Editing Truth (section)

===Kripke's semantics<!--Linked from 'Semantic theory of truth'-->===
{{Main|Semantic theory of truth}}
[[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 "<nowiki>'</nowiki>''The barn is big'' is true' is true".
* Defining truth for all sentences that predicate truth or falsity of a member of the second set. Imagine this process repeated infinitely, so that truth is defined for ''The barn is big''; then for "''The barn is big'' is true"; then for "<nowiki>'</nowiki>''The barn is big'' is true' is true", and so on.

Truth never gets defined for sentences like ''This sentence is false'', since it was not in the original subset and does not predicate truth of any sentence in the original or any subsequent set. In Kripke's terms, these are "ungrounded." Since these sentences are never assigned either truth or falsehood even if the process is carried out infinitely, Kripke's theory implies that some sentences are neither true nor false. This contradicts the [[principle of bivalence]]: every sentence must be either true or false. Since this principle is a key premise in deriving the [[liar paradox]], the paradox is dissolved.<ref>Kripke, Saul. "Outline of a Theory of Truth", Journal of Philosophy, 72 (1975), 690–716</ref>

[[Kripke semantics|Kripke's semantics]] are related to the use of [[topos|topoi]] and other concepts from [[category theory]] in the study of [[mathematical logic]].<ref>{{Cite book |last=Goldblatt |first=Robert |title=Topoi, the categorial analysis of logic |date=1983 |publisher=Sole distributors for the U.S.A. and Canada, Elsevier North-Holland |isbn=0-444-86711-2 |edition=revised |location=Amsterdam |oclc=9622076}}</ref> They provide a choice of formal semantics for [[intuitionistic logic]].