 Further, if propositions are sets of possible worlds, belief is construed as a relation between individuals and propositions and sentences of the form ‘A believes that P’ assert that the individual A stands in the belief relation to the proposition expressed by ‘P’, then for any necessarily equivalent sentences ‘P’ and ‘Q’, ‘A believes that P’ and ‘A believes that Q’ cannot differ in truth value. This means that, for example, if ‘A believes that 1+1=2’ is true, so is ‘A believes that there is no greatest natural number’. These consequences of the view that propositions are sets of possible worlds were appreciated early on; and theorists made a variety of attempts to make these consequences seem less unpalatable. Despite these valiant efforts, many philosophers viewed these consequences as a sign that there was something very wrong with the view that propositions are sets of possible worlds. Thus, philosophers were open to an account of propositions that individuated propositions more finely than the possible worlds account. As we have seen, the structured proposition account is just such an account.