  Possible world semantics, of course, uses the concept of a possible world to give substance to the idea of alternative extensions and alternative domains of quantification. (Possible world semantics can be traced most clearly back to the work of Carnap (1947), its basic development culminating in the work of Hintikka (1957, 1961), Bayart (1958, 1959), and Kripke (1959, 1963a, 1963b).[5]) Similar to Tarskian semantics, a possible world interpretation M of a modal language ℒ specifies a nonempty set D, although thought of now as the set of “possible individuals” of M. Also as in Tarskian semantics, M assigns each term τ of ℒ a referent aτ in D.[6] Additionally however, M specifies a set W, the set of “possible worlds” of M, one of which is designated its “actual world”, and each world w in W is assigned its own domain of quantification, d(w) ⊆ D, intuitively, the set of individuals that exist in w.[7] To capture the idea of both the actual and possible extensions of a predicate, M assigns to each n-place predicate π a function Mπ — the intension of π — that, for each possible world w, returns the extension Mπ(w) of π at w: a truth value, if n = 0; a set of individuals, if n = 1; and a set of n-tuples of individuals, if n &gt; 1.[8] We can thus rigorously define a “possible extension” of a predicate π to be any of its w-extensions Mπ(w), for any world w. 