The 6‐49 Lottery Paradox

Abstract

The paradox of the preface and the lottery paradox are paradoxes of practical certainty sharing certain features. The paradox of the lottery argues that rational agents are at once practically certain that each ticket in a lottery will lose but also practically certain some ticket will win. The paradox of the preface argues that rational agents are at once practically certain that all facts in a written volume are true, yet are also practically certain that some fact is wrong. A difference between real lotteries and prefaces is that a winning lottery ticket is generally an intended feature of the lottery, whereas incorrect facts are generally unintended.Despite these similarities, Pollock gives a novel argument suggesting that the preface paradox warrants qualitatively different treatment from the lottery, using as a rationale the differences between real lotteries and prefaces. This draws a clear line between the work of Pollock and the work of Kyburg, both of whom have had a prominent influence in recent thinking on nonmonotonic reasoning in AI.This note shows there are real lotteries with the formal structure of the preface paradox and possibly prefaces with the formal structure of lotteries. The surprising conclusion is that within Pollock's framework, the treatment of any problem with a formal structure resembling the lottery (or the preface) depends on the process by which winning tickets (or publishing errors) are generated. The rationales given by Pollock seem to be unrelated to the actual mechanisms implemented.