GRICE E CARAPELLE: LINGUAGGIO E META-LINGUAGGIO

 Apparently, David Hilbert was the first to use the prefix meta(from the Greek over) in the sense we use it in metalanguage, metatheory, and now metasystem. He introduced the term metamathematics to denote a mathematical theory of mathematical proof. In terms of our control scheme, Hilbert's MST has a non-trivial representation: a mapping of proofs in the form of usual mathematical texts (in a natural language with formulas) on the set of texts in a formal logical language which makes it possible to treat proofs as precisely defined mathematical objects. This done, the rest is as usual: the controlled system is a mathematician who proves theorems; the controlling person is a metamathematician who translates texts into the formal logical language and controls the work of the mathematician by checking the validity of his proofs and, possibly mechanically generating proofs in a computer. The emergence of the metamathematician is an MST. 9. OBJECT-LANGUAGE AND META-LANGUAGE. Since we have agreed not to employ semantically closed languages, we have to use two different languages in discussing the problem of the definition of truth and, more generally, any problems in the field of semantics. The first of these languages is the language which is "talked about" and which is the subject- matter of the whole discussion; the definition of truth which we are seeking   350 PHILOSOPHYAND PHENOMENOLOGICARLESEARCH applies to the sentences of this language. The second is the language in which we "talk about" the first language, and in terms of which we wish, in particular, to construct the definition of truth for the first language. We shall refer to the first language as "the object-language,"and to the second as "the meta-language." It should be noticed that these terms "object-language" and "meta- language" have only a relative sense. If, for instance, we become inter- ested in the notion of truth applying to sentences, not of our original object-language, but of its meta-language, the latter becomes automatically the object-language of our discussion; and in order to define truth for this language, we have to go to a new meta-language-so to speak, to a meta- language of a higher level. In this way we arrive at a whole hierarchy of languages. The vocabulary of the meta-language is to a large extent determined by previously stated conditions under which a definition of truth will be considered materially adequate. This definition, as we recall, has to imply all equivalences of the form (T): (T) X is true if, and only if, p. The definition itself and all the equivalences implied by it are to be formulated in the meta-language. On the other hand, the symbol 'p' in (T) stands for an arbitrary sentence of our object-language.