 To see why this is so, imagine that a million lottery tickets have been sold. Each player purchased only a single ticket, and exactly one of the players is certain to win. Now imagine further that an informed observer has a distinct belief about each of the million individual players that that particular player will lose. According to thesis T, each of those million beliefs is rational. For example, if Sue is one of the players, then according to T the observer’s belief that Sue will lose is rational because it is rational for the observer to have a (very) high credence in the proposition that Sue will lose. Since, however, it is certain that someone will win, it is also rational for the observer to believe that some player will win. It is not rational, however, to have contradictory beliefs, so it is not rational for the observer to believe that no player will win. This implies, however, that rational belief is not closed under conjunction, for the proposition that no player will win just is the conjunction of all of the propositions that say of some individual player that they will lose.