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 mos

This theory meets the requirements imposed on definitions that they do not create new valid statements which are not equivalent to those which could be derived before. However, this account of definitions does not produce circular definitions which are completely eliminable (for reasons discussed above). Gupta argues that the idea behind eliminability is respected insofar as the meaning of the concept can be completely expressed by the definiens. When self-reference is introduced the revision process yields unstable results for theses sentences. This means that they never remain in the interpretation of T at all stages. Gupta asserts that some self-referencing Tarski biconditionals, when read materially, may fail to be valid if they are unstable. He argues that this shows that a fundamental difference exists between reading these biconditionals as definitionally equivalent or materially equivalent, since the former come out valid. This implies that equivocation between these two similar structures points to a way out of the liar paradox. Read materially, the biconditional countenances the substitution which leads to the paradox. However, read as equivalences they merely state the definition of the concept and imply no contradiction. The material reading fails to respect the hypothetical character of the biconditional.

The strengthened liar builds on the idea that any semantic concept can be expressed by an extended natural language. It creates a disjunctive sentence which states of itself that it is either not true or k, where k is any semantic concept built by the universality theorist to cover self-reference cases. This shows that semantic concepts beget further instances of self-reference. This sets a higher standard for the universality theorist. They must create semantic concepts that can apply both to the problematic sentences, as well as the further paradoxes which arise from those applications.

Some topics in this essay:
Revision Theories, Strong Kleene, Contextual Theories, , Fixed-point Theories, Kripke Woodruff, extension anti-extension, fixed-point theories, concept truth, revision theories, natural languages, liar paradox, applied natural languages, val functions, contextual theories, tarski biconditionals, semantic concepts, solution applied natural, “is true” predicate, strong kleene jump, theories regard concept,