developed for formal languages, such as formal logic. Here he restricted it in this way: no language could contain its own truth predicate, that is, the expression is true could only apply to sentences in some other language. The latter he called an object language, the language being talked about. (It may, in turn, have a truth predicate that can be applied to sentences in still another language.) The reason for his restriction was that languages that contain their own truth predicate will contain paradoxical sentences such as, "This sentence is not true". As a result, Tarski held that the semantic theory could not be applied to any natural language, such as English, because they contain their own truth predicates. Donald Davidson used it as the foundation of his truth-conditional semantics and linked it to radical interpretation in a form of coherentism. Bertrand Russell is credited with noticing the existence of such paradoxes even in the best symbolic formations of mathematics in his day, in particular the paradox that came to be named after him, Russell's paradox. Russell and Whitehead attempted to solve these problems in Principia Mathematica by putting statements into a hierarchy of types, wherein a statement cannot refer to itself, but only to statements lower in the hierarchy. This in turn led to new orders of difficulty regarding the precise natures of types and the structures of conceptually possible type systems that have yet to be resolved to this day. Kripke's semantics Kripke's theory of truth (named after Saul Kripke) contends that a natural language can in fact contain its own truth predicate without giving rise to contradiction. He showed how to construct one as follows: Beginning with a subset of sentences of a natural language that contains no occurrences of the expression "is true" (or "is false"). So The barn is big is included in the subset, but not "The barn is big is true", nor problematic sentences such as "This sentence is false". Defining truth just for the sentences in that subset. Extending the definition of truth to include sentences that predicate truth or falsity of one of the original subset of sentences. So "The barn is big is true" is now included, but not either "This sentence is false" nor "'The barn is big is true' is true". Defining truth for all sentences that predicate truth or falsity of a member of the second