JE argues that the most reasonable reading of Hume’s argument against miracles is as follows. Say we have seen a long succession of some event A occurring. Further stipulate that in all known instances of A, it turns out that A was also B. On JE’s reading of Hume, this makes the statement ‘All As are Bs’ a presumptive law of nature.[iv] This yields the conclusion that we should assign the likelihood that the next A will also be a B to 1. In other words, we should possess absolute certainty that the next occurrence will not be a violation of our presumed law of nature. Put into Bayesian terms[v], this can be expressed as follows:[vi]
Equation 1: P [M | t (M)] = P [t(M) | M] P (M) / P [t (M)]
I should stress before proceeding that while I am familiar with Bayes Theorem, I am hardly an expert. This review may therefore be read provisionally as “This is how things appear to someone who is familiar with, but far from an expert on, Bayes theorem.”
Before proceeding to Hume’s argument, a few features about Equation 1 should be noted. First is the obvious point that this equation is a ratio. The closer the blue portion is to 1, the greater the probability that a miracle (‘M’) occurred. The blue portion will be closer to 1 as the numerator (the green and red portions) are closer themselves to 1. The blue portion will also be closer to 1 as the denominator of the ratio (the orange portion) is closer to 0.[vii] So what the person pressing an argument from miracles will want to claim is that the green portion (the likelihood of testimony occurring when a miracle happened) is high, that the red portion (the intrinsic or a priori probability of miracles) is either high or at least not insuperably low, and that the a priori probability of testimony to miracles is low. (See footnote vii.)
Hume’s argument against miracles is focused on the red portion of the equation.[viii] Returning to the second paragraph of this review, Hume would argue that if all known As are Bs, then we should assign the probability of the next A being B as 1. Recast in terms of the Bayesian formula, if all known events are ~M (not miracles) we should assign the probability of the next event as ~M at 1. So P (M) is effectively 0! This, of course, means that P [M | t (M)] is ineluctably going to be 0 as well, no matter what the other values are.
Put in such stark terms, JE notes sardonically “little wonder then that those of Hume’s contemporaries who had a less crude view of how induction works found no merit in Hume’s “proof”.[ix] JE notes that Hume’s view of induction, at least as interpreted by JE, is “both descriptively inadequate to scientific practice, and it is stultifying to scientific inquiry”.[x] In essence, scientific practice appears to do a better job of meeting JE’s proposed criteria for any adequate epistemology: it provides some means by which possible exceptions to purported laws of nature could be evidentially supported, while also taking into account background knowledge which is not easily overthrown.[xi] JE notes, for example, that although particle physicists have observed zillions of protons, none have been observed to decay, nonetheless those scientists do not think they have adequate inductive evidence to assert that the probability of such decay is zero.
Given that P (M) cannot be reasonably set at zero, is it possible that it be insuperably low? Hume claims that no testimony is sufficient to establish a miracle, unless the falsity of that testimony is more improbable than the occurrence of the miracle. Put in Bayesian terms, the proponent of miracles would thus have to claim that
Equation 2: P [M | t(M) & K] > P [~M | t(M) & K]
Equation 3: P [M | t (M)] / P [~M | t(M)] is > 1.[xiii]
JE then discusses a fallback position. Assuming that a miracle could be proven, Hume raises an objection which we might term ‘the argument from religious diversity’. That is, for a given miracle to provide evidence for a specific religion (say, the Resurrection as supporting Christianity), Bayesian reasoning requires that we calculate not only the prior probability of the Resurrection but its conditional probability on the assumption that any other (non-Christian) religion is true, further weighted by the prior probabilities of those religions. And here we run into a problem: as noted in footnote v below, these are not objective statistical probabilities but degrees of belief or what we might term ‘creedence’ functions. And hence they are ultimately subjective. JE notes, however, that similar problems plague scientific theorizing. JE draws a further parallel between theory confirmation in science and in adjudicating among religions in noting that incremental confirmation, while perhaps not what theists would prefer, could possibly (although, given JE’s agnosticism, improbably) result in a high degree of confirmation for theism (and even for specific theisms—say Christianity over Islam).[xvii]
Despite his trenchant criticisms of Hume’s argument, JE notes that he and Hume are incredulous about miracle claims.[xviii] JE opines that, in his opinion, the circumstances of religious believers are such that we should suspect that they do not (and likely never will) be sufficiently reliable so as to overcome the low prior probability of a miracle occurring. To his credit, however, he acknowledges that he is not able to defend this thesis in great detail[xix] and that he is not, like Hume, cloaking a personal opinion in an eloquent-sounding but ultimately (in JE’s opinion) ‘abject failure’ of philosophizing.[xx]
In sum, JE does a fine job of showing where Hume’s argument against miracles falls short. Hume arguably wanted to develop a knock-down rule against miracles by arbitrarily assigning a nil prior probability to a miracles occurring. Such an approach is contrary to theory adjudication in science and in other areas[xxi]. Further, the same factors that guide theory adjudication in science can, at least in principle, guide adjudication between metaphysical / theological systems.[xxii] The theist can thus consider that there is, at least based on JE’s reading of Hume’s argument, no ‘God killer’ argument to be found. Indeed, it appears that there is nothing in principle which makes metaphysical claims any less justifiable[xxiii] than scientific claims.
Apologetics 315 Book Reviewer Latter Day Inkling is a U.S.-based research psychologist for the military. He is especially interested in epistemology and natural theology.
[i]“ I contend that Hume's argument is largely derivative, almost wholly without merit where it is original, and worst of all, reveals the impoverishment of his treatment of inductive reasoning.”-Kindle edition, Location 9
[ii] “In criticizing Hume's argument against miracles. I have occasionally been subjected to a kind of reverse inquisition: since I attack Hume, must I not have some hidden agenda of Christian apologetics? I find such inquisitions profoundly distasteful since they deflect attention from the real issues. I am not averse, however, to laying my cards on the table. I find much that is valuable in the Judeo-Christian heritage, but I find nothing attractive, either intellectually or emotionally, in the theological doctrines of Christianity.”-Kindle edition, Location 27
[iii] “Any epistemology that does not allow for the possibility that evidence, whether from eyewitness testimony or from some other source, can establish the credibility of it UFO landing, it walking on water, or a resurrection is inadequate. At the same time, of course, an adequate epistemology should deliver the conclusion that in most (all?) actual cases, when all the evidence is weighed up, little credibility should be given to such events. Hume's account of inductive reasoning is incapable of satisfying these dual demands.”-Kindle edition, Location 75
[iv] This makes this an epistemological rather than an ontological matter. That is, we are justified in assuming that ‘All As are B’s’ is a law of nature, and thus uniform (never violated). This allows JE to avoid assigning the question begging interpretation of Hume as arguing that laws of nature are never violated, so that violations never occur.
[v] It must be kept in mind that the probabilities assigned here are not statistical probabilities. It is not as if, say, we have been able to create worlds and see what proportion of those worlds do (not) contain miracles and testimonies of miracles and calculate the percentages of that set of worlds in which miracles and/or testimonies of miracles do (not) happen. Rather “a number of Hume’s contemporaries, such as Price, understood Hume’s claims as being about quantifiable degrees of belief or credibility, the quantification being subject to the constraints of the probability calculus.”-Kindle edition, Location 437
[vi] I apologize for my nonstandard notation. While there seems to be general agreement about the basic structure of Bayes’ Theorem, different authors prefer to use slightly different symbols. I have proceeded in a way that struck me as slightly more intuitive, and provided color coding to help illustrate the portions of the equation that are at issue at any given juncture.
[vii] This last point can be confusing: isn’t the orange portion concerned with the probability of testimony occurring? Yes, but it is an a priori probability. If the likelihood that the testimony to miracles would occur was very high whether or not miracles actually occurred, then the evidential value of miracles is useless. Look at it this way: testimonial evidence is good evidence only to the extent that we should expect such testimony to occur if the miracle actually happened, and we would not expect such testimony to occur if the miracle did not happen.
[viii] Not that Hume explicitly cast his argument in Bayesian terms. Earman is a Bayesian and argues (or perhaps rather asserts) that Bayes Theorem is appropriate for analyzing Hume’s argument, especially given that proto-Bayesian formulations of this topic were extant in Hume’s lifetime.
[ix] Kindle edition, Location 409
[x] Kindle edition, Location 525
[xi] One way of thinking about this is as follows, and is discussed at some point by JE. While he is inclined to discount testimony to alien visitations, he is also not inclined to assign such events a likelihood of zero. He therefore acknowledges that in his view the likelihood of such events is small (based on his background knowledge of the world) but not utterly impossible.
[xii] Here one sees the point of Richard Swinburne’s cumulative approach to religious belief. As he argues in multiple places, including The Resurrection of God Incarnate and (more simply) in Was Jesus God? if the background knowledge includes the evidences of natural theology then the probability required to believe in a miracle is lower than it would be without such background knowledge.
[xiii] Because K is a constant (assumed on both sides of the equation) it can be dropped out.
[xiv] Equation 4: P [M | t (M)] / P [~M | t(M)] = P [t(M)|M] / P [t(M) | ~M] x P (M) / P (~M). I have kept the color coding across Equations 3 and 4 consistent to allow the studious reader to crosswalk the relevant portions.
[xv] “The effects of multiple testimonies to the same event were given a systematic Bayesian analysis by Charles Babbage in his Ninth Bridgelvater Treatise (1838). The claimed upshot of his discussion is this: "If independent witnesses can he found, who speak the truth more frequently than falsehood, it is always possible to assign a number of independent witnesses, the improbability of the falsehood of whose concurring testimonies shall be greater than that of the improbability of the miracle itself'. “-Kindle edition, Location 917
[xvi] The more reliable each eyewitness is, the fewer eyewitnesses are needed to overcome a given a priori (presumably low) probability of a miracle occurring. The more unreliable a witness is (given the requirement that each eyewitness is more reliable than unreliable) it is still possible, given enough eyewitnesses, that the low prior probability of a miracle be ‘overcome’ (i.e., that the miracle be adequately evidenced).
[xvii] Nor is the fact that God is unseen a barrier to justified belief in theism. JE notes multiple unseen entities (e.g., quarks) that scientists believe in. Such parallels are also made throughout the works of Richard Swinburne.
[xviii] “Like Hume, I do not think that a man of sense should give much credence to such tales, although unlike Hume 1 do not think that there are valid principles of inductive reasoning to show that such tales are never, in principle, to be credited.”-Kindle edition, Location 994
[xix] But I do believe, in a way that I cannot articulate in detail, that these cases are in fact relevantly similar to the case of faith healing where there is it palpable atmosphere of collective hysteria that renders the participants unable to achieve the minimal reliability condition-indeed, one might even say that it necessary condition for being a sincere participant in a faith healing meeting is the suspension of critical faculties essential to accurate reporting.”-Kindle edition, Location 998
[xx] I was powerfully reminded of similar, straightforward-admissions of personal preference for non-theistic worldviews made by Thomas Nagel. See his Mind and Cosmos pages 10 (‘an ungrounded assumption of my own…I lack the sensus divinitatis) and 23 (‘an ungrounded intellectual preference’).
[xxi] See Chapter 14 ‘The Indian Prince’, Kindle edition, Location 554
[xxii] Many theists will be put off by the idea of running their religious beliefs through Bayesian calculations. Someone sympathetic to Reformed Epistemology will, of course, be unfazed.
[xxiii] Again, let me stress: JE does think religious claims are less justified (at least at present) than scientific claims. But this is not an ‘in principle’ objection.