which would need to be demonstrated as doing so in other texts (and without circular argument).

-Richard Carrier

Understanding the Spirit vs. the Letter of Probability

My apologies for there being so little activity on Civitas Humana for a long while now. I have been insanely busy and stressed lately (both professionally and personally), and I have only recently been able to catch up with blogging.

I’ll begin pumping some fresh blood into Civ by discussing Bayes’ theorem, the resurrection of Jesus, and why I think that, even with a non-zero prior (which is still very, very low) for the resurrection event, Paul’s letters and the Gospels are far too weak of consequent evidence to offset more probable (naturalistic or mundane) explanations for the same data.

Over a decade ago (March 2006) secular New Testament scholar Bart D. Ehrman debated Christian philosopher and theologian William Lane Craig about the evidence for the resurrection of Jesus (the transcript of the debate can be read here). Overall, the debate left me with the impression that Ehrman made a better case for naturalistic or non-paranormal explanations being more probable than a veridical resurrection event, with regards to the origin of the resurrection belief among Jesus’ disciples and the first generation of Christians. But there was one area where I think Craig scored a technical, though relatively minor point against Ehrman (as will be discussed below), and this was with regards to how Ehrman was defining a miracle event and conflating prior probability with posterior probability.

Lowder’s Summary of Craig’s First Rebuttal

Jeff Lowder on the Secular Outpost has made a useful summary of Craig’s critique of how Ehrman was defining a miracle event, and the arguments Ehrman had presented (in earlier publications) for why historians cannot argue that a miracle is the most probable explanation for a past event. This summary can be read in Lowder’s post “William Lane Craig’s Critique of Bart Ehrman on the Probability of Miracles.” Since the ensuing discussion involves Bayesian reasoning, you can read my essay “History, Probability, and Miracles” for a basic overview of how Bayesian logic works, if you are unfamiliar with the theorem.

Lowder begins by listing two published statements by Ehrman, which were quoted by Craig during the debate (bolding is my own):

(1) “Because historians can only establish what probably happened, and a miracle of this nature is highly improbable, the historian cannot say it probably occurred.”
(The Historical Jesus, part 2, page 50)
(2) “Since historians can establish only what probably happened in the past, they cannot show that miracles happened, since this would involve a contradiction — that the most improbable event is the most probable.”
(The New Testament: A Historical Introduction, page 229)
In his response to these statements of Ehrman, Craig critiques his line of reasoning by arguing that Ehrman is conflating prior probability with posterior probability. The odds that a given individual may resurrect from the dead could, indeed, be very, very low. But if there is very, very good evidence that such a resurrection event has occurred, it may offset the low prior, and even outweigh alternative explanations, to degree such that Pr (R/B & E) > 0.5 (perhaps even by a wide margin, e.g., +0.9).
Here is what Craig states in his own words (bolding is my own):
“In other words, in calculating the probability of Jesus’ resurrection, the only factor he [Ehrman] considers is the intrinsic probability of the resurrection alone [Pr(R/B)]. He just ignores all of the other factors. And that’s just mathematically fallacious. The probability of the resurrection could still be very high even though the Pr(R/B) alone is terribly low.”
Lowder likewise offers his own interpretation of Ehrman’s two quotations, and here is what he states regarding the first:
“I am inclined to interpret (1) as the following claim: (1′) Pr(R/B) is so low that it is impossible, even in theory, for there to be sufficient evidence to confer a high final epistemic probability on R, i.e., Pr(R/B & E) > 0.5.”

I would argue that to describe a miracle as an event that cannot be probable, even in theory, one would need to assign its prior probability a value of zero. And this is the same conclusion that Lowder reaches, when he states (bolding is my own):

The only way to reconcile (1′) with BT would be to assign Pr(R/B) a value of zero. If Pr(R/B) = 0, then it follows from BT that Pr(R/B&E;)=0. So, on the basis of (1) alone, as Craig has quoted Ehrman, I think it is premature to assume that Ehrman ‘just ignores all of the other factors.’ Maybe he does do that, but the quotation provided in (1) doesn’t show that. What I can say is that either Ehrman ignores all of the other factors or Ehrman assumes that historians must assign Pr(R/B) a value of zero. If the latter, then I think that is false.”

With regards to Ehrman’s second quotation, Lowder briefly states:

“Turning to (2), I don’t have much to say, other than I think Craig is 100% correct when he says that Ehrman ‘Confuses Pr (R/B & E) with Pr (R/B).’”

My Thoughts on Ehrman’s Quotes

At the end of this essay, I will make some suggestions for improving Ehrman’s arguments (which in spirit I think are correct, even if they may be formally invalid at parts). That said, I agree with Lowder’s conclusions on both accounts, for at least three reasons:

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A Brief Status Update for the Summer

I know it’s been a long time since Francis Adams or I have added new content to Civ, but we will be active again soon. We have both been very busy lately, and so I want to share this announcement that I just wrote on Κέλσος about what we’ve been up to. I hope everyone is having a good summer, and expect new content at some point in the near future.


I have been super busy with both academic and personal work lately, and so I just want to give a brief status update about my (apparent) absence from the blog.

First off, if you don’t see new posts from me, that doesn’t mean that I am not adding new content. I regularly add footnotes and new material to old essays (as well as answering comments), and so this is a heavily tailored blog. I tend to write long essays on specific topics, rather than short blog posts, and to beef them up over time.

Most of my page views come from Google searches and not new posts, and so the blog is still getting a lot of new material out there. I want to clarify this, especially since I have some people supporting me on Patreon, and I don’t want to give the impression that I’m not active on the…

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Jeff Lowder, “The VICTIMs of Christian Apologetics”

I’ve been taking a break from blogging lately, in order to focus on my dissertation work, but a recent counter-apologetics video was posted that is worth mention. I have been co-authoring a critical review of apologists Norman Geisler and Frank Turek’s I Don’t Have Enough Faith to Be an Atheist, with Jeff Lowder, founder of the Internet Infidels and blogger at the Secular Outpost.

Both Geisler and Turek repeat the slogan “the atheist must borrow X from theism” (e.g., logic, morality, agency, mathematics, etc.) in a dozen of variations, all making assertions about metaphysical issues that allegedly cannot be explained under the hypothesis of naturalism, and require the existence of God. Of course, actual naturalist philosophers, such as Jack Ritche in Understanding Naturalism, have also thought about such things, and written philosophical resources extensively discussing issues of physicalism, logic, and value. But what is nice is that Lowder provides a detailed, point-by-point refutation against arguments of this sort in a two and half hour video, based on slides he used to prepare for a public debate with Turek.

A lot of the things that Lowder discusses have overlap with my metaphysics series–“Thinking about the ‘Metaphysics’ in Metaphysical Naturalism”–here on Civitas Humana, where I have discussed issues of cosmology, agency, and value from a secular and naturalistic point of view. Lowder’s video supplements this material quite nicely, so check it out (and take notes)!

-Matthew Ferguson

Thomas Aquinas on Divine Simplicity and Richard Dawkins’ Ultimate Boeing 747 Gambit

I have been discussing the theology of Thomas Aquinas in recent posts on this blog, including an extensive rebuttal to Aquinas’ Five Ways of proving God’s existence in this previous essay. The Five Ways belong to question 2 of the first part of Summa Theologica, and in this post I am going to discuss the content under question 3. In question 3, Aquinas writes about the theological attribute of divine simplicity. The discussion there is relevant to an argument that atheist Richard Dawkins made about a decade ago in The God Delusion, termed the Ultimate Boeing 747 gambit.

Boeing 747

Dawkins’ argument is a play on the notion of a “tornado sweeping through a junkyard to assemble a Boeing 747,” which is used by creationists to mischaracterize the probability of abiogenesis and evolution. Allegedly, the odds of complex life emerging by chance should be as rare as a tornado passing through a junkyard and assembling a Boeing 747. Dawkins’ response, however, is to turn this argument on its head. If life is too complex to have emerged by chance, then what are the odds that a complex deity, with all of the intelligence needed to design life, would just happen to exist by chance as the uncaused creator of the universe, in order to create life? Dawkins argues that the unexplained complexity of this designer poses a greater question than the problem that it seeks to solve. Rather, God is the Ultimate Boeing 747, in that the odds of such a being just happening to exist is much improbable than the more simple explanations of abiogenesis and evolution.

This argument did not jive well with many theologians, however, and both Alvin Plantinga (response here) and William Lane Craig (response here) wrote a rebuttal to it. In their responses both Plantinga and Craig appeal to Aquinas’ conception of divine simplicity to argue that Dawkins does not have a correct understanding of theology. Below is my response to their counter-arguments, and why I do not think that they have correctly characterized the complexity described by the Ultimate Boeing 747 gambit.

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A Minimalist Definition of Naturalism

(This blog essay no long represents my current views. I will post a new essay on the subject, at some point in the future. -MWF 2/5/19)

I have been away from blogging here on Civitas Humana for a while, due to being busy with graduate work as part of my Ph.D. program. Thankfully, I passed my dissertation prospectus and advanced to Ph.D. candidacy last quarter, and so now I can dedicate more time to research and blogging.

I am going to start posting again here on Civ by beginning with a relatively short discussion of my definition of metaphysical naturalism. I have discussed some of the conceptual and ontological ways of defining both the “natural” and the “supernatural” in a couple of my previous essays on this blog (see here and here). In those essays I discuss criteria such as physicalism, reductionism, uniformity, and teleology. I think that all of these criteria are useful for articulating some of the ways that we differentiate the natural from the supernatural, but recently I have started to think that an even more minimal definition of naturalism is sufficient to deny one particular supernatural concept, namely the existence of God.


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Big History: An Introduction

What is big history? This emergent and interdisciplinary field, enriched and pioneered by Dr. David Christian of Macquarie University, encourages a more holistic understanding of human events than does the traditional study of history. While historians are concerned with understanding the past in context, and considering cause and effect in human terms, big historians are concerned with understanding the past not only in its immediate human historical setting, but in the context of scientific and physical laws of nature as well. If history is written by the victor, then big history is written in the stars themselves.

Screenshot 2015-12-20 at 1.39.32 PMDr. Christian, bolstered by the support of philanthropist Bill Gates, first injected big history into the public sector with a 2011 TED Talk, providing an 18 minute overview of world history. In this sensational talk, which has garnered more than 5 million views since its publication, Dr. Christian identifies the basic principles of big history, including the concept of Goldilocks conditions and the various “thresholds” of complexity that we observe in the universe. At various moments in the cosmic past, Christian states, certain Goldilocks conditions have come about, in which “not too little, and not too much” of certain components — usually energy or mass — have allowed the universe to reach states of increasing complexity.

Starting at the Big Bang and the first moment of time itself, Christian traces the cause-and-effect of each moment and identifies these thresholds. He highlights the six universal thresholds of complexity as follows:

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