Editing Truth (section)

==Formal theories==
===Logic===
{{Main|Logical truth|Criteria of truth|Truth value}}

[[Logic]] is concerned with the patterns in [[reason]] that can help tell if a [[proposition]] is true or not. Logicians use [[formal language]]s to express the truths they are concerned with, and as such there is only truth under some [[interpretation (logic)|interpretation]] or truth within some [[logical system]].<ref>{{Cite book |last=Novaes |first=Catarina Dutilh |url=https://books.google.com/books?id=5ZV7AAAAQBAJ |title=Formal Languages in Logic: A Philosophical and Cognitive Analysis |date=2012 |publisher=Cambridge University Press |isbn=978-1-107-02091-7 |language=en}}</ref>

A logical truth (also called an analytic truth or a necessary truth) is a statement that is true in all logically possible worlds<ref>[[Ludwig Wittgenstein]], ''[[Tractatus Logico-Philosophicus]]''.</ref> or under all possible interpretations, as contrasted to a ''[[fact]]'' (also called a ''[[Analytic-synthetic distinction|synthetic claim]]'' or a ''[[Necessary and sufficient condition|contingency]]''), which is only true in this [[World (philosophy)|world]] as it has historically unfolded. A proposition such as "If p and q, then p" is considered to be a logical truth because of the meaning of the [[symbol (formal)|symbols]] and [[well-formed formula|words]] in it and not because of any fact of any particular world. They are such that they could not be untrue.

[[Degree of truth|Degrees]] of [[truth values|truth]] in logic may be represented using two or more discrete values, as with [[principle of bivalence|bivalent logic]] (or [[Boolean logic|binary logic]]), [[three-valued logic]], and other forms of [[finite-valued logic]].<ref>{{cite book| last=Kretzmann| first=Norman| title=William of Sherwood's Treatise on Syncategorematic Words| chapter=IV, section<nowiki>=</nowiki>2. 'Infinitely Many' and 'Finitely Many'| publisher=University of Minnesota Press| year=1968| chapter-url=https://books.google.com/books?id=fW5rlSy-5D8C&pg=PA42| isbn=978-0-8166-5805-3}}</ref><ref>{{cite book| last=Smith| first=Nicholas J.J.| chapter=Article 2.6| title=Many-Valued Logics| publisher=Routledge| year=2010| chapter-url=https://www-personal.usyd.edu.au/~njjsmith/papers/smith-many-valued-logics.pdf| access-date=2018-05-25| archive-date=2018-04-08| archive-url=https://web.archive.org/web/20180408200831/http://www-personal.usyd.edu.au/~njjsmith/papers/smith-many-valued-logics.pdf| url-status=live}}</ref> Truth in logic can be represented using numbers comprising a [[continuous or discrete variable|continuous]] range, typically between 0 and 1, as with [[fuzzy logic]] and other forms of [[infinite-valued logic]].<ref>{{cite book| title=The Development of Modern Logic| last1=Mancosu| first1=Paolo| last2=Zach| first2=Richard| last3=Badesa| first3=Calixto| chapter=9. The Development of Mathematical Logic from Russell to Tarski 1900-1935" §7.2 "Many-valued logics| publisher=Oxford University Press| pages=418–420| year=2004| url=https://books.google.com/books?id=0jXavKsArnIC| isbn=978-0-19-972272-3}}</ref><ref>{{cite web| last=Garrido| first=Angel| title=A Brief History of Fuzzy Logic| publisher=Revista EduSoft| year=2012| url=https://www.edusoft.ro/brain/index.php/brain/article/viewFile/308/390| access-date=2018-05-25| archive-date=2018-05-17| archive-url=https://web.archive.org/web/20180517152622/https://www.edusoft.ro/brain/index.php/brain/article/viewFile/308/390| url-status=live}}, Editorial</ref> In general, the concept of representing truth using more than two values is known as [[many-valued logic]].<ref>{{cite book| last=Rescher| first=Nicholas| publisher=Humanities Press Synthese Library volume 17| year=1968| doi=10.1007/978-94-017-3546-9_6| title=Topics in Philosophical Logic|pages = 54–125|isbn = 978-90-481-8331-9| chapter=Many-Valued Logic}}</ref>

===Mathematics===
{{anchor|Truth_in_mathematics}} 
{{Main|Model theory|Proof theory}}

There are two main approaches to truth in mathematics. They are the ''[[model theory|model theory of truth]]'' and the ''[[proof theory|proof theory of truth]]''.<ref>Penelope Maddy; ''Realism in Mathematics''; Series: Clarendon Paperbacks; Paperback: 216 pages; Publisher: Oxford University Press, US (1992); 978-0-19-824035-8.</ref>

Historically, with the nineteenth century development of [[Boolean algebra (logic)|Boolean algebra]], mathematical models of logic began to treat "truth", also represented as "T" or "1", as an arbitrary constant. "Falsity" is also an arbitrary constant, which can be represented as "F" or "0". In [[propositional logic]], these symbols can be manipulated according to a set of [[axioms]] and [[rules of inference]], often given in the form of [[truth table]]s.

In addition, from at least the time of [[Hilbert's program]] at the turn of the twentieth century to the proof of [[Gödel's incompleteness theorems]] and the development of the [[Church–Turing thesis]] in the early part of that century, true statements in mathematics were [[logical positivism|generally assumed]] to be those statements that are provable in a formal axiomatic system.<ref>Elliott Mendelson; ''Introduction to Mathematical Logic''; Series: Discrete Mathematics and Its Applications; Hardcover: 469 pages; Publisher: Chapman and Hall/CRC; 5 edition (August 11, 2009); 978-1-58488-876-5.</ref>

The works of [[Kurt Gödel]], [[Alan Turing]], and others shook this assumption, with the development of statements that are true but cannot be proven within the system.<ref>''See, e.g.,'' Chaitin, Gregory L., ''The Limits of Mathematics'' (1997) esp. 89 ''ff''.</ref> Two examples of the latter can be found in [[Hilbert's problems]]. Work on [[Hilbert's 10th problem]] led in the late twentieth century to the construction of specific [[Diophantine equations]] for which it is undecidable whether they have a solution,<ref>M. Davis. "Hilbert's Tenth Problem is Unsolvable." ''American Mathematical Monthly'' 80, pp. 233–269, 1973</ref> or even if they do, whether they have a finite or infinite number of solutions. More fundamentally, [[Hilbert's first problem]] was on the [[continuum hypothesis]].<ref>Yandell, Benjamin H.. ''The Honors Class. Hilbert's Problems and Their Solvers'' (2002).</ref> Gödel and [[Paul Cohen (mathematician)|Paul Cohen]] showed that this hypothesis cannot be proved or disproved using the standard [[axiom]]s of [[set theory]].<ref>Chaitin, Gregory L., ''The Limits of Mathematics'' (1997) 1–28, 89 ''ff''.</ref> In the view of some, then, it is equally reasonable to take either the continuum hypothesis or its negation as a new axiom.

Gödel thought that the ability to perceive the truth of a mathematical or logical proposition is a matter of [[logical intuition|intuition]], an ability he admitted could be ultimately beyond the scope of a formal theory of logic or mathematics<ref>{{cite web| last=Ravitch| first=Harold| title=On Gödel's Philosophy of Mathematics| year=1998| url=http://www.friesian.com/goedel/chap-2.htm| access-date=2018-05-25| archive-date=2018-02-28| archive-url=https://web.archive.org/web/20180228005628/http://friesian.com/goedel/chap-2.htm| url-status=live}}</ref><ref>{{cite web| last=Solomon| first=Martin| title=On Kurt Gödel's Philosophy of Mathematics| year=1998| url=http://calculemus.org/lect/07logika/godel-solomon.html| access-date=2018-05-25| archive-date=2016-03-04| archive-url=https://web.archive.org/web/20160304030146/http://www.calculemus.org/lect/07logika/godel-solomon.html| url-status=live}}</ref> and perhaps best considered in the realm of human [[comprehension (logic)|comprehension]] and communication. But he commented, "The more I think about language, the more it amazes me that people ever understand each other at all".<ref>{{cite book| last=Wang| first=Hao| title=A Logical Journey: From Gödel to Philosophy| publisher=The MIT Press| year=1997| url=https://books.google.com/books?isbn=0262261251}} (A discussion of Gödel's views on [[logical intuition]] is woven throughout the book; the quote appears on page 75.)</ref>

===Tarski's semantics===
{{main|Semantic theory of truth}}

[[Tarski's theory of truth]] (named after [[Alfred Tarski]]) was 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 [[Liar paradox|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 (philosopher)|Donald Davidson]] used it as the foundation of his [[truth-conditional semantics]] and linked it to [[radical interpretation]] in a form of [[coherentism]].<ref>{{Citation |last=Hodges |first=Wilfrid |title=Tarski's Truth Definitions |date=2022 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/tarski-truth/ |access-date=2025-04-04 |edition=Winter 2022 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref>

[[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 [[Alfred North Whitehead|Whitehead]] attempted to solve these problems in ''[[Principia Mathematica]]'' by putting statements into a hierarchy of [[type theory|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 system]]s that have yet to be resolved to this day.<ref>{{Cite book |last=Link |first=Godehard |url=https://books.google.com/books?id=Xg6QpedPpcsC |title=One Hundred Years of Russell's Paradox: Mathematics, Logic, Philosophy |date=2004 |publisher=Walter de Gruyter |isbn=978-3-11-017438-0 |language=en}}</ref>

===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]].