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<span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Relations to logical conjunction and disjunction subsection</span> </button> <ul id="toc-Relations_to_logical_conjunction_and_disjunction-sublist" class="vector-toc-list"> <li id="toc-Infinite_domain_of_discourse" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinite_domain_of_discourse"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Infinite domain of discourse</span> </div> </a> <ul id="toc-Infinite_domain_of_discourse-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algebraic_approaches_to_quantification" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algebraic_approaches_to_quantification"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Algebraic approaches to quantification</span> </div> </a> <ul id="toc-Algebraic_approaches_to_quantification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Notation</span> </div> </a> <ul id="toc-Notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Order_of_quantifiers_(nesting)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Order_of_quantifiers_(nesting)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Order of quantifiers (nesting)</span> </div> </a> <ul id="toc-Order_of_quantifiers_(nesting)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equivalent_expressions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equivalent_expressions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Equivalent expressions</span> </div> </a> <button aria-controls="toc-Equivalent_expressions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Equivalent expressions subsection</span> </button> <ul id="toc-Equivalent_expressions-sublist" class="vector-toc-list"> <li id="toc-Quantifier_elimination" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantifier_elimination"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Quantifier elimination</span> </div> </a> <ul id="toc-Quantifier_elimination-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Range_of_quantification" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Range_of_quantification"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Range of quantification</span> </div> </a> <ul id="toc-Range_of_quantification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_semantics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formal_semantics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Formal semantics</span> </div> </a> <ul id="toc-Formal_semantics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Paucal,_multal_and_other_degree_quantifiers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Paucal,_multal_and_other_degree_quantifiers"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Paucal, multal and other degree quantifiers</span> </div> </a> <ul id="toc-Paucal,_multal_and_other_degree_quantifiers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_quantifiers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_quantifiers"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Other quantifiers</span> </div> </a> <ul id="toc-Other_quantifiers-sublist" class="vector-toc-list"> </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">10</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">11</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">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-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">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> 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</label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Quantifier (logic)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 37 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-37" 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">37 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%83%D9%85%D9%85" title="مكمم – Arabic" lang="ar" hreflang="ar" data-title="مكمم" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D1%80" title="Квантор – Bulgarian" lang="bg" hreflang="bg" data-title="Квантор" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Quantificador_(l%C3%B2gica)" title="Quantificador (lògica) – Catalan" lang="ca" hreflang="ca" data-title="Quantificador (lògica)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kvantifik%C3%A1tor" title="Kvantifikátor – Czech" lang="cs" hreflang="cs" data-title="Kvantifikátor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kvantor" title="Kvantor – Danish" lang="da" hreflang="da" data-title="Kvantor" 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/Quantor" title="Quantor – German" lang="de" hreflang="de" data-title="Quantor" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kvantifikatsioon_(loogika)" title="Kvantifikatsioon (loogika) – Estonian" lang="et" hreflang="et" data-title="Kvantifikatsioon (loogika)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cuantificador" title="Cuantificador – Spanish" lang="es" hreflang="es" data-title="Cuantificador" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kvantizanto" title="Kvantizanto – Esperanto" lang="eo" hreflang="eo" data-title="Kvantizanto" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbatzaile_(logika)" title="Zenbatzaile (logika) – Basque" lang="eu" hreflang="eu" data-title="Zenbatzaile (logika)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B3%D9%88%D8%B1_(%D9%85%D9%86%D8%B7%D9%82)" title="سور (منطق) – Persian" lang="fa" hreflang="fa" data-title="سور (منطق)" 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/Quantification_(logique)" title="Quantification (logique) – French" lang="fr" hreflang="fr" data-title="Quantification (logique)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%BE%D5%A1%D5%B6%D5%BF%D5%B8%D6%80%D5%B6%D5%A5%D6%80" title="Քվանտորներ – Armenian" lang="hy" hreflang="hy" data-title="Քվանտորներ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Logi%C4%8Dki_kvantifikator" title="Logički kvantifikator – Croatian" lang="hr" hreflang="hr" data-title="Logički kvantifikator" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Quantifikanto_(logiko)" title="Quantifikanto (logiko) – Ido" lang="io" hreflang="io" data-title="Quantifikanto (logiko)" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kuantifer_(logika)" title="Kuantifer (logika) – Indonesian" lang="id" hreflang="id" data-title="Kuantifer (logika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Quantificator" title="Quantificator – Interlingua" lang="ia" hreflang="ia" data-title="Quantificator" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Quantificatore" title="Quantificatore – Italian" lang="it" hreflang="it" data-title="Quantificatore" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9B%D7%9E%D7%AA" title="כמת – Hebrew" lang="he" hreflang="he" data-title="כמת" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D1%80" title="Квантор – Kazakh" lang="kk" hreflang="kk" data-title="Квантор" 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%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D1%80" title="Квантор – Kyrgyz" lang="ky" hreflang="ky" data-title="Квантор" 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/Kvantor_(logika)" title="Kvantor (logika) – Hungarian" lang="hu" hreflang="hu" data-title="Kvantor (logika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pengkuantitian" title="Pengkuantitian – Malay" lang="ms" hreflang="ms" data-title="Pengkuantitian" 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/Kwantor_(logica)" title="Kwantor (logica) – Dutch" lang="nl" hreflang="nl" data-title="Kwantor (logica)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kvantor" title="Kvantor – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kvantor" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kwantyfikator" title="Kwantyfikator – Polish" lang="pl" hreflang="pl" data-title="Kwantyfikator" 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/Quantifica%C3%A7%C3%A3o" title="Quantificação – Portuguese" lang="pt" hreflang="pt" data-title="Quantificação" 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%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D1%80" title="Квантор – Russian" lang="ru" hreflang="ru" data-title="Квантор" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Logical_quantifier" title="Logical quantifier – Simple English" lang="en-simple" hreflang="en-simple" data-title="Logical quantifier" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kvantifik%C3%A1tor_(logika)" title="Kvantifikátor (logika) – Slovak" lang="sk" hreflang="sk" data-title="Kvantifikátor (logika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Kvantifikator_(logika)" title="Kvantifikator (logika) – Serbian" lang="sr" hreflang="sr" data-title="Kvantifikator (logika)" 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/Kvantifikator" title="Kvantifikator – Swedish" lang="sv" hreflang="sv" data-title="Kvantifikator" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Niceleme_(mant%C4%B1k)" title="Niceleme (mantık) – Turkish" lang="tr" hreflang="tr" data-title="Niceleme (mantık)" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D1%80" title="Квантор – Ukrainian" lang="uk" hreflang="uk" data-title="Квантор" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C6%B0%E1%BB%A3ng_t%E1%BB%AB_(logic)" title="Lượng từ (logic) – Vietnamese" lang="vi" hreflang="vi" data-title="Lượng từ (logic)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%87%8F%E5%8C%96_(%E6%95%B8%E7%90%86%E9%82%8F%E8%BC%AF)" title="量化 (數理邏輯) – Cantonese" lang="yue" hreflang="yue" data-title="量化 (數理邏輯)" 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/%E9%87%8F%E5%8C%96_(%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91)" title="量化 (数理逻辑) – Chinese" lang="zh" hreflang="zh" data-title="量化 (数理逻辑)" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q592911#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul 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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"></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 use of "for all" and "there exists"</div> <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">For other uses, see <a href="/wiki/Quantifier_(disambiguation)" class="mw-redirect mw-disambig" title="Quantifier (disambiguation)">Quantifier (disambiguation)</a>.</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-Lead_rewrite plainlinks metadata ambox ambox-style ambox-lead_rewrite" 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">The article's <a href="/wiki/Wikipedia:Manual_of_Style/Lead_section" title="Wikipedia:Manual of Style/Lead section">lead section</a> <b>may need to be rewritten</b>.<span class="hide-when-compact"> Please help <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Quantifier_(logic)&action=edit">improve the lead</a> and read the <a href="/wiki/Wikipedia:Manual_of_Style/Lead_section" title="Wikipedia:Manual of Style/Lead section">lead layout guide</a>.</span> <span class="date-container"><i>(<span class="date">August 2022</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> <p>In <a href="/wiki/Mathematical_logic" title="Mathematical logic">logic</a>, a <b>quantifier</b> is an operator that specifies how many individuals in the <a href="/wiki/Domain_of_discourse" title="Domain of discourse">domain of discourse</a> satisfy an <a href="/wiki/Open_formula" title="Open formula">open formula</a>. For instance, the <a href="/wiki/Universal_quantifier" class="mw-redirect" title="Universal quantifier">universal quantifier</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc1a1a9c4c0f8d5df989c98aa2773ed657c5937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \forall }"></span> in the <a href="/wiki/First_order_logic" class="mw-redirect" title="First order logic">first order</a> formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall xP(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall xP(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25873948fc98344950ea1b91f88dd52239cf9c87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.507ex; height:2.843ex;" alt="{\displaystyle \forall xP(x)}"></span> expresses that everything in the domain satisfies the property denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>. On the other hand, the <a href="/wiki/Existential_quantifier" class="mw-redirect" title="Existential quantifier">existential quantifier</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ed842b6b90b2fdd825320cf8e5265fa937b583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \exists }"></span> in the formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists xP(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists xP(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4bb6d8a3cf6275a71b7183604aa81e8ba7edb50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.507ex; height:2.843ex;" alt="{\displaystyle \exists xP(x)}"></span> expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest <a href="/wiki/Scope_(logic)" title="Scope (logic)">scope</a> is called a quantified formula. A quantified formula must contain a <a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">bound variable</a> and a <a href="/wiki/Subformula" class="mw-redirect" title="Subformula">subformula</a> specifying a property of the referent of that variable. </p><p>The most commonly used quantifiers are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc1a1a9c4c0f8d5df989c98aa2773ed657c5937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \forall }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ed842b6b90b2fdd825320cf8e5265fa937b583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \exists }"></span>. These quantifiers are standardly defined as <a href="/wiki/Dual_(mathematics)" class="mw-redirect" title="Dual (mathematics)">duals</a>; in <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>, they are interdefinable using <a href="/wiki/Negation" title="Negation">negation</a>. They can also be used to define more complex quantifiers, as in the formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \exists xP(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \exists xP(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/833054c01cd2f7c59c1978d1e259bb39661dad53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.057ex; height:2.843ex;" alt="{\displaystyle \neg \exists xP(x)}"></span> which expresses that nothing has the property <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>. Other quantifiers are only definable within <a href="/wiki/Second_order_logic" class="mw-redirect" title="Second order logic">second order logic</a> or <a href="/wiki/Higher_order_logic" class="mw-redirect" title="Higher order logic">higher order logics</a>. Quantifiers have been generalized beginning with the work of <a href="/wiki/Andrzej_Mostowski" title="Andrzej Mostowski">Mostowski</a> and <a href="/wiki/Per_Lindstr%C3%B6m" title="Per Lindström">Lindström</a>. </p><p>In a first-order logic statement, quantifications in the same type (either universal quantifications or existential quantifications) can be exchanged without changing the meaning of the statement, while the exchange of quantifications in different types changes the meaning. As an example, the only difference in the definition of <a href="/wiki/Uniform_continuity#Definition_of_uniform_continuity" title="Uniform continuity">uniform continuity</a> and <a href="/wiki/Uniform_continuity#Definition_of_(ordinary)_continuity" title="Uniform continuity">(ordinary) continuity</a> is the order of quantifications. </p><p>First order quantifiers approximate the meanings of some <a href="/wiki/Natural_language" title="Natural language">natural language</a> quantifiers such as "some" and "all". However, many natural language quantifiers can only be analyzed in terms of <a href="/wiki/Generalized_quantifier" title="Generalized quantifier">generalized quantifiers</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Relations_to_logical_conjunction_and_disjunction">Relations to logical conjunction and disjunction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantifier_(logic)&action=edit&section=1" title="Edit section: Relations to logical conjunction and disjunction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a finite domain of discourse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=\{a_{1},...a_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=\{a_{1},...a_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c779a53bc75eba219f3d7381e64baf40e4513748" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.216ex; height:2.843ex;" alt="{\displaystyle D=\{a_{1},...a_{n}\}}"></span>, the universally quantified formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in D\;P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>D</mi> <mspace width="thickmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in D\;P(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7086afa8ac2968632459cca0f8361453cb0fcfb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.917ex; height:2.843ex;" alt="{\displaystyle \forall x\in D\;P(x)}"></span> is equivalent to the <a href="/wiki/Logical_conjunction" title="Logical conjunction">logical conjunction</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(a_{1})\land ...\land P(a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∧</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(a_{1})\land ...\land P(a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2734ffb11427ea744a9ecce4d21e31234718b355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.044ex; height:2.843ex;" alt="{\displaystyle P(a_{1})\land ...\land P(a_{n})}"></span>. Dually, the existentially quantified formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x\in D\;P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo>∈</mo> <mi>D</mi> <mspace width="thickmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\in D\;P(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50c5615e22c7ddfce6590a525ce6d360def62bba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.917ex; height:2.843ex;" alt="{\displaystyle \exists x\in D\;P(x)}"></span> is equivalent to the <a href="/wiki/Logical_disjunction" title="Logical disjunction">logical disjunction</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(a_{1})\lor ...\lor P(a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∨</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>∨</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(a_{1})\lor ...\lor P(a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398b8b2ed1f2c0b2a810f3763dfd6a6727379244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.044ex; height:2.843ex;" alt="{\displaystyle P(a_{1})\lor ...\lor P(a_{n})}"></span>. For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{0,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f582e459b750046ded8517cf592c0b5adb5247ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.546ex; height:2.843ex;" alt="{\displaystyle B=\{0,1\}}"></span> is the set of <a href="/wiki/Binary_digit" class="mw-redirect" title="Binary digit">binary digits</a>, the formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in B\;x=x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>B</mi> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in B\;x=x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/897cb0bbd8aaeef361260bc701be6c087e23e39b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.684ex; height:2.676ex;" alt="{\displaystyle \forall x\in B\;x=x^{2}}"></span> abbreviates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=0^{2}\land 1=1^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∧</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=0^{2}\land 1=1^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/213aa12aef76d4b403166a3fa68715db7e292216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.538ex; height:2.676ex;" alt="{\displaystyle 0=0^{2}\land 1=1^{2}}"></span>, which evaluates to <i>true</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Infinite_domain_of_discourse">Infinite domain of discourse</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantifier_(logic)&action=edit&section=2" title="Edit section: Infinite domain of discourse"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the following statement (<i>using dot notation for multiplication</i>): </p> <dl><dd>1 · 2 = 1 + 1, and 2 · 2 = 2 + 2, and 3 · 2 = 3 + 3, ..., and 100 · 2 = 100 + 100, and ..., etc.</dd></dl> <p>This has the appearance of an <i>infinite <a href="/wiki/Logical_conjunction" title="Logical conjunction">conjunction</a></i> of propositions. From the point of view of <a href="/wiki/Formal_language" title="Formal language">formal languages</a>, this is immediately a problem, since <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a> rules are expected to generate <a href="/wiki/Finite_set" title="Finite set">finite</a> words. </p><p>The example above is fortunate in that there is a <a href="/wiki/Algorithm" title="Algorithm">procedure</a> to generate all the conjuncts. However, if an assertion were to be made about every <a href="/wiki/Irrational_number" title="Irrational number">irrational number</a>, there would be no way to enumerate all the conjuncts, since irrationals cannot be enumerated. A succinct, equivalent formulation which avoids these problems uses <i>universal quantification</i>: </p> <dl><dd>For each <a href="/wiki/Natural_number" title="Natural number">natural number</a> <i>n</i>, <i>n</i> · 2 = <i>n</i> + <i>n</i>.</dd></dl> <p>A similar analysis applies to the <a href="/wiki/Disjunction_(logic)" class="mw-redirect" title="Disjunction (logic)">disjunction</a>, </p> <dl><dd>1 is equal to 5 + 5, or 2 is equal to 5 + 5, or 3 is equal to 5 + 5, ... , or 100 is equal to 5 + 5, or ..., etc.</dd></dl> <p>which can be rephrased using <i>existential quantification</i>: </p> <dl><dd>For some <a href="/wiki/Natural_number" title="Natural number">natural number</a> <i>n</i>, <i>n</i> is equal to 5+5.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Algebraic_approaches_to_quantification">Algebraic approaches to quantification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantifier_(logic)&action=edit&section=3" title="Edit section: Algebraic approaches to quantification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is possible to devise <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebras</a> whose <a href="/wiki/Model_theory" title="Model theory">models</a> include <a href="/wiki/Formal_language" title="Formal language">formal languages</a> with quantification, but progress has been slow<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="The text near this tag may need clarification or removal of jargon. (October 2016)">clarification needed</span></a></i>]</sup> and interest in such algebra has been limited. Three approaches have been devised to date: </p> <ul><li><a href="/wiki/Relation_algebra" title="Relation algebra">Relation algebra</a>, invented by <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a>, and developed by <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a>, <a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Ernst Schröder</a>, <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>, and Tarski's students. Relation algebra cannot represent any formula with quantifiers nested more than three deep. Surprisingly, the models of relation algebra include the <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a> <a href="/wiki/ZFC" class="mw-redirect" title="ZFC">ZFC</a> and <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>;</li> <li><a href="/wiki/Cylindric_algebra" title="Cylindric algebra">Cylindric algebra</a>, devised by <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>, <a href="/wiki/Leon_Henkin" title="Leon Henkin">Leon Henkin</a>, and others;</li> <li>The <a href="/wiki/Polyadic_algebra" title="Polyadic algebra">polyadic algebra</a> of <a href="/wiki/Paul_Halmos" title="Paul Halmos">Paul Halmos</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantifier_(logic)&action=edit&section=4" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The two most common quantifiers are the universal quantifier and the existential quantifier. The traditional symbol for the universal quantifier is "<a href="/wiki/%E2%88%80" class="mw-redirect" title="∀">∀</a>", a rotated letter "<a href="/wiki/A" title="A">A</a>", which stands for "for all" or "all". The corresponding symbol for the existential quantifier is "<a href="/wiki/%E2%88%83" class="mw-redirect" title="∃">∃</a>", a rotated letter "<a href="/wiki/E" title="E">E</a>", which stands for "there exists" or "exists".<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><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>An example of translating a quantified statement in a natural language such as English would be as follows. Given the statement, "Each of Peter's friends either likes to dance or likes to go to the beach (or both)", key aspects can be identified and rewritten using symbols including quantifiers. So, let <i>X</i> be the set of all Peter's friends, <i>P</i>(<i>x</i>) the <a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">predicate</a> "<i>x</i> likes to dance", and <i>Q</i>(<i>x</i>) the predicate "<i>x</i> likes to go to the beach". Then the above sentence can be written in formal notation as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall {x}{\in }X,(P(x)\lor Q(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∈</mo> </mrow> <mi>X</mi> <mo>,</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall {x}{\in }X,(P(x)\lor Q(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/220e36039ee73749e6b45c467775f26ee7a1386e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.44ex; height:2.843ex;" alt="{\displaystyle \forall {x}{\in }X,(P(x)\lor Q(x))}"></span>, which is read, "for every <i>x</i> that is a member of <i>X</i>, <i>P</i> applies to <i>x</i> <a href="/wiki/Inclusive_or" class="mw-redirect" title="Inclusive or">or</a> <i>Q</i> applies to <i>x</i>". </p><p>Some other quantified expressions are constructed as follows, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists {x}\,P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mspace width="thinmathspace" /> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists {x}\,P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b227c46ed5c44401a56548d410afa22009209231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.755ex; height:2.176ex;" alt="{\displaystyle \exists {x}\,P}"></span><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>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall {x}\,P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mspace width="thinmathspace" /> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall {x}\,P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d6fc1e1ac7362750e5b5d09f67d44f4eccd204c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.755ex; height:2.176ex;" alt="{\displaystyle \forall {x}\,P}"></span></dd></dl> <p>for a formula <i>P</i>. These two expressions (using the definitions above) are read as "there exists a friend of Peter who likes to dance" and "all friends of Peter like to dance", respectively. Variant notations include, for set <i>X</i> and set members <i>x</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigvee _{x}P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </munder> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigvee _{x}P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535cab302719f9699a629ed04643251f5ca240cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:4.714ex; height:5.509ex;" alt="{\displaystyle \bigvee _{x}P}"></span>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\exists {x})P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo stretchy="false">)</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\exists {x})P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d02413093ef269af0bbeb4a7904101c6c57bd32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.177ex; height:2.843ex;" alt="{\displaystyle (\exists {x})P}"></span><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>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\exists x\ .\ P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mtext> </mtext> <mo>.</mo> <mtext> </mtext> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\exists x\ .\ P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a98f239b7ef34ff8bd6edbcb2bd2fe9f6fba5383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.372ex; height:2.843ex;" alt="{\displaystyle (\exists x\ .\ P)}"></span>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x\ \cdot \ P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mtext> </mtext> <mo>⋅</mo> <mtext> </mtext> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\ \cdot \ P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f02646fa8fd1404511d35080af16071c84291ac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.208ex; height:2.176ex;" alt="{\displaystyle \exists x\ \cdot \ P}"></span>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\exists x:P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo>:</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\exists x:P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab23b47a4c97536f6249e91ffd13b23e6c3f22e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.114ex; height:2.843ex;" alt="{\displaystyle (\exists x:P)}"></span>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists {x}(P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists {x}(P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97d483ed1b63980df7352507a306afed5a0d8ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.177ex; height:2.843ex;" alt="{\displaystyle \exists {x}(P)}"></span><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>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists _{x}\,P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∃</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists _{x}\,P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7f75779ccc84a5bee61d4b3ba49ae00f5cc3f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.598ex; height:2.509ex;" alt="{\displaystyle \exists _{x}\,P}"></span>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists {x}{,}\,P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mspace width="thinmathspace" /> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists {x}{,}\,P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/635ed14eb6168f36ea87716cf7c4e01be896127d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.402ex; height:2.509ex;" alt="{\displaystyle \exists {x}{,}\,P}"></span>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists {x}{\in }X\,P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∈</mo> </mrow> <mi>X</mi> <mspace width="thinmathspace" /> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists {x}{\in }X\,P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef5701065bab130e781cf02fffd4195fd8be44e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.285ex; height:2.176ex;" alt="{\displaystyle \exists {x}{\in }X\,P}"></span>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists \,x{:}X\,P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mspace width="thinmathspace" /> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>:</mo> </mrow> <mi>X</mi> <mspace width="thinmathspace" /> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists \,x{:}X\,P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/269db1d6ab264492bac58e4ad8629ed139080c6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.769ex; height:2.176ex;" alt="{\displaystyle \exists \,x{:}X\,P}"></span></dd></dl> <p>All of these variations also apply to universal quantification. Other variations for the universal quantifier are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge _{x}P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </munder> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge _{x}P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b888a93297af78548faf3b448aedf5445c9413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:4.714ex; height:5.509ex;" alt="{\displaystyle \bigwedge _{x}P}"></span><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Give an example citation for each notation (January 2021)">citation needed</span></a></i>]</sup>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge xP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋀</mo> <mi>x</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge xP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df2d86df96603a23288fd687bc642ae2f1b73fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.044ex; height:3.843ex;" alt="{\displaystyle \bigwedge xP}"></span><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>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x)\,P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)\,P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8d6562b4b4361a4a10a6559f69e3b8c6dde19e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.272ex; height:2.843ex;" alt="{\displaystyle (x)\,P}"></span><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></dd></dl> <p>Some versions of the notation explicitly mention the range of quantification. The range of quantification must always be specified; for a given mathematical theory, this can be done in several ways: </p> <ul><li>Assume a fixed domain of discourse for every quantification, as is done in <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>,</li> <li>Fix several domains of discourse in advance and require that each variable have a declared domain, which is the <i>type</i> of that variable. This is analogous to the situation in <a href="/wiki/Type_system" title="Type system">statically typed</a> <a href="/wiki/Computer_programming" title="Computer programming">computer programming</a> languages, where variables have declared types.</li> <li>Mention explicitly the range of quantification, perhaps using a symbol for the set of all objects in that domain (or the <a href="/wiki/Type_(type_theory)" class="mw-redirect" title="Type (type theory)">type</a> of the objects in that domain).</li></ul> <p>One can use any variable as a quantified variable in place of any other, under certain restrictions in which <i>variable capture</i> does not occur. Even if the notation uses typed variables, variables of that type may be used. </p><p>Informally or in natural language, the "∀<i>x</i>" or "∃<i>x</i>" might appear after or in the middle of <i>P</i>(<i>x</i>). Formally, however, the phrase that introduces the dummy variable is placed in front. </p><p>Mathematical formulas mix symbolic expressions for quantifiers with natural language quantifiers such as, </p> <dl><dd>For every natural number <i>x</i>, ...</dd> <dd>There exists an <i>x</i> such that ...</dd> <dd>For at least one <i>x, ....</i></dd></dl> <p>Keywords for <a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">uniqueness quantification</a> include: </p> <dl><dd>For exactly one natural number <i>x</i>, ...</dd> <dd>There is one and only one <i>x</i> such that ....</dd></dl> <p>Further, <i>x</i> may be replaced by a <a href="/wiki/Pronoun" title="Pronoun">pronoun</a>. For example, </p> <dl><dd>For every natural number, its product with 2 equals to its sum with itself.</dd> <dd>Some natural number is prime.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Order_of_quantifiers_(nesting)"><span id="Order_of_quantifiers_.28nesting.29"></span>Order of quantifiers (nesting)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantifier_(logic)&action=edit&section=5" title="Edit section: Order of quantifiers (nesting)"><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">See also: <a href="/wiki/Quantifier_shift" title="Quantifier shift">Quantifier shift</a></div> <p>The order of quantifiers is critical to meaning, as is illustrated by the following two propositions: </p> <dl><dd>For every natural number <i>n</i>, there exists a natural number <i>s</i> such that <i>s</i> = <i>n</i><sup>2</sup>.</dd></dl> <p>This is clearly true; it just asserts that every natural number has a square. The meaning of the assertion in which the order of quantifiers is reversed is different: </p> <dl><dd>There exists a natural number <i>s</i> such that for every natural number <i>n</i>, <i>s</i> = <i>n</i><sup>2</sup>.</dd></dl> <p>This is clearly false; it asserts that there is a single natural number <i>s</i> that is the square of <i>every</i> natural number. This is because the syntax directs that any variable cannot be a function of subsequently introduced variables. </p><p>A less trivial example from <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> regards the concepts of <a href="/wiki/Uniform_continuity" title="Uniform continuity">uniform</a> and <a href="/wiki/Continuous_function" title="Continuous function">pointwise</a> continuity, whose definitions differ only by an exchange in the positions of two quantifiers. A function <i>f</i> from <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers"><b>R</b></a> to <b>R</b> is called </p> <ul><li>Pointwise continuous if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \varepsilon >0\;\forall x\in \mathbb {R} \;\exists \delta >0\;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mi>δ</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>h</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ</mi> <mspace width="thinmathspace" /> <mo stretchy="false">⇒</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0\;\forall x\in \mathbb {R} \;\exists \delta >0\;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/069da946a9a722095ed95a1fd05c3944459ce86e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.419ex; height:2.843ex;" alt="{\displaystyle \forall \varepsilon >0\;\forall x\in \mathbb {R} \;\exists \delta >0\;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )}"></span></li> <li>Uniformly continuous if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in \mathbb {R} \;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mi>δ</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>h</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ</mi> <mspace width="thinmathspace" /> <mo stretchy="false">⇒</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in \mathbb {R} \;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/764c59627e05f792419978bb01dd62645a886767" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.419ex; height:2.843ex;" alt="{\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in \mathbb {R} \;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )}"></span></li></ul> <p>In the former case, the particular value chosen for <i>δ</i> can be a function of both <i>ε</i> and <i>x</i>, the variables that precede it. In the latter case, <i>δ</i> can be a function only of <i>ε</i> (i.e., it has to be chosen independent of <i>x</i>). For example, <i>f</i>(<i>x</i>) = <i>x</i><sup>2</sup> satisfies pointwise, but not uniform continuity (its slope is unbound). In contrast, interchanging the two initial universal quantifiers in the definition of pointwise continuity does not change the meaning. </p><p>As a general rule, swapping two adjacent universal quantifiers with the same <a href="/wiki/Scope_(logic)" title="Scope (logic)">scope</a> (or swapping two adjacent existential quantifiers with the same scope) doesn't change the meaning of the formula (see <a href="/wiki/Prenex_normal_form#Example" title="Prenex normal form">Example here</a>), but swapping an existential quantifier and an adjacent universal quantifier may change its meaning. </p><p>The maximum depth of nesting of quantifiers in a formula is called its "<a href="/wiki/Quantifier_rank" title="Quantifier rank">quantifier rank</a>". </p> <div class="mw-heading mw-heading2"><h2 id="Equivalent_expressions">Equivalent expressions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantifier_(logic)&action=edit&section=6" title="Edit section: Equivalent expressions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>D</i> is a domain of <i>x</i> and <i>P</i>(<i>x</i>) is a predicate dependent on object variable <i>x</i>, then the universal proposition can be expressed as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\!\in \!D\;P(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="negativethinmathspace" /> <mo>∈</mo> <mspace width="negativethinmathspace" /> <mi>D</mi> <mspace width="thickmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\!\in \!D\;P(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4545696d0811ab5ec0802f0b3f18b6c3679a9b0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.789ex; height:2.843ex;" alt="{\displaystyle \forall x\!\in \!D\;P(x).}"></span></dd></dl> <p>This notation is known as restricted or relativized or <a href="/wiki/Bounded_quantifier" title="Bounded quantifier">bounded quantification</a>. Equivalently one can write, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\;(x\!\in \!D\to P(x)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>x</mi> <mspace width="negativethinmathspace" /> <mo>∈</mo> <mspace width="negativethinmathspace" /> <mi>D</mi> <mo stretchy="false">→</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\;(x\!\in \!D\to P(x)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6e3b976de52bd3a23f5777b79b481376d8f719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.542ex; height:2.843ex;" alt="{\displaystyle \forall x\;(x\!\in \!D\to P(x)).}"></span></dd></dl> <p>The existential proposition can be expressed with bounded quantification as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x\!\in \!D\;P(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="negativethinmathspace" /> <mo>∈</mo> <mspace width="negativethinmathspace" /> <mi>D</mi> <mspace width="thickmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\!\in \!D\;P(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0dc37edf253087f70dbaedd99bdfae103de164f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.789ex; height:2.843ex;" alt="{\displaystyle \exists x\!\in \!D\;P(x),}"></span></dd></dl> <p>or equivalently </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x\;(x\!\in \!\!D\land P(x)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>x</mi> <mspace width="negativethinmathspace" /> <mo>∈</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mi>D</mi> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\;(x\!\in \!\!D\land P(x)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd90ca2dcc913d32faed506ba05b781dfab6381d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.124ex; height:2.843ex;" alt="{\displaystyle \exists x\;(x\!\in \!\!D\land P(x)).}"></span></dd></dl> <p>Together with negation, only one of either the universal or existential quantifier is needed to perform both tasks: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (\forall x\!\in \!D\;P(x))\equiv \exists x\!\in \!D\;\neg P(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="negativethinmathspace" /> <mo>∈</mo> <mspace width="negativethinmathspace" /> <mi>D</mi> <mspace width="thickmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≡</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="negativethinmathspace" /> <mo>∈</mo> <mspace width="negativethinmathspace" /> <mi>D</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (\forall x\!\in \!D\;P(x))\equiv \exists x\!\in \!D\;\neg P(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ba0f046027ab4a7bdea9274d9f47f68d9fae5d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.94ex; height:2.843ex;" alt="{\displaystyle \neg (\forall x\!\in \!D\;P(x))\equiv \exists x\!\in \!D\;\neg P(x),}"></span></dd></dl> <p>which shows that to disprove a "for all <i>x</i>" proposition, one needs no more than to find an <i>x</i> for which the predicate is false. Similarly, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (\exists x\!\in \!D\;P(x))\equiv \forall x\!\in \!D\;\neg P(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="negativethinmathspace" /> <mo>∈</mo> <mspace width="negativethinmathspace" /> <mi>D</mi> <mspace width="thickmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≡</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="negativethinmathspace" /> <mo>∈</mo> <mspace width="negativethinmathspace" /> <mi>D</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (\exists x\!\in \!D\;P(x))\equiv \forall x\!\in \!D\;\neg P(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe522dea2eee3b7bd005ee19a1012f4e5f836c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.94ex; height:2.843ex;" alt="{\displaystyle \neg (\exists x\!\in \!D\;P(x))\equiv \forall x\!\in \!D\;\neg P(x),}"></span></dd></dl> <p>to disprove a "there exists an <i>x</i>" proposition, one needs to show that the predicate is false for all <i>x</i>. </p><p>In <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>, every formula is <a href="/wiki/Logically_equivalent" class="mw-redirect" title="Logically equivalent">logically equivalent</a> to a formula in <a href="/wiki/Prenex_normal_form" title="Prenex normal form">prenex normal form</a>, that is, a string of quantifiers and bound variables followed by a quantifier-free formula. </p> <div class="mw-heading mw-heading3"><h3 id="Quantifier_elimination">Quantifier elimination</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantifier_(logic)&action=edit&section=7" title="Edit section: Quantifier elimination"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Quantifier_elimination" title="Quantifier elimination">Quantifier elimination</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Quantifier_elimination&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <p><a href="/wiki/Quantifier_elimination" title="Quantifier elimination">Quantifier elimination</a> is a concept of simplification used in <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, <a href="/wiki/Model_theory" title="Model theory">model theory</a>, and <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a>. Informally, a quantified statement "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab833914405cde960b3b9af3feaa9e4fef96ffa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.622ex; height:2.176ex;" alt="{\displaystyle \exists x}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8619532e44ee1ccae3ab03405a6885260d09ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="{\displaystyle \ldots }"></span>" can be viewed as a question "When is there an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8619532e44ee1ccae3ab03405a6885260d09ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="{\displaystyle \ldots }"></span>?", and the statement without quantifiers can be viewed as the answer to that question.<sup id="cite_ref-FOOTNOTEBrown2002_8-0" class="reference"><a href="#cite_note-FOOTNOTEBrown2002-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>One way of classifying <a href="/wiki/Well-formed_formula" title="Well-formed formula">formulas</a> is by the amount of quantification. Formulas with less <a class="mw-selflink-fragment" href="#Nesting">depth of quantifier alternation</a> are thought of as being simpler, with the quantifier-free formulas as the simplest. </p> A <a href="/wiki/Logical_theory" class="mw-redirect" title="Logical theory">theory</a> has quantifier elimination if for every formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, there exists another formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{QF}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> <mi>F</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{QF}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8e52ee8fb776f7c919339cfc8eab2006b3f1e6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.251ex; height:2.343ex;" alt="{\displaystyle \alpha _{QF}}"></span> without quantifiers that is <a href="/wiki/Logical_equivalence" title="Logical equivalence">equivalent</a> to it (<a href="/wiki/Modulo_(jargon)" class="mw-redirect" title="Modulo (jargon)">modulo</a> this theory).</div></div> <div class="mw-heading mw-heading2"><h2 id="Range_of_quantification">Range of quantification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantifier_(logic)&action=edit&section=8" title="Edit section: Range of quantification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every quantification involves one specific variable and a <a href="/wiki/Domain_of_discourse" title="Domain of discourse">domain of discourse</a> or <b>range of quantification</b> of that variable. The range of quantification specifies the set of values that the variable takes. In the examples above, the range of quantification is the set of natural numbers. Specification of the range of quantification allows us to express the difference between, say, asserting that a predicate holds for some natural number or for some <a href="/wiki/Real_number" title="Real number">real number</a>. Expository conventions often reserve some variable names such as "<i>n</i>" for natural numbers, and "<i>x</i>" for real numbers, although relying exclusively on naming conventions cannot work in general, since ranges of variables can change in the course of a mathematical argument. </p><p>A universally quantified formula over an empty range (like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\!\in \!\varnothing \;x\neq x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="negativethinmathspace" /> <mo>∈</mo> <mspace width="negativethinmathspace" /> <mi class="MJX-variant">∅</mi> <mspace width="thickmathspace" /> <mi>x</mi> <mo>≠</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\!\in \!\varnothing \;x\neq x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7280e13b110aea8d78161a8d45cac5fdae5d9ea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.9ex; height:2.676ex;" alt="{\displaystyle \forall x\!\in \!\varnothing \;x\neq x}"></span>) is always <a href="/wiki/Vacuously_true" class="mw-redirect" title="Vacuously true">vacuously true</a>. Conversely, an existentially quantified formula over an empty range (like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x\!\in \!\varnothing \;x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="negativethinmathspace" /> <mo>∈</mo> <mspace width="negativethinmathspace" /> <mi class="MJX-variant">∅</mi> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\!\in \!\varnothing \;x=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54f354c5d11b17614d40c003e3c9583b83376577" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.9ex; height:2.176ex;" alt="{\displaystyle \exists x\!\in \!\varnothing \;x=x}"></span>) is always false. </p><p>A more natural way to restrict the domain of discourse uses <i>guarded quantification</i>. For example, the guarded quantification </p> <dl><dd>For some natural number <i>n</i>, <i>n</i> is even and <i>n</i> is prime</dd></dl> <p>means </p> <dl><dd>For some <a href="/wiki/Even_number" class="mw-redirect" title="Even number">even number</a> <i>n</i>, <i>n</i> is prime.</dd></dl> <p>In some <a href="/wiki/Mathematical_theory" class="mw-redirect" title="Mathematical theory">mathematical theories</a>, a single domain of discourse fixed in advance is assumed. For example, in <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>, variables range over all sets. In this case, guarded quantifiers can be used to mimic a smaller range of quantification. Thus in the example above, to express </p> <dl><dd>For every natural number <i>n</i>, <i>n</i>·2 = <i>n</i> + <i>n</i></dd></dl> <p>in Zermelo–Fraenkel set theory, one would write </p> <dl><dd>For every <i>n</i>, if <i>n</i> belongs to <b>N</b>, then <i>n</i>·2 = <i>n</i> + <i>n</i>,</dd></dl> <p>where <b>N</b> is the set of all natural numbers. </p> <div class="mw-heading mw-heading2"><h2 id="Formal_semantics">Formal semantics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantifier_(logic)&action=edit&section=9" title="Edit section: Formal semantics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematical semantics is the application of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> to study the meaning of expressions in a formal language. It has three elements: a mathematical specification of a class of objects via <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a>, a mathematical specification of various <a href="/wiki/Semantic_domain" title="Semantic domain">semantic domains</a> and the relation between the two, which is usually expressed as a function from syntactic objects to semantic ones. This article only addresses the issue of how quantifier elements are interpreted. The syntax of a formula can be given by a syntax tree. A quantifier has a <a href="/wiki/Scope_(logic)" title="Scope (logic)">scope</a>, and an occurrence of a variable <i>x</i> is <a href="/wiki/Free_variable" class="mw-redirect" title="Free variable">free</a> if it is not within the scope of a quantification for that variable. Thus in </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x(\exists yB(x,y))\vee C(y,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mi>y</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x(\exists yB(x,y))\vee C(y,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/136f694d97fb7a22d3999e94dc8f3b5b525245ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.649ex; height:2.843ex;" alt="{\displaystyle \forall x(\exists yB(x,y))\vee C(y,x)}"></span></dd></dl> <p>the occurrence of both <i>x</i> and <i>y</i> in <i>C</i>(<i>y</i>, <i>x</i>) is free, while the occurrence of <i>x</i> and <i>y</i> in <i>B</i>(<i>y</i>, <i>x</i>) is bound (i.e. non-free). </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:QuantifierScopes_svg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/QuantifierScopes_svg.svg/350px-QuantifierScopes_svg.svg.png" decoding="async" width="350" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/QuantifierScopes_svg.svg/525px-QuantifierScopes_svg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/QuantifierScopes_svg.svg/700px-QuantifierScopes_svg.svg.png 2x" data-file-width="407" data-file-height="230" /></a><figcaption>Syntax tree of the formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x(\exists yB(x,y))\vee C(y,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mi>y</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x(\exists yB(x,y))\vee C(y,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/136f694d97fb7a22d3999e94dc8f3b5b525245ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.649ex; height:2.843ex;" alt="{\displaystyle \forall x(\exists yB(x,y))\vee C(y,x)}"></span>, illustrating scope and variable capture. Bound and free variable occurrences are colored in red and green, respectively.</figcaption></figure> <p>An <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">interpretation</a> for <a href="/wiki/First-order_predicate_calculus" class="mw-redirect" title="First-order predicate calculus">first-order predicate calculus</a> assumes as given a domain of individuals <i>X</i>. A formula <i>A</i> whose free variables are <i>x</i><sub>1</sub>, ..., <i>x</i><sub>n</sub> is interpreted as a <a href="/wiki/Boolean_function" title="Boolean function">Boolean</a>-valued function <i>F</i>(<i>v</i><sub>1</sub>, ..., <i>v</i><sub><i>n</i></sub>) of <i>n</i> arguments, where each argument ranges over the domain <i>X</i>. Boolean-valued means that the function assumes one of the values <b>T</b> (interpreted as truth) or <b>F</b> (interpreted as falsehood). The interpretation of the formula </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x_{n}A(x_{1},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>A</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x_{n}A(x_{1},\ldots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb8446639a2a6dc8de5abbba022a7e8a100f105" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.503ex; height:2.843ex;" alt="{\displaystyle \forall x_{n}A(x_{1},\ldots ,x_{n})}"></span></dd></dl> <p>is the function <i>G</i> of <i>n</i>-1 arguments such that <i>G</i>(<i>v</i><sub>1</sub>, ..., <i>v</i><sub><i>n</i>-1</sub>) = <b>T</b> if and only if <i>F</i>(<i>v</i><sub>1</sub>, ..., <i>v</i><sub><i>n</i>-1</sub>, <i>w</i>) = <b>T</b> for every <i>w</i> in <i>X</i>. If <i>F</i>(<i>v</i><sub>1</sub>, ..., <i>v</i><sub><i>n</i>-1</sub>, <i>w</i>) = <b>F</b> for at least one value of <i>w</i>, then <i>G</i>(<i>v</i><sub>1</sub>, ..., <i>v</i><sub><i>n</i>-1</sub>) = <b>F</b>. Similarly the interpretation of the formula </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x_{n}A(x_{1},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>A</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x_{n}A(x_{1},\ldots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8dd6667a3d1825e8e5d474ecb95e9a358d22ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.503ex; height:2.843ex;" alt="{\displaystyle \exists x_{n}A(x_{1},\ldots ,x_{n})}"></span></dd></dl> <p>is the function <i>H</i> of <i>n</i>-1 arguments such that <i>H</i>(<i>v</i><sub>1</sub>, ..., <i>v</i><sub><i>n</i>-1</sub>) = <b>T</b> if and only if <i>F</i>(<i>v</i><sub>1</sub>, ..., <i>v</i><sub><i>n</i>-1</sub>, <i>w</i>) = <b>T</b> for at least one <i>w</i> and <i>H</i>(<i>v</i><sub>1</sub>, ..., <i>v</i><sub><i>n</i>-1</sub>) = <b>F</b> otherwise. </p><p>The semantics for <a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">uniqueness quantification</a> requires first-order predicate calculus with equality. This means there is given a distinguished two-placed predicate "="; the semantics is also modified accordingly so that "=" is always interpreted as the two-place equality relation on <i>X</i>. The interpretation of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists !x_{n}A(x_{1},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mo>!</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>A</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists !x_{n}A(x_{1},\ldots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b15b5c2fe62a6b15ea712099a5d1b0e0b8ea125c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.15ex; height:2.843ex;" alt="{\displaystyle \exists !x_{n}A(x_{1},\ldots ,x_{n})}"></span></dd></dl> <p>then is the function of <i>n</i>-1 arguments, which is the logical <i>and</i> of the interpretations of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x_{n}A(x_{1},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>A</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x_{n}A(x_{1},\ldots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8dd6667a3d1825e8e5d474ecb95e9a358d22ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.503ex; height:2.843ex;" alt="{\displaystyle \exists x_{n}A(x_{1},\ldots ,x_{n})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall y,z{\big (}A(x_{1},\ldots ,x_{n-1},y)\wedge A(x_{1},\ldots ,x_{n-1},z)\implies y=z{\big )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>A</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>A</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>y</mi> <mo>=</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall y,z{\big (}A(x_{1},\ldots ,x_{n-1},y)\wedge A(x_{1},\ldots ,x_{n-1},z)\implies y=z{\big )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5628c521d35488d5f1e8d8ab62ea3ab4dfde78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:57.496ex; height:3.176ex;" alt="{\displaystyle \forall y,z{\big (}A(x_{1},\ldots ,x_{n-1},y)\wedge A(x_{1},\ldots ,x_{n-1},z)\implies y=z{\big )}.}"></span></dd></dl> <p>Each kind of quantification defines a corresponding <a href="/wiki/Closure_operator" title="Closure operator">closure operator</a> on the set of formulas, by adding, for each free variable <i>x</i>, a quantifier to bind <i>x</i>.<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> For example, the <i>existential closure</i> of the <a href="/wiki/Open_formula" title="Open formula">open formula</a> <i>n</i>>2 ∧ <i>x</i><sup><i>n</i></sup>+<i>y</i><sup><i>n</i></sup>=<i>z</i><sup><i>n</i></sup> is the closed formula ∃<i>n</i> ∃<i>x</i> ∃<i>y</i> ∃<i>z</i> (<i>n</i>>2 ∧ <i>x</i><sup><i>n</i></sup>+<i>y</i><sup><i>n</i></sup>=<i>z</i><sup><i>n</i></sup>); the latter formula, when interpreted over the positive integers, is known to be false by <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>. As another example, equational axioms, like <i>x</i>+<i>y</i>=<i>y</i>+<i>x</i>, are usually meant to denote their <i>universal closure</i>, like ∀<i>x</i> ∀<i>y</i> (<i>x</i>+<i>y</i>=<i>y</i>+<i>x</i>) to express <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutativity</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Paucal,_multal_and_other_degree_quantifiers"><span id="Paucal.2C_multal_and_other_degree_quantifiers"></span>Paucal, multal and other degree quantifiers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantifier_(logic)&action=edit&section=10" title="Edit section: Paucal, multal and other degree quantifiers"><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">See also: <a href="/wiki/Fubini%27s_theorem" title="Fubini's theorem">Fubini's theorem</a> and <a href="/wiki/Measurable" class="mw-redirect" title="Measurable">measurable</a></div> <p>None of the quantifiers previously discussed apply to a quantification such as </p> <dl><dd>There are many integers <i>n</i> < 100, such that <i>n</i> is divisible by 2 or 3 or 5.</dd></dl> <p>One possible interpretation mechanism can be obtained as follows: Suppose that in addition to a semantic domain <i>X</i>, we have given a <a href="/wiki/Probability_measure" title="Probability measure">probability measure</a> P defined on <i>X</i> and cutoff numbers 0 < <i>a</i> ≤ <i>b</i> ≤ 1. If <i>A</i> is a formula with free variables <i>x</i><sub>1</sub>,...,<i>x</i><sub><i>n</i></sub> whose interpretation is the function <i>F</i> of variables <i>v</i><sub>1</sub>,...,<i>v</i><sub><i>n</i></sub> then the interpretation of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists ^{\mathrm {many} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∃</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">y</mi> </mrow> </mrow> </msup> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>A</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists ^{\mathrm {many} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c47fb8ee2146d01a5cf91d4e1cc4bf327f0a7883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.391ex; height:2.843ex;" alt="{\displaystyle \exists ^{\mathrm {many} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})}"></span></dd></dl> <p>is the function of <i>v</i><sub>1</sub>,...,<i>v</i><sub><i>n</i>-1</sub> which is <b>T</b> if and only if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\geq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mi>w</mi> <mo>:</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>≥</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\geq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/789b8d8d984c0f8d564b694213500b16ed00448d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.618ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\geq b}"></span></dd></dl> <p>and <b>F</b> otherwise. Similarly, the interpretation of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists ^{\mathrm {few} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∃</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">w</mi> </mrow> </mrow> </msup> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>A</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists ^{\mathrm {few} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed17b8efefe0ecd2b8bbee21baf4567a37d1247b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.947ex; height:3.176ex;" alt="{\displaystyle \exists ^{\mathrm {few} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})}"></span></dd></dl> <p>is the function of <i>v</i><sub>1</sub>,...,<i>v</i><sub><i>n</i>-1</sub> which is <b>F</b> if and only if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\leq a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mi>w</mi> <mo>:</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>≤</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\leq a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/695f8bf544096a784aba3821611df0ef29fc573b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.111ex; height:2.843ex;" alt="{\displaystyle 0<\operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\leq a}"></span></dd></dl> <p>and <b>T</b> otherwise. </p> <div class="mw-heading mw-heading2"><h2 id="Other_quantifiers">Other quantifiers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantifier_(logic)&action=edit&section=11" title="Edit section: Other quantifiers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A few other quantifiers have been proposed over time. In particular, the solution quantifier,<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><sup class="reference nowrap"><span title="Page / location: 28">: 28 </span></sup> noted § (<a href="/wiki/Section_sign" title="Section sign">section sign</a>) and read "those". For example, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\S n\in \mathbb {N} \quad n^{2}\leq 4\right]=\{0,1,2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mi mathvariant="normal">§</mi> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="1em" /> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mn>4</mn> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[\S n\in \mathbb {N} \quad n^{2}\leq 4\right]=\{0,1,2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d23c1dea39d57915c78be8af2410258de45cccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.536ex; height:4.843ex;" alt="{\displaystyle \left[\S n\in \mathbb {N} \quad n^{2}\leq 4\right]=\{0,1,2\}}"></span></dd></dl> <p>is read "those <i>n</i> in <b>N</b> such that <i>n</i><sup>2</sup> ≤ 4 are in {0,1,2}." The same construct is expressible in <a href="/wiki/Set-builder_notation" title="Set-builder notation">set-builder notation</a> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{n\in \mathbb {N} :n^{2}\leq 4\}=\{0,1,2\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>:</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{n\in \mathbb {N} :n^{2}\leq 4\}=\{0,1,2\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f81f13cc32bd091ad3d2cf4bf24f8385998346" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.511ex; height:3.176ex;" alt="{\displaystyle \{n\in \mathbb {N} :n^{2}\leq 4\}=\{0,1,2\}.}"></span></dd></dl> <p>Contrary to the other quantifiers, § yields a set rather than a formula.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Some other quantifiers sometimes used in mathematics include: </p> <ul><li>There are infinitely many elements such that...</li> <li>For all but finitely many elements... (sometimes expressed as "for <a href="/wiki/Almost_all" title="Almost all">almost all</a> elements...").</li> <li>There are uncountably many elements such that...</li> <li>For all but countably many elements...</li> <li>For all elements in a set of positive measure...</li> <li>For all elements except those in a set of measure zero...</li></ul> <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=Quantifier_(logic)&action=edit&section=12" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Term_logic" title="Term logic">Term logic</a>, also called Aristotelian logic, treats quantification in a manner that is closer to natural language, and also less suited to formal analysis. Term logic treated <i>All</i>, <i>Some</i> and <i>No</i> in the 4th century BC, in an account also touching on the <a href="/wiki/Alethic_modalities" class="mw-redirect" title="Alethic modalities">alethic modalities</a>. </p><p>In 1827, <a href="/wiki/George_Bentham" title="George Bentham">George Bentham</a> published his <i>Outline of a New System of Logic: With a Critical Examination of Dr. Whately's Elements of Logic</i>, describing the principle of the quantifier, but the book was not widely circulated.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:De_Morgan_Augustus.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/De_Morgan_Augustus.jpg/170px-De_Morgan_Augustus.jpg" decoding="async" width="170" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/De_Morgan_Augustus.jpg/255px-De_Morgan_Augustus.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/De_Morgan_Augustus.jpg/340px-De_Morgan_Augustus.jpg 2x" data-file-width="958" data-file-height="1182" /></a><figcaption><a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a> (1806–1871) was the first to use "quantifier" in the modern sense.</figcaption></figure> <p><a href="/wiki/Sir_William_Hamilton,_9th_Baronet" title="Sir William Hamilton, 9th Baronet">William Hamilton</a> claimed to have coined the terms "quantify" and "quantification", most likely in his Edinburgh lectures c. 1840. <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a> confirmed this in 1847, but modern usage began with De Morgan in 1862 where he makes statements such as "We are to take in both <i>all</i> and <i>some-not-all</i> as quantifiers".<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>, in his 1879 <i><a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a></i>, was the first to employ a quantifier to bind a variable ranging over a <a href="/wiki/Domain_of_discourse" title="Domain of discourse">domain of discourse</a> and appearing in <a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">predicates</a>. He would universally quantify a variable (or relation) by writing the variable over a dimple in an otherwise straight line appearing in his diagrammatic formulas. Frege did not devise an explicit notation for existential quantification, instead employing his equivalent of ~∀<i>x</i>~, or <a href="/wiki/Contraposition" title="Contraposition">contraposition</a>. Frege's treatment of quantification went largely unremarked until <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>'s 1903 <i>Principles of Mathematics</i>. </p><p>In work that culminated in Peirce (1885), <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a> and his student <a href="/w/index.php?title=Oscar_Howard_Mitchell&action=edit&redlink=1" class="new" title="Oscar Howard Mitchell (page does not exist)">Oscar Howard Mitchell</a> independently invented universal and existential quantifiers, and <a href="/wiki/Bound_variable" class="mw-redirect" title="Bound variable">bound variables</a>. Peirce and Mitchell wrote Π<sub>x</sub> and Σ<sub>x</sub> where we now write ∀<i>x</i> and ∃<i>x</i>. Peirce's notation can be found in the writings of <a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Ernst Schröder</a>, <a href="/wiki/Leopold_Loewenheim" class="mw-redirect" title="Leopold Loewenheim">Leopold Loewenheim</a>, <a href="/wiki/Thoralf_Skolem" title="Thoralf Skolem">Thoralf Skolem</a>, and Polish logicians into the 1950s. Most notably, it is the notation of <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a>'s landmark 1930 paper on the <a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">completeness</a> of <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>, and 1931 paper on the <a href="/wiki/G%C3%B6del%27s_incompleteness_theorem" class="mw-redirect" title="Gödel's incompleteness theorem">incompleteness</a> of <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>. </p><p>Peirce's approach to quantification also influenced <a href="/wiki/William_Ernest_Johnson" title="William Ernest Johnson">William Ernest Johnson</a> and <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a>, who invented yet another notation, namely (<i>x</i>) for the universal quantification of <i>x</i> and (in 1897) ∃<i>x</i> for the existential quantification of <i>x</i>. Hence for decades, the canonical notation in philosophy and mathematical logic was (<i>x</i>)<i>P</i> to express "all individuals in the domain of discourse have the property <i>P</i>," and "(∃<i>x</i>)<i>P</i>" for "there exists at least one individual in the domain of discourse having the property <i>P</i>." Peano, who was much better known than Peirce, in effect diffused the latter's thinking throughout Europe. Peano's notation was adopted by the <i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> of <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Whitehead</a> and <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell</a>, <a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Quine</a>, and <a href="/wiki/Alonzo_Church" title="Alonzo Church">Alonzo Church</a>. In 1935, <a href="/wiki/Gentzen" class="mw-redirect" title="Gentzen">Gentzen</a> introduced the ∀ symbol, by analogy with Peano's ∃ symbol. ∀ did not become canonical until the 1960s. </p><p>Around 1895, Peirce began developing his <a href="/wiki/Existential_graph" title="Existential graph">existential graphs</a>, whose variables can be seen as tacitly quantified. Whether the shallowest instance of a variable is even or odd determines whether that variable's quantification is universal or existential. (Shallowness is the contrary of depth, which is determined by the nesting of negations.) Peirce's graphical logic has attracted some attention in recent years by those researching <a href="/w/index.php?title=Heterogeneous_reasoning&action=edit&redlink=1" class="new" title="Heterogeneous reasoning (page does not exist)">heterogeneous reasoning</a> and <a href="/wiki/Logical_graph" class="mw-redirect" title="Logical graph">diagrammatic inference</a>. </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=Quantifier_(logic)&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.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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data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon 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.csm.ornl.gov/~sheldon/ds/sec1.6.html">"Predicates and Quantifiers"</a>. <i>www.csm.ornl.gov</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-04</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.csm.ornl.gov&rft.atitle=Predicates+and+Quantifiers&rft_id=https%3A%2F%2Fwww.csm.ornl.gov%2F~sheldon%2Fds%2Fsec1.6.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantifier+%28logic%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</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://www.whitman.edu/mathematics/higher_math_online/section01.02.html">"1.2 Quantifiers"</a>. <i>www.whitman.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-04</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.whitman.edu&rft.atitle=1.2+Quantifiers&rft_id=https%3A%2F%2Fwww.whitman.edu%2Fmathematics%2Fhigher_math_online%2Fsection01.02.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantifier+%28logic%29" 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 id="CITEREFK.R._Apt1990" class="citation book cs1">K.R. Apt (1990). "Logic Programming". In Jan van Leeuwen (ed.). <i>Formal Models and Semantics</i>. Handbook of Theoretical Computer Science. Vol. B. Elsevier. pp. <span class="nowrap">493–</span>574. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-88074-7" title="Special:BookSources/0-444-88074-7"><bdi>0-444-88074-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Logic+Programming&rft.btitle=Formal+Models+and+Semantics&rft.series=Handbook+of+Theoretical+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E493-%3C%2Fspan%3E574&rft.pub=Elsevier&rft.date=1990&rft.isbn=0-444-88074-7&rft.au=K.R.+Apt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantifier+%28logic%29" class="Z3988"></span> Here: p.497</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="CITEREFSchwichtenbergWainer2009" class="citation book cs1">Schwichtenberg, Helmut; Wainer, Stanley S. (2009). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1017/cbo9781139031905"><i>Proofs and Computations</i></a>. Cambridge: Cambridge University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fcbo9781139031905">10.1017/cbo9781139031905</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-139-03190-5" title="Special:BookSources/978-1-139-03190-5"><bdi>978-1-139-03190-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proofs+and+Computations&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2009&rft_id=info%3Adoi%2F10.1017%2Fcbo9781139031905&rft.isbn=978-1-139-03190-5&rft.aulast=Schwichtenberg&rft.aufirst=Helmut&rft.au=Wainer%2C+Stanley+S.&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1017%2Fcbo9781139031905&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantifier+%28logic%29" 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 id="CITEREFJohn_E._Hopcroft_and_Jeffrey_D._Ullman1979" class="citation book cs1">John E. Hopcroft and Jeffrey D. Ullman (1979). <i>Introduction to Automata Theory, Languages, and Computation</i>. Reading/MA: Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-02988-X" title="Special:BookSources/0-201-02988-X"><bdi>0-201-02988-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Automata+Theory%2C+Languages%2C+and+Computation&rft.place=Reading%2FMA&rft.pub=Addison-Wesley&rft.date=1979&rft.isbn=0-201-02988-X&rft.au=John+E.+Hopcroft+and+Jeffrey+D.+Ullman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantifier+%28logic%29" class="Z3988"></span> Here: p.p.344</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHans_Hermes1973" class="citation book cs1">Hans Hermes (1973). <i>Introduction to Mathematical Logic</i>. Hochschultext (Springer-Verlag). London: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3540058192" title="Special:BookSources/3540058192"><bdi>3540058192</bdi></a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1431-4657">1431-4657</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mathematical+Logic&rft.place=London&rft.series=Hochschultext+%28Springer-Verlag%29&rft.pub=Springer&rft.date=1973&rft.issn=1431-4657&rft.isbn=3540058192&rft.au=Hans+Hermes&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantifier+%28logic%29" class="Z3988"></span> Here: Def. II.1.5</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGlebskiiKoganLiogon'kiiTalanov1972" class="citation journal cs1">Glebskii, Yu. V.; Kogan, D. I.; Liogon'kii, M. I.; Talanov, V. A. (1972). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007/bf01071084">"Range and degree of realizability of formulas in the restricted predicate calculus"</a>. <i>Cybernetics</i>. <b>5</b> (2): <span class="nowrap">142–</span>154. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01071084">10.1007/bf01071084</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0011-4235">0011-4235</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121409759">121409759</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Cybernetics&rft.atitle=Range+and+degree+of+realizability+of+formulas+in+the+restricted+predicate+calculus&rft.volume=5&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E142-%3C%2Fspan%3E154&rft.date=1972&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121409759%23id-name%3DS2CID&rft.issn=0011-4235&rft_id=info%3Adoi%2F10.1007%2Fbf01071084&rft.aulast=Glebskii&rft.aufirst=Yu.+V.&rft.au=Kogan%2C+D.+I.&rft.au=Liogon%27kii%2C+M.+I.&rft.au=Talanov%2C+V.+A.&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1007%2Fbf01071084&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantifier+%28logic%29" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEBrown2002-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBrown2002_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBrown2002">Brown 2002</a>.</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">in general, for a quantifer <b>Q</b>, closure makes sense only if the order of <b>Q</b> quantification does not matter, i.e. if <b>Q</b><i>x</i> <b>Q</b><i>y</i> <i>p</i>(<i>x</i>,<i>y</i>) is equivalent to <b>Q</b><i>y</i> <b>Q</b><i>x</i> <i>p</i>(<i>x</i>,<i>y</i>). This is satisfied for <b>Q</b> ∈ {∀,∃}, cf. <a href="#Order_of_quantifiers_(nesting)">#Order of quantifiers (nesting)</a> above.</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"><a href="/wiki/E._C._R._Hehner" class="mw-redirect" title="E. C. R. Hehner">Hehner, Eric C. R.</a>, 2004, <a rel="nofollow" class="external text" href="http://www.cs.utoronto.ca/~hehner/aPToP"><i>Practical Theory of Programming</i></a>, 2nd edition, p. 28</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Hehner (2004) uses the term "quantifier" in a very general sense, also including e.g. <a href="/wiki/Summation" title="Summation">summation</a>.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">George Bentham, <i>Outline of a new system of logic: with a critical examination of Dr. Whately's Elements of Logic</i> (1827); Thoemmes; Facsimile edition (1990) <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/1-85506-029-9" title="Special:BookSources/1-85506-029-9">1-85506-029-9</a></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPetersWesterståhl2006" class="citation book cs1">Peters, Stanley; Westerståhl, Dag (2006-04-27). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PrYUDAAAQBAJ&pg=PA34"><i>Quantifiers in Language and Logic</i></a>. Clarendon Press. pp. 34–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-929125-0" title="Special:BookSources/978-0-19-929125-0"><bdi>978-0-19-929125-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantifiers+in+Language+and+Logic&rft.pages=34-&rft.pub=Clarendon+Press&rft.date=2006-04-27&rft.isbn=978-0-19-929125-0&rft.aulast=Peters&rft.aufirst=Stanley&rft.au=Westerst%C3%A5hl%2C+Dag&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPrYUDAAAQBAJ%26pg%3DPA34&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantifier+%28logic%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantifier_(logic)&action=edit&section=15" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Jon_Barwise" title="Jon Barwise">Barwise, Jon</a>; and <a href="/wiki/John_Etchemendy" title="John Etchemendy">Etchemendy, John</a>, 2000. <i>Language Proof and Logic</i>. CSLI (University of Chicago Press) and New York: Seven Bridges Press. A gentle introduction to <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> by two first-rate logicians.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown2002" class="citation web cs1">Brown, Christopher W. (July 31, 2002). <a rel="nofollow" class="external text" href="https://www.usna.edu/CS/qepcadweb/B/QE.html">"What is Quantifier Elimination"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">Aug 30,</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=What+is+Quantifier+Elimination&rft.date=2002-07-31&rft.aulast=Brown&rft.aufirst=Christopher+W.&rft_id=https%3A%2F%2Fwww.usna.edu%2FCS%2Fqepcadweb%2FB%2FQE.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantifier+%28logic%29" class="Z3988"></span></li> <li><a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege, Gottlob</a>, 1879. <i><a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a></i>. Translated in <a href="/wiki/Jean_van_Heijenoort" title="Jean van Heijenoort">Jean van Heijenoort</a>, 1967. <i>From Frege to Gödel: A Source Book on Mathematical Logic, 1879-1931</i>. Harvard University Press. The first appearance of quantification.</li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert, David</a>; and <a href="/wiki/Wilhelm_Ackermann" title="Wilhelm Ackermann">Ackermann, Wilhelm</a>, 1950 (1928). <i><a href="/wiki/Principles_of_Mathematical_Logic" title="Principles of Mathematical Logic">Principles of Mathematical Logic</a></i>. Chelsea. Translation of <i>Grundzüge der theoretischen Logik</i>. Springer-Verlag. The 1928 first edition is the first time quantification was consciously employed in the now-standard manner, namely as binding variables ranging over some fixed domain of discourse. This is the defining aspect of <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>.</li> <li><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, C. S.</a>, 1885, "On the Algebra of Logic: A Contribution to the Philosophy of Notation, <i>American Journal of Mathematics</i>, Vol. 7, pp. 180–202. Reprinted in Kloesel, N. <i>et al.</i>, eds., 1993. <i>Writings of C. S. Peirce, Vol. 5</i>. Indiana University Press. The first appearance of quantification in anything like its present form.</li> <li><a href="/wiki/Hans_Reichenbach" title="Hans Reichenbach">Reichenbach, Hans</a>, 1975 (1947). <i>Elements of Symbolic Logic</i>, Dover Publications. The quantifiers are discussed in chapters §18 "Binding of variables" through §30 "Derivations from Synthetic Premises".</li> <li>Westerståhl, Dag, 2001, "Quantifiers," in Goble, Lou, ed., <i>The Blackwell Guide to Philosophical Logic</i>. Blackwell.</li> <li>Wiese, Heike, 2003. <i>Numbers, language, and the human mind</i>. Cambridge University Press. <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/0-521-83182-2" title="Special:BookSources/0-521-83182-2">0-521-83182-2</a>.</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=Quantifier_(logic)&action=edit&section=16" 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 #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/quantification" class="extiw" title="wiktionary:quantification">quantification</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Quantifier">"Quantifier"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Quantifier&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DQuantifier&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantifier+%28logic%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20000301144835/http://www.math.hawaii.edu/~ramsey/Logic/ForAll.html">"<span class="cs1-kern-left"></span>"For all" and "there exists" topical phrases, sentences and expressions"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.math.hawaii.edu/~ramsey/Logic/ForAll.html">the original</a> on March 1, 2000.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=%22For+all%22+and+%22there+exists%22+topical+phrases%2C+sentences+and+expressions&rft_id=http%3A%2F%2Fwww.math.hawaii.edu%2F~ramsey%2FLogic%2FForAll.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantifier+%28logic%29" class="Z3988"></span>. From College of Natural Sciences, <a href="/wiki/University_of_Hawaii_at_Manoa" class="mw-redirect" title="University of Hawaii at Manoa">University of Hawaii at Manoa</a>.</li> <li><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>: <ul><li>Shapiro, Stewart (2000). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/logic-classical/">"Classical Logic"</a> (Covers syntax, model theory, and metatheory for first order logic in the natural deduction style.)</li> <li>Westerståhl, Dag (2005). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/generalized-quantifiers/">"Generalized quantifiers"</a></li></ul></li> <li>Peters, Stanley; Westerståhl, Dag (2002). <a rel="nofollow" class="external text" href="http://www.stanford.edu/group/nasslli/courses/peters-wes/PWbookdraft2-3.pdf">"Quantifiers"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120716183245/http://www.stanford.edu/group/nasslli/courses/peters-wes/PWbookdraft2-3.pdf">Archived</a> 2012-07-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul> <div class="navbox-styles"><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 dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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4em">Common <a href="/wiki/Fallacy" title="Fallacy">fallacies</a> (<a href="/wiki/List_of_fallacies" title="List of fallacies">list</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_fallacy" title="Formal fallacy">Formal</a></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><th scope="row" class="navbox-group" style="width:1%">In <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional logic</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/Affirming_a_disjunct" title="Affirming a disjunct">Affirming a disjunct</a></li> <li><a href="/wiki/Affirming_the_consequent" title="Affirming the consequent">Affirming the consequent</a></li> <li><a href="/wiki/Denying_the_antecedent" title="Denying the antecedent">Denying the antecedent</a></li> <li><a href="/wiki/Argument_from_fallacy" title="Argument from fallacy">Argument from fallacy</a></li> <li><a href="/wiki/Masked-man_fallacy" title="Masked-man fallacy">Masked man</a></li> <li><a href="/wiki/Mathematical_fallacy" title="Mathematical fallacy">Mathematical fallacy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">In <a class="mw-selflink selflink">quantificational logic</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/Existential_fallacy" title="Existential fallacy">Existential</a></li> <li><a href="/wiki/Affirming_the_consequent" title="Affirming the consequent">Illicit conversion</a></li> <li><a href="/wiki/Proof_by_example" title="Proof by example">Proof by example</a></li> <li><a href="/wiki/Quantifier_shift" title="Quantifier shift">Quantifier shift</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Syllogistic_fallacy" class="mw-redirect" title="Syllogistic fallacy">Syllogistic fallacy</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/Affirmative_conclusion_from_a_negative_premise" title="Affirmative conclusion from a negative premise">Affirmative conclusion from a negative premise</a></li> <li><a href="/wiki/Negative_conclusion_from_affirmative_premises" title="Negative conclusion from affirmative premises">Negative conclusion from affirmative premises</a></li> <li><a href="/wiki/Fallacy_of_exclusive_premises" title="Fallacy of exclusive premises">Exclusive premises</a></li> <li><a href="/wiki/Existential_fallacy" title="Existential fallacy">Existential</a></li> <li><a href="/wiki/Modal_scope_fallacy" title="Modal scope fallacy">Necessity</a></li> <li><a href="/wiki/Fallacy_of_four_terms" title="Fallacy of four terms">Four terms</a></li> <li><a href="/wiki/Illicit_major" title="Illicit major">Illicit major</a></li> <li><a href="/wiki/Illicit_minor" title="Illicit minor">Illicit minor</a></li> <li><a href="/wiki/Fallacy_of_the_undistributed_middle" title="Fallacy of the undistributed middle">Undistributed middle</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/Informal_fallacy" title="Informal fallacy">Informal</a></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><th scope="row" class="navbox-group" style="width:1%">Equivocation</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/Equivocation" title="Equivocation">Equivocation</a></li> <li><a href="/wiki/False_equivalence" title="False equivalence">False equivalence</a></li> <li><a href="/wiki/False_attribution" title="False attribution">False attribution</a></li> <li><a href="/wiki/Quoting_out_of_context" title="Quoting out of context">Quoting out of context</a></li> <li><a href="/wiki/Loki%27s_Wager" class="mw-redirect" title="Loki's Wager">Loki's Wager</a></li> <li><a href="/wiki/No_true_Scotsman" title="No true Scotsman">No true Scotsman</a></li> <li><a href="/wiki/Reification_(fallacy)" title="Reification (fallacy)">Reification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Question-begging</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/Circular_reasoning" title="Circular reasoning">Circular reasoning</a> / <a href="/wiki/Begging_the_question" title="Begging the question">Begging the question</a></li> <li><a href="/wiki/Loaded_language" title="Loaded language">Loaded language</a> <ul><li><a href="/wiki/Leading_question" title="Leading question">Leading question</a></li></ul></li> <li><a href="/wiki/Double-barreled_question" title="Double-barreled question">Compound question</a> / <a href="/wiki/Loaded_question" title="Loaded question">Loaded question</a> / <a href="/wiki/Complex_question" title="Complex question">Complex question</a></li> <li><a href="/wiki/No_true_Scotsman" title="No true Scotsman">No true Scotsman</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Correlative-based_fallacies" title="Correlative-based fallacies">Correlative-based</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/False_dilemma" title="False dilemma">False dilemma</a> <ul><li><a href="/wiki/Nirvana_fallacy" title="Nirvana fallacy">Perfect solution</a></li></ul></li> <li><a href="/wiki/Denying_the_correlative" title="Denying the correlative">Denying the correlative</a></li> <li><a href="/wiki/Suppressed_correlative" title="Suppressed correlative">Suppressed correlative</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fallacies_of_illicit_transference" title="Fallacies of illicit transference">Illicit transference</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/Fallacy_of_composition" title="Fallacy of composition">Composition</a></li> <li><a href="/wiki/Fallacy_of_division" title="Fallacy of division">Division</a></li> <li><a href="/wiki/Ecological_fallacy" title="Ecological fallacy">Ecological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><i><a href="/wiki/Secundum_quid" title="Secundum quid">Secundum quid</a></i></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/Accident_(fallacy)" title="Accident (fallacy)">Accident</a></li> <li><a href="/wiki/Converse_accident" title="Converse accident">Converse accident</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Faulty_generalization" title="Faulty generalization">Faulty generalization</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/Anecdotal_evidence" title="Anecdotal evidence">Anecdotal evidence</a></li> <li><a href="/wiki/Sampling_bias" title="Sampling bias">Sampling bias</a> <ul><li><a href="/wiki/Cherry_picking" title="Cherry picking">Cherry picking</a></li> <li><a href="/wiki/McNamara_fallacy" title="McNamara fallacy">McNamara</a></li></ul></li> <li><a href="/wiki/Base_rate_fallacy" title="Base rate fallacy">Base rate</a> / <a href="/wiki/Conjunction_fallacy" title="Conjunction fallacy">Conjunction</a></li> <li><a href="/wiki/Double_counting_(fallacy)" title="Double counting (fallacy)">Double counting</a></li> <li><a href="/wiki/Argument_from_analogy" title="Argument from analogy">False analogy</a></li> <li><a href="/wiki/Slothful_induction" title="Slothful induction">Slothful induction</a></li> <li><a href="/wiki/Overwhelming_exception" title="Overwhelming exception">Overwhelming exception</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ambiguity" title="Ambiguity">Ambiguity</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/Fallacy_of_accent" title="Fallacy of accent">Accent</a></li> <li><a href="/wiki/False_precision" title="False precision">False precision</a></li> <li><a href="/wiki/Moving_the_goalposts" title="Moving the goalposts">Moving the goalposts</a></li> <li><a href="/wiki/Quoting_out_of_context" title="Quoting out of context">Quoting out of context</a></li> <li><a href="/wiki/Slippery_slope" title="Slippery slope">Slippery slope</a></li> <li><a href="/wiki/Sorites_paradox" title="Sorites paradox">Sorites paradox</a></li> <li><a href="/wiki/Syntactic_ambiguity" title="Syntactic ambiguity">Syntactic ambiguity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Questionable_cause" title="Questionable cause">Questionable cause</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/Animistic_fallacy" title="Animistic fallacy">Animistic</a> <ul><li><a href="/wiki/Furtive_fallacy" title="Furtive fallacy">Furtive</a></li></ul></li> <li>Correlation implies causation <ul><li><i><a href="/wiki/Correlation_does_not_imply_causation" title="Correlation does not imply causation">Cum hoc</a></i></li> <li><i><a href="/wiki/Post_hoc_ergo_propter_hoc" title="Post hoc ergo propter hoc">Post hoc</a></i></li></ul></li> <li><a href="/wiki/Gambler%27s_fallacy" title="Gambler's fallacy">Gambler's</a> <ul><li><a href="/wiki/Inverse_gambler%27s_fallacy" title="Inverse gambler's fallacy">Inverse</a></li></ul></li> <li><a href="/wiki/Regression_fallacy" title="Regression fallacy">Regression</a></li> <li><a href="/wiki/Fallacy_of_the_single_cause" title="Fallacy of the single cause">Single cause</a></li> <li><a href="/wiki/Slippery_slope" title="Slippery slope">Slippery slope</a></li> <li><a href="/wiki/Texas_sharpshooter_fallacy" title="Texas sharpshooter fallacy">Texas sharpshooter</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Appeals</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/Appeal_to_the_law" title="Appeal to the law">Law/Legality</a></li> <li><a href="/wiki/Appeal_to_the_stone" title="Appeal to the stone">Stone</a> / <a href="/wiki/Proof_by_assertion" title="Proof by assertion">Proof by assertion</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Appeal_to_consequences" title="Appeal to consequences">Consequences</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><i><a href="/wiki/Argumentum_ad_baculum" title="Argumentum ad baculum">Argumentum ad baculum</a></i></li> <li><a href="/wiki/Wishful_thinking" title="Wishful thinking">Wishful thinking</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Appeal_to_emotion" title="Appeal to emotion">Emotion</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/Think_of_the_children" title="Think of the children">Children</a></li> <li><a href="/wiki/Appeal_to_fear" title="Appeal to fear">Fear</a></li> <li><a href="/wiki/Appeal_to_flattery" title="Appeal to flattery">Flattery</a></li> <li><a href="/wiki/Appeal_to_novelty" title="Appeal to novelty">Novelty</a></li> <li><a href="/wiki/Appeal_to_pity" title="Appeal to pity">Pity</a></li> <li><a href="/wiki/Appeal_to_ridicule" title="Appeal to ridicule">Ridicule</a></li> <li><a href="/wiki/In-group_favoritism" title="In-group favoritism">In-group favoritism</a></li> <li><a href="/wiki/Invented_here" title="Invented here">Invented here</a> / <a href="/wiki/Not_invented_here" title="Not invented here">Not invented here</a></li> <li><a href="/wiki/Island_mentality" title="Island mentality">Island mentality</a></li> <li><a href="/wiki/Appeal_to_loyalty" title="Appeal to loyalty">Loyalty</a></li> <li><a href="/wiki/Parade_of_horribles" title="Parade of horribles">Parade of horribles</a></li> <li><a href="/wiki/Appeal_to_spite" class="mw-redirect" title="Appeal to spite">Spite</a></li> <li><a href="/wiki/Flag-waving" title="Flag-waving">Stirring symbols</a></li> <li><a href="/wiki/Wisdom_of_repugnance" title="Wisdom of repugnance">Wisdom of repugnance</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/Genetic_fallacy" title="Genetic fallacy">Genetic fallacy</a></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><th id="Ad_hominem18" scope="row" class="navbox-group" style="width:1%"><i><a href="/wiki/Ad_hominem" title="Ad hominem">Ad hominem</a></i></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/Appeal_to_motive" title="Appeal to motive">Appeal to motive</a></li> <li><a href="/wiki/Association_fallacy" title="Association fallacy">Association</a> <ul><li><i><a href="/wiki/Reductio_ad_Hitlerum" title="Reductio ad Hitlerum">Reductio ad Hitlerum</a></i> <ul><li><a href="/wiki/Godwin%27s_law" title="Godwin's law">Godwin's law</a></li></ul></li> <li><i><a href="/wiki/Red-baiting" title="Red-baiting">Reductio ad Stalinum</a></i></li></ul></li> <li><a href="/wiki/Bulverism" title="Bulverism">Bulverism</a></li> <li><a href="/wiki/Poisoning_the_well" title="Poisoning the well">Poisoning the well</a></li> <li><a href="/wiki/Tone_policing" title="Tone policing">Tone</a></li> <li><i><a href="/wiki/Tu_quoque" title="Tu quoque">Tu quoque</a></i></li> <li><a href="/wiki/Whataboutism" title="Whataboutism">Whataboutism</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/Argument_from_authority" title="Argument from authority">Authority</a> <ul><li><a href="/wiki/Appeal_to_accomplishment" title="Appeal to accomplishment">Accomplishment</a></li> <li><i><a href="/wiki/Ipse_dixit" title="Ipse dixit">Ipse dixit</a></i></li> <li><a href="/wiki/Argumentum_ad_lazarum" title="Argumentum ad lazarum">Poverty</a> / <a href="/wiki/Argumentum_ad_crumenam" title="Argumentum ad crumenam">Wealth</a></li></ul></li> <li><a href="/wiki/Etymological_fallacy" title="Etymological fallacy">Etymology</a></li> <li><a href="/wiki/Appeal_to_nature" title="Appeal to nature">Nature</a></li> <li><a href="/wiki/Appeal_to_tradition" title="Appeal to tradition">Tradition</a> / <a href="/wiki/Appeal_to_novelty" title="Appeal to novelty">Novelty</a> <ul><li><a href="/wiki/Chronological_snobbery" title="Chronological snobbery">Chronological snobbery</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="/wiki/Fallacy_of_relevance" class="mw-redirect" title="Fallacy of relevance">fallacies<br /> of relevance</a></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><th id="Arguments13" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Argument" title="Argument">Arguments</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><i><a href="/wiki/Ad_nauseam" title="Ad nauseam">Ad nauseam</a></i> <ul><li><a href="/wiki/Sealioning" title="Sealioning">Sealioning</a></li></ul></li> <li><a href="/wiki/Argument_from_anecdote" title="Argument from anecdote">Argument from anecdote</a></li> <li><a href="/wiki/Argument_from_silence" title="Argument from silence">Argument from silence</a></li> <li><a href="/wiki/Argument_to_moderation" title="Argument to moderation">Argument to moderation</a></li> <li><i><a href="/wiki/Argumentum_ad_populum" title="Argumentum ad populum">Argumentum ad populum</a></i></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/Clich%C3%A9" title="Cliché">Cliché</a></li> <li><i><a href="/wiki/The_Four_Great_Errors" title="The Four Great Errors">The Four Great Errors</a></i></li> <li><a href="/wiki/I%27m_entitled_to_my_opinion" title="I'm entitled to my opinion">I'm entitled to my opinion</a></li> <li><i><a href="/wiki/Irrelevant_conclusion" title="Irrelevant conclusion">Ignoratio elenchi</a></i></li> <li><a href="/wiki/Invincible_ignorance_fallacy" title="Invincible ignorance fallacy">Invincible ignorance</a></li> <li><a href="/wiki/Moralistic_fallacy" title="Moralistic fallacy">Moralistic</a> / <a href="/wiki/Naturalistic_fallacy" title="Naturalistic fallacy">Naturalistic</a></li> <li><a href="/wiki/Motte-and-bailey_fallacy" title="Motte-and-bailey fallacy">Motte-and-bailey fallacy</a></li> <li><a href="/wiki/Psychologist%27s_fallacy" title="Psychologist's fallacy">Psychologist's fallacy</a></li> <li><a href="/wiki/Rationalization_(psychology)" title="Rationalization (psychology)">Rationalization</a></li> <li><a href="/wiki/Red_herring" title="Red herring">Red herring</a> <ul><li><a href="/wiki/Two_wrongs_make_a_right" class="mw-redirect" title="Two wrongs make a right">Two wrongs make a right</a></li></ul></li> <li><a href="/wiki/Special_pleading" title="Special pleading">Special pleading</a></li> <li><a href="/wiki/Straw_man" title="Straw man">Straw man</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Fallacies" title="Category:Fallacies">Category</a></li></ul> </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="Mathematical_logic326" style="padding:3px"><table class="nowraplinks 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:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic326" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical 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="Traditional95" 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 class="mw-selflink selflink">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 href="/wiki/Formal_system" title="Formal system">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 class="mw-selflink selflink">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 href="/wiki/Formal_system" title="Formal system">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 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href="/wiki/Intension" title="Intension">Intension</a></li> <li><a href="/wiki/Logical_form_(linguistics)" title="Logical form (linguistics)">Logical form</a></li> <li><a href="/wiki/Presupposition" title="Presupposition">Presupposition</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Reference" title="Reference">Reference</a></li> <li><a href="/wiki/Scope_(formal_semantics)" title="Scope (formal semantics)">Scope</a></li> <li><a href="/wiki/Speech_act" title="Speech act">Speech act</a></li> <li><a href="/wiki/Syntax%E2%80%93semantics_interface" title="Syntax–semantics interface">Syntax–semantics interface</a></li> <li><a href="/wiki/Truth-conditional_semantics" title="Truth-conditional semantics">Truth conditions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Topics</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 scope="row" class="navbox-group" style="width:1%">Areas</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/Anaphora_(linguistics)" title="Anaphora (linguistics)">Anaphora</a></li> <li><a href="/wiki/Ambiguity" title="Ambiguity">Ambiguity</a></li> <li><a href="/wiki/Binding_(linguistics)" title="Binding (linguistics)">Binding</a></li> <li><a href="/wiki/Conditional_sentence" title="Conditional sentence">Conditionals</a></li> <li><a href="/wiki/Definiteness" title="Definiteness">Definiteness</a></li> <li><a href="/wiki/Disjunction" class="mw-redirect" title="Disjunction">Disjunction</a></li> <li><a href="/wiki/Evidentiality" title="Evidentiality">Evidentiality</a></li> <li><a href="/wiki/Focus_(linguistics)" title="Focus (linguistics)">Focus</a></li> <li><a href="/wiki/Indexicality" title="Indexicality">Indexicality</a></li> <li><a href="/wiki/Lexical_semantics" title="Lexical semantics">Lexical semantics</a></li> <li><a href="/wiki/Linguistic_modality" class="mw-redirect" title="Linguistic modality">Modality</a></li> <li><a href="/wiki/Negation" title="Negation">Negation</a></li> <li><a href="/wiki/Propositional_attitudes" class="mw-redirect" title="Propositional attitudes">Propositional attitudes</a></li> <li><a href="/wiki/Tense%E2%80%93aspect%E2%80%93mood" title="Tense–aspect–mood">Tense–aspect–mood</a></li> <li><a class="mw-selflink selflink">Quantification</a></li> <li><a href="/wiki/Vagueness" title="Vagueness">Vagueness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Phenomena</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/Antecedent-contained_deletion" title="Antecedent-contained deletion">Antecedent-contained deletion</a></li> <li><a href="/wiki/Cataphora" title="Cataphora">Cataphora</a></li> <li><a href="/wiki/Coercion_(linguistics)" title="Coercion (linguistics)">Coercion</a></li> <li><a href="/wiki/Conservativity" title="Conservativity">Conservativity</a></li> <li><a href="/wiki/Counterfactuals" class="mw-redirect" title="Counterfactuals">Counterfactuals</a></li> <li><a href="/wiki/Crossover_effects" title="Crossover effects">Crossover effects</a></li> <li><a href="/wiki/Cumulativity_(linguistics)" title="Cumulativity (linguistics)">Cumulativity</a></li> <li><a href="/wiki/De_dicto_and_de_re" title="De dicto and de re">De dicto and de re</a></li> <li><a href="/wiki/De_se" title="De se">De se</a></li> <li><a href="/wiki/Deontic_modality" title="Deontic modality">Deontic modality</a></li> <li><a href="/wiki/Discourse_relation" title="Discourse relation">Discourse relations</a></li> <li><a href="/wiki/Donkey_anaphora" class="mw-redirect" title="Donkey anaphora">Donkey anaphora</a></li> <li><a href="/wiki/Epistemic_modality" title="Epistemic modality">Epistemic modality</a></li> <li><a href="/wiki/Exhaustivity" title="Exhaustivity">Exhaustivity</a></li> <li><a href="/wiki/Faultless_disagreement" title="Faultless disagreement">Faultless disagreement</a></li> <li><a href="/wiki/Free_choice_inference" title="Free choice inference">Free choice inferences</a></li> <li><a href="/wiki/Givenness" title="Givenness">Givenness</a></li> <li><a href="/wiki/Homogeneity_(linguistics)" class="mw-redirect" title="Homogeneity (linguistics)">Homogeneity (linguistics)</a></li> <li><a href="/wiki/Hurford_disjunction" title="Hurford disjunction">Hurford disjunction</a></li> <li><a href="/wiki/Inalienable_possession" title="Inalienable possession">Inalienable possession</a></li> <li><a href="/wiki/Intersective_modifier" title="Intersective modifier">Intersective modification</a></li> <li><a href="/wiki/Logophoricity" title="Logophoricity">Logophoricity</a></li> <li><a href="/wiki/Mirativity" title="Mirativity">Mirativity</a></li> <li><a href="/wiki/Modal_subordination" title="Modal subordination">Modal subordination</a></li> <li><a href="/wiki/Opaque_context" title="Opaque context">Opaque contexts</a></li> <li><a href="/wiki/Performative_utterance" title="Performative utterance">Performatives</a></li> <li><a href="/wiki/Polarity_item" title="Polarity item">Polarity items</a></li> <li><a href="/wiki/Privative_adjective" title="Privative adjective">Privative adjectives</a></li> <li><a href="/wiki/Quantificational_variability_effect" title="Quantificational variability effect">Quantificational variability effect</a></li> <li><a href="/wiki/Responsive_predicate" title="Responsive predicate">Responsive predicate</a></li> <li><a href="/wiki/Rising_declarative" title="Rising declarative">Rising declaratives</a></li> <li><a href="/wiki/Scalar_implicature" title="Scalar implicature">Scalar implicature</a></li> <li><a href="/wiki/Sloppy_identity" title="Sloppy identity">Sloppy identity</a></li> <li><a href="/wiki/Subsective_modifier" title="Subsective modifier">Subsective modification</a></li> <li><a href="/wiki/Subtrigging" title="Subtrigging">Subtrigging</a></li> <li><a href="/wiki/Telicity" title="Telicity">Telicity</a></li> <li><a href="/wiki/Temperature_paradox" title="Temperature paradox">Temperature paradox</a></li> <li><a href="/wiki/Veridicality" title="Veridicality">Veridicality</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formalism</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 scope="row" class="navbox-group" style="width:1%">Formal systems</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/Alternative_semantics" title="Alternative semantics">Alternative semantics</a></li> <li><a href="/wiki/Categorial_grammar" title="Categorial grammar">Categorial grammar</a></li> <li><a href="/wiki/Combinatory_categorial_grammar" title="Combinatory categorial grammar">Combinatory categorial grammar</a></li> <li><a href="/wiki/Discourse_representation_theory" title="Discourse representation theory">Discourse representation theory (DRT)</a></li> <li><a href="/wiki/Dynamic_semantics" title="Dynamic semantics">Dynamic semantics</a></li> <li><a href="/wiki/Generative_grammar" title="Generative grammar">Generative grammar</a></li> <li><a href="/wiki/Glue_semantics" title="Glue semantics">Glue semantics</a></li> <li><a href="/wiki/Inquisitive_semantics" title="Inquisitive semantics">Inquisitive semantics</a></li> <li><a href="/wiki/Intensional_logic" title="Intensional logic">Intensional logic</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Mereology" title="Mereology">Mereology</a></li> <li><a href="/wiki/Montague_grammar" title="Montague grammar">Montague grammar</a></li> <li><a href="/wiki/Segmented_discourse_representation_theory" class="mw-redirect" title="Segmented discourse representation theory">Segmented discourse representation theory (SDRT)</a></li> <li><a href="/wiki/Situation_semantics" title="Situation semantics">Situation semantics</a></li> <li><a href="/wiki/Supervaluationism" title="Supervaluationism">Supervaluationism</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li> <li><a href="/wiki/Type_theory_with_records" title="Type theory with records">TTR</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-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autonomy_of_syntax" title="Autonomy of syntax">Autonomy of syntax</a></li> <li><a href="/wiki/Context_set" class="mw-redirect" title="Context set">Context set</a></li> <li><a href="/wiki/Continuation" title="Continuation">Continuation</a></li> <li><a href="/wiki/Conversational_scoreboard" title="Conversational scoreboard">Conversational scoreboard</a></li> <li><a href="/wiki/Downward_entailing" title="Downward entailing">Downward entailing</a></li> <li><a href="/wiki/Existential_closure" title="Existential closure">Existential closure</a></li> <li><a href="/wiki/Function_application" title="Function application">Function application</a></li> <li><a href="/wiki/Meaning_postulate" title="Meaning postulate">Meaning postulate</a></li> <li><a href="/wiki/Monad_(functional_programming)" title="Monad (functional programming)">Monads</a></li> <li><a href="/wiki/Plural_quantification" title="Plural quantification">Plural quantification</a></li> <li><a href="/wiki/Possible_world" title="Possible world">Possible world</a></li> <li><a href="/wiki/Quantifier_raising" class="mw-redirect" title="Quantifier raising">Quantifier raising</a></li> <li><a href="/wiki/Quantization_(linguistics)" title="Quantization (linguistics)">Quantization</a></li> <li><a href="/wiki/Question_under_discussion" title="Question under discussion">Question under discussion</a></li> <li><a href="/wiki/Semantic_parsing" title="Semantic parsing">Semantic parsing</a></li> <li><a href="/wiki/Squiggle_operator" title="Squiggle operator">Squiggle operator</a></li> <li><a href="/wiki/Strawson_entailment" title="Strawson entailment">Strawson entailment</a></li> <li><a href="/wiki/Strict_conditional" title="Strict conditional">Strict conditional</a></li> <li><a href="/wiki/Type_shifter" title="Type shifter">Type shifter</a></li> <li><a href="/wiki/Universal_grinder" title="Universal grinder">Universal grinder</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</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/Cognitive_semantics" title="Cognitive semantics">Cognitive semantics</a></li> <li><a href="/wiki/Computational_semantics" title="Computational semantics">Computational semantics</a></li> <li><a href="/wiki/Distributional_semantics" title="Distributional semantics">Distributional semantics</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Formal grammar</a></li> <li><a href="/wiki/Inferentialism" class="mw-redirect" title="Inferentialism">Inferentialism</a></li> <li><a href="/wiki/Logic_translation" title="Logic translation">Logic translation</a></li> <li><a href="/wiki/Linguistics_wars" title="Linguistics wars">Linguistics wars</a></li> <li><a href="/wiki/Philosophy_of_language" title="Philosophy of language">Philosophy of language</a></li> <li><a href="/wiki/Pragmatics" title="Pragmatics">Pragmatics</a></li> <li><a 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