 For discussion of the application of the framework of possible world semantics to natural language, see David Lewis (1970). The intension of a sentence—i.e., the proposition that sentence expresses, on the present view—will then be a function from worlds to truth-values. In particular, it will be that function which returns the truth-value true for every world with respect to which that sentence is true, and false otherwise. The intension of a simple predicate like “is red” will be a function from worlds to the function from objects to truth-values which, for each world, returns the truth-value true if the thing in question is red, and returns the truth-value false otherwise. In effect, possible worlds semantics takes the meanings of expressions to be functions from worlds to the values which would be assigned by a theory of reference to those expressions at the relevant world: in that sense, intensions are a kind of “extra layer” on top of the theory of reference.