I didn’t realize until I came to WordPress to write this post that my last post was more than a week ago! I had originally intended to write this last Friday or Saturday. All future statements about forthcoming blog posts should be taken in the same general vein as the Scriptural statement “Behold, I am coming soon.”
Hume states in his Enquiry that no testimony is sufficient to establish a miracle as fact, unless the testimony is such that its falsehood would be more miraculous than the miracle in question. This could be interpreted in various ways, but in order to be a true statement it must mean something like this: that given a choice between two mutually exclusive choices (e.g. the falsehood of certain testimony or the occurrence of an alleged miracle), the rational thing to do is to side with the more probable choice. It’s almost tautalogical in a way; the more probable is more probable than the less probable (which, by the way, reminds of a brand of shampoo that once advertised itself as costing “less than the more expensive brands”).
Now, Hume evidently intends this principle to show that it is irrational to accept the occurrence of any alleged miracle. The idea, presumably, is that testimony is never so strong that it its falsehood would be less probable than supposed miraculous occurrences.
I bring this up because the flaw in Hume’s argument is directly pertinent to the question I asked at the end of the last post. Specifically, Hume’s application of his principle seems plausible on the surface because there is a tendency to consider only the prior probability when judging the probability of, for example, the falsehood of a certain statement. Clearly, if we simply consider in general all the claims that people make, the prior probability of a given claim being false is relatively high. It is less than one-half, since people say more true things than false things, but it is higher than, say, 0.1 percent, which is still much more than the prior probability of, say, water turning to wine. And so the principle implies that someone’s claim that water turned to wine cannot be accepted because it is more probable that the claim is false than that the event actually occurred.
The problem with this, of course, is the consideration of prior probabilities only. By this reasoning, no testimony could ever be accepted if the event being described had a low prior probability. If I read in the newspaper that John Smith won the lottery last week, I could reason to myself that I should conclude rather that the newspaper is in error than that Smith in fact won the lottery, since the chance of Smith winning the lottery is considerably lower than the chance of any given statement in a newspaper being in error. But no one does this, and it is evident that it would be a mistake to do so. The reason that it is a mistake is the one just stated, although this may not be obvious at first. So what does the right analysis look like?
When judging a claim, the right question to ask is not what is the simple probability of the event claimed, but what is the probability of the event given the particular evidence for the event. Even when people realize this fact, they tend to not see (or to ignore) the additional fact that the claim itself constitutes evidence for the event. This error frequently arises when people want to raise doubts about things like the historicity of Scripture; that is, they imply that the baseline assumption is that the events narrated are not historical, and that outside evidence must be accumulated that outweighs the assumption before the events can rationally be accepted as historical. People say things like this (to make up an example): “There is no historical evidence that Caesar Augustus put out a decree of enrollment.” But the truth of the matter is that the Gospel of Luke is itself historical evidence that Caesar Augustus put out a decree of enrollment. How strong evidence it constitutes would need to be considered, of course, but one cannot simply ignore, as people tend to, the fact that it does constitute evidence.
What is most crucial for the original question about extraordinary claims, however, is that not only do claims of events constitute evidence for the events, but also, in general, claims of less probable events constitute stronger evidence. In other words, extraordinary claims are extraordinary evidence. To use the example I gave in the last post, the claimed event of a person flipping a coin and getting eight heads in a row does indeed require stronger evidence than the claim of getting three heads in a row; but the claim itself is stronger evidence. Similarly, the reason the newspaper can be believed when it states that John Smith won the lottery is that, although the average probability of a generic statement in the newspaper being erroneous is greater than the probability of Smith’s winning the lottery, nevertheless the probability of the paper’s claim that Smith won the lottery being erroneous is less than the probability of Smith’s winning the lottery.
This principle might seem counterintuitive, but it is demonstrably true. In particular cases, it can be demonstrated empirically, at least in theory. Given sufficient quantities of newspaper accounts, the claim in the previous paragraph could be shown simply by counting and fact-checking. In general, one reason for the truth of the principle can be seen from Bayes’ Theorem. This theorem relates the conditional and the prior probability of a given event, and is (in one form) as follows: The probability of A, given B, equals the probability of B, given A, times the (prior) probability of A, divided by the (prior) probability of B. Among other things, this theorem is used by many e-mail spam filters to calculate the probability that a particular e-mail is spam, given that it contains the words that it does. In our case, the theorem works like this: The probability that a given event occurred, given someone’s claim that it did, equals the probability of the person’s making the claim, given that the event occurred, times the prior probability of the event occurring, divided by the prior probability of the person’s making the claim. But–and this is crucial–the prior probability of the person’s making the claim is less for less probable events. Now, this prior probability forms the denominator of the fraction, and therefore as it decreases the overall fraction (the probabilty of the event, given the claim of it) increases. The overall change will depend on how the other values change as well, of course, but this is sufficient to show that claims of less probable things constitute stronger evidence; namely, because claims of less probable things have lower prior probabilities. For example, it is less probable that I would claim to have flipped eight heads in a row than three heads in a row (we can ignore here the fact that I might find the latter too unremarkable to bother stating it in the first place), simply because it is less likely to have happened. Thus, when I do make the claim, it provides stronger evidence. It is important to note that there cannot be a simple direct proportion, since that would make all claims equally likely; the exact relation varies from case to case and needs to be judged from other factors.
The point that claims of less probable things have a lower prior probability is also empirically evident. For instance, it is much more common to hear someone say that they recently bought a new car than to hear someone say they recently bought a new pet dragon. The fact that we tend to believe the former and not the latter does not show that the latter is not stronger evidence, as I claimed above; rather, it shows that the overall judgment, which takes into account the low prior probability of someone buying a pet dragon, yields a lower probability.
Does any of this make any sense?
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