Editing Liar paradox (section)

===Alfred Tarski===
[[Alfred Tarski]] diagnosed the paradox as arising only in languages that are "semantically closed", by which he meant a language in which it is possible for one sentence to predicate truth (or falsehood) of another sentence in the same language (or even of itself). To avoid self-contradiction, it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) only of languages at a lower level. So, when one sentence refers to the truth-value of another, it is semantically higher. The sentence referred to is part of the "object language", while the referring sentence is considered to be a part of a "meta-language" with respect to the object language. It is legitimate for sentences in "languages" higher on the semantic hierarchy to refer to sentences lower in the "language" hierarchy, but not the other way around. This prevents a system from becoming self-referential.

However, this system is incomplete. One would like to be able to make statements such as "For every statement in level ''α'' of the hierarchy, there is a statement at level ''α''+1 which asserts that the first statement is false." This is a true, meaningful statement about the hierarchy that Tarski defines, but it refers to statements at every level of the hierarchy, so it must be above every level of the hierarchy, and is therefore not possible within the hierarchy (although bounded versions of the sentence are possible).<ref name=Kripke.1975>{{cite conference |last1=Kripke |first1=Saul |author-link=Saul Kripke |date=1975-11-06 |title=Outline of a theory of truth |conference=Seventy-Second Annual Meeting American Philosophical Association, Eastern Division |publisher=Journal of Philosophy |volume=72 |number=19 |pages=690–716 |doi=10.2307/2024634 |jstor=2024634 }}</ref><ref name=PlatoProject.LiarParadox.TarskiHier>{{cite web |url=https://plato.stanford.edu/entries/liar-paradox/#TarsHierLang |title=Liar Paradox: Section 4.3.1 Tarski's hierarchy of languages |access-date=2021-01-16 |date=2016-12-12 |orig-date=January 20, 2011 |first1=Jc |last1=Beall |first2=Michael |last2=Glanzberg |first3=David |last3=Ripley |archive-date=2021-01-12 |archive-url=https://web.archive.org/web/20210112011051/https://plato.stanford.edu/entries/liar-paradox/#TarsHierLang |url-status=live }}</ref> [[Saul Kripke]] is credited with identifying this incompleteness in Tarski's hierarchy in his highly cited paper "Outline of a theory of truth,"<ref name=PlatoProject.LiarParadox.TarskiHier/> and it is recognized as a general problem in hierarchical languages.<ref name=PlatoProject.LiarParadox.TarskiHier/><ref>{{cite book |first1=Michael |last1=Glanzberg |chapter=Complexity and Hierarchy in Truth Predicates|publisher=Springer|location= Dordrecht |title=Unifying the Philosophy of Truth|series= Logic, Epistemology, and the Unity of Science |volume=36 |date=2015-06-17 |pages=211–243 |doi=10.1007/978-94-017-9673-6_10 |isbn=978-94-017-9672-9 }}</ref>