The lottery paradox, the preface paradox and rational belief

Abstract

Traditionally rationality has been analysed in rather puristic terms; thus rational acceptance has been presented as unsullied by the demands of competing claims -- the only demand admitted generally being truth ('Do not have false beliefs'). Such a view leads us to the straightforward rejection of the thesis that a. rule of detachment forprobability statements is sufficient to explicate rational acceptance; since such a rule leads, apparently unavoidably, to the lottery paradox. (Lottery Paradox: Accept only those propositions whose probability is shown to be greater than N people enter a lottery, therefore the probability of an individual losing is this goes for each separately and so we may accept that each will lose, and so that all will lose. But we know that this is false.). The appeal of this rather contemptuous treatment diminishes in the face of the Preface Paradox. (Preface Paradox: A man writes the following, eminently reasonable, lines: Each of the propositions I assert in this book I believe to be true; but I am also sure that some will be proved false.). If we reason as before we have to accept the impossibility of rational belief. The two paradoxes are examined in detail and their consequences spelt out in Chapter One; giving us two alternatives:(1) To show, despite appearances, that neither set of beliefs is inconsistent, or (2) To show some difference between the two paradoxes that enables the traditional view of rationality to separate them. (1) is rejected, and (2) in the course of the same argument, in Chapters Two and Three, where we formulate a criterion for the consistency of sets of beliefs, defend it against apparent counter-examples, (versions of Moore's Paradox) and demonstrate that both sets of beliefs are inconsistent. This despite attempts by some, notably Kyburg, to show the opposite. If we are to avoid concluding rationality bankrupt, and yet maintain our original reaction to the rule of detachment must do two things: (a) reject the rule of detachment on grounds other than the Lottery paradox. (b) give an account of rationality that will accommodate the Preface paradox. In Chapter Four we justify (a) by considering the asymmetries that can be shown to exist between the syntax of the classical probability calculus and the syntax of confirmation in ordinary language, and by pointing to the difficulties encountered in giving an adequate semantics to such a calculus when cast in the role of s calculus of confirmation. Finally, in Chapter Five, we present a more complex account of rationality, capable of accommodating the Preface Paradox, one which takes seriously the diverse needs of human beings.