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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”.