Base-Rate Neglect in the News

It has been a while since I have thought about the fallacy of base-rate neglect. I did not even think about it when I was recently talking to someone about the reliability (or lack thereof) of tests for SARS-CoV2 antibodies. The piece by Todd Haugh and Suneal Bedi published in the New York Times today (linked above) is a useful reminder.

But it seems to me that Haugh and Bedi do not state their example clearly enough (perhaps because of editorial pruning). I would state it this way: Suppose that a test for SARS-CoV2 antibodies has a sensitivity of 90%. This means that it gives a positive result to 90% of subjects taking it who actually have the antibodies. Suppose also that it has a specificity of 90%. This means that it gives a negative result to 90% of subjects taking it who don't have antibodies. Suppose also that the incidence of antibodies to the virus in the population is (as the writers estimate it to be) 5 percent. And suppose, finally, that 2,000 randomly selected people take the test. If our sample is perfectly representative, 100 of the people taking the test will have the antibodies; and of these, 90 will get a positive result. Of the other 1,900 people taking the test, the ones who don’t have the antibodies, 1710 will get a negative result. But this means—and this is the most significant part—the remaining 190 will get a (false) positive result.

This is significant because it means that, of the 280 people in total who get a positive test result, more than two thirds will not have antibodies. In more general terms, the lower the base rate of what you are testing for, the higher the ratio of false positives to true positives, and therefore the less reliable a positive test result is. If the base rate is 10% then out of our representative sample of 2,000 tests there will be 180 true positives and 180 false positives. So, assuming 90% specificity and sensitivity of a test (which, as I gather, is far better than what any test now available can offer), the base rate has to be above 10% for a positive test result to be more reliable than a coin toss. (A coin toss as to whether a given positive result is true or false, that is, not a coin toss as to whether a given person has antibodies.)

These figures, of course, only describe a mathematical model. One assumption of the model is that persons taking the test are representative of the whole population with regard to the incidence of antibodies among them. This is not necessarily the case. In fact, it is not even probable; rather, people who have had symptoms that they attribute to COVID-19 are more likely to take the test than people who have not. Consequently, the incidence of antibodies to the virus will be higher among persons taking the test than the base rate of the total population. How much higher? I don’t know, and I don’t know how one would go about estimating such a thing.

Quantum Subterfuge

An account of the double-slit experiment by a former professor of mathematical physics is supposed to show the necessity of a paradoxical conclusion, but under examination it shows only the logical confusions of its author.

A few years ago, wanting to gain some knowledge of quantum mechanics, I started reading Quantum Theory: A Very Short Introduction by John Polkinghorne, formerly Professor of Mathematical Physics in the University of Cambridge (Oxford University Press, 2002). I recall being exasperated by the book on my first attempt to read it, but I did not recall the reason until more recently, when I reread Polkinghorne’s commentary on the double-slit experiment.

Polkinghorne opens his exposition by quoting a comment of Richard Feynman on the experiment:

In reality it contains the only mystery. We cannot make the mystery go away by ‘explaining’ how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics. [Polkinghorne, p. 22; no citation of source provided]

I recall thinking, when I first read this: “Excellent! Now all I have to do is pay very close attention to the account of the experiment, and I shall understand the basic peculiarities of all quantum mechanics.” This expectation was disappointed.

Describing the physical setup and the results of the experiment is not difficult. Electrons or other quantum particles are fired at a barrier in which there is a pair of adjacent slits. On the far side of the slits is a screen that detects the impacts of the particles. Classical physics predicts that the particles will impinge on the screen in a scattering pattern, with two areas of greatest intensity located directly across from the two slits. But in fact what emerges is an interference pattern, with a single area of greatest intensity located across from the midpoint of the two slits, exactly as if one were sending waves of some sort through the slits.

One might try to explain this on classical-physical lines by supposing that the electrons, though individually particulate in nature, behave in a wave-like fashion when they are shot together in a stream: it is the stream of electrons, not the individual electron, that behaves like a wave. It is true that one cannot observe wave behavior in an individual electron but only in a collection of electrons. But that does not mean that the wave behavior can be explained on classical lines as an effect of physical interaction among electrons in a stream, for the fact is that the electrons form an interference pattern even when they are fired at the slits one at a time. So either one must suppose that the behavior of each electron is influenced by the path taken by its predecessors, or one must attribute wave properties to each electron.

Things get even weirder when the experiment is set up so as to allow the option of detecting which slit the particles pass through. In so-called “quantum eraser” experiments (q.v., Wikipedia), photons passing through circular polarizers form a scattering pattern or an interference pattern according to whether the polarization that distinguishes which slit they went through is preserved or “erased” by a second, diagonal polarizer. And with so-called the “delayed choice” quantum eraser experiment (q.v., Wikipedia again), things get even weirder. But those are other stories, not covered in Polkinghorne’s book. Polkinghorne finds weirdness enough in the original plain double-slit experiment. He writes:

Electrons arriving one by one is particlelike behaviour; the resulting collective interference pattern is wavelike behaviour. But there is something much more interesting than that to be said. We can probe a little deeper into what is going on by asking the question, When an indivisible single electron is traversing the apparatus, through which slit does it pass in order to get to the detector screen? Let us suppose that it went through the top slit, A. If that were the case, the lower slit, B, was really irrelevant and it might just as well have been temporarily closed up. But, with only A open, the electron would not be most likely to arrive at the midpoint of the far screen, but instead it would be most likely to end up at the point opposite A. Since this is not the case [emphasis mine], we conclude that the electron could not have gone through A. Standing the argument on its head, we conclude that the electron could not have gone through B either. What then was happening? That great and good man, Sherlock Holmes, was fond of saying that when you have eliminated the impossible, whatever remains must have been the case, however improbable it may seem to be. Applying this Holmesian principle leads us to the conclusion that the indivisible electron went through both slits (pp. 23–24, latter emphasis in original).

Pay attention to the words “this is not the case” and try to identify their antecedent: what is not the case? The only thing mentioned in the preceding sentences that can in any clear sense be said to be the case or not the case is that the electron in question struck the screen at the point opposite slit A. But it would be insane to say casually that this is not the case, since there is nothing in the preceding stipulations that would justify such a conclusion. No, what Polkinghorne seems to mean by “this is not the case” is that it is not the case that the electron would be “most likely” to arrive at the point opposite slit A.

(Note: The remainder of this post has been extensively revised since I first posted it. The paragraphs that immediately follow analyze and criticize Polkinghorne’s argument in a very detailed fashion. Readers whose patience or interest is tried by such a treatment may profit by skipping down to the paragraph just before the first graph, in which I restate my criticisms by means of an analogy with an argument whose defects are much easier to recognize.)

Let “E” designate a randomly selected electron that is fired at the slits and that strikes the screen on the far side of them. The first part of Polkinghorne’s reasoning can then be summed up as follows:

1. If E passes through slit A, then E is most likely to strike the screen at a point opposite A.
2. E is not most likely to strike the screen at a point opposite A.
Therefore,
3. E does not pass through A.
   

The argument appears to be impeccable as far as its logical form is concerned. If that is so then the only question to be raised about its cogency is whether both premises are true. But in fact, reflection on the premises reveals an ambiguity that defeats the argument.

Consider premise 2 first. Given the setup of the experiment, the only evidence that we have for attributing to a randomly selected electron a probability of hitting one or another part of the screen is the interference pattern that emerges on the screen.  That pattern shows the highest incidence of impacts at the midpoint between the two slits. From this fact we can conclude that a randomly selected electron that strikes the screen is most likely to do so at the midpoint and not opposite either slit. This reasoning justifies Polkinghorne’s second premise.

Now consider the first premise. E was defined as an electron randomly selected from among all the electrons that reach the screen. But premise 1 concerns an electron that is randomly selected from among those that have passed through slit A. The pattern on the screen provides no evidence whatever relevant to a conclusion about the most likely point of arrival of such an electron as that. The only way to get evidence relevant to a conclusion about a randomly selected electron that has passed through slit A is either to block off slit B or to use a device that distinguishes the impacts of electrons that have passed through A from the impacts of electrons that have passed through B, as in the quantum eraser experiments. It is established that if we do either of these things then no interference pattern emerges. If we block off slit B, the highest incidence of impacts is opposite slit A, and if a device is used that distinguishes the electrons passing through A from those passing through B, then there will be an area of highest incidence opposite each slit. Under such conditions, Polkinghorne’s first premise is true. But his second premise is either false or irrelevant to the conclusion—false if it concerns an electron that has passed through slit A; irrelevant if it concerns an electron whose place of passage is undetermined.

Of course, this is not Polkinghorne’s entire argument, but only one half of its preliminary part. The second half of the preliminary part is the repetition of this argument with “slit B” taking the place of “slit A” in the premises and the conclusion.

4. If E passes through slit B, then E is most likely to strike the screen at a point opposite B.
5. E is not most likely to strike the screen at a point opposite B.
Therefore,
6. The electron did not go through B.
   

That part of the argument is, obviously, just as futile with “B” in it as it was with “A,” but let it stand for the moment, so that we can consider the would-be clinching part of the argument: the supposedly Holmesian conclusion that the electron passed through both slits. Polkinghorne’s stated reasoning is:

3. E does not pass through slit A.
6. E does not pass through slit B.
Therefore,
7. E passes through both slits.

There is nothing Holmesian about such a conclusion at all: Sherlock Holmes would not mistake such a bald non sequitur for a deduction. Nor does it take the intellect of Sherlock Holmes to see that what follows from lines 3 and 6 is:

E does not pass through either slit.

Of course, this is not the desired conclusion at all. Not only does Polkinghorne fail to establish his desired intermediate conclusions (3 and 6); he has set up his entire argument to establish the wrong final conclusion. The undesired conclusion can be avoided if we go back to re-write the previous two subordinate arguments by inserting the qualifier “solely” before the phrases “through slit A” and “through slit B,” and adding a further premise, “E passes through at least one slit.” But the fact remains that the intermediate conclusions 3 and 6 are not established or even made in the slightest degree credible by any of Polkinghorne’s reasoning.

The defects in Polkinghorne’s argument can be brought out by means of an analogy. A graph of the distribution of heights among adults in the United States looks like this (this graph and the two that follow are taken from this Web page by John D. Cook Consulting):

Graph 1: height distribution of all adults in US

The numbers along the bottom represent height in inches. The midpoint of the peak is around 67 inches. Let R be a randomly selected adult resident of the United States. According to this graph, R is most likely to be 67 inches tall. So R is not most likely to be, say, 64 inches tall, or 70 inches tall.

But R may, and indeed (setting aside rare cases of indeterminate sex) must, be either female or male. Suppose that R is female. The distribution of heights for adult females has a peak around 64 inches:

Graph 2: height distribution of adult females in the US

So if R is female, R is most likely to be about 64 inches tall. By contrast, the distribution for adult males has a peak around 70 inches.

Graph 3: height distribution of adult males in the US

So if R is male, R is most likely to be about 70 inches tall. Now imagine that, with these facts in hand, statistician John Jokinghorne presents us with the following argument:

1. If R is female, then R is most likely to be about 64 inches tall (by graph 2).
2. R is not most likely to be about 64 inches tall (by graph 1).
Therefore,
3. R is not female (from 1 and 2).
4. If R is male, then R is most likely to be about 70 inches tall (by graph 3).
5. R is not most likely to be about 70 inches tall (by graph 1).
Therefore,
6. R is not male (from 4 and 5).
Therefore,
7. R is both female and male (supposedly Holmesian conclusion from 3 and 6).

Clearly, all three of the conclusions in this argument are non sequiturs. Conclusion 3 does not follow from (1) and (2), because adding the supposition that R is female, as in (1), makes (2) false or irrelevant to (3). The same applies to the relation of premises 4 and 5 to conclusion 6. And the would-be Holmesian conclusion is of stupefying inconsequence. One may think that Polkinghorne’s argument cannot be as bad as Jokinghorne’s, because it is not so obviously bogus; but logically considered, it is every bit as bad. Its logical defects are exactly analogous. They just happen to be less conspicuous because of the more recondite subject matter.

One last observation: Presumably, Polkinghorne intends his argument to establish something not just about some randomly selected electron in the experiment but about every electron in the experiment, namely that it passes through both slits. The analogous conclusion of Jokinghorne’s argument would be:

8. Every American adult is both female and male.

If Jokinghorne’s argument does not incline you to accept this conclusion (and it shouldn’t), then neither should Polkinghorne’s argument incline you to accept his conclusion about the double-slit experiment.

There may be compelling reasons in quantum mechanics to say that each electron goes through both slits, but whatever those reasons may be, Polkinghorne fails to state them. Making a popular exposition of quantum mechanics requires making the reasoning that leads to its paradoxical conclusions clear. Instead of this, Polkinghorne’s book offers the kind of confused thinking that can at best produce only incomprehension and that at worst produces the false belief that one has understood something when in fact one has merely participated in the author’s own confusions.

More on Thinking Probabilistically

We typically use the plural noun “probabilities” only when speaking of events that are potentially repeatable, like throws of a pair of dice. But the notion of probability has another aspect, namely the degree of strength of belief warranted by evidence. This seems to apply, at least potentially, to the question of divine existence. But one may doubt whether the “God” about which some reason probabilistically can be identified with the God worshiped and served in any actual religion.

According to this page, these actually work

My previous entry addressed, but—characteristically, I confess—did not answer, the question “Is the existence of God a matter of probabilities?” I wish now that I had used the singular form of the noun “probability” rather than the plural, as the latter has associations that I don’t welcome. The plural form “probabilities” tends to suggest numerical values or measures of probability, which in turn (and this is the most unwelcome part) suggests the sort of case in which an event of a specific, repeatable type occurs under specific conditions—for instance, the event of a hand of five playing cards containing a pair, given that the five cards are dealt randomly from a deck of 52. Even if we are speaking, say, of the probability that candidate So and So will win the upcoming election, which is not a repeatable event-type but a single occurrence, we may consider that outcome as belonging to a type specifiable more or less broadly according to country, locale, time period, type of office, characteristics of the candidate, and so on; and we can then calculate the chances accordingly.

But what if we are speaking of the probability of a possible fact that is not an instance of a repeatable type? Discussions of the existence of God would be a case of this. The idea of assigning the existence of God to some type of repeatable event seems senseless. Perhaps some diligent analytic metaphysician somewhere has reckoned the probability of divine existence as the proportion of God-made possible worlds to Godless ones; but I don’t care to take account of all conceivable products of academic invention. If the concept of probability applies only to repeatable event-types, and if, as seems plain, the existence of God is not an event of a repeatable type, then the answer to the question “Is the existence of God a matter of probability?” is a flat and rather uninteresting “No.”

But the concept of probability is not restricted to such cases. When Bishop Butler remarked in the “Introduction” to his Analogy of Religion (1736) that “to us, probability is the very guide of life,” he was not referring to the calculus of chance, which was then in its infancy. He was speaking of probability in contrast with absolute certainty, and of the condition of finite intellects in contrast with that of an infinite one:

Probable evidence is essentially distinguished from demonstrative by this, that it admits of degrees; and all variety of them, from the highest moral certainty, to the very lowest presumption. . . .

Probable evidence, in its very nature, affords but an imperfect kind of information, and is to be considered as relative only to beings of limited capacities. For nothing which is the possible object of knowledge, whether past, present, or future, can be probable to an infinite Intelligence; since it cannot but be discerned absolutely, as it is in itself, certainly true, or certainly false. But, to us, probability is the very guide of life.

Probability is our guide in life because our knowledge of the world is, by our nature, limited. To follow probability in the pertinent sense is not to reckon odds but to weigh what Butler calls “presumptions,” or reasons for belief. There is more to probability than mere chance. As Ian Hacking remarks in his historical study The Emergence of Probability,

Probability has two aspects. It is connected with the degree of belief warranted by evidence, and it is connected with the tendency, displayed by some chance devices, to produce stable relative frequencies.

Hacking dubs the first aspect of probability the “epistemological” (from Greek epistēmē, “knowledge”) and the other the “aleatory” (from Latin ālea, “die” or, by derivation, “game of chance”). I think “epistemic” is a more widely used term for the former, although, since it is belief and not knowledge that is in question, “doxic” (from Greek doxa, “belief”) would be more apt. Whatever the terminology, and however we may try to understand the relation between these two aspects of probability, it is the doxic or epistemic aspect that is pertinent when the existence of God is treated probabilistically. The fundamental thought is not that we can calculate the chance that God exists as we can the chance of getting a certain result from throwing a pair of dice, say, but that some degree of strength of belief that God exists is warranted by the evidence available to us.

The question “Is the existence of God a matter of probability?” is a question about a question. It concerns how the question “Does God exist?” may be answered—what sort of thing one has to do, or may do, to answer it. Anyone who assumes that the question must be, or may be, answered by weighing what Butler terms “probable” evidence (meaning empirical evidence, as contrasted with the “demonstrative” evidence of proofs a priori) assumes that the answer to the first question is “Yes”—that the existence of God is a matter of probability.

Most writers who argue for atheism seem to make this assumption. They typically argue either that there is no evidence that God exists or that there is evidence that God does not exist. It seems to go without saying for them that to answer the question of God’s existence otherwise than by evaluating the available evidence would be incompatible with intellectual integrity. For instance, Richard Dawkins entitles one chapter of his book The God Delusion “The God Hypothesis” and another “Why There Almost Certainly Is No God.” For Dawkins, to treat belief in God as a “hypothesis” is what it means to take the proposition “God exists” seriously as a contender for truth. As for the probabilistic qualification “almost certainly,” it is not for him a sign of weakness but a point of strength, as it shows that he, like any good scientist and in contrast to the great majority of theistic believers, founds his opinion in the matter on where the preponderance of evidence lies. “What matters,” he says at one point, “is not whether God is disprovable (he isn’t) but whether his existence is probable”; which, of course, it isn’t, according to him.

I am inclined to agree, in a certain guarded fashion, with Dawkins that the existence of God is not probable—not, however, because it is improbable, as he thinks, but because it is not a matter of probability at all. I said in my previous entry that it is not easy to defend this claim. This evoked some interesting comments from Tommi Uschanov, who does not share my sense of difficulty on this point. The following two observations, which, he says, “have been presented often in Wittgensteinian philosophy of religion, by O. K. Bouwsma or D. Z. Phillips, for instance,” he finds “do the work so well that nothing more needs to be said”:

1) If someone has lived his life atheistically or otherwise irreligiously through a wrong assessment of probabilities, due to an innate lack of talent for mathematics and statistics, this would seem to mean that God condemns him to perdition through a failure to endow him with sufficient talent to make the required calculations. But this is obviously contrary to the moral teaching of the religion itself. And indeed to the whole official self-image of the religion.

2) The importation of the probabilistic way of speaking to properly religious language makes this language (not unintelligible, which would be the positivist critique, but) uproariously funny.

For instance, . . . Psalm 23 does not say: “The Lord is probably my shepherd; I probably shall not want. . . .” [Other examples follow.]

The first argument seems to me an effective objection to anyone who, like William Lane Craig, uses probabilistic arguments to defend the reasonableness of Christianity; but only because Christianity, at least in some of its varieties, holds the non-acceptance of Christian doctrine to be a sin subject to divine retribution. There are, of course, interpretations of Christianity that reject this belief, but it has been a part of Christian doctrine historically and is, so far as I know, not found in any other major religion. In any case, it is not a part of theistic belief per se. The objection, therefore, tells only against probabilistic defenses of some varieties Christianity and not to probabilistic approaches to the question of divine existence in general. Further, the objection seems to be just a variant of the ancient one that if God makes human beings sinful that he cannot justly punish them for their sins: so if he makes someone inept at forming beliefs, he cannot justly punish that person for failing to arrive at the right beliefs. In any case, the most that this objection can show is that it is imprudent for a Christian to try to make probabilistic arguments for the existence of God. It doesn’t show that there is anything inherently wrong with doing this in general or with treating the question of God’s existence probabilistically in the first place.

Uschanov’s second argument may seem even less effective, as it can be rebutted on several grounds. For one thing, to make a probabilistic argument means only that the premises from which one argues provide reasons to accept one’s conclusion without entailing it with logical necessity. It does not mean that the conclusion has to include a probabilistic qualifier. For instance, if I know that Smith fell into a piece of industrial machinery and was ground to bits, and I conclude on that basis that he is dead, I am reasoning probabilistically; that does not mean that I am obliged to say only, “Smith is probably dead.” In such a case, my premise warrants my conclusion with moral certainty, which is certainty beyond a reasonable doubt (though not beyond all logically conceivable doubt). For another thing, if someone tries to show that there is sufficient empirical evidence to conclude that God exists, it does not follow that she is bound to import probabilistic language into her religious practices, such as prayers, or to rephrase scriptural passages to include such language. Finally, to advance a probabilistic argument for belief in the existence of God does not commit one to holding that theistic believers should base their belief on such a justification. One might offer the argument purely for the purpose of refuting skeptical doubts of God’s existence and showing that theistic belief is rationally warranted. (As I said in my reply to Tommi’s comment, William Lane Craig seems to be trying to do something parallel to this, but specifically for certain Christian doctrines, not for bare theism.)

With all that said, I think that there is at least potentially more to Uschanov’s objection (or to the sources from which he draws it) than such replies recognize. The point of the objection, as I understand it, is not to argue, “To defend theism probabilistically commits you to saying things like these; these things are patently ridiculous; therefore, it is misconceived to defend theism probabilistically.” At least, I think that the objection is much more effective if it is taken differently, as an attempt to bring out something incoherent in the probabilistic approach to divine existence precisely by taking it seriously. It is as if one were to say: “You want to treat the existence of God as a matter of probability? Fine! Let’s do that consistently and see what happens!”

The suggestion, in other words—at least, this is the suggestion that I derive from the objection as stated—is that if you adopt a probabilistic approach to the question of God’s existence, the “God” that you reason about, no matter whether your conclusion is theistic or atheistic, will be a philosophical fetish or idol and not that which is worshiped and served in any of the world’s religions. Probabilistic reasoning and religious practice are not two different ways of relating oneself to the same entity; rather, one is a way of relating oneself to God, if God exists, and the other is a way of relating oneself to a figment of the intellect mistakenly called by the same name. To put the point another way, a possible object of religious devotion is not a possible object of probabilistic reasoning.

That, at any rate, is the idea that Uschanov’s comment suggests to me. I think it can also be taken as a development of the objection that Duncan Richter was making in the blog entry that I discussed in my previous entry here, when he said that a probabilistic approach to the question of divine existence “treats God as the same kind of thing as a fluke gust of wind, i.e. something whose odds we might calculate or at least estimate, i.e. as something natural, however super.” If the objection can be satisfactorily worked out, it should be applicable to polytheistic religions as well as to monotheisms—or rather, not to the religions, but to probabilistic treatments of the question of the existence of their gods. It may even be applicable to probabilistic would-be defenses of revealed religion, such as that offered by Craig, who incorporates scripture into his evidence base.

I find it an attractive idea, but I don’t entirely trust it, and I certainly don’t have a defense of it ready. So, once again, I close with unfinished business.

REFERENCES

Joseph Butler, Analogy of Religion, Natural and Revealed, to the Constitution and Course of Nature, ed. by G. R. Crooks (New York: Harper and Brothers, 1860), p. 84.

Richard Dawkins, The God Delusion, paperback ed. (Boston and New York: Houghton Mifflin, 2008), p. 77.

Ian Hacking, The Emergence of Probability, 2nd ed. (Cambridge: Cambridge University Press, 2006), p. 1.

Is the Existence of God a Matter of Probabilities?

To treat the question whether God exists as a matter of probabilities seems to some people completely natural and to some utterly perverse. Believers and non-believers are found in both camps. I agree with Duncan Richter in finding such a way of thinking deeply wrongheaded, but I find his attempt to say what is wrong with it unsatisfactory.

I was delighted to find my previous two posts (1, 2) on ancient polytheism and the concept of evidence cited and discussed by Duncan Richter in his blog Language on Holiday. Richter’s discussion includes a parenthetical remark that approaches some lines of thought that I have pursued. He remarks that arguments for the existence of God that are founded on empirical observations, whether concerning religious experience, miracles, or design, all try to establish their conclusion as a matter of probability. (For the sake of simplicity, I shall in this entry equate questions of the existence of a divine being with the question of the existence of God, i.e., an inherently unique deity, leaving polytheism out of account.) He says of this way of thinking:

It is a logical and ethical mistake, an error in grammar and theology, to think of the existence of God as a question of probabilities. This might become clearer if one tried to calculate the odds, although I think people have done this and not achieved the clarity I have in mind. In case it isn’t clear, it’s a mistake because it treats God as the same kind of thing as a fluke gust of wind, i.e. something whose odds we might calculate or at least estimate, i.e. as something natural, however super. To think of God this way is to misunderstand what believers believe in a way that is both simply wrong (that isn’t what they believe) and insulting (it is to treat God as something less than what they believe). This is complicated by the fact that some believers (or “believers”) are idolaters in just this way, but that isn’t the kind of belief that interests me. There’s also the question whether non-believers like me should care about the alleged badness of insulting God, but we can at least respect the feelings of believers. And I think we can respect the concept of God, too, and want to do justice to it.

I suspect that the passage was written with some haste and impatience, for two reasons: first, it is rather long and contentious, not to say blustery, for a merely parenthetical remark; and second, saying exactly what is wrong with treating the existence of God as a matter of probability is no easy matter—or so, at any rate, say I. In this piece, I will give reasons for finding Richter’s presentation of the case unsatisfactory. I hope, though I dare not promise, to make a stronger case of my own in a subsequent entry to this blog.

Richter finds fault with probabilistic discourse about theism in two respects. One concerns the way in which it treats theistic believers. According to him, it insults them by treating God as “something less than what they believe [in].” He also implies that it fails to “respect their feelings.” Now it is possible that I am missing something here, but to me such claims seem simply irrelevant. If—and this is a large “if”—there is no fundamental conceptual error inherent in inquiring whether probability favors the existence or the non-existence of God, then I can see no compelling reason why those making such inquiries should care in the least whether they hurt the feelings of theistic believers or denigrate the object of their beliefs. At most, such considerations would be reasons to pursue such inquiries out of public hearing, so that they not offend the delicate ears of believers. But one could just as credibly argue that it is insulting to believers to assume that their sensibilities require this kind of protection. In any case, if they do, then it’s hard cheese for them and nothing more.

So it seems to me that Richter’s would-be ethical objection can be set aside. The entire weight of his objection must rest on its logical and grammatical part—“grammatical” here in Wittgenstein’s sense of concerning what one can intelligibly say and under what conditions. I believe that if this element of the objection could be satisfactorily articulated, the ethical aspect would emerge by itself. In fact, if I may mix the terms of the later Wittgenstein with the phrasing of the earlier, I would say that on this point grammar and ethics are one: if we could understand exactly what is so perverse about talking probabilistically about the existence of God, we would not distinguish a logical an ethical objections. Richter seems to me to move, or at least to face, in this direction when he describes theistic believers who take the question of divine existence to be a matter of probability as “idolaters,” a term that implies perversion of both intellect and will; but to make such a heavy charge stick would require an argument than I, for one, have not got at the ready.

What, then, is wrong with trying to assess the probability of divine existence? Richter holds that to do so “treats God as the same kind of thing as a fluke gust of wind, i.e. something whose odds we might calculate or at least estimate, i.e. as something natural, however super.” To take the first point first: what are the conditions under which we can calculate or at least estimate the odds of something? Richter may be assuming that we can do this only when we are talking about a type of event that occurs and recurs unpredictably under certain specifiable conditions, such a gust of wind of such and such a character that occurs a certain number of times in a certain location over a certain period of time. Given such specifications, we can observe a sample of cases and calculate the relative frequency of the event in question. The larger the sample that we have observed, the more confidently can we identify this relative frequency with the probability of a gust of wind occurring under the specified conditions.

Obviously, none of this is applicable to the existence of God, since that is not a repeatable event. So, if those who think of the existence of God probabilistically operate with a frequentist interpretation of probability, then they are hamstrung from the outset. But, of course, they do nothing of the sort; or at least, they need not do so. Here is the philosopher and Christian apologist—and, if Richter’s assessment is just, idolater—William Lane Craig on his website Reasonable Christianity answering a correspondent who is perplexed by the application of probabilistic terms to the question of the existence of God. Craig’s correspondent understands probability not in terms of relative frequency but according to what is known as the classical interpretation of probability, in which probability values are equated with the ratio of the number of cases in which a certain event occurs to the total number of possible cases. But Craig’s reply is equally applicable to the frequentist interpretation:

Probabilities are always relative to some background information. . . . Now the atheist says God’s existence is improbable. You should immediately ask, ‘Improbable relative to what?’ What is the background information? . . . The interesting question is whether God’s existence is probable relative to the full scope of the evidence.

Had you asked that question of your friend, it would have been evident that he is considering no background information at all! He seems to be talking about a sort of absolute probability of God’s existence Pr (G) in abstraction from any background information B and specific evidence E. That’s a pointless exercise. He seems to be imagining all the possible deities that could exist and asking, “What are the chances apriori that a certain one of these exists?” How silly! That’s like inquiring about the absolute probability that a certain person, for example, you, exists, given the infinite number of possible persons there could be. Nobody is interested in such absolute probabilities, if there even are such things. What we want to know, rather, is the probability of your existence or God’s existence relative to our background information and specific evidence: Pr (G|E & B).

Craig operates with a subjectivist or, as it is widely known, Bayesian interpretation of probability. On this interpretation, the values that are assigned to probabilities of events represent degrees of confidence in the occurrence of those events. Such assignments do not require that the events be repeated or repeatable at all: one can attribute a degree of probability to any event whatever, even the existence of God (or of a god of some specific description). (“Event” here is a technical term in probability theory for that to which a probability value is assigned and is not contrasted with “fact” or “state of affairs.”)

The competition among interpretations of probability is a vast and complicated issue, into which I don’t propose to enter any farther. My point here is simply that, if one holds there to be a confusion inherent in treating the existence of God as a matter of probability, one cannot support that claim by simply assuming an interpretation of probability that requires a repeated event or a countable set of possible outcomes, as there are interpretations of probability that don’t require those things. To Richter’s remark that to talk of the existence of God in probabilistic terms treats it as “something whose odds we might calculate or at least estimate,” Craig would reply, or anyway could reply, “Yes; so what?” So, for that matter, could Richard Dawkins.

I can imagine one of these probabilists saying to Richter (and, for that matter, to me): “I suspect that the reason why you dislike this talk of the probability of God’s existence is that it seems to kill all the existential drama and to make the business of believing or not believing in God out to be a matter purely of the intellect. But, look you, I am not touching at all on the question of what moves people to believe or disbelieve in God, or what difference their belief or lack of belief makes to their lives. I am just assuming that when we ask, ‘Does God exist?’, we are posing a genuine and well-formed question—one that has a correct answer. The correct answer is either ‘Yes, God exists’ or ‘No, God does not exist.’ To determine which is the correct answer, one has to determine where the preponderance of evidence lies. To do this is to assess the probability of the proposition ‘God exists.’”

I do not think that this argument is unanswerable, but I do think that to answer it is not easy. In any case, I leave the task for a later post.