prior probability

Thus far, I have surveyed “the good” and “the bad” sides of David Hume’s famous argument against miracles (see here and here). That leaves “the ugly”: Hume’s circular definition of what a miracle is.

To the point, for Hume “[a] miracle is a violation of the laws of nature” (Hume, Of Miracles, para. 12; cf. Voltaire 1764/1901, p. 272). Alas, the problem with Hume’s definition is that it is circular! Why is this definition a circular one? Because unlike human laws, which are defied and disregarded all the time, “laws of nature” consist of patterns or regularities that can never be set aside or suspended. In other words, a violation of a law of nature has, by Hume’s own definition, a probability of zero!

To see why, ask yourself two deeper questions: (1) what is a “law of nature” (see here, for example), and (2) what does it mean to “violate” or disobey such a thing? Although Hume himself uses the term “laws of nature” in several different senses (see here), at a minimum a law of nature in the traditional Newtonian sense (see, e.g., “Newton’s Three Laws of Motion“) generally refers to some uniform or regular pattern of behavior. On this view of “natural law”, a miracle must consist of an unforeseen departure from or an unexpected interruption of such a pattern.

By way of illustration, consider Newton’s First Law: “Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.”(Translation: “Every object perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.”) In plain English, this law tells us that objects tend to “keep on doing what they’re doing” until they are acted upon by an outside force. The dirty clothes in your hamper, for example, will remain there until someone grabs them and throws them in the washing machine.

Maybe you agree with Hume’s circular definition of miracles. After all, what if your clothes did wash themselves? That would be a true miracle! But from a purely logical perspective, the problem with Hume’s definition is that he’s rigged the game ex ante. To see why, recall Hume’s two-part probability test for miracles: 1st, given some evidence that a miracle took place, Hume would have us assign two separate probability values to the evidence — the probability p₁ that the miracle really happened and the probability p₂ that the evidence is either mistaken or fraudulent or otherwise defective — and 2nd, Hume would have us compare p₁ and p₂: you should believe in the miracle only if p₁ > p₂.

The problem, however, is this: under Hume’s natural-law-violation definition of miracles, p₁ = 0, no amount of evidence of whatever kind would ever be sufficient to establish the occurrence of a miracle, a truly dogmatic and closed-minded position to take, especially when we have so many reports of miracles from so many different sources! Nevertheless, that said, it takes a theory to beat a theory, so in my next post, I will present an alternative method for evaluating incredible claims of miracles.

***

As I mentioned at the end of my previous post, David Hume’s argument against miracles has two big blind spots. One is the unknown probability problem; the other is the reference class problem. To appreciate the significance of these logical troubles, recall Hume’s two-part probability test for miracles: 1st, given some evidence that a miracle took place, Hume would have us assign two separate probability values to the evidence — the probability p₁ that the miracle really happened and the probability p₂ that the evidence is either mistaken or fraudulent or otherwise defective — and 2nd, Hume would have us compare p₁ and p₂: you should believe in the miracle only if p₁ > p₂.

But what if the probability values for p₁ and p₂ don’t add up to 1? By way of illustration, what if p₁ = 0.01 and p₂ = 0.1? (Or in plain English, what if you believe there is only a 1% chance that the miracle in question really took place, and at the same time, you also believe there is only a 10% chance the evidence of the miracle is reliable?) In this case, what do we do with the missing 89% of the probability space in this example, i.e. the “unknown probability”?! Alas, Hume does not say, and Hume’s silence, in a nutshell, is what I mean by the unknown probability problem.

The reference class problem, however, is an even bigger problem for Hume, for how do we figure out what numerical values to assign to p₁ and p₂ in the first place? Stated formally, in deciding whether a particular miracle X is real or not, what is the reference class of X? After all, X could be a member of a wide variety of classes or groups, and the probability of X will thus differ depending on how the class or group of X‘s is defined. (See, e.g., James Franklin (2011), The objective Bayesian conceptualisation of proof and reference class problems, Sydney Law Review, Vol. 33, pp. 545–561.)

Airplane crashes, for example, have been in the news lately, especially since the fatal midair collision at Reagan National Airport on January 30th of this year, but at the same time, we are also being told that aviation safety is at an all-time high. This disparity between the number of news reports and the true level of aviation safety provides a textbook example of the reference class problem, for to estimate the probability of an aviation disaster, one might use the frequency of crashes of all aircraft (civilian and military), the frequency of crashes of a particular model or type of aircraft (e.g. helicopters), or the frequency of airplane accidents in the last five, ten, or y years.

To recap, Hume’s argument against miracles is caught between the Scylla of the unknown probability problem — i.e. the possibility that p₁ and p₂ do not add up to 1 — and the Charybdis of the reference class problem — i.e. the difficulty of deciding what class or group of events to use when calculating p₁ and p₂. Can either of these Humean probability monsters be tamed? Stay tuned, for before I try to rescue Hume from this metaphorical Strait of Messina, I want to take the Scottish essayist to task for his circular definition of “miracles”.

As I mentioned in my previous post, this week I will offer my own lawyerly take on David Hume’s influential argument against the possibility of miracles: the good, the bad, and the ugly. Let’s start out on a positive note, shall we, with “the good”? (Rest assured, I will consider “the bad” and “the ugly” sides of Hume’s anti-miracle essay in subsequent posts.)

Simply put, what I like the most about Hume’s argument is his lawyerlike and probabilistic approach to the problem of miracles. For Hume, whenever you read or hear a report of a miracle or some other remarkable event — whether it be a UFO sighting, a levitating saint, or Lazarus’ resurrection from the dead — you should weigh the evidence based on your own experience and common sense. Specifically, Hume proposes a two-part probabilistic test for weighing the evidence.

In summary, the first part of Hume’s probabilistic test requires us to assign two separate probability values to the evidence as follows: the probability p₁ that the miracle really happened and the probability p₂ that the evidence is either mistaken or fraudulent or otherwise defective. The second part of Hume’s test is to compare p₁ and p₂. To the point, you should believe in the miracle only if p₁ > p₂.

Moreover, Hume’s exhortation to weigh the evidence is exactly what juries do when deciding law cases. In fact, Hume’s simple probabilistic test reminds me a lot of my colleagues Ron Allen and Mike Pardo’s “storytelling” account of legal proof. (See Ronald J. Allen and Michael S. Pardo (2008), Juridical proof and the best explanation, Law and Philosophy, Vol. 27, pp. 223–268, which is available here. See also Allen & Pardo (2019), Relative plausibility and its critics, International Journal of Evidence & Proof, Vol. 23, pp. 5–59, available here.)

Among other things, Allen and Pardo’s account of legal proof not only makes intuitive sense; their storytelling approach also parallels Hume’s probabilistic two-part test for miracles:

“The proof process involves two stages: (1) the generation of potential explanations of the evidence and events, and (2) a comparisons of the [plaintiff’s and defendant’s competing] explanations in light of the applicable standard of proof. In general, the process depends on the parties to obtain evidence and to offer what they consider to be the best explanation (or explanations) that support their respective cases.” (Allen & Pardo 2019)

In other words, when jurors are deciding cases, what they are really doing is comparing the competing stories of the plaintiff and the defendant, and it is the party who offers the best explanation of the evidence — i.e. the party who tells the best story — who wins. Sound familiar?

This storytelling account of legal proof — like Hume’s probabilistic test for miracles — also has two major virtues. One is that it makes the most intuitive and descriptive sense: jurors do tend to weigh the evidence presented at a trial holistically, and they tend to favor the side whose story is more persuasive, or to borrow Allen and Pardo’s terminology, juries usually prefer the side whose story offers the best explanation of the evidence. The other virtue of the storytelling approach is an aesthetic one: its elegant simplicity. Why is simplicity a virtue? Because more simple or parsimonious explanations are almost always better than complex or convoluted ones, a fundamental principle called “Occam’s Razor”.

Nevertheless, Hume’s probabilistic test, along with Allen and Pardo’s storytelling approach to legal proof, has two big blind spots: (1) the reference class problem, and (2) the unknown probability problem. Stay tuned: I will explain both of these fatal flaws and present an alternative approach to the problem of miracles in my next few posts.

Last week, we surveyed David Hume’s influential argument against the possibility of miracles. (See here, here, and here.) This week, I want to offer my own take on Hume’s argument — the good, the bad, and the ugly — but before I do, I want to begin by making a full disclosure: I will for the most part avoid consulting or mentioning (except in today’s introductory post) any of secondary and even tertiary literature that already exists on this topic.

For starters, the scholarly literature is massive. A quick search for “Hume on miracles” in Google Scholar, for example, generates over 67,000 discrete entries! (Here, for example, is an early response to Hume by one William Adams.) So, what can I contribute to the problem of miracles? In a word, I will use my background knowledge and experience as a law professor — one with a deep interest in economics, history, and probability theory — to offer my own “legal proof” perspective on Hume’s argument.

In the meantime, below are some fairly recent scholarly works that students of Hume and the problem of miracles may wish to consult for future reference:

• ANTI-HUME: John Earman, Hume’s Abject Failure: The Argument Against Miracles, Oxford University Press (2000).
• PRO-HUME: Robert J. Fogelin, A Defense of Hume on Miracles, Princeton University Press (2003).
• FURTHER READING (HONORABLE MENTIONS): William L. Vanderburgh, David Hume on Miracles, Evidence, and Probability, Lexington Books (2019) and Peter Millican, Twenty Questions about Hume’s “Of Miracles”, Royal Institute of Philosophy Supplements, Vol. 68 (2011), pp. 151-192.

The artist Roberta Flack, whose music made an indelible mark on my childhood, died last month at the age of 88; here is her New York Times obituary.

Having defined what a miracle is (see here) and having established the relationship between evidence and probability (here), Hume is now ready to finally unveil his novel argument against miracles. To the point, for Hume “no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavours to establish …” (Hume, Of Miracles, Para. 13). In plain English, what Hume is saying here is that even when we have direct evidence of a miracle, such as eyewitness testimony, our inquiry is not over. We still have to weigh the evidence. Specifically, we must consider not only the probability that the evidence is reliable or true but also the probability that it is defective or false, or in the immortal words of David Hume himself:

“When any one tells me, that he saw a dead man restored to life, I immediately consider with myself, whether it be more probable, that this person should either deceive or be deceived, or that the fact, which he relates, should really have happened. I weigh the one miracle against the other; and according to the superiority, which I discover, I pronounce my decision, and always reject the greater miracle.” (Hume, Of Miracles, Para. 13, emphasis added)

Hume thus proposes a simple two-part probabilistic test for evaluating reports of miracles. The first part works as follows: if someone, for example, tells you X — that they saw a UFO or were abducted by aliens — you need to consider two separate probabilities: A and B, where A is the probability that X, the remarkable or unusual event in question, took place — i.e., how likely is it, given your own experience and common sense, that the UFO sighting or alien abduction really occurred? — and where B is the probability that the report is either mistaken or fraudulent or otherwise defective — or in Hume’s words, the probability “that its falsehood would be more miraculous, than the fact, which it endeavours to establish” (ibid.).

Next, after assigning probability values to both of these logical possibilities (A and B), the second and last part of Hume’s test is to compare both probabilities. According to Hume’s probabilistic logic, you should prefer the possibility whose probability value is greater. (Or, put another way, only if the probability of B is somehow smaller than that of A should you believe in X.)

This post concludes my review of Part 1 of Hume’s essay “Of Miracles” (paragraphs 1 to 13). Starting next week, I will present a “Bayesian” critique of Hume’s solution to the problem of miracles and then turn my attention to Part 2 of his essay (paragraphs 14 to 41). In the meantime, I would like you to think about the following question: is there any way a report of a miracle or other remarkable event could ever pass Hume’s probabilistic test?

David Hume finally gets around to miracles in Paragraph 12 of his famous essay on this subject (see here), where he writes: “A miracle is a violation of the laws of nature” (Hume, Of Miracles, para. 12; cf. Voltaire 1764/1901, p. 272). For Hume, the textbook example of a miracle is the resurrection of Lazurus of Bethany in the Gospel of John (see John 11:1–44), or in the immortal words of the Scottish essayist, “… it is a miracle, that a dead man should come to life; because that has never been observed, in any age or country” (para. 12).

But is Hume’s definition a circular one? As my colleagues Timothy McGrew and Robert Larmer explain in their 2010 essay on “Miracles” for The Stanford Encyclopedia of Philosophy (available here), Hume’s miracle definition raises many more difficult questions than it answers. For starters, how many “laws” of nature are there? What do these so-called natural laws consist of? And who is the “lawgiver”, so to speak? If it’s God or some other deity, then why can’t the lawmaker violate his own laws, and what does it mean to “violate” a law of nature anyways? Consider again the resurrection of Lazurus. However remarkable or rare this feat is, which natural law in particular did Jesus violate when he performed this miracle?

In any case, if we have multiple firsthand or eyewitness reports of Jesus raising Lazarus from the dead (direct evidence, not just hearsay!), why should we follow Hume in rejecting this testimony out of hand? That is, why should we assume that a law of nature has been violated whenever something strange, unusual, or unexpected has occurred? As we shall see in my next post, Hume’s answer to these questions is an ingenious one!

The Scottish Enlightenment figure David Hume (pictured above) makes four important preliminary observations about the relationship between evidence and probability in the first part his essay “Of Miracles” (paragraphs 3 to 8). For reference, I will restate Hume’s main points as follows:

• First off, Hume describes belief as a continuous variable, not a binary one, because one’s belief about a “matter of fact” — i.e. the probability that the fact might, or might not, be true — can vary from very high (i.e. near certainty) to very low (i.e. total disbelief), or in the immmortal words of David Hume: “in our reasonings concerning matter of fact, there are all imaginable degrees of assurance, from the highest certainty to the lowest species of moral evidence” (Hume, Of Miracles, para. 3, emphasis added).
• From this fundamental premise (i.e. belief is a continuous variable), Hume concludes that the actual level of your degree of belief about a disputed fact should correspond to the amount of relevant evidence available to you: “A wise man, therefore, proportions his belief to the evidence” (para. 4, emphasis added). And he then illustrates this relationship between belief and evidence with a specific example: “A hundred instances or experiments on one side, and fifty on another, afford a doubtful expectation of any event; though a hundred uniform experiments, with only one that is contradictory, reasonably beget a pretty strong degree of assurance” (ibid.).
• Next, Hume observes that in ordinary life the most common type of evidence we use when making our probability judgments (i.e. when determining how strong or weak our degree of belief should be) are not scientific experiments or clinical trials. The most common forms of evidence we rely on are historical reports and eyewitness testimony: “there is no species of reasoning more common, more useful, and even necessary to human life, than that which is derived from the testimony of men, and the reports of eye-witnesses and spectators” (para. 5).
• Lastly (and most importantly), Hume says that we judge the credibility of such evidence (i.e. historical reports or eyewitness testimony) in light of our own understanding of how the world works: “the ultimate standard, by which we determine all disputes, that may arise concerning them [eyewitnesses testimony], is always derived from experience and observation” (para. 6, emphasis added). In other words, we don’t automatically accept a person’s testimony at face value. Instead, we ask ourselves whether his testimony is consistent with our own personal experience of the world and, I might add, with common sense:

“The reason, why we place any credit in witnesses and historians, is not derived from any connexion, which we perceive à priori, between testimony and reality, but because we are accustomed to find a conformity between them.” (para. 8)

To be continued …

Thus far this week, I have restated Hume’s “hearsay argument” against transubstantiation (see here) and surveyed some possible exceptions to the hearsay rule in law that might be application to the case of transubstantiation (here). But what is my position regarding this controversy? Am I with Hume or Augustine?

To the point, I would apply the “Principle of Indifference” — an idea from the domain of probability theory — to the question of transubstantiation and the theology of Jesus’ Last Supper. This principle, which can be traced back to John Maynard Keynes’s Treatise on Probability (1921, pp. 41-64), applies when two conditions are met: (i) you are considering two or more mutually-exclusive hypotheses, such as the possibility of transubstantiation during the celebration of the Eucharist, but only one of which can be true, and (ii) you have no direct evidence to favor one possibility over the others. In that case, when you have no direct evidence to favor one outcome over another, you should assign equal probability to each hypothesis.

Or in the words of Lord Keynes (ibid., p. 42) himself:

“The Principle of Indifference asserts that if there is no known reason for predicating of our subject one rather than another of several alternatives, then relatively to such knowledge the assertions of each of these alternatives have an equal probability. Thus equal probabilities must be assigned to each of several arguments, if there is an absence of positive ground for assigning unequal ones.”

In closing, although transubstantiation has all the hallmarks of a self-sealing conspiracy theory (e.g., emotional appeal, unfalsifiability, and in-group/out-group dynamics between true believers and heretics), at the same time both of the conditions mentioned above apply to the case at hand, since we have no direct evidence one way or another telling us whether transubstantiation is true or not. Contra Hume, then, I would start any discussion of the Last Supper by assigning a 0.5 probability to the possibility that transubstantiation might be true. Keep this conclusion in mind as we consider Hume’s other arguments against the possibility of miracles … (to be continued)

Previously, I described David Hume’s restatement of John Tillotson’s anti-transubstantiation argument in the form of a logical syllogism. In summary, Hume’s syllogism is this: there is no direct evidence that transubstantiation really occurs during the sacrament of Communion; instead, the only evidence we have that this doctrine might be true are some second-hand statements during the Patristic period, such as the sermons of Saint Augustine. Second-hand hearsay evidence, however, is always weaker or less reliable than direct evidence; therefore, transubstantiation cannot be true. But as I noted at the end of my previous post, this syllogism is incomplete, for Messrs Hume and Tillotson fail to consider any of the standard exceptions to the hearsay rule.

By way of illustration, the Federal Rules of Evidence (see here or here) recognize no less than 23 exceptions to the hearsay rule, including excited utterances, business records, and death-bed or dying declarations. Do any of these hearsay exceptions apply to the problem of transubstantiation? As it happens, there are at least three exceptions to the hearsay rule that might be applicable to the case at hand: statements in ancient documents; market reports and commercial publications; and statements in learned treatises, periodicals, and pamphlets. (For reference, see Rules 803(16), 803(17), and 803(18) of the Federal Rules of Evidence.)

Consider, for example, the collection of sermons of Saint Augustine of Hippo (pictured below) published under the title “Sermons 230-270B on the Liturgical Seasons.” Specifically, in Sermon 234 (418 A.D.) Saint Augustine declares, without citing any supporting evidence, that the bread consecrated in the Eucharist becomes the body of Christ: “The faithful know what I’m talking about; they know Christ in the breaking of bread. It isn’t every loaf of bread, you see, but the one receiving Christ’s blessing, that becomes the body of Christ.” Is Augustine speaking metaphorically here, or is he being literal? Either way, along with Ambrose (AD 340–397), Jerome (347–420), and Pope Gregory I (540–604), Augustine (354–430) is considered one of the Great Church Fathers, so this sermon could easily fall under the exception for an ancient document or a learned treatise — or perhaps even a commercial publication!

But that said, just because a particular piece of evidence regarding transubstantiation — such as a statement in an ancient sermon by a Church father — would be admissible in a court of law under one of the exceptions to the hearsay rule does not by itself mean that transubstantiation is true. We still have to weigh the evidence (i.e. decide how trustworthy it is), draw reasonable inferences from it, and make up our own minds. In my next post, I will turn to another religious figure, the Rev Thomas Bayes (!), and explain why we should (contra Hume and Tillotson) start any discussion of transubstantiation by assigning a 0.5 probability to the possibility that this doctrine might be true.