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class="vector-toc-list"> <li id="toc-Formal_language" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formal_language"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Formal language</span> </div> </a> <ul id="toc-Formal_language-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Deductive_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Deductive_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Deductive system</span> </div> </a> <ul id="toc-Deductive_system-sublist" class="vector-toc-list"> <li id="toc-Proof_system" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Proof_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.1</span> <span>Proof system</span> </div> </a> <ul id="toc-Proof_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_semantics_of_logical_system" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Formal_semantics_of_logical_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.2</span> <span>Formal semantics of logical system</span> </div> </a> <ul id="toc-Formal_semantics_of_logical_system-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>External links</span> </div> </a> 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Available in 32 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-32" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">32 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Formales_System" title="Formales System – Alemannic" lang="gsw" hreflang="gsw" data-title="Formales System" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%A7%D9%85_%D8%B4%D9%83%D9%84%D9%8A" title="نظام شكلي – Arabic" lang="ar" hreflang="ar" data-title="نظام شكلي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Formal_sistem" title="Formal sistem – Azerbaijani" lang="az" hreflang="az" data-title="Formal sistem" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sistema_formal" title="Sistema formal – Catalan" lang="ca" hreflang="ca" data-title="Sistema formal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BB%C4%95_%D1%82%D1%8B%D1%82%C4%83%D0%BC" title="Формаллĕ тытăм – Chuvash" lang="cv" hreflang="cv" data-title="Формаллĕ тытăм" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Formelt_system" title="Formelt system – Danish" lang="da" hreflang="da" data-title="Formelt system" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Formales_System" title="Formales System – German" lang="de" hreflang="de" data-title="Formales System" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CF%85%CF%80%CE%B9%CE%BA%CF%8C_%CF%83%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1" title="Τυπικό σύστημα – Greek" lang="el" hreflang="el" data-title="Τυπικό σύστημα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_formal" title="Sistema formal – Spanish" lang="es" hreflang="es" data-title="Sistema formal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B3%D8%AA%DA%AF%D8%A7%D9%87_%D8%B5%D9%88%D8%B1%DB%8C" title="دستگاه صوری – Persian" lang="fa" hreflang="fa" data-title="دستگاه صوری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_formel" title="Système formel – French" lang="fr" hreflang="fr" data-title="Système formel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Siostam_foirmeil" title="Siostam foirmeil – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Siostam foirmeil" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%98%95%EC%8B%9D_%EC%B2%B4%EA%B3%84" title="형식 체계 – Korean" lang="ko" hreflang="ko" data-title="형식 체계" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D6%84%D5%BD%D5%AB%D5%B8%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF_%D5%B4%D5%A5%D5%A9%D5%B8%D5%A4" title="Աքսիոմատիկ մեթոդ – Armenian" lang="hy" hreflang="hy" data-title="Աքսիոմատիկ մեթոդ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Formala_sistemo" title="Formala sistemo – Ido" lang="io" hreflang="io" data-title="Formala sistemo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_formale" title="Sistema formale – Italian" lang="it" hreflang="it" data-title="Sistema formale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%B4%D1%8B_%D0%B6%D2%AF%D0%B9%D0%B5" title="Формалды жүйе – Kazakh" lang="kk" hreflang="kk" data-title="Формалды жүйе" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D0%BC%D0%B5%D1%82%D0%BE%D0%B4" title="Аксиоматикалык метод – Kyrgyz" lang="ky" hreflang="ky" data-title="Аксиоматикалык метод" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Logikai_kalkulus" title="Logikai kalkulus – Hungarian" lang="hu" hreflang="hu" data-title="Logikai kalkulus" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Формален систем – Macedonian" lang="mk" hreflang="mk" data-title="Формален систем" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sistem_formal" title="Sistem formal – Malay" lang="ms" hreflang="ms" data-title="Sistem formal" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Formeel_systeem" title="Formeel systeem – Dutch" lang="nl" hreflang="nl" data-title="Formeel systeem" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%BD%A2%E5%BC%8F%E4%BD%93%E7%B3%BB" title="形式体系 – Japanese" lang="ja" hreflang="ja" data-title="形式体系" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/System_formalny" title="System formalny – Polish" lang="pl" hreflang="pl" data-title="System formalny" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sistema_formal" title="Sistema formal – Portuguese" lang="pt" hreflang="pt" data-title="Sistema formal" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Формальная система – Russian" lang="ru" hreflang="ru" data-title="Формальная система" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sistemi_formal" title="Sistemi formal – Albanian" lang="sq" hreflang="sq" data-title="Sistemi formal" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D1%87%D0%BA%D0%B8_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Логички систем – Serbian" lang="sr" hreflang="sr" data-title="Логички систем" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Formellt_system" title="Formellt system – Swedish" lang="sv" hreflang="sv" data-title="Formellt system" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Формальна система – Ukrainian" lang="uk" hreflang="uk" data-title="Формальна система" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%BD%A2%E5%BC%8F%E7%B3%BB%E7%B5%B1" title="形式系統 – Cantonese" lang="yue" hreflang="yue" data-title="形式系統" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BD%A2%E5%BC%8F%E7%B3%BB%E7%B5%B1" title="形式系統 – Chinese" lang="zh" 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Deductive_system&redirect=no" class="mw-redirect" title="Deductive system">Deductive system</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical model for deduction or proof systems</div> <p>A <b>formal system</b> is an <a href="/wiki/Abstract_structure" title="Abstract structure">abstract structure</a> and <a href="/wiki/Formalism_(philosophy_of_mathematics)" title="Formalism (philosophy of mathematics)">formalization</a> of an <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic system</a> used for <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deducing</a>, using <a href="/wiki/Rule_of_inference" title="Rule of inference">rules of inference</a>, <a href="/wiki/Theorem" title="Theorem">theorems</a> from <a href="/wiki/Axioms" class="mw-redirect" title="Axioms">axioms</a> by a set of <a href="/wiki/Rule_of_inference" title="Rule of inference">inference rules</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>In 1921, <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> proposed to use formal systems as the foundation of knowledge in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The term formalism is sometimes a rough synonym for <i>formal system</i>, but it also refers to a given style of <a href="/wiki/Notation" class="mw-redirect" title="Notation">notation</a>, for example, <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>'s <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Concepts">Concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formal_system&action=edit&section=1" title="Edit section: Concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Formal_languages.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Formal_languages.svg/300px-Formal_languages.svg.png" decoding="async" width="300" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Formal_languages.svg/450px-Formal_languages.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Formal_languages.svg/600px-Formal_languages.svg.png 2x" data-file-width="250" data-file-height="230" /></a><figcaption>This diagram shows the <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntactic entities</a> that may be constructed from <a href="/wiki/Formal_language" title="Formal language">formal languages</a>. The symbols and <a href="/wiki/String_(computer_science)" title="String (computer science)">strings of symbols</a> may be broadly divided into <a href="/wiki/Nonsense" title="Nonsense">nonsense</a> and <a href="/wiki/Well-formed_formula" title="Well-formed formula">well-formed formulas</a>. A formal language can be thought of as identical to the set of its well-formed formulas, which may be broadly divided into <a href="/wiki/Theorem" title="Theorem">theorems</a> and non-theorems.</figcaption></figure> <p>A formal system has the following:<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <ul><li><a href="/wiki/Formal_language" title="Formal language">Formal language</a>, which is a set of <a href="/wiki/Well-formed_formula" title="Well-formed formula">well-formed formulas</a>, which are strings of <a href="/wiki/Symbol_(formal)" title="Symbol (formal)">symbols</a> from an <a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">alphabet</a>, formed by a <a href="/wiki/Formal_grammar" title="Formal grammar">formal grammar</a> (consisting of <a href="/wiki/Production_(computer_science)" title="Production (computer science)">production rules</a> or <a href="/wiki/Formation_rule" title="Formation rule">formation rules</a>).</li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">Deductive system</a>, deductive apparatus, or <a href="/wiki/Proof_calculus" title="Proof calculus">proof system</a>, which has <a href="/wiki/Rule_of_inference" title="Rule of inference">rules of inference</a> that take <a href="/wiki/Axiom" title="Axiom">axioms</a> and infers <a href="/wiki/Theorem" title="Theorem">theorems</a>, both of which are part of the formal language.</li></ul> <p>A formal system is said to be <a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">recursive</a> (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are <a href="/wiki/Decidable_set" class="mw-redirect" title="Decidable set">decidable sets</a> or <a href="/wiki/Recursively_enumerable_set" class="mw-redirect" title="Recursively enumerable set">semidecidable sets</a>, respectively. </p> <div class="mw-heading mw-heading3"><h3 id="Formal_language">Formal language</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formal_system&action=edit&section=2" title="Edit section: Formal language"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline 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.mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:Formal_languages" title="Category:Formal languages"><i>a series</i></a> on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Formal_languages" class="mw-redirect" title="Formal languages">Formal languages</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.2em;"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:#ddddff;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Key concepts</div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a class="mw-selflink selflink">Formal system</a></li> <li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Syntax_(logic)" title="Syntax (logic)">Syntax</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics (logic)</a></li> <li><a href="/wiki/Semantics_(computer_science)" title="Semantics (computer science)">Semantics (programming languages)</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Formal grammar</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Well-formed formula</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata theory</a></li> <li><a href="/wiki/Regular_expression" title="Regular expression">Regular expression</a></li> <li><a href="/wiki/Production_(computer_science)" title="Production (computer science)">Production</a></li> <li><a href="/wiki/Ground_expression" title="Ground expression">Ground expression</a></li> <li><a href="/wiki/Atomic_formula" title="Atomic formula">Atomic formula</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content" style="padding-top:0.2em;"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddddff;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Applications</div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Formal_methods" title="Formal methods">Formal methods</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate logic</a></li> <li><a href="/wiki/Mathematical_notation" title="Mathematical notation">Mathematical notation</a></li> <li><a href="/wiki/Natural_language_processing" title="Natural language processing">Natural language processing</a></li> <li><a href="/wiki/Programming_language_theory" title="Programming language theory">Programming language theory</a></li> <li><a href="/wiki/Computational_linguistics" title="Computational linguistics">Computational linguistics</a></li> <li><a href="/wiki/Syntax_analysis" class="mw-redirect" title="Syntax analysis">Syntax analysis</a></li> <li><a href="/wiki/Formal_verification" title="Formal verification">Formal verification</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output 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.navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Formal_languages" title="Template:Formal languages"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Formal_languages&action=edit&redlink=1" class="new" title="Template talk:Formal languages (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Formal_languages" title="Special:EditPage/Template:Formal languages"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Formal_language" title="Formal language">Formal language</a>, <a href="/wiki/Formal_grammar" title="Formal grammar">Formal grammar</a>, <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">Syntax (logic)</a>, and <a href="/wiki/Logical_form" title="Logical form">Logical form</a></div> <p>A <a href="/wiki/Formal_language" title="Formal language">formal language</a> is a language that is defined by a formal system. Like languages in <a href="/wiki/Linguistics" title="Linguistics">linguistics</a>, formal languages generally have two aspects: </p> <ul><li>the <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a> is what the language looks like (more formally: the set of possible expressions that are valid utterances in the language)</li> <li>the <a href="/wiki/Semantics_of_logic" title="Semantics of logic">semantics</a> are what the utterances of the language mean (which is formalized in various ways, depending on the type of language in question)</li></ul> <p>Usually only the <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a> of a formal language is considered via the notion of a <a href="/wiki/Formal_grammar" title="Formal grammar">formal grammar</a>. The two main categories of formal grammar are that of <a href="/wiki/Generative_grammar" title="Generative grammar">generative grammars</a>, which are sets of rules for how strings in a language can be written, and that of <a href="/wiki/Analytic_grammar" class="mw-redirect" title="Analytic grammar">analytic grammars</a> (or reductive grammar<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>), which are sets of rules for how a string can be analyzed to determine whether it is a member of the language. </p> <div class="mw-heading mw-heading3"><h3 id="Deductive_system">Deductive system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formal_system&action=edit&section=3" title="Edit section: Deductive system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Cleanup plainlinks metadata ambox ambox-style ambox-Cleanup" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section may <b>require <a href="/wiki/Wikipedia:Cleanup" title="Wikipedia:Cleanup">cleanup</a></b> to meet Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>. The specific problem is: <b>This section needs better organization and more citations.</b><span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Formal_system" title="Special:EditPage/Formal system">improve this section</a> if you can.</span> <span class="date-container"><i>(<span class="date">October 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Inference" title="Inference">Inference</a>, <a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a>, and <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">Deductive reasoning</a></div> <p>A <i>deductive system</i>, also called a <i>deductive apparatus</i>,<sup id="cite_ref-:1_8-0" class="reference"><a href="#cite_note-:1-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> consists of the <a href="/wiki/Axiom" title="Axiom">axioms</a> (or <a href="/wiki/Axiom_schema" title="Axiom schema">axiom schemata</a>) and <a href="/wiki/Rules_of_inference" class="mw-redirect" title="Rules of inference">rules of inference</a> that can be used to <a href="/wiki/Formal_proof" title="Formal proof">derive</a> <a href="/wiki/Theorem" title="Theorem">theorems</a> of the system.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Such deductive systems preserve <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deductive</a> qualities in the <a href="/wiki/Formula_(mathematical_logic)" class="mw-redirect" title="Formula (mathematical logic)">formulas</a> that are expressed in the system. Usually the quality we are concerned with is <a href="/wiki/Truth" title="Truth">truth</a> as opposed to falsehood. However, other <a href="/wiki/Modal_logic" title="Modal logic">modalities</a>, such as <a href="/wiki/Theory_of_justification" class="mw-redirect" title="Theory of justification">justification</a> or <a href="/wiki/Belief" title="Belief">belief</a> may be preserved instead. </p><p>In order to sustain its deductive integrity, a <i>deductive apparatus</i> must be definable without reference to any <a href="/wiki/Intended_interpretation" class="mw-redirect" title="Intended interpretation">intended interpretation</a> of the language. The aim is to ensure that each line of a <a href="/wiki/Mathematical_proof" title="Mathematical proof">derivation</a> is merely a <a href="/wiki/Logical_consequence" title="Logical consequence">logical consequence</a> of the lines that precede it. There should be no element of any <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">interpretation</a> of the language that gets involved with the deductive nature of the system. </p><p>The <a href="/wiki/Logical_consequence" title="Logical consequence">logical consequence</a> (or entailment) of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model. Often the formal system will be the basis for or even identified with a larger <a href="/wiki/Theory" title="Theory">theory</a> or field (e.g. <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>) consistent with the usage in modern mathematics such as <a href="/wiki/Model_theory" title="Model theory">model theory</a>.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="This section doesn't really do a group job stating what an entailment actually is. (September 2017)">clarification needed</span></a></i>]</sup> </p><p>An example of a deductive system would be the rules of inference and <a href="/wiki/First-order_logic#Equality_and_its_axioms" title="First-order logic">axioms regarding equality</a> used in <a href="/wiki/First-order_logic" title="First-order logic">first order logic</a>. </p><p>The two main types of deductive systems are proof systems and formal semantics.<sup id="cite_ref-:1_8-1" class="reference"><a href="#cite_note-:1-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Proof_system">Proof system</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formal_system&action=edit&section=4" title="Edit section: Proof system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Proof_system" class="mw-redirect" title="Proof system">Proof system</a> and <a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></div> <p>Formal proofs are sequences of <a href="/wiki/Well-formed_formula" title="Well-formed formula">well-formed formulas</a> (or WFF for short) that might either be an <a href="/wiki/Axiom" title="Axiom">axiom</a> or be the product of applying an inference rule on previous WFFs in the proof sequence. The last WFF in the sequence is recognized as a <a href="/wiki/Theorem#Theorems_in_logic" title="Theorem">theorem</a>. </p><p>Once a formal system is given, one can define the set of theorems which can be proved inside the formal system. This set consists of all WFFs for which there is a proof. Thus all axioms are considered theorems. Unlike the grammar for WFFs, there is no guarantee that there will be a <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decision procedure</a> for deciding whether a given WFF is a theorem or not. </p><p>The point of view that generating formal proofs is all there is to mathematics is often called <i><a href="/wiki/Formalism_(philosophy_of_mathematics)" title="Formalism (philosophy of mathematics)">formalism</a></i>. <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> founded <a href="/wiki/Metamathematics" title="Metamathematics">metamathematics</a> as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called a <i><a href="/wiki/Metalanguage" title="Metalanguage">metalanguage</a></i>. The metalanguage may be a natural language, or it may be partially formalized itself, but it is generally less completely formalized than the formal language component of the formal system under examination, which is then called the <i>object language</i>, that is, the object of the discussion in question. The notion of <i>theorem</i> just defined should not be confused with <i>theorems about the formal system</i>, which, in order to avoid confusion, are usually called <a href="/wiki/Metatheorem" title="Metatheorem">metatheorems</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Formal_semantics_of_logical_system"><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a> of logical system</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formal_system&action=edit&section=5" title="Edit section: Formal semantics of logical system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a>, <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation (logic)</a>, and <a href="/wiki/Model_theory" title="Model theory">Model theory</a></div> <p>A <i>logical system</i> is a deductive system (most commonly <a href="/wiki/First-order_logic" title="First-order logic">first order logic</a>) together with additional <a href="/wiki/Non-logical_axioms" class="mw-redirect" title="Non-logical axioms">non-logical axioms</a>. According to <a href="/wiki/Model_theory" title="Model theory">model theory</a>, a logical system may be given <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">interpretations</a> which describe whether a given <a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">structure</a> - the mapping of formulas to a particular meaning - satisfies a well-formed formula. A structure that satisfies all the axioms of the formal system is known as a <a href="/wiki/Model_(model_theory)" class="mw-redirect" title="Model (model theory)">model</a> of the logical system. </p><p>A logical system is: </p> <ul><li><a href="/wiki/Soundness" title="Soundness">Sound</a>, if each well-formed formula that can be inferred from the axioms is satisfied by every model of the logical system.</li> <li><a href="/wiki/Completeness_(logic)#Semantic_completeness" title="Completeness (logic)">Semantically complete</a>, if each well-formed formula that is satisfied by every model of the logical system can be inferred from the axioms.</li></ul> <p>An example of a logical system is <a href="/wiki/Peano_axioms#First-order_theory_of_arithmetic" title="Peano axioms">Peano arithmetic</a>. The standard model of arithmetic sets the <a href="/wiki/Domain_of_discourse" title="Domain of discourse">domain of discourse</a> to be the <a href="/wiki/Nonnegative_integer" class="mw-redirect" title="Nonnegative integer">nonnegative integers</a> and gives the symbols their usual meaning.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> There are also <a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">non-standard models of arithmetic</a>. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formal_system&action=edit&section=6" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Formalism_(philosophy_of_mathematics)" title="Formalism (philosophy of mathematics)">Formalism (philosophy of mathematics)</a> and <a href="/wiki/Formal_logical_systems" class="mw-redirect" title="Formal logical systems">Formal logical systems</a></div> <p>Early logic systems includes Indian logic of <a href="/wiki/P%C4%81%E1%B9%87ini" title="Pāṇini">Pāṇini</a>, syllogistic logic of Aristotle, propositional logic of Stoicism, and Chinese logic of <a href="/wiki/Gongsun_Long" title="Gongsun Long">Gongsun Long</a> (c. 325–250 BCE) . In more recent times, contributors include <a href="/wiki/George_Boole" title="George Boole">George Boole</a>, <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a>, and <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>. <a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a> was developed in 19th century <a href="/wiki/Europe" title="Europe">Europe</a>. </p><p><a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> instigated a <a href="/wiki/Formalism_(philosophy_of_mathematics)" title="Formalism (philosophy of mathematics)">formalist</a> movement called <a href="/wiki/Hilbert%27s_program" title="Hilbert's program">Hilbert’s program</a> as a proposed solution to the <a href="/wiki/Foundational_crisis_of_mathematics" class="mw-redirect" title="Foundational crisis of mathematics">foundational crisis of mathematics</a>, that was eventually tempered by <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a>.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/QED_manifesto" title="QED manifesto">QED manifesto</a> represented a subsequent, as yet unsuccessful, effort at formalization of known mathematics. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formal_system&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid 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width="32" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Complex-adaptive-system.jpg/48px-Complex-adaptive-system.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Complex-adaptive-system.jpg/64px-Complex-adaptive-system.jpg 2x" data-file-width="2100" data-file-height="1500" /></span></span></span><span class="portalbox-link"><a href="/wiki/Portal:Systems_science" title="Portal:Systems science">Systems science portal</a></span></li><li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/18px-Socrates.png" decoding="async" width="18" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/27px-Socrates.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/36px-Socrates.png 2x" data-file-width="326" data-file-height="500" 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class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Substitution_instance" class="mw-redirect" title="Substitution instance">Substitution instance</a> – Concept in logic<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory (mathematical logic)</a> – Set of sentences in a formal language</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formal_system&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output 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a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/formal-system">"Formal system | Logic, Symbols & Axioms | Britannica"</a>. <i>www.britannica.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-10-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.britannica.com&rft.atitle=Formal+system+%7C+Logic%2C+Symbols+%26+Axioms+%7C+Britannica&rft_id=https%3A%2F%2Fwww.britannica.com%2Ftopic%2Fformal-system&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFormal+system" class="Z3988"></span></span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZach2003" class="citation book cs1">Zach, Richard (31 July 2003). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/spr2016/entries/hilbert-program">"Hilbert's Program"</a>. <i>Hilbert's Program, Stanford Encyclopedia of Philosophy</i>. Metaphysics Research Lab, Stanford University.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Hilbert%27s+Program&rft.btitle=Hilbert%27s+Program%2C+Stanford+Encyclopedia+of+Philosophy&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2003-07-31&rft.aulast=Zach&rft.aufirst=Richard&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Fspr2016%2Fentries%2Fhilbert-program&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFormal+system" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://planetmath.org/formalsystem">"formal system"</a>. <i>planetmath.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-10-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=planetmath.org&rft.atitle=formal+system&rft_id=https%3A%2F%2Fplanetmath.org%2Fformalsystem&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFormal+system" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRapaport2010" class="citation web cs1">Rapaport, William J. (25 March 2010). <a rel="nofollow" class="external text" href="https://cse.buffalo.edu/~rapaport/formalsystems">"Syntax & Semantics of Formal Systems"</a>. <i>University of Buffalo</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=University+of+Buffalo&rft.atitle=Syntax+%26+Semantics+of+Formal+Systems&rft.date=2010-03-25&rft.aulast=Rapaport&rft.aufirst=William+J.&rft_id=https%3A%2F%2Fcse.buffalo.edu%2F~rapaport%2Fformalsystems&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFormal+system" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Definition:Formal_System">"Definition:Formal System - ProofWiki"</a>. <i>proofwiki.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-10-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=proofwiki.org&rft.atitle=Definition%3AFormal+System+-+ProofWiki&rft_id=https%3A%2F%2Fproofwiki.org%2Fwiki%2FDefinition%3AFormal_System&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFormal+system" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Reductive grammar: (<i>computer science</i>) A set of syntactic rules for the analysis of strings to determine whether the strings exist in a language. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://encyclopedia2.thefreedictionary.com/reductive+grammar">"Sci-Tech Dictionary McGraw-Hill Dictionary of Scientific and Technical Terms"</a> (6th ed.). McGraw-Hill.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Sci-Tech+Dictionary+McGraw-Hill+Dictionary+of+Scientific+and+Technical+Terms&rft.edition=6th&rft.pub=McGraw-Hill&rft_id=http%3A%2F%2Fencyclopedia2.thefreedictionary.com%2Freductive%2Bgrammar&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFormal+system" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Reliable_sources" title="Wikipedia:Reliable sources"><span title="The material near this tag may rely on an unreliable source. (February 2015)">unreliable source?</span></a></i>]</sup> About the Author Compiled by The Editors of the McGraw-Hill Encyclopedia of Science & Technology (New York, NY) an in-house staff who represents the cutting-edge of skill, knowledge, and innovation in science publishing. <a rel="nofollow" class="external autonumber" href="https://www.amazon.com/McGraw-Hill-Dictionary-Scientific-Technical-Terms/dp/007042313X#productDescription_secondary_view_pageState_1423332836958">[1]</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"> "There are two classes of formal-language definition compiler-writing schemes. The productive <a href="/wiki/Formal_grammar" title="Formal grammar">grammar</a> approach is the most common. A productive grammar consists primarrly of a set of rules that describe a method of generating all possible strings of the language. The reductive or <a href="/wiki/Formal_grammar#Analytic_grammars" title="Formal grammar">analytical grammar</a> technique states a set of rules that describe a method of analyzing any string of characters and deciding whether that string is in the language." <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://bitsavers.informatik.uni-stuttgart.de/pdf/sri/arc/rulifson/A_Tree_Meta_For_The_XDS_940_Appendix_D_Apr68.pdf">"<b>The TREE-META Compiler-Compiler System: A Meta Compiler System for the Univac 1108 and General Electric 645</b>, University of Utah Technical Report RADC-TR-69-83. C. Stephen Carr, David A. Luther, Sherian Erdmann"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">5 January</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+TREE-META+Compiler-Compiler+System%3A+A+Meta+Compiler+System+for+the+Univac+1108+and+General+Electric+645%2C+University+of+Utah+Technical+Report+RADC-TR-69-83.+C.+Stephen+Carr%2C+David+A.+Luther%2C+Sherian+Erdmann&rft_id=http%3A%2F%2Fbitsavers.informatik.uni-stuttgart.de%2Fpdf%2Fsri%2Farc%2Frulifson%2FA_Tree_Meta_For_The_XDS_940_Appendix_D_Apr68.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFormal+system" class="Z3988"></span></span> </li> <li id="cite_note-:1-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Definition:Deductive_Apparatus">"Definition:Deductive Apparatus - ProofWiki"</a>. <i>proofwiki.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-10-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=proofwiki.org&rft.atitle=Definition%3ADeductive+Apparatus+-+ProofWiki&rft_id=https%3A%2F%2Fproofwiki.org%2Fwiki%2FDefinition%3ADeductive_Apparatus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFormal+system" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaye1991" class="citation book cs1">Kaye, Richard (1991). "1. The Standard Model". <i>Models of Peano arithmetic</i>. Oxford: Clarendon Press. p. 10. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198532132" title="Special:BookSources/9780198532132"><bdi>9780198532132</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=1.+The+Standard+Model&rft.btitle=Models+of+Peano+arithmetic&rft.place=Oxford&rft.pages=10&rft.pub=Clarendon+Press&rft.date=1991&rft.isbn=9780198532132&rft.aulast=Kaye&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFormal+system" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formal_system&action=edit&section=9" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Raymond_M._Smullyan" class="mw-redirect" title="Raymond M. Smullyan">Raymond M. Smullyan</a>, 1961. <i>Theory of Formal Systems: Annals of Mathematics Studies</i>, Princeton University Press (April 1, 1961) 156 pages <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08047-X" title="Special:BookSources/0-691-08047-X">0-691-08047-X</a></li> <li><a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Stephen Cole Kleene</a>, 1967. <i>Mathematical Logic</i> Reprinted by Dover, 2002. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-42533-9" title="Special:BookSources/0-486-42533-9">0-486-42533-9</a></li> <li><a href="/wiki/Douglas_Hofstadter" title="Douglas Hofstadter">Douglas Hofstadter</a>, 1979. <i><a href="/wiki/G%C3%B6del,_Escher,_Bach:_An_Eternal_Golden_Braid" class="mw-redirect" title="Gödel, Escher, Bach: An Eternal Golden Braid">Gödel, Escher, Bach: An Eternal Golden Braid</a></i> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-02656-2" title="Special:BookSources/978-0-465-02656-2">978-0-465-02656-2</a>. 777 pages.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formal_system&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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href="https://commons.wikimedia.org/wiki/Category:Formal_systems" class="extiw" title="commons:Category:Formal systems">Formal systems</a> at Wikimedia Commons</li> <li>Encyclopædia Britannica, <a rel="nofollow" class="external text" href="https://www.britannica.com/eb/article-9034889/formal-system">Formal system</a> definition, 2007.</li> <li>Daniel Richardson, <a rel="nofollow" class="external text" href="http://cs.bath.ac.uk/pb/EMCL/DS/DS-Ref-2011/c19.pdf">Formal systems, logic and semantics</a></li> <li>William J. Rapaport, <a rel="nofollow" class="external text" href="https://cse.buffalo.edu/~rapaport/formalsystems">Syntax & Semantics of Formal Systems</a></li> <li>PlanetMath, <a rel="nofollow" class="external text" href="https://planetmath.org/FormalSystem">Formal System</a></li> <li>Pr∞fWiki, <a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Definition:Formal_System">Definition:Formal System</a></li> <li>Pr∞fWiki, <a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Definition:Deductive_Apparatus">Definition:Deductive Apparatus</a></li> <li>Encyclopedia of Mathematics, <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Formal_system">Formal system</a></li> <li>Peter Suber, <a rel="nofollow" class="external text" href="http://www.earlham.edu/~peters/courses/logsys/machines.htm">Formal Systems and Machines: An Isomorphism</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110524103726/http://www.earlham.edu/~peters/courses/logsys/machines.htm">Archived</a> 2011-05-24 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, 1997.</li> <li>Ray Taol, <a rel="nofollow" class="external text" href="https://cs.lmu.edu/~ray/notes/formalsystems/">Formal Systems</a></li> <li><a rel="nofollow" class="external text" href="http://www.cs.indiana.edu/~port/teach/641/formal.sys.haug.html">What is a Formal System?</a>: Some quotes from John Haugeland's `Artificial Intelligence: The Very Idea' (1985), pp. 48–64.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output 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logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a class="mw-selflink selflink">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_logic" title="Abstract logic">Abstract logic</a></li> <li><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Concrete_category" title="Concrete category">Concrete</a>/<a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li> <li><a href="/wiki/Category_of_sets" title="Category of sets">Category of sets</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">History of logic</a></li> <li><a href="/wiki/History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a> <ul><li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li> <li><a href="/wiki/Logicism" title="Logicism">Logicism</a></li> <li><a href="/wiki/Mathematical_object" title="Mathematical object">Mathematical object</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Supertask" title="Supertask">Supertask</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Systems_science" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Systems_science" title="Template:Systems science"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Systems_science" title="Template talk:Systems science"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Systems_science" title="Special:EditPage/Template:Systems science"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Systems_science" style="font-size:114%;margin:0 4em"><a href="/wiki/Systems_science" title="Systems science">Systems science</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/System" title="System">System</a><br />types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Systems_art" title="Systems art">Art</a></li> <li><a href="/wiki/Biological_system" title="Biological system">Biological</a></li> <li><a href="/wiki/Complex_system" title="Complex system">Complex</a></li> <li><a href="/wiki/Coupled_human%E2%80%93environment_system" title="Coupled human–environment system">Coupled human–environment</a></li> <li><a href="/wiki/Ecosystem" title="Ecosystem">Ecological</a></li> <li><a href="/wiki/Economic_system" title="Economic system">Economic</a></li> <li><a href="/wiki/Information_system" title="Information system">Information</a></li> <li><a href="/wiki/Multi-agent_system" title="Multi-agent system">Multi-agent</a></li> <li><a href="/wiki/Nervous_system" title="Nervous system">Nervous</a></li> <li><a href="/wiki/Recommender_system" title="Recommender system">Recommender</a></li> <li><a href="/wiki/Social_system" title="Social system">Social</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Doubling_time" title="Doubling time">Doubling time</a></li> <li><a href="/wiki/Twelve_leverage_points" title="Twelve leverage points">Leverage points</a></li> <li><a href="/wiki/Limiting_factor" title="Limiting factor">Limiting factor</a></li> <li><a href="/wiki/Negative_feedback" title="Negative feedback">Negative feedback</a></li> <li><a href="/wiki/Positive_feedback" title="Positive feedback">Positive feedback</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theoretical<br />fields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li> <li><a href="/wiki/Cybernetics" title="Cybernetics">Cybernetics</a></li> <li><a href="/wiki/Earth_system_science" title="Earth system science">Earth system science</a></li> <li><a href="/wiki/Living_systems" title="Living systems">Living systems</a></li> <li><a href="/wiki/Sociotechnical_system" title="Sociotechnical system">Sociotechnical system</a></li> <li><a href="/wiki/Systemics" title="Systemics">Systemics</a></li> <li><a href="/wiki/Urban_metabolism" title="Urban metabolism">Urban metabolism</a></li> <li><a href="/wiki/World-systems_theory" title="World-systems theory">World-systems theory</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Systems_analysis" title="Systems analysis">Analysis</a></li> <li><a href="/wiki/Systems_biology" title="Systems biology">Biology</a></li> <li><a href="/wiki/System_dynamics" title="System dynamics">Dynamics</a></li> <li><a href="/wiki/Systems_ecology" title="Systems ecology">Ecology</a></li> <li><a href="/wiki/Systems_engineering" title="Systems engineering">Engineering</a></li> <li><a href="/wiki/Systems_neuroscience" title="Systems neuroscience">Neuroscience</a></li> <li><a href="/wiki/Systems_pharmacology" title="Systems pharmacology">Pharmacology</a></li> <li><a href="/wiki/Systems_philosophy" title="Systems philosophy">Philosophy</a></li> <li><a href="/wiki/Systems_psychology" title="Systems psychology">Psychology</a></li> <li><a href="/wiki/Systems_theory" title="Systems theory">Theory</a> (<a href="/wiki/Systems_thinking" title="Systems thinking">Systems thinking</a>)</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_systems_scientists" title="List of systems scientists">Scientists</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Russell_L._Ackoff" title="Russell L. Ackoff">Russell L. Ackoff</a></li> <li><a href="/wiki/Victor_Aladjev" title="Victor Aladjev">Victor Aladjev</a></li> <li><a href="/wiki/William_Ross_Ashby" class="mw-redirect" title="William Ross Ashby">William Ross Ashby</a></li> <li><a href="/wiki/Ruzena_Bajcsy" title="Ruzena Bajcsy">Ruzena Bajcsy</a></li> <li><a href="/wiki/B%C3%A9la_H._B%C3%A1n%C3%A1thy" title="Béla H. Bánáthy">Béla H. Bánáthy</a></li> <li><a href="/wiki/Gregory_Bateson" title="Gregory Bateson">Gregory Bateson</a></li> <li><a href="/wiki/Anthony_Stafford_Beer" class="mw-redirect" title="Anthony Stafford Beer">Anthony Stafford Beer</a></li> <li><a href="/wiki/Richard_E._Bellman" title="Richard E. Bellman">Richard E. Bellman</a></li> <li><a href="/wiki/Ludwig_von_Bertalanffy" title="Ludwig von Bertalanffy">Ludwig von Bertalanffy</a></li> <li><a href="/wiki/Margaret_Boden" title="Margaret Boden">Margaret Boden</a></li> <li><a href="/wiki/Alexander_Bogdanov" title="Alexander Bogdanov">Alexander Bogdanov</a></li> <li><a href="/wiki/Kenneth_E._Boulding" title="Kenneth E. Boulding">Kenneth E. Boulding</a></li> <li><a href="/wiki/Murray_Bowen" title="Murray Bowen">Murray Bowen</a></li> <li><a href="/wiki/Kathleen_Carley" title="Kathleen Carley">Kathleen Carley</a></li> <li><a href="/wiki/Mary_Cartwright" title="Mary Cartwright">Mary Cartwright</a></li> <li><a href="/wiki/C._West_Churchman" title="C. West Churchman">C. West Churchman</a></li> <li><a href="/wiki/Manfred_Clynes" title="Manfred Clynes">Manfred Clynes</a></li> <li><a href="/wiki/George_Dantzig" title="George Dantzig">George Dantzig</a></li> <li><a href="/wiki/Edsger_W._Dijkstra" title="Edsger W. Dijkstra">Edsger W. Dijkstra</a></li> <li><a href="/wiki/Fred_Emery" title="Fred Emery">Fred Emery</a></li> <li><a href="/wiki/Heinz_von_Foerster" title="Heinz von Foerster">Heinz von Foerster</a></li> <li><a href="/wiki/Stephanie_Forrest" title="Stephanie Forrest">Stephanie Forrest</a></li> <li><a href="/wiki/Jay_Wright_Forrester" title="Jay Wright Forrester">Jay Wright Forrester</a></li> <li><a href="/wiki/Barbara_J._Grosz" title="Barbara J. Grosz">Barbara Grosz</a></li> <li><a href="/wiki/Charles_A._S._Hall" title="Charles A. S. Hall">Charles A. S. Hall</a></li> <li><a href="/wiki/Mike_Jackson_(systems_scientist)" title="Mike Jackson (systems scientist)">Mike Jackson</a></li> <li><a href="/wiki/Lydia_Kavraki" title="Lydia Kavraki">Lydia Kavraki</a></li> <li><a href="/wiki/James_J._Kay" title="James J. Kay">James J. Kay</a></li> <li><a href="/wiki/Faina_Mihajlovna_Kirillova" class="mw-redirect" title="Faina Mihajlovna Kirillova">Faina M. Kirillova</a></li> <li><a href="/wiki/George_Klir" title="George Klir">George Klir</a></li> <li><a href="/wiki/Allenna_Leonard" title="Allenna Leonard">Allenna Leonard</a></li> <li><a href="/wiki/Edward_Norton_Lorenz" title="Edward Norton Lorenz">Edward Norton Lorenz</a></li> <li><a href="/wiki/Niklas_Luhmann" title="Niklas Luhmann">Niklas Luhmann</a></li> <li><a href="/wiki/Humberto_Maturana" title="Humberto Maturana">Humberto Maturana</a></li> <li><a href="/wiki/Margaret_Mead" title="Margaret Mead">Margaret Mead</a></li> <li><a href="/wiki/Donella_Meadows" title="Donella Meadows">Donella Meadows</a></li> <li><a href="/wiki/Mihajlo_D._Mesarovic" title="Mihajlo D. Mesarovic">Mihajlo D. Mesarovic</a></li> <li><a href="/wiki/James_Grier_Miller" title="James Grier Miller">James Grier Miller</a></li> <li><a href="/wiki/Radhika_Nagpal" title="Radhika Nagpal">Radhika Nagpal</a></li> <li><a href="/wiki/Howard_T._Odum" title="Howard T. Odum">Howard T. Odum</a></li> <li><a href="/wiki/Talcott_Parsons" title="Talcott Parsons">Talcott Parsons</a></li> <li><a href="/wiki/Ilya_Prigogine" title="Ilya Prigogine">Ilya Prigogine</a></li> <li><a href="/wiki/Qian_Xuesen" title="Qian Xuesen">Qian Xuesen</a></li> <li><a href="/wiki/Anatol_Rapoport" title="Anatol Rapoport">Anatol Rapoport</a></li> <li><a href="/wiki/John_Seddon" title="John Seddon">John Seddon</a></li> <li><a href="/wiki/Peter_Senge" title="Peter Senge">Peter Senge</a></li> <li><a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a></li> <li><a href="/wiki/Katia_Sycara" title="Katia Sycara">Katia Sycara</a></li> <li><a href="/wiki/Eric_Trist" title="Eric Trist">Eric Trist</a></li> <li><a href="/wiki/Francisco_Varela" title="Francisco Varela">Francisco Varela</a></li> <li><a href="/wiki/Manuela_M._Veloso" title="Manuela M. Veloso">Manuela M. Veloso</a></li> <li><a href="/wiki/Kevin_Warwick" title="Kevin Warwick">Kevin Warwick</a></li> <li><a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a></li> <li><a href="/wiki/Jennifer_Wilby" title="Jennifer Wilby">Jennifer Wilby</a></li> <li><a href="/wiki/Anthony_Wilden" title="Anthony Wilden">Anthony Wilden</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Systems_theory_in_anthropology" title="Systems theory in anthropology">Systems theory in anthropology</a></li> <li><a href="/wiki/Systems_theory_in_archaeology" title="Systems theory in archaeology">Systems theory in archaeology</a></li> <li><a href="/wiki/Systems_theory_in_political_science" title="Systems theory in political science">Systems theory in political science</a></li></ul> </div></td></tr><tr><th 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proof">Mathematical proof</a></li> <li><a href="/wiki/Model_theory" title="Model theory">Model theory</a></li> <li><a href="/wiki/Mathematical_constructivism" class="mw-redirect" title="Mathematical constructivism">Mathematical constructivism</a></li> <li><a href="/wiki/Modal_logic" title="Modal logic">Modal logic</a></li> <li><a href="/wiki/List_of_mathematical_logic_topics" title="List of mathematical logic topics">List of mathematical logic topics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#fdf;;width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Set_(mathematics)" title="Set (mathematics)">Set</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive set theory</a></li> <li><a href="/wiki/Set_theory#Axiomatic_set_theory" title="Set theory">Axiomatic 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