bookmark_borderExtraordinary Claims Require Extraordinary Evidence (ECREE), Part 2: Is ECREE False? A Reply to William Lane Craig

In my last post, I offered a Bayesian interpretation of the principle, “extraordinary claims require extraordinary evidence” (ECREE). William Lane Craig, however, disagrees with ECREE. In a response to philosopher Stephen Law, Craig wrote this.

This sounds so commonsensical, doesn’t it? But in fact it is demonstrably false. Probability theorists studying what sort of evidence it would take to establish a highly improbable event came to realize that if you just weigh the improbability of the event against the reliability of the testimony, we’d have to be sceptical of many commonly accepted claims. Rather what’s crucial is the probability that we should have the evidence we do if the extraordinary event had not occurred.3 This can easily offset any improbability of the event itself. In the case of the resurrection of Jesus, for example, this means that we must also ask, “What is the probability of the facts of the empty tomb, the post-mortem appearances, and the origin of the disciples’ belief in Jesus’ resurrection, if the resurrection had not occurred?” It is highly, highly, highly, improbable that we should have that evidence if the resurrection had not occurred.
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[3] See the very nice account by S. L. Zabell, “The Probabilistic Analysis of Testimony,” Journal of Statistical Planning and Inference 20 (1988): 327-54.

I agree with Craig that it would be incorrect to “just weigh the improbability of the event against the reliability of the testimony.” I also agree with Craig that “the probability that we should have the evidence we do if the extraordinary event had not occurred … can easily offset any improbability of the event itself.” I disagree with Craig, however, regarding his interpretation that ECREE requires that we ignore that probability. This can be seen using Bayes’s Theorem (BT).

Let B represent our background information; E represent our evidence to be explained; H be an explanatory hypothesis, and ~H be the falsity of H. Here is one form of BT:

As I argued in my last post, an “extraordinary claim” is an explanatory hypothesis which is extremely improbable, conditional upon background information alone, i.e., Pr(H | B) <<<  0.5. And “extraordinary evidence” can be interpreted as the requirement that a hypothesis’s explanatory power is proportionally high enough to offset its prior improbability (the “extraordinary claim”). Here I offer an even more precise definition.

It follows from BT that H will have a high epistemic probability on the evidence B and E:

just in case it has a greater overall balance of prior probability and explanatory power than its denial:


With that inequality in mind, let’s return to Craig’s objection to ECREE. Here again is the relevant portion of his objection:

Probability theorists studying what sort of evidence it would take to establish a highly improbable event came to realize that if you just weigh the improbability of the event against the reliability of the testimony, we’d have to be sceptical of many commonly accepted claims. Rather what’s crucial is the probability that we should have the evidence we do if the extraordinary event had not occurred.

It seems, then, that Craig’s objection to ECREE is based upon an interpretation of ECREE which requires that we only consider the “extraordinary claim,” i.e., Pr(H | B). If that interpretation is correct, then I will join Craig in rejecting ECREE. But is it correct?

In mathematical notation, “the probability that we should have the evidence we do if the extraordinary event had not occurred” is Pr(E | B & ~H). But now consider again the inequality used to define extraordinary evidence.

The expression, Pr(E | B & ~H), is literally right there, in the numerator on the right-hand side. It appears, then, that Craig’s objection is based upon a misinterpretation of ECREE. For the same reason, Craig’s reason that ECREE would cause us “to be sceptical of many commonly accepted claims” is therefore misplaced.

I could be wrong, but I suspect there are two factors which contributed to this misinterpretation. First, many skeptics have used ECREE in connection with (or as support for) Hume’s argument against miracles. While I’m inclined to agree with John Earman that Hume’s argument is highly overrated–i.e., it may be the case that BT does not provide Hume with the support many skeptics think it provides–this is not of obvious relevance to ECREE. ECREE, like BT, is not dependent on Hume.

The other factor which may have contributed to the misinterpretation is the definition of “extraordinary claim;” Craig may disagree with the criteria skeptics have used to determine whether a claim is extraordinary. I think it is helpful to use probabilistic notation to clarify the issue. Again, I proposed that an “extraordinary claim” is
an explanatory hypothesis which is extremely improbable, conditional upon background information alone, i.e., Pr(H | B) <<<  0.5. Let’s assume, for the sake of argument, that definition is wrong. Instead, define an “extraordinary claim” as any explanatory hypothesis H which has a prior probability below some number x, i..e., Pr(H | B) < x, where x can be any real number between 0 and 1. Here’s the point. X can be any real number between 0 and 1. It doesn’t matter which value one chooses, since BT can accommodate all probability values. In terms of calculating the final probability of H, Pr(H | E & B), we use the same formula–BT–regardless of whether H is an extraordinary claim. From a mathematical perspective, it makes no difference whatsoever whether we label a claim “extraordinary” or “ordinary.” We can use BT to assess the epistemic probabilities of both types of claims.

bookmark_borderOpen Question to Theists: Do You Condone the Use of the Phrase a “Murder of Atheists”?

I just learned about this. Apparently the apologist who runs the site www.truefreethinker.com has described Geisler’s response to The Empty Tomb: Jesus Beyond the Grave as a “murder of atheists.” (See here and here.) To be clear, the author is not calling for the murder of atheists. Rather, he says,

I am employing the term “murder” in relation to the group of atheists who have team [sic] up to produce a failure of a book-the term “murder” in this sense is taken from referring to a group of crows a [sic] “a murder of crows.”

If you are a Jew, would you feel if a non-Jew referred to a group of Jewish authors as a “murder of Jews”? If you are a Christian, how would you feel if an atheist referred to, say, the contributors to The Blackwell Companion to Natural Theology as a “murder of Christians”? Do you condone the use of the word “murder” in this context?

To be clear, the author’s clarification notwithstanding, I find the use of the word “murder” in this context to be deeply offensive and very unhelpful for having genuine, respectful dialogue between theists and nontheists. I hope theists who read this will consider condemning this site’s incendiary language on their own blogs.

Update 21-Jun-12:

Steve Hays has responded to this post, questioning why I think Christians should condemn this sort of language.

Over at The Secular Outpost, Jeff Lowder has been calling on Christians to denounce what he regards as mistreatment of atheists. But it’s unclear why Jeff thinks we should do this. Does he think we ought to condemn these (alleged) offenses because they violate Christian ethics? Yet Jeff doesn’t believe in Christian ethics. It’s as if Jeff approves of Christians expressing disapproval on the basis of an ethical code that Jeff disapproves of. 

In reply, notice two things. First, Hays refers to “what he [Lowder] regards as mistreatment of atheists” and “(alleged) offenses” without actually acknowledging the “mistreatment” is actual, not “alleged.” Does he deny that these instances are mistreatment? Is he so opposed to atheists that he is unwilling to condemn mistreatment, even when he agrees it is mistreatment? Is there another reason?

Second, Hays seems (?) to assume that if two people (P1 and P2) accept contradictory normative ethical systems, but both systems agree that an action A is wrong (even if for different reasons), it’s unreasonable for P1 to ask P2 to condemn A. I find that bizarre. If they both agree that A is wrong, even if for different reasons, then surely they can both condemn A. If P2 claims to believe that A is wrong, then P2 already has a reason to condemn; P2 should condemn A because P2 believes A is wrong.

bookmark_borderExtraordinary Claims Require Extraordinary Evidence (ECREE), Part 1: The Bayesian Interpretation of ECREE

If you read this blog, chances are that you very familiar with the slogan, popularized by the late Carl Sagan, that “Extraordinary claims require extraordinary evidence.” What I want to do is to offer a Bayesian interpretation and defense of that slogan. In order to make this a ‘self-contained’ post, I will need to repeat some of what I’ve written elsewhere. Epistemic Probability

In this article, when I refer to probability I shall be adopting the epistemic interpretation of probability. The epistemic probability of a statement is a measure of the probability that a statement is true, given some stock of knowledge. In other words, epistemic probability measures a person’s degree of belief in a statement, given some body of evidence. The epistemic probability of a statement can vary from person to person and from time to time (based upon what knowledge a given person had at a given time).[1] For example, the epistemic personal probability that a factory worker Joe will get a pay raise might be different for Joe than it is for Joe’s supervisor, due to differences in their knowledge.

Two Types of Evidence and Corresponding Probabilities

Let us divide the evidence relevant to naturalism and theism into two categories. First, certain items of evidence function as “odd” facts that need to be explained. Second, other items of evidence are background evidence, which determine the prior probability of rival theories and partially determine their explanatory power.

These two types of evidence have two probabilistic counterparts which are useful for evaluating explanatory hypotheses: (1) the prior probability and (2) the explanatory power of a hypothesis H. (1) is a measure of how likely H is to occur based on background information B alone, whether or not E is true. As for (2), this measures the ability of a hypothesis (combined with background evidence B) to predict (i.e., make probable) an item of evidence. [2]

Notation

Let us proceed, then, to defining some basic probabilistic notation.

B: background evidence

E: the evidence to be explained

H: an explanatory hypothesis

Ri: the rival explanatory hypotheses to H

Pr(x): the probability of x

Pr(x | y): the probability of x conditional upon y

Next, let us define the following conditional probabilities.

Pr(H | B) = the prior probability of H with respect to B—a measure of how likely His to occur at all, whether or not E is true.

Pr(Ri | B) = the prior probability of Riwith respect to B—a measure of how likely Ri is to occur at all, whether or not E is true.

Pr(E | H & B) = the explanatory power of H—a measure of the degree to which the hypothesis Hpredicts the data E given B.

Pr(E | Ri & B) = the explanatory power of Ri—a measure of the degree to which Ri predicts E given B.

Pr(H | E& B) = the final probability that His true conditional upon the total evidence Band E.

The Probabilistic Interpretation of “Extraordinary Claims” Using this notation, it is possible to provide a mathematically rigorous definition of an extraordinary claim. An extraordinary claim may be defined as an explanatory hypothesis which is extremely improbable, conditional upon background information alone, i.e., Pr(H | B) <<< 0.5. Because we are using the epistemic interpretation of probability, it follows that what counts as an “extraordinary claim
” may vary from person to person and from time to time. For example, a healing miracle attributed to the Virgin Mary will be a very extraordinary claim to naturalists (and perhaps? non-Catholics), but not as extraordinary (or not extraordinary at all) to Catholics. With “extraordinary claim” defined, let us now define what constitutes extraordinary evidence. Before we do, however, we must first review Bayes’s Theorem. Bayes’s Theorem

Bayes’s Theorem is a mathematical formula which can be used to represent the effect of new information upon our degree of belief in a hypothesis. In its general form, Bayes’s Theorem may be expressed as follows:

With Bayes’s Theorem defined, we are now in a position to offer a mathematically rigorous definition of “extraordinary evidence.”

The Bayesian Interpretation of “Extraordinary Evidence”

It follows from Bayes’s Theorem that H will have a high final epistemic probability on the evidence B and E:

just in case it has a greater overall balance of prior probability and explanatory power than do its alternatives collectively:

Thus, “extraordinary evidence” can be interpreted as the requirement that a hypothesis’s explanatory power is proportionally high enough to offset its prior improbability (the “extraordinary claim”). Proposal for Both Theists and Naturalists Regarding Extraordinary Claims Because background information plays such a crucial role in determining the prior probability of any hypothesis, extraordinary or not, one thing that both theists and naturalists could do to improve communication with one another is to explicitly identify the propositions which make up their background information. Doing so will not magically resolve their disagreements, but it will greatly improve the chances that the two parties are at least understanding one another. For an example of what this might look like, see any of the individual arguments for naturalism in my series on evidential arguments for naturalism.

Notes [1] Brian Skyrms, Choice & Chance: An Introduction to Inductive Logic (4th ed., Belmont: Wadsworth, 2000), 23. [2] I owe these definitions to Robert Greg Cavin in private correspondence.

bookmark_borderLINK: Leah Libresco of Unequally Yoked Converts from Atheism to Catholicism

I had never heard of her before this, but Leah Libresco of the blog Unequally Yoked has converted from atheism to Catholicism. It appears a big factor in her conversion was her belief that the Moral Law is a person.

I hope that nontheists will (1) not commit the “No True Scotsman” fallacy and deny she was ever an atheist; and (2) be nice to her.

LINK (HT: Paul Fidalgo)

Update: Apparently her conversion is now being reported by MSNBC News.

bookmark_borderCan Theists Be Moral?

That’s a pretty silly question, isn’t it? I would argue that it is about as silly as the question, “Can Atheists Be Moral?” Even fundamentalist Christian philosophers grant that atheists can know moral principles and behave according to those principles. If someone wishes to deny that theists or atheists can have morals, it seems the burden of proof should fall on them to offer some reason why they could not have morals.