 Incidentally, Carnap (1947: 26), when introducing truth-values as extensions of sentences, is guided by essentially the same idea. Namely, he points out a strong analogy between extensions of predicators and truth values of sentences. Carnap considers a wide class of designating expressions (“designators”) among which there are predicate expressions (“predicators”), functional expressions (“functors”), and some others. Applying the well-known technique of interpreting sentences as predicators of degree 0, he generalizes the fact that two predicators of degree \(n\) (say, \(P\) and \(Q)\) have the same extension if and only if \(\forall x_1\forall x_2 \ldots \forall x_n(Px_1 x_2\ldots x_n \leftrightarrow Qx_1 x_2\ldots x_n)\) holds. Then, analogously, two sentences (say, \(p\) and \(q)\), being interpreted as predicators of degree 0, must have the same extension if and only if \(p\leftrightarrow q\) holds, that is if and only if they are equivalent. And then, Carnap remarks, it seems quite natural to take truth values as extensions for sentences.