Date of this Version
Published in Transactions of the Nebraska Academy of Sciences, Volume 2 (1973).
1. There are two main reasons to use a metalanguage, when we analyse a given informal language (a) of science. The first reason is to avoid semantic antinomies of the liar type. The concept of a metalanguage, widely used since Tarski (1956: 152-268), proposed to split up the normal informal scientific language (a) into an object language (b) and a metalanguage (c). Only within the metalanguage (c) we can speak about the object language (b). Ordinary informal languages are according to Tarski "closed languages." Closed languages make no differences between semantic expressions such as "true," which refer to expressions of the object language. If, for example, we assume the statement "all decision makers are liars" within the informal language (a) then whatever a decision maker says, a contradiction will follow. If he says the truth according to the earlier statement he lies and if he lies, then he says the truth. Semantic expressions belong therefore in such a stratified language system to the metalanguage, which contains two parts: the critical one by Tarski called the semantical and the translational part in which the object language or object theory is repeated solely by terms of the metalanguage. Thus the first reason to introduce a metalanguage was given by the definition of truth, or the truth of a statement or proposition S.
Similar difficulties arose within metamathematics, when Hilbert tried to analyse the concept of provability. This second reason is more a syntactic one, notwithstanding the fact that any semantic definition of truth demands the definition of proofs. If a statement asserts its own unprovability, then it follows, that this statement is provable if unprovable, and if unprovable then it is provable. Since this holds for any richer formal system according to Godel (1931: 176), we have to use a metalanguage, when 'provable' belon&, to his metalanguage. The results of Tarski's and Hilbert's introduction of metalanguages show clearly (i) that any analysis of especially scientific languages which uses critical expressions has to use a metalanguage. Critical expressions are linguistic expressions, which refer to other linguistic expressions, such as true, false, provable (ii) that the separation of the object theory as well as its formalizatiGJ1 and axiomatization is from the beginning a highly artificial procedure. The object language (b) can preserve only approximatively all the characteristic features of the informal theory (a), is therefore not an exact mapping of the informal theory (a) into the object theory (b), hence a more arbitrarily and artificially reconstructed object theory (b). This is important to understand the difference between mathematical object theories or between cognitive axiomatized object theories (b) and the actual informal theories (a) of mathematical textbooks or of informal cognitive scientific theories (a). In most cases the analyst does not look back any longer at the informal theory (a) if he builds up his object theory (b). Therefore philosophy of science cannot analyse the factual given form of science (a) but rather a highly idealized, formalized and by axiomatization separated paradigm. (iii) The most important result of any meta theoretical analysis of sciellce, is, that according to Tarski's remarks, there are far more critical expressions in our language, than true, provable (Leinfellner, 1973).