This section introduces the liar paradox and shows how it leads to seemingly absurd conclusions. These conclusions illustrate that the concept of truth may be problematic. A problem occurs when standard Tarski correspondence biconditionals (which are usually given as the paradigm scheme of what it means for something to be true) produce contradictions in cases of self-reference. Since self-reference occurs in languages with even the most basic expressive powers, the problem will extend to most languages. Meeting the challenge set by the liar paradox seems to require preserving the classical and intuitive logical inferences which the Tarski biconditionals warrant, while leaving the unproblematic applications of the "is true- predicate intact.
Tarski developed a hierarchy of interpretations of the truth-predicate, each of which could not apply to applications of itself. Thus, to say of a sentence which contained the "is true- predicate for that level that it was true would require an extension of the language to a new level. This creates a meta-truth-predicate in the new level which applies to all the sentences which are true at lower levels. This solution was the standard answer to the problem for most of the twentieth century.
Kripke and Woodruff showed that it was possible to create three-value languages which contain truth-predicates which can range over the entire system, i.e. truth-predicates which do not require an extension to the meta-language to apply to their own applications. Theories with this feature are called fixed-point theories. However, though the fixed-point theories seem to point at an answer to the problem, the solution cannot be applied to natural languages because the expressive power of the fixed-point languages is too weak. This leaves the challenge unmet because, although formal versions of the liar paradox exist, it is the natural language truth-predicate which is ultimately trying to be made sense of.
