There once was a man who wrote a book. He was very careful in his reasoning, and was confident of each claim that he made. With some display of pride, he showed the book to a friend (who happened to be a probability theorist). He was dismayed when the friend observed that any book that long and that interesting was almost certain to contain at least one falsehood. Thus it was not reasonable to believe that all of the claims made in the book were true. If it were reasonable to believe each claim then it would be reasonable to believe that the book contained no falsehoods, so it could not be reasonable to believe each claim. Furthermore, because there was no way to pick out some of the claims as being more problematic than others, there could be no reasonable way of withholding assent to some but not others. “Therefore,” concluded his friend, “you are not justified in believing anything you asserted in the book.” This is the paradox of the preface (so named because in the original version the author confesses in the preface that his book probably contains a falsehood). The paradox of the preface is more than a curiosity. It has been used by some philosophers to argue that the set of one's warranted beliefs need not be deductively consistent, and by others to argue that you should not befriend probability theorists. If (Al) is to be a correct acceptance rule it must be capable of explaining what is involved in the paradox of the preface. The lottery paradox and the paradox of the preface seem superficially similar, so it might be supposed that a resolution of one will automatically generate a resolution of the other in some trivial manner. But in fact, the opposite is true. It is the principle of collective defeat that makes possible the resolution of the lottery paradox, but it is the principle of collective defeat that is responsible for the creation of the paradox of the preface.
The Psychological Dimension of the Lottery Paradox
