Tarski, in his important book Der Walzrheitshegriff in den formalisierten Sprachen, has shown that the words "true" and "false", as applied to the sentences of a given language, always require another language, of higher order, for their adequate definition. The conception of a hierarchy of languages is involved in the theory of types, which, in some form, is necessary for the solution of the paradoxes; it plays an important part in Carnap's work as well as in Tarski's. I suggested it in my introduction to Wittgenstein's Tractatus, as an escape from his theory that syntax can only be "shown", not expressed in words. The arguments for the necessity of a hierarchy of languages are overwhelming, and I shall henceforth assume their validity.•
* These arguments are derived from the paradoxes; their applicability to the words "true" and "false" is derived from the paradox of the liar.