An airtight Dutch book
your subjective probability function is concentrated at finitely many points, you can be enticed into an infinite sequence of simple and conditional bets whose cumulative effect will be that you sustain a net loss in every possible situation. Now it is surely unreasonable to suppose that our degrees of rational belief are concentrated at finitely many points, for it would be foolhardy to imagine that we know so very much that we are able to rule out, with probability one, all but finitely many possible ways the world might be. Assuming, as is often done, that vulnerability to a Dutch book2 is a sure sign of irrationality, we find ourselves driven to the cheerless conclusion that, in situations in which there can be infinitely many bets over an unbounded utility scale, no rational plan of action is available. Let me emphasize that the system of bets we shall examine provides more than just an arrangement in which you sustain a net loss with probability one. The change in your utility will be defined and negative in every possible situation, where 'possible situation' is understood as liberally as possible: A set of sentences describes a possible situation if it is consistent by the sentential calculus. There is no failure of compactness here,3 no use of infinitary logic - the conjunctions and disjunctions we employ are all 1 An equivalent condition is that the range of the probability function is a finite set of numbers Another equivalent is that there is a finite set of sentences such that every sentence is equivalent, with probability one, to a member of the finite set.