prior probability

In our previous post, we presented Kenny Easwaran’s vivid description of the paradox of the preface. Briefly, the paradox is this: when a scholar writes up an academic paper, he would like to believe that every claim or proposition in his paper is true. But at the same, that same scholar may add a statement or disclaimer (usually in the acknowledgements section of his paper) accepting responsibility for any error or errors that may appear in the body of the paper. Hence the paradox: if all the claims and propositions in the paper are true, the statement in the preface or acknowledgements section is false; but if the preface/acknowledgements section is true, then there must a claim or proposition in the body of the paper that is false.

In the next part of his beautiful paper, Easwaran presents the standard Bayesian solution to this paradox: the ingenious idea of “degrees of belief.” Simply put, a scholar’s belief in the truth of a claim or proposition is not binary, is not all or nothing; instead, his belief in the truth of claim x or proposition y may range anywhere from 0 to 1; in other words, our beliefs may vary in degrees of strength or weakness; our beliefs may come in shades of grey. So, how does the Bayesian notion of “degrees of belief” solve the paradox? Through the axioms of probability. By way of example, let’s say a scholar has written a paper containing two claims or propositions (Claim A and Claim B)–a very short paper indeed!–and further assume that the scholar’s degree of belief in each claim/proposition is only 0.51 (i.e., the scholar believes that it is only more likely than not that each proposition or claim is true). If the truth of the first claim (Claim A) is independent of the truth of the second claim (Claim B), this strange state of affairs means that there is a high probability that at least one of the claims or propositions might, in fact, be false. (Why? Because when two probabilistic events are independent, the probability of both occurring is P(A and B) = P(A) times P(B).) This ingenious device (degrees of belief) thus solves the paradox of the preface: it is consistent for the scholar to believe in the truth of his claims and to believe that one of those claims might turn out to be false. In our next post, however, we will consider some objections to the Bayesian solution.

Kenny Easwaran, a philosopher at Texas A&M, recently published in the journal Nous this beautiful paper on Bayesian probabilities (hat tip: Brian Leiter). Among other things, Easwaran’s paper contains the best and most succinct explanation of the “paradox of the preface” we’ve ever read. Here it is (edited by us for clarity):

Dr. Truthlove … has just written an extensively researched book, and she believes every claim in the body of the book. However, she is also aware of the history of other books on the same subject, and knows that every single one of them has turned out to contain some false claims, despite the best efforts of their authors. Thus, one of the claims she makes, in the preface of the book, is to the effect that the body of this book too, like all the others, surely contains at least one false claim. She believes that too. She notices a problem. At least one of her beliefs is false. Either some claim from the body of the book (all of which she believes) is false, or else the claim from the preface (which she also believes) is. So she knows that she’s doing something that she hates–believing a false claim. At the same time, she notices a benefit. At least one of her beliefs is true! Either the claim from the preface is true, or all of the claims in the body of the book are true.

We shall have more things to say about this original paper in the days ahead …