  Think of the function \(f_{X}\) as the intensional meaning of the formula \(X\)—indeed, think of it as the proposition expressed by the formula (relative to a particular model, of course). At a state \(\Gamma\),  \(f_{X}(\Gamma )\) is a truth value—think of this as the extensional meaning of \(X\) at that state. This is a way of thinking that goes back to Frege, who concluded that the denotation of a sentence should be a truth value, but the sense should be a proposition. He was a little vague about what constituted a proposition—the formalization just presented provides a natural mathematical entity to serve the purpose, and was explicitly proposed for this purpose by Carnap. It should be clear that the mathematical structure does, in a general way, capture some part of Frege’s idea. Incidentally we could, with no loss, replace the function \(f_{X}\) on states with the set \(\{ \Gamma \in \bG \mid f_{X}(\Gamma) = \textit{true}\}\). The function \(f_{X}\) is simply the characteristic function of this set. Sets like these are commonly referred to as propositions in the modal logic community. In a technical sense, then, Frege’s ideas on this particular topic have become common currency.