 Because we wish the extensions of expressions to vary from possible world to possible world (at least in some cases), it is natural to associate with each expression a function from possible worlds to extensions appropriate to that sort of expression. Thus we associate with names, functions from possible worlds to individuals; with n-place predicates functions from possible worlds to sets of n-tuples; and with sentences, functions from possible worlds to truth values. Such functions from possible worlds to extensions of the appropriate sort are often called intensions of the expressions in question, and the term ‘intension’ shall be used this way throughout the present  work.[5]  Now since most think that the extensions of sentences are truth values, as indicated above, the intension of a sentence is a function from possible worlds to truth values. Intuitively, it maps a world to true if the sentence is true at that world. Thus the intension of a sentence can be seen as the primary bearer of truth and falsity at a world: the sentence has the truth value it has at the world in virtue of its intension mapping that world to that truth value. Further, modal operators were typically construed as operating on the intensions of the sentences they embed, and so those intensions could plausibly be thought of as possessing modal properties. Since propositions were traditionally held to be the primary bearers of truth and falsity and the bearers of modal properties, it was natural for possible world semanticists to identify propositions with functions from possible worlds to truth values (sentential intensions), or equivalently, sets of possible worlds (the set of possible worlds at which the sentence in question is true). Indeed, this identification was thought by many to vindicate the previously mysterious notion of a  proposition.[6]  Possible worlds were apparently needed for the model theory of modal logic anyway; why not build propositions out of them?