Dissolving the Wine/Water Paradox

Abstract

Von Mises’ wine/water paradox has long served as an argument against the Principle of Indifference. A solution to the paradox is proposed, with a view toward resolving general difficulties in applying the principle. The Principle of Indifference (PI), an artifact of the classical theory of probability, says that we should assign equal probability to any mutually exclusive and jointly exhaustive set of possible outcomes, iff we have insufficient reason to consider any one of these outcomes more or less likely than any other. Despite its intuitive appeal and its formative role in the field of probability, PI has fallen into serious disrepute among probability theorists. Perhaps the most common and compelling argument against PI is that it leads to irresolvable paradox. Beginning with Bertrand ([1889]), authors have attempted to show that applying PI to certain kinds of examples leads to contradiction. Opinions are divided as to which of these so-called Bertrand paradoxes can be resolved simply by making PI more precise, but critics and defenders of the principle agree that not all of them can be. One particular paradox, von Mises’ wine/water paradox, plays a curiously pivotal role in this discussion. Everyone seems to agree that it has no solution. Van Fraassen ([1989]) claims that the paradox signals the ‘ultimate defeat’ of the Principle of Indifference, nullifying the ‘Pyrrhic victory’ won by Poincare ´ ([1912]) and Jaynes ([1973]) over other Bertrand paradoxes. Gillies ([2000]) calls it the one ‘fatal’ objection to PI, and Oakes ([1986]) uses it to confirm that ‘the classical conception of probability cannot withstand more than casual examination’. Even Jaynes ([1973]) and Schlesinger ([1991]), lonely defenders of PI, throw up their hands at this paradox, arguing that it simply stakes out the limitations of the principle. The fact that so many critics rely on this one example to discredit PI is in itself cause for some suspicion. At the very least, it suggests that dissolving the wine/water paradox would score a major victory for PI. I hope to supply this victory.