Truth as Translation – Part A

Abstract

According to Tarski's Convention T, the adequacy of a truth definition is (implicitly) defined relatively to a translation mapping from the object language to the metalanguage; the translation mapping itself is left unspecified. This paper restates Convention T in a form in which the relativity to translation is made explicit. The notion of an interpreted language is introduced, and a corresponding notion of a translation between interpreted languages is defined. The latter definition is stated both in an algebraic version, and in an equivalent possible worlds version. It is a consequence of our definition that translation is indeterminate in certain cases. Finally, we give an application of our revised version of Convention T and show that interpreted languages exist, which allow for vicious self-reference but which nevertheless contain their own truth predicate. This is possible if only truth is based on a nonstandard translation mapping by which, e.g., the Liar sentence is translated to its own negation. In this part of the paper this existence result is proved only for languages without quantifiers; in Part B the result will be extended to first-order languages.