 So far, the apparatus is extensional, which, besides providing only two possible sentence-meanings, TRUE and FALSE, imposes severe limitations on the range of concepts we can express. Suppose that the Scottish clothing company Pringle has a monopoly on the manufacture of woollen sweaters, and makes sweaters of no other material. Then a garment is a woollen sweater iff it is a Pringle sweater, meaning that woollen(sweater) and pringle(sweater) are the same function of type ib, and these two terms for that function are everywhere interchangeable in the type-theoretic language. Then modal operators such as ‘it is contingent that’ cannot be in the language, since interchanging woollen(sweater) and pringle(sweater) within their scope should sometimes lead to change of truth-value, but cannot if the two expressions receive the same meaning in the semantics. For example, ‘it is contingent that every Pringle sweater is woollen’ is true, but ‘it is contingent that every woollen sweater is woollen’ is false. Therefore the concept of contingency has no adequate representation in the type-theoretic (boldface) language.