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class="vector-toc-list"> <li id="toc-List_of_common_logical_connectives" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#List_of_common_logical_connectives"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>List of common logical connectives</span> </div> </a> <ul id="toc-List_of_common_logical_connectives-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History_of_notations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#History_of_notations"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>History of notations</span> </div> </a> <ul id="toc-History_of_notations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Redundancy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Redundancy"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Redundancy</span> </div> </a> <ul id="toc-Redundancy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Natural_language" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Natural_language"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Natural language</span> </div> </a> <ul id="toc-Natural_language-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Order_of_precedence" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Order_of_precedence"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Order of precedence</span> </div> </a> <ul id="toc-Order_of_precedence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Table_and_Hasse_diagram" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Table_and_Hasse_diagram"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Table and Hasse diagram</span> </div> </a> <ul id="toc-Table_and_Hasse_diagram-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-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 Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Computer_science" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computer_science"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Computer science</span> </div> </a> <ul id="toc-Computer_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Set_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Set_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Set theory</span> </div> </a> <ul id="toc-Set_theory-sublist" class="vector-toc-list"> </ul> </li> </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">7</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">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-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">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </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">Logical connective</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 39 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-39" 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">39 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/%D8%B1%D8%A7%D8%A8%D8%B7%D8%A9_%D9%85%D9%86%D8%B7%D9%82%D9%8A%D8%A9" title="رابطة منطقية – Arabic" lang="ar" hreflang="ar" data-title="رابطة منطقية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/M%C9%99ntiqi_%C9%99m%C9%99liyyat" title="Məntiqi əməliyyat – Azerbaijani" lang="az" hreflang="az" data-title="Məntiqi əməliyyat" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Логическа операция – Bulgarian" lang="bg" hreflang="bg" data-title="Логическа операция" 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/Connectiva_l%C3%B2gica" title="Connectiva lògica – Catalan" lang="ca" hreflang="ca" data-title="Connectiva lògica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Logisk_konnektiv" title="Logisk konnektiv – Danish" lang="da" hreflang="da" data-title="Logisk konnektiv" 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/Logische_Verkn%C3%BCpfung" title="Logische Verknüpfung – German" lang="de" hreflang="de" data-title="Logische Verknüpfung" 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/Konnektor_(keeleteadus)" title="Konnektor (keeleteadus) – Estonian" lang="et" hreflang="et" data-title="Konnektor (keeleteadus)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9B%CE%BF%CE%B3%CE%B9%CE%BA%CE%AD%CF%82_%CF%83%CF%85%CE%BD%CE%B1%CF%81%CF%84%CE%AE%CF%83%CE%B5%CE%B9%CF%82" title="Λογικές συναρτήσεις – Greek" lang="el" hreflang="el" data-title="Λογικές συναρτήσεις" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Conet%C3%AEv_l%C3%B2gic" title="Conetîv lògic – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Conetîv lògic" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Conectiva_l%C3%B3gica" title="Conectiva lógica – Spanish" lang="es" hreflang="es" data-title="Conectiva lógica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Lokailu_logiko" title="Lokailu logiko – Basque" lang="eu" hreflang="eu" data-title="Lokailu logiko" 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%B1%D8%A7%D8%A8%D8%B7_%D9%85%D9%86%D8%B7%D9%82%DB%8C" title="رابط منطقی – Persian" lang="fa" hreflang="fa" data-title="رابط منطقی" 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/Connecteur_logique" title="Connecteur logique – French" lang="fr" hreflang="fr" data-title="Connecteur logique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%85%BC%EB%A6%AC_%EC%97%B0%EC%82%B0" title="논리 연산 – Korean" lang="ko" hreflang="ko" data-title="논리 연산" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8F%D6%80%D5%A1%D5%B4%D5%A1%D5%A2%D5%A1%D5%B6%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A3%D5%B8%D6%80%D5%AE%D5%B8%D5%B2%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Տրամաբանական գործողություն – Armenian" lang="hy" hreflang="hy" data-title="Տրամաբանական գործողություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Operator_logika" title="Operator logika – Indonesian" lang="id" hreflang="id" data-title="Operator 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/R%C3%B6ka%C3%B0ger%C3%B0" title="Rökaðgerð – Icelandic" lang="is" hreflang="is" data-title="Rökaðgerð" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Connettivo_logico" title="Connettivo logico – Italian" lang="it" hreflang="it" data-title="Connettivo logico" 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%A7%D7%A9%D7%A8_%D7%9C%D7%95%D7%92%D7%99" title="קשר לוגי – Hebrew" lang="he" hreflang="he" data-title="קשר לוגי" 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%9F%D1%80%D0%BE%D0%BF%D0%BE%D0%B7%D0%B8%D1%86%D0%B8%D1%8F%D0%BB%D1%8B%D2%9B_%D3%A9%D0%B7%D0%B0%D1%80%D0%B0_%D2%9B%D0%B0%D1%82%D1%8B%D0%BD%D0%B0%D1%81" title="Пропозициялық өзара қатынас – Kazakh" lang="kk" hreflang="kk" data-title="Пропозициялық өзара қатынас" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Logikai_m%C5%B1velet" title="Logikai művelet – Hungarian" lang="hu" hreflang="hu" data-title="Logikai művelet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D1%87%D0%BA%D0%B0_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Логичка операција – Macedonian" lang="mk" hreflang="mk" data-title="Логичка операција" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pengoperasi_logik" title="Pengoperasi logik – Malay" lang="ms" hreflang="ms" data-title="Pengoperasi logik" 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-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%AB%96%E7%90%86%E6%BC%94%E7%AE%97" title="論理演算 – Japanese" lang="ja" hreflang="ja" data-title="論理演算" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Logisk_konstant" title="Logisk konstant – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Logisk konstant" 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/Funktor_zdaniotw%C3%B3rczy" title="Funktor zdaniotwórczy – Polish" lang="pl" hreflang="pl" data-title="Funktor zdaniotwórczy" 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/Conectivo_l%C3%B3gico" title="Conectivo lógico – Portuguese" lang="pt" hreflang="pt" data-title="Conectivo lógico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Conector_logic" title="Conector logic – Romanian" lang="ro" hreflang="ro" data-title="Conector logic" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Логическая операция – Russian" lang="ru" hreflang="ru" data-title="Логическая операция" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/V%C3%BDrokov%C3%A1_spojka" title="Výroková spojka – Slovak" lang="sk" hreflang="sk" data-title="Výroková spojka" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Logisk_operator" title="Logisk operator – Swedish" lang="sv" hreflang="sv" data-title="Logisk operator" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8F%E0%AE%B0%E0%AE%A3_%E0%AE%87%E0%AE%A3%E0%AF%88%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%BF" title="ஏரண இணைப்பி – Tamil" lang="ta" hreflang="ta" data-title="ஏரண இணைப்பி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B8%94%E0%B8%B3%E0%B9%80%E0%B8%99%E0%B8%B4%E0%B8%99%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%95%E0%B8%A3%E0%B8%A3%E0%B8%81%E0%B8%B0" title="ตัวดำเนินการตรรกะ – Thai" lang="th" hreflang="th" data-title="ตัวดำเนินการตรรกะ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%BC%D0%B0%D0%BB%D2%B3%D0%BE%D0%B8_%D0%BC%D0%B0%D0%BD%D1%82%D0%B8%D2%9B%D3%A3" title="Амалҳои мантиқӣ – Tajik" lang="tg" hreflang="tg" data-title="Амалҳои мантиқӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Mant%C4%B1k_ba%C4%9Flac%C4%B1" title="Mantık bağlacı – Turkish" lang="tr" hreflang="tr" data-title="Mantık bağlacı" 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%9B%D0%BE%D0%B3%D1%96%D1%87%D0%BD%D0%B8%D0%B9_%D1%81%D0%BF%D0%BE%D0%BB%D1%83%D1%87%D0%BD%D0%B8%D0%BA" title="Логічний сполучник – Ukrainian" lang="uk" hreflang="uk" data-title="Логічний сполучник" 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data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <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">Symbol connecting sentential formulas in logic</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 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="font-size: 130%; margin: 6px 0px 6px 0px; background: #ddf;"><a class="mw-selflink selflink">Logical connectives</a></th></tr><tr><td class="sidebar-content"> <table style="width:100%;border-collapse:collapse;border-spacing:0px 0px;border:none;line-height:1.3em;"><tbody><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Negation" title="Negation">NOT</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"></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 -A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6daf6db742ace65252b589963f7e7a07603ccb56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.551ex; height:2.343ex;" alt="{\displaystyle -A}"></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 {\overline {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efef0e89bdc77f6a848764195ef5b9d9bfcc6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:3.009ex;" alt="{\displaystyle {\overline {A}}}"></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 \sim A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79bf96c247833282e773fae43602343150c1665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.196ex; height:2.176ex;" alt="{\displaystyle \sim A}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_conjunction" title="Logical conjunction">AND</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></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 A\cdot B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cdot B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a90e903f21f11a0f4ab3caca1e6943ba7a9849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.186ex; height:2.176ex;" alt="{\displaystyle A\cdot B}"></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 AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}"></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 A\&B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi mathvariant="normal">&</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\&B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f65ee34a8896390e0d1f193c137d9eb64815c1a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.315ex; height:2.176ex;" alt="{\displaystyle A\&B}"></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 A\&\&B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi mathvariant="normal">&</mi> <mi mathvariant="normal">&</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\&\&B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab3517dd859d7eaa8f3f1656c8125f99ada1470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.123ex; height:2.176ex;" alt="{\displaystyle A\&\&B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Sheffer_stroke" title="Sheffer stroke">NAND</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A{\overline {\land }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∧</mo> <mo accent="false">¯</mo> </mover> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\overline {\land }}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0167d56141342887a74d56a036e6fbbad7172b0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.172ex; height:2.843ex;" alt="{\displaystyle A{\overline {\land }}B}"></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 A\uparrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↑</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\uparrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5723e9ef44d446f4410c273b056d7c7c8e6f2564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.96ex; height:2.509ex;" alt="{\displaystyle A\uparrow B}"></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 A\mid B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∣</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mid B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1400973074b4691cc0638a68118716a2b218fce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.444ex; height:2.843ex;" alt="{\displaystyle A\mid B}"></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 {\overline {A\cdot B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A\cdot B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/225f35bb78e90b9126458f1bc6bf1ed3f0724bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.301ex; height:3.009ex;" alt="{\displaystyle {\overline {A\cdot B}}}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_disjunction" title="Logical disjunction">OR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"></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 A+B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4279cdbd3cb8ec4c3423065d9a7d83a82cfc89e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A+B}"></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 A\mid B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∣</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mid B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1400973074b4691cc0638a68118716a2b218fce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.444ex; height:2.843ex;" alt="{\displaystyle A\mid B}"></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 A\parallel B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∥</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\parallel B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd0239f9b74f7ef1520ba4e30454b06e695289" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.96ex; height:2.843ex;" alt="{\displaystyle A\parallel B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_NOR" title="Logical NOR">NOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A{\overline {\lor }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∨</mo> <mo accent="false">¯</mo> </mover> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\overline {\lor }}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591afbca3e984765b18abb189f4bb1b88116c400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.172ex; height:2.843ex;" alt="{\displaystyle A{\overline {\lor }}B}"></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 A\downarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↓</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\downarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c5e77260e67880093dafe958880ea02f5026164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.96ex; height:2.509ex;" alt="{\displaystyle A\downarrow B}"></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 {\overline {A+B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A+B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08840f8e2022f127fc459d801a8f8ce93f65f55a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.462ex; height:3.176ex;" alt="{\displaystyle {\overline {A+B}}}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/XNOR_gate" title="XNOR gate">XNOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\ {\text{XNOR}}\ B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>XNOR</mtext> </mrow> <mtext> </mtext> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\ {\text{XNOR}}\ B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54fbdc2cf9fad58dda134c0c9baa55a4712b3955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.673ex; height:2.176ex;" alt="{\displaystyle A\ {\text{XNOR}}\ B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> └ <a href="/wiki/Logical_biconditional" title="Logical biconditional">equivalent</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\equiv B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≡</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\equiv B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b933daba3ef47ec3b4f3097ea6e741b85149707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\equiv B}"></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 A\Leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇔</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\Leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce08fffb4d36ba12921b8b3e06228887015b2b8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\Leftrightarrow B}"></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 A\leftrightharpoons B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇋</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightharpoons B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8639312e9e120cd65c98fc48a6d5256d57288c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftrightharpoons B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Exclusive_or" title="Exclusive or">XOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A{\underline {\lor }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∨</mo> <mo>_</mo> </munder> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\underline {\lor }}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803e1413692912954b90e99694e10c728e27a153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:5.06ex; height:3.176ex;" alt="{\displaystyle A{\underline {\lor }}B}"></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 A\oplus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\oplus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0512d6bdd29ff000dea0bf68b853618dcaabc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\oplus B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> └nonequivalent</td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\not \equiv B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≢</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\not \equiv B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0339ed15cd58f7263c5eec8e5628168aa6006200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.676ex;" alt="{\displaystyle A\not \equiv B}"></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 A\not \Leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⇎</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\not \Leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47bfd97641b05a7f7fc0bcd02e83fa6532c62bb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\not \Leftrightarrow B}"></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 A\nleftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>↮</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\nleftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926467de6fd4709bdbc59c3168a21298cdf0d26c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\nleftrightarrow B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Material_conditional" title="Material conditional">implies</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\Rightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇒</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\Rightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e560143d45c97e6387c7c3aa90e9d7745002228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\Rightarrow B}"></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 A\supset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊃</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\supset B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee952838d8b3e67045072a8f2b71e7fc0467dea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\supset B}"></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 A\rightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\rightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23efef033def56a67de7ded823f14626de26d174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\rightarrow B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Converse_(logic)" title="Converse (logic)">converse</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\Leftarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇐</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\Leftarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8569d34530d97f701080546ca0f20c0defadf8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\Leftarrow B}"></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 A\subset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}"></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 A\leftarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">←</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456da65c891c438fea04d7e40283b67d600fe92d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftarrow B}"></span></td></tr></tbody></table></td> </tr><tr><th class="sidebar-heading" style="background: #eef; text-align: center;"> Related concepts</th></tr><tr><td class="sidebar-content"> <div class="hlist" style="line-height:1.3em;"><ul><li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li><li><a href="/wiki/First-order_logic" title="First-order logic">Predicate logic</a></li><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li><li><a href="/wiki/Truth_table" title="Truth table">Truth table</a></li><li><a href="/wiki/Truth_function" title="Truth function">Truth function</a></li><li><a href="/wiki/Boolean_function" title="Boolean function">Boolean function</a></li><li><a href="/wiki/Functional_completeness" title="Functional completeness">Functional completeness</a></li><li><a href="/wiki/Scope_(logic)" title="Scope (logic)">Scope (logic)</a></li></ul></div></td> </tr><tr><th class="sidebar-heading" style="background: #eef; text-align: center;"> Applications</th></tr><tr><td class="sidebar-content"> <div class="hlist"><ul><li><a href="/wiki/Logic_gate" title="Logic gate">Digital logic</a></li><li><a href="/wiki/Programming_language" title="Programming language">Programming languages</a></li><li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li><li><a href="/wiki/Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li></ul></div></td> </tr><tr><td class="sidebar-below hlist" style="background: #eef; text-align: center;"> <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:Logical_connectives" title="Category:Logical connectives">Category</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Logical_connectives_sidebar" title="Template:Logical connectives sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Logical_connectives_sidebar&action=edit&redlink=1" class="new" title="Template talk:Logical connectives sidebar (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Logical_connectives_sidebar" title="Special:EditPage/Template:Logical connectives sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Logical_connectives_Hasse_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/300px-Logical_connectives_Hasse_diagram.svg.png" decoding="async" width="300" height="424" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/450px-Logical_connectives_Hasse_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/600px-Logical_connectives_Hasse_diagram.svg.png 2x" data-file-width="744" data-file-height="1052" /></a><figcaption><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> of logical connectives.</figcaption></figure> <p>In <a href="/wiki/Mathematical_logic" title="Mathematical logic">logic</a>, a <b>logical connective</b> (also called a <b>logical operator</b>, <b>sentential connective</b>, or <b>sentential operator</b>) is a <a href="/wiki/Logical_constant" title="Logical constant">logical constant</a>. Connectives can be used to connect logical formulas. For instance in the <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a> of <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a>, the <a href="/wiki/Binary_relation" title="Binary relation">binary</a> connective <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 \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"></span> can be used to join the two <a href="/wiki/Atomic_formula" title="Atomic formula">atomic formulas</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}"> <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> 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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span>, rendering the complex 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 P\lor Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\lor Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2bc60d4b9ff5ec772fec5c2ef72a39536d4323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\lor Q}"></span>. </p><p>Common connectives include <a href="/wiki/Negation" title="Negation">negation</a>, <a href="/wiki/Disjunction" class="mw-redirect" title="Disjunction">disjunction</a>, <a href="/wiki/Logical_conjunction" title="Logical conjunction">conjunction</a>, <a href="/wiki/Material_conditional" title="Material conditional">implication</a>, and <a href="/wiki/Logical_biconditional" title="Logical biconditional">equivalence</a>. In standard systems of <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>, these connectives are <a href="/wiki/Semantics_of_logic" title="Semantics of logic">interpreted</a> as <a href="/wiki/Truth_function" title="Truth function">truth functions</a>, though they receive a variety of alternative interpretations in <a href="/wiki/Nonclassical_logic" class="mw-redirect" title="Nonclassical logic">nonclassical logics</a>. Their classical interpretations are similar to the meanings of natural language expressions such as <a href="/wiki/English_language" title="English language">English</a> "not", "or", "and", and "if", but not identical. Discrepancies between natural language connectives and those of classical logic have motivated nonclassical approaches to natural language meaning as well as approaches which pair a classical <a href="/wiki/Formal_semantics_(natural_language)" title="Formal semantics (natural language)">compositional semantics</a> with a robust <a href="/wiki/Pragmatics" title="Pragmatics">pragmatics</a>. </p><p>A logical connective is similar to, but not equivalent to, a syntax commonly used in programming languages called a <a href="/wiki/Conditional_operator" title="Conditional operator">conditional operator</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template noprint noexcerpt Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:NOTRS" class="mw-redirect" title="Wikipedia:NOTRS"><span title="This claim needs references to better sources. (March 2021)">better source needed</span></a></i>]</sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_connective&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Formal_language" title="Formal language">formal languages</a>, truth functions are represented by unambiguous symbols. This allows logical statements to not be understood in an ambiguous way. These symbols are called <i>logical connectives</i>, <i>logical operators</i>, <i>propositional operators</i>, or, in <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>, <i><a href="/wiki/Truth_function" title="Truth function">truth-functional</a> connectives</i>. For the rules which allow new well-formed formulas to be constructed by joining other well-formed formulas using truth-functional connectives, see <a href="/wiki/Well-formed_formula" title="Well-formed formula">well-formed formula</a>. </p><p>Logical connectives can be used to link zero or more statements, so one can speak about <i><a href="/wiki/Arity" title="Arity"><span class="texhtml mvar" style="font-style:italic;">n</span>-ary</a> logical connectives</i>. The <a href="/wiki/Boolean_algebra" title="Boolean algebra">boolean</a> constants <i>True</i> and <i>False</i> can be thought of as zero-ary operators. Negation is a 1-ary connective, and so on. </p> <table class="floatright" style="margin-left:2em; margin-bottom:1ex; text-align:center; border: 1px solid #a2a9b1; color: black; padding:0.2em; background-color: #f8f9fa; font-size:90%;"> <tbody><tr> <th colspan="2">Symbol, name </th> <th colspan="5">Truth <br />table </th> <th>Venn <br /><small>diagram</small> </th></tr> <tr> <th colspan="8">Zeroary connectives (constants) </th></tr> <tr> <td>⊤</td> <td style="text-align:left;"><a href="/wiki/Truth" title="Truth">Truth</a>/<a href="/wiki/Tautology_(logic)" title="Tautology (logic)">tautology</a> </td> <td> </td> <td colspan="2"> </td> <td colspan="2">1 </td> <td><span typeof="mw:File"><a href="/wiki/File:Red_Square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Red_Square.svg/32px-Red_Square.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Red_Square.svg/48px-Red_Square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Red_Square.svg/64px-Red_Square.svg.png 2x" data-file-width="525" data-file-height="525" /></a></span> </td></tr> <tr> <td>⊥</td> <td style="text-align:left;"><a href="/wiki/False_(logic)" title="False (logic)">Falsity</a>/<a href="/wiki/Contradiction" title="Contradiction">contradiction</a> </td> <td> </td> <td colspan="2"> </td> <td colspan="2">0 </td> <td><span typeof="mw:File"><a href="/wiki/File:Blank_Square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Blank_Square.svg/32px-Blank_Square.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Blank_Square.svg/48px-Blank_Square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Blank_Square.svg/64px-Blank_Square.svg.png 2x" data-file-width="525" data-file-height="525" /></a></span> </td></tr> <tr> <th colspan="8">Unary connectives </th></tr> <tr> <td colspan="2"> </td> <td style="background-color:#ffff66;"><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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> = </td> <td colspan="2" style="background-color:#ffff66;">0 </td> <td colspan="2" style="background-color:#ffff66;">1 </td> <td> </td></tr> <tr> <td></td> <td style="text-align:left;">Proposition <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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> </td> <td> </td> <td colspan="2">0 </td> <td colspan="2">1 </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/32px-Venn01.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/48px-Venn01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/64px-Venn01.svg.png 2x" data-file-width="280" data-file-height="280" /></a></span> </td></tr> <tr> <td>¬</td> <td style="text-align:left;"><a href="/wiki/Negation" title="Negation">Negation</a> </td> <td> </td> <td colspan="2">1 </td> <td colspan="2">0 </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn10.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Venn10.svg/32px-Venn10.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Venn10.svg/48px-Venn10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Venn10.svg/64px-Venn10.svg.png 2x" data-file-width="280" data-file-height="280" /></a></span> </td></tr> <tr> <th colspan="9">Binary connectives </th></tr> <tr style="background-color:#ffff66; text-align:center;"> <td><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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> = </td> <td>0 </td> <td>0 </td> <td>1 </td> <td>1 </td></tr> <tr style="background-color:#ffff66; text-align:center;"> <td><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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> = </td> <td>0 </td> <td>1 </td> <td>0 </td> <td>1 </td></tr> <tr> <td></td> <td style="text-align:left;">Proposition <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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> </td> <td></td> <td>0</td> <td>0</td> <td>1</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/File:Venn0101.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Venn0101.svg/40px-Venn0101.svg.png" decoding="async" width="40" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Venn0101.svg/60px-Venn0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Venn0101.svg/80px-Venn0101.svg.png 2x" data-file-width="380" data-file-height="280" /></a></span> </td></tr> <tr> <td></td> <td style="text-align:left;">Proposition <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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> </td> <td></td> <td>0</td> <td>1</td> <td>0</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/File:Venn0011.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Venn0011.svg/40px-Venn0011.svg.png" decoding="async" width="40" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Venn0011.svg/60px-Venn0011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Venn0011.svg/80px-Venn0011.svg.png 2x" data-file-width="380" data-file-height="280" /></a></span> </td></tr> <tr> <td>∧</td> <td style="text-align:left;"><a href="/wiki/Logical_conjunction" title="Logical conjunction">Conjunction</a> </td> <td></td> <td>0</td> <td>0</td> <td>0</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/File:Venn0001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/40px-Venn0001.svg.png" decoding="async" width="40" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/60px-Venn0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/80px-Venn0001.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span> </td></tr> <tr> <td>↑</td> <td style="text-align:left;"><a href="/wiki/Sheffer_stroke" title="Sheffer stroke">Alternative denial</a> </td> <td></td> <td>1</td> <td>1</td> <td>1</td> <td>0</td> <td><span typeof="mw:File"><a href="/wiki/File:Venn1110.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Venn1110.svg/40px-Venn1110.svg.png" decoding="async" width="40" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Venn1110.svg/60px-Venn1110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Venn1110.svg/80px-Venn1110.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span> </td></tr> <tr> <td>∨</td> <td style="text-align:left;"><a href="/wiki/Logical_disjunction" title="Logical disjunction">Disjunction</a> </td> <td></td> <td>0</td> <td>1</td> <td>1</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/File:Venn0111.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/40px-Venn0111.svg.png" decoding="async" width="40" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/60px-Venn0111.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/80px-Venn0111.svg.png 2x" data-file-width="380" data-file-height="280" /></a></span> </td></tr> <tr> <td>↓</td> <td style="text-align:left;"><a href="/wiki/Logical_NOR" title="Logical NOR">Joint denial</a> </td> <td></td> <td>1</td> <td>0</td> <td>0</td> <td>0</td> <td><span typeof="mw:File"><a href="/wiki/File:Venn1000.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Venn1000.svg/40px-Venn1000.svg.png" decoding="async" width="40" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Venn1000.svg/60px-Venn1000.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Venn1000.svg/80px-Venn1000.svg.png 2x" data-file-width="380" data-file-height="280" /></a></span> </td></tr> <tr> <td>→</td> <td style="text-align:left;"><a href="/wiki/Material_conditional" title="Material conditional">Material conditional</a> </td> <td></td> <td>1</td> <td>1</td> <td>0</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/File:Venn1011.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Venn1011.svg/40px-Venn1011.svg.png" decoding="async" width="40" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Venn1011.svg/60px-Venn1011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Venn1011.svg/80px-Venn1011.svg.png 2x" data-file-width="380" data-file-height="280" /></a></span> </td></tr> <tr> <td><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 \nleftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nleftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce85ed756bc5a6cdf0f62892f57a6a1f96803ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nleftrightarrow }"></span></td> <td style="text-align:left;"><a href="/wiki/Exclusive_or" title="Exclusive or">Exclusive or</a> </td> <td></td> <td>0</td> <td>1</td> <td>1</td> <td>0</td> <td><span typeof="mw:File"><a href="/wiki/File:Venn0110.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/40px-Venn0110.svg.png" decoding="async" width="40" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/60px-Venn0110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/80px-Venn0110.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span> </td></tr> <tr> <td>↔</td> <td style="text-align:left;"><a href="/wiki/Logical_biconditional" title="Logical biconditional">Biconditional</a> </td> <td></td> <td>1</td> <td>0</td> <td>0</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/File:Venn1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/40px-Venn1001.svg.png" decoding="async" width="40" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/60px-Venn1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/80px-Venn1001.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span> </td></tr> <tr> <td>←</td> <td style="text-align:left;"><a href="/wiki/Converse_implication" class="mw-redirect" title="Converse implication">Converse implication</a> </td> <td></td> <td>1</td> <td>0</td> <td>1</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/File:Venn1101.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Venn1101.svg/40px-Venn1101.svg.png" decoding="async" width="40" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Venn1101.svg/60px-Venn1101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Venn1101.svg/80px-Venn1101.svg.png 2x" data-file-width="380" data-file-height="280" /></a></span> </td></tr> <tr> <td colspan="8" style="text-align:center;"><a href="/wiki/Truth_function#Table_of_binary_truth_functions" title="Truth function">More information</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="List_of_common_logical_connectives">List of common logical connectives</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_connective&action=edit&section=2" title="Edit section: List of common logical connectives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Commonly used logical connectives include the following ones.<sup id="cite_ref-chao2023_2-0" class="reference"><a href="#cite_note-chao2023-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <ul><li><a href="/wiki/Negation" title="Negation">Negation (not)</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 \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></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 \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> (prefix) in which <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></span> is the most modern and widely used, 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 \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span> is used by many people too;</li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">Conjunction (and)</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 \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></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 \&}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \&}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7e8142574f96000e13827167bdfdd69e38076f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \&}"></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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> (prefix) in which <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 \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> is the most modern and widely used;</li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">Disjunction (or)</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 \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> (prefix) in which <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 \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span> is the most modern and widely used;</li> <li><a href="/wiki/Material_conditional" title="Material conditional">Implication (if...then)</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 \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"></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 \supset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊃</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27bfe0828a2ed4c9c6b70987a85c02a1f005843c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \supset }"></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 \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> (prefix) in which <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 \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"></span> is the most modern and widely used, 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 \supset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊃</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27bfe0828a2ed4c9c6b70987a85c02a1f005843c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \supset }"></span> is used by many people too;</li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">Equivalence (if and only if)</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 \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></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 \subset \!\!\!\supset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊂</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mo>⊃</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subset \!\!\!\supset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e0e7433de8951430cafd5d13f6cf17286dcaa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.455ex; height:1.843ex;" alt="{\displaystyle \subset \!\!\!\supset }"></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 \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></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 \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"></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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> (prefix) in which <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 \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></span> is the most modern and widely used, 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 \subset \!\!\!\supset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊂</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mo>⊃</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subset \!\!\!\supset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e0e7433de8951430cafd5d13f6cf17286dcaa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.455ex; height:1.843ex;" alt="{\displaystyle \subset \!\!\!\supset }"></span> may be also a good choice compared to <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 \supset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊃</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27bfe0828a2ed4c9c6b70987a85c02a1f005843c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \supset }"></span> denoting implication just 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 \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></span> to <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 \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"></span>.</li></ul> <p>For example, the meaning of the statements <i>it is raining</i> (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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>) and <i>I am indoors</i> (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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) is transformed, when the two are combined with logical connectives: </p> <ul><li>It is <i><b>not</b></i> raining (<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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b198c79234d926cbee42c0f271d903ea55dc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.72ex; height:2.009ex;" alt="{\displaystyle \neg p}"></span>);</li> <li>It is raining <i><b>and</b></i> I am indoors (<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\wedge q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∧</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\wedge q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adbba104865105a200e955131cec7f3dcc7faef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\wedge q}"></span>);</li> <li>It is raining <i><b>or</b></i> I am indoors (<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\lor q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∨</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\lor q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18488a87bb6ae7abe9649f7e195980041c1c53dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\lor q}"></span>);</li> <li><i><b>If</b></i> it is raining, <i><b>then</b></i> I am indoors (<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\rightarrow q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\rightarrow q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e3f9e5de9aaead8d19411bb3ad0dc490a50ec69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\rightarrow q}"></span>);</li> <li><i><b>If</b></i> I am indoors, <i><b>then</b></i> it is raining (<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 q\rightarrow p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\rightarrow p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c4b7a404e70f10b9e0734cf1d64777619bf271" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.853ex; height:2.176ex;" alt="{\displaystyle q\rightarrow p}"></span>);</li> <li>I am indoors <i><b>if and only if</b></i> it is raining (<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\leftrightarrow q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\leftrightarrow q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85f8399d94c0a05764a2961ddee15ee8ac2a8ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\leftrightarrow q}"></span>).</li></ul> <p>It is also common to consider the <i>always true</i> formula and the <i>always false</i> formula to be connective (in which case they are <a href="/wiki/Nullary" class="mw-redirect" title="Nullary">nullary</a>). </p> <ul><li><a href="/wiki/Truth" title="Truth">True</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 \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }"></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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> (prefix), or <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 \mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06cc73e47284b51d2ab60d333176c2366a333e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathrm {T} }"></span>;</li> <li><a href="/wiki/False_(logic)" title="False (logic)">False</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 \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }"></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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></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 O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> (prefix), or <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 \mathrm {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {F} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd88ce01cebb719c991531bf9a76dacc204f6d1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.518ex; height:2.176ex;" alt="{\displaystyle \mathrm {F} }"></span>.</li></ul> <p>This table summarizes the terminology: </p> <table class="wikitable" style="margin:1em auto; text-align:left;"> <tbody><tr> <th>Connective </th> <th>In English </th> <th>Noun for parts </th> <th>Verb phrase </th></tr> <tr> <th>Conjunction </th> <td>Both A and B </td> <td>conjunct </td> <td>A and B are conjoined </td></tr> <tr> <th>Disjunction </th> <td>Either A or B, or both </td> <td>disjunct </td> <td>A and B are disjoined </td></tr> <tr> <th>Negation </th> <td>It is not the case that A </td> <td>negatum/negand </td> <td>A is negated </td></tr> <tr> <th>Conditional </th> <td>If A, then B </td> <td>antecedent, consequent </td> <td>B is implied by A </td></tr> <tr> <th>Biconditional </th> <td>A if, and only if, B </td> <td>equivalents </td> <td>A and B are equivalent </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="History_of_notations">History of notations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_connective&action=edit&section=3" title="Edit section: History of notations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Negation: the symbol <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></span> appeared in <a href="/wiki/Arend_Heyting" title="Arend Heyting">Heyting</a> in 1930<sup id="cite_ref-heyting1930_3-0" class="reference"><a href="#cite_note-heyting1930-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> (compare to <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege</a>'s symbol ⫟ in his <a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a><sup id="cite_ref-frege1879a_5-0" class="reference"><a href="#cite_note-frege1879a-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>); the symbol <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 \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span> appeared in <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell</a> in 1908;<sup id="cite_ref-autogenerated222_6-0" class="reference"><a href="#cite_note-autogenerated222-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> an alternative notation is to add a horizontal line on top of the formula, as in <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 {\overline {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3884eccbf972d435831da42dd4154e192d550e16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.032ex; width:1.316ex; height:2.676ex;" alt="{\displaystyle {\overline {p}}}"></span>; another alternative notation is to use a <a href="/wiki/Prime_(symbol)" title="Prime (symbol)">prime symbol</a> as in <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"> <msup> <mi>p</mi> <mo>′</mo> </msup> </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/40e623e3163571a220ed60ecb31aa78c24104b85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.944ex; height:2.843ex;" alt="{\displaystyle p'}"></span>.</li> <li>Conjunction: the symbol <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 \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> appeared in Heyting in 1930<sup id="cite_ref-heyting1930_3-1" class="reference"><a href="#cite_note-heyting1930-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> (compare to <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano</a>'s use of the set-theoretic notation of <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</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 \cap }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cap }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4e886e6f5a28a33e073fb108440c152ecfe2d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cap }"></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>); the symbol <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 \&}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \&}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7e8142574f96000e13827167bdfdd69e38076f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \&}"></span> appeared at least in <a href="/wiki/Moses_Sch%C3%B6nfinkel" title="Moses Schönfinkel">Schönfinkel</a> in 1924;<sup id="cite_ref-autogenerated1924_8-0" class="reference"><a href="#cite_note-autogenerated1924-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> the symbol <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 \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"></span> comes from <a href="/wiki/George_Boole" title="George Boole">Boole</a>'s interpretation of logic as an <a href="/wiki/Elementary_algebra" title="Elementary algebra">elementary algebra</a>.</li> <li>Disjunction: the symbol <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 \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span> appeared in <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell</a> in 1908<sup id="cite_ref-autogenerated222_6-1" class="reference"><a href="#cite_note-autogenerated222-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> (compare to <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano</a>'s use of the set-theoretic notation of <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</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 \cup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∪</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ff7d0293ad19b43524a133ae5129f3d71f2040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cup }"></span>); the symbol <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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> is also used, in spite of the ambiguity coming from the fact that the <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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> of ordinary <a href="/wiki/Elementary_algebra" title="Elementary algebra">elementary algebra</a> is an <a href="/wiki/Exclusive_or" title="Exclusive or">exclusive or</a> when interpreted logically in a two-element <a href="/wiki/Boolean_ring" title="Boolean ring">ring</a>; punctually in the history 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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> together with a dot in the lower right corner has been used by <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce</a>.<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></li> <li>Implication: the symbol <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 \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"></span> appeared in <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a> in 1918;<sup id="cite_ref-hilbert1918_10-0" class="reference"><a href="#cite_note-hilbert1918-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 76">: 76 </span></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 \supset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊃</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27bfe0828a2ed4c9c6b70987a85c02a1f005843c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \supset }"></span> was used by Russell in 1908<sup id="cite_ref-autogenerated222_6-2" class="reference"><a href="#cite_note-autogenerated222-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> (compare to Peano's Ɔ the inverted C); <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 \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></span> appeared in <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki</a> in 1954.<sup id="cite_ref-bourbaki1954a_11-0" class="reference"><a href="#cite_note-bourbaki1954a-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li> <li>Equivalence: the symbol <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 \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"></span> in <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege</a> in 1879;<sup id="cite_ref-frege1879b_12-0" class="reference"><a href="#cite_note-frege1879b-12"><span class="cite-bracket">[</span>12<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 \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></span> in Becker in 1933 (not the first time and for this see the following);<sup id="cite_ref-becker1933_13-0" class="reference"><a href="#cite_note-becker1933-13"><span class="cite-bracket">[</span>13<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 \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></span> appeared in <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki</a> in 1954;<sup id="cite_ref-bourbaki1954b_14-0" class="reference"><a href="#cite_note-bourbaki1954b-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> other symbols appeared punctually in the history, such 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 \supset \subset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊃⊂</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supset \subset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e744935a1dc666f388c853967cf5f812c2ace5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.616ex; height:1.843ex;" alt="{\displaystyle \supset \subset }"></span> in <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gentzen</a>,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<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 \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span> in Schönfinkel<sup id="cite_ref-autogenerated1924_8-1" class="reference"><a href="#cite_note-autogenerated1924-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> or <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 \subset \supset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊂⊃</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subset \supset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9dc56fa9c4547807a5ad92bec8bd59c82a34aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.616ex; height:1.843ex;" alt="{\displaystyle \subset \supset }"></span> in Chazal, <sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></li> <li>True: the symbol <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> comes from <a href="/wiki/George_Boole" title="George Boole">Boole</a>'s interpretation of logic as an <a href="/wiki/Elementary_algebra" title="Elementary algebra">elementary algebra</a> over the <a href="/wiki/Two-element_Boolean_algebra" title="Two-element Boolean algebra">two-element Boolean algebra</a>; other notations include <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 \mathrm {V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {V} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/664209da7650f00b3507efe25f89aeff9783146c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {V} }"></span> (abbreviation for the Latin word "verum") to be found in Peano in 1889.</li> <li>False: the symbol <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> comes also from Boole's interpretation of logic as a ring; other notations include <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 \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span> (rotated <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 \mathrm {V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {V} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/664209da7650f00b3507efe25f89aeff9783146c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {V} }"></span>) to be found in Peano in 1889.</li></ul> <p>Some authors used letters for connectives: <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 {u.} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">u</mi> <mo>.</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {u.} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54375714c371ec37041df13a1773966ff903703b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:1.676ex;" alt="{\displaystyle \operatorname {u.} }"></span> for conjunction (German's "und" for "and") 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 \operatorname {o.} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">o</mi> <mo>.</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {o.} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f241c418c21b6c63cead2eb35d2eb69b195bf1f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:1.676ex;" alt="{\displaystyle \operatorname {o.} }"></span> for disjunction (German's "oder" for "or") in early works by Hilbert (1904);<sup id="cite_ref-hilbert1904_17-0" class="reference"><a href="#cite_note-hilbert1904-17"><span class="cite-bracket">[</span>17<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 Np}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Np}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48d86ac82f9845ca629f9b3496c0d82908fb60f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.233ex; height:2.509ex;" alt="{\displaystyle Np}"></span> for negation, <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 Kpq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mi>p</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Kpq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abc986b8b9fc6a19ddd831ee224ee22d3bfc14d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.305ex; height:2.509ex;" alt="{\displaystyle Kpq}"></span> for conjunction, <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 Dpq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>p</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Dpq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9894093c67e1c3ee00f8b45964066e0a6d74f5c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.163ex; height:2.509ex;" alt="{\displaystyle Dpq}"></span> for alternative denial, <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 Apq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>p</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Apq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b704493c936083991cd74c2f28967ad1d1fd0dd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.982ex; height:2.509ex;" alt="{\displaystyle Apq}"></span> for disjunction, <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 Cpq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>p</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Cpq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d77892e3efc7c577e93c08774120a94c786148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.005ex; height:2.509ex;" alt="{\displaystyle Cpq}"></span> for implication, <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 Epq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mi>p</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Epq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b2ada8e6588fa4b4932f6b6263025e5d223914" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.015ex; height:2.509ex;" alt="{\displaystyle Epq}"></span> for biconditional in <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Łukasiewicz</a> in 1929. </p> <div class="mw-heading mw-heading3"><h3 id="Redundancy">Redundancy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_connective&action=edit&section=4" title="Edit section: Redundancy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Such a logical connective as <a href="/wiki/Converse_implication" class="mw-redirect" title="Converse implication">converse implication</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 \leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">←</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0fb4bce772117bbaf55b7ca1539ceff9ae218c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftarrow }"></span>" is actually the same as <a href="/wiki/Material_conditional" title="Material conditional">material conditional</a> with swapped arguments; thus, the symbol for converse implication is redundant. In some logical calculi (notably, in <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>), certain essentially different compound statements are <a href="/wiki/Logical_equivalence" title="Logical equivalence">logically equivalent</a>. A less <a href="/wiki/Triviality_(mathematics)" title="Triviality (mathematics)">trivial</a> example of a redundancy is the classical equivalence between <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 p\vee q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg p\vee q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcd38fdfa90c931b98234caec5f6d4060f92123c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.372ex; height:2.343ex;" alt="{\displaystyle \neg p\vee q}"></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 p\to q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\to q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fccb3827df1efe7930f9d1febd15d2359971b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\to q}"></span>. Therefore, a classical-based logical system does not need the conditional operator "<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 \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"></span>" 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 \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></span>" (not) 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 \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span>" (or) are already in use, or may use the "<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 \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"></span>" only as a <a href="/wiki/Syntactic_sugar" title="Syntactic sugar">syntactic sugar</a> for a compound having one negation and one disjunction. </p><p>There are sixteen <a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a> associating the input <a href="/wiki/Truth_value" title="Truth value">truth values</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}"> <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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> with four-digit <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> outputs.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> These correspond to possible choices of binary logical connectives for <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>. Different implementations of classical logic can choose different <a href="/wiki/Functional_completeness" title="Functional completeness">functionally complete</a> subsets of connectives. </p><p>One approach is to choose a <i>minimal</i> set, and define other connectives by some logical form, as in the example with the material conditional above. The following are the <a href="/wiki/Functional_completeness#Minimal_functionally_complete_operator_sets" title="Functional completeness">minimal functionally complete sets of operators</a> in classical logic whose arities do not exceed 2: </p> <dl><dt>One element</dt> <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 \{\uparrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">↑</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\uparrow \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32a3f2caaa359634df304ad5964711507b5a2ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{\uparrow \}}"></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 \{\downarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">↓</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\downarrow \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a680fc5f220df2ab4fd6c8a144fa44cb8d5aad4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{\downarrow \}}"></span>.</dd> <dt>Two elements</dt> <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 \{\vee ,\neg \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>∨</mo> <mo>,</mo> <mi mathvariant="normal">¬</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\vee ,\neg \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2e9c0ba8d7e6d25eb7b7c42514a8e5455533aa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:2.843ex;" alt="{\displaystyle \{\vee ,\neg \}}"></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 \{\wedge ,\neg \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>∧</mo> <mo>,</mo> <mi mathvariant="normal">¬</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\wedge ,\neg \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cea2d696984c094e742f17b4a0980812c34c5311" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:2.843ex;" alt="{\displaystyle \{\wedge ,\neg \}}"></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 \{\to ,\neg \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mi mathvariant="normal">¬</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\to ,\neg \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce75e2f3daac9cec837fe974c6afd887701b5a64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.233ex; height:2.843ex;" alt="{\displaystyle \{\to ,\neg \}}"></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 \{\gets ,\neg \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">←</mo> <mo>,</mo> <mi mathvariant="normal">¬</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\gets ,\neg \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e85dd5c575acd2855f93a08b102eabb6a2c2086" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.233ex; height:2.843ex;" alt="{\displaystyle \{\gets ,\neg \}}"></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 \{\to ,\bot \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mi mathvariant="normal">⊥</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\to ,\bot \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55ebcc40157f1c4fb9352bcda77afdcfe135b11b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.491ex; height:2.843ex;" alt="{\displaystyle \{\to ,\bot \}}"></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 \{\gets ,\bot \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">←</mo> <mo>,</mo> <mi mathvariant="normal">⊥</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\gets ,\bot \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fcc3d747c0f79a70db6878511e5008e48df1c95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.491ex; height:2.843ex;" alt="{\displaystyle \{\gets ,\bot \}}"></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 \{\to ,\nleftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo>↮</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\to ,\nleftrightarrow \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2d41b0afc20bb0ee130b470c495068a8771adb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle \{\to ,\nleftrightarrow \}}"></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 \{\gets ,\nleftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">←</mo> <mo>,</mo> <mo>↮</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\gets ,\nleftrightarrow \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25a8500d440a083dec2165e932ec595f2bd2190a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle \{\gets ,\nleftrightarrow \}}"></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 \{\to ,\nrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo>↛</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\to ,\nrightarrow \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ead9c6cca0d49e461ddc6772d82b6d6d367b4d61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle \{\to ,\nrightarrow \}}"></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 \{\to ,\nleftarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo>↚</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\to ,\nleftarrow \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daab405ca48bd7c7e387d635272447b59dd88a0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle \{\to ,\nleftarrow \}}"></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 \{\gets ,\nrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">←</mo> <mo>,</mo> <mo>↛</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\gets ,\nrightarrow \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5983419b30c5273441d0dc24b634b8206fe1c678" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle \{\gets ,\nrightarrow \}}"></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 \{\gets ,\nleftarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">←</mo> <mo>,</mo> <mo>↚</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\gets ,\nleftarrow \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54bb952bacbeef29cd08883ae5eb3619f32b103a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle \{\gets ,\nleftarrow \}}"></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 \{\nrightarrow ,\neg \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>↛</mo> <mo>,</mo> <mi mathvariant="normal">¬</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\nrightarrow ,\neg \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b20fab34e5fb82fefa38d5066ad53924c3bce448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.233ex; height:2.843ex;" alt="{\displaystyle \{\nrightarrow ,\neg \}}"></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 \{\nleftarrow ,\neg \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>↚</mo> <mo>,</mo> <mi mathvariant="normal">¬</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\nleftarrow ,\neg \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43288ebe50fa63d70ef9157560434678ca37ba48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.233ex; height:2.843ex;" alt="{\displaystyle \{\nleftarrow ,\neg \}}"></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 \{\nrightarrow ,\top \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>↛</mo> <mo>,</mo> <mi mathvariant="normal">⊤</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\nrightarrow ,\top \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8a953e72b9d747fc5b8ce20875c0851ec5029f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.491ex; height:2.843ex;" alt="{\displaystyle \{\nrightarrow ,\top \}}"></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 \{\nleftarrow ,\top \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>↚</mo> <mo>,</mo> <mi mathvariant="normal">⊤</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\nleftarrow ,\top \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5206ca5d7c500efe3089b24d9808cde1dd705db7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.491ex; height:2.843ex;" alt="{\displaystyle \{\nleftarrow ,\top \}}"></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 \{\nrightarrow ,\leftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>↛</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\nrightarrow ,\leftrightarrow \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/653c466be112e74751723a7139883dc2d5e7d4a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle \{\nrightarrow ,\leftrightarrow \}}"></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 \{\nleftarrow ,\leftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>↚</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\nleftarrow ,\leftrightarrow \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8bc42ae74459b1c57a928407c5921e33cb8e597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle \{\nleftarrow ,\leftrightarrow \}}"></span>.</dd> <dt>Three elements</dt> <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 \{\lor ,\leftrightarrow ,\bot \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo>,</mo> <mi mathvariant="normal">⊥</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\lor ,\leftrightarrow ,\bot \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c4d581f0cfe84e08c450cc07d3f2fb33fb1842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.075ex; height:2.843ex;" alt="{\displaystyle \{\lor ,\leftrightarrow ,\bot \}}"></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 \{\lor ,\leftrightarrow ,\nleftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo>,</mo> <mo>↮</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\lor ,\leftrightarrow ,\nleftrightarrow \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/634408661bb912f434b14ec6561f21d74b2af415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.591ex; height:2.843ex;" alt="{\displaystyle \{\lor ,\leftrightarrow ,\nleftrightarrow \}}"></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 \{\lor ,\nleftrightarrow ,\top \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>∨</mo> <mo>,</mo> <mo>↮</mo> <mo>,</mo> <mi mathvariant="normal">⊤</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\lor ,\nleftrightarrow ,\top \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acf8252f4312b6a20a7f0f2ca495aed3f7cb861e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.075ex; height:2.843ex;" alt="{\displaystyle \{\lor ,\nleftrightarrow ,\top \}}"></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 \{\land ,\leftrightarrow ,\bot \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>∧</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo>,</mo> <mi mathvariant="normal">⊥</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\land ,\leftrightarrow ,\bot \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b45580e38a2f5365ddb6aa6c736d061dd71b601" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.075ex; height:2.843ex;" alt="{\displaystyle \{\land ,\leftrightarrow ,\bot \}}"></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 \{\land ,\leftrightarrow ,\nleftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>∧</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo>,</mo> <mo>↮</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\land ,\leftrightarrow ,\nleftrightarrow \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d83f94b1b629f429470d329fc4c35af42068f797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.591ex; height:2.843ex;" alt="{\displaystyle \{\land ,\leftrightarrow ,\nleftrightarrow \}}"></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 \{\land ,\nleftrightarrow ,\top \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>∧</mo> <mo>,</mo> <mo>↮</mo> <mo>,</mo> <mi mathvariant="normal">⊤</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\land ,\nleftrightarrow ,\top \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b7dfe68db84b36a082a02fe126099bba736ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.075ex; height:2.843ex;" alt="{\displaystyle \{\land ,\nleftrightarrow ,\top \}}"></span>.</dd></dl> <p>Another approach is to use with equal rights connectives of a certain convenient and functionally complete, but <i>not minimal</i> set. This approach requires more propositional <a href="/wiki/Axiom" title="Axiom">axioms</a>, and each equivalence between logical forms must be either an <a href="/wiki/Axiom" title="Axiom">axiom</a> or provable as a theorem. </p><p>The situation, however, is more complicated in <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>. Of its five connectives, {∧, ∨, →, ¬, ⊥}, only negation "¬" can be reduced to other connectives (see <a href="/wiki/False_(logic)#False,_negation_and_contradiction" title="False (logic)">False (logic) § False, negation and contradiction</a> for more). Neither conjunction, disjunction, nor material conditional has an equivalent form constructed from the other four logical connectives. </p> <div class="mw-heading mw-heading2"><h2 id="Natural_language">Natural language</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_connective&action=edit&section=5" title="Edit section: Natural language"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard logical connectives of classical logic have rough equivalents in the grammars of natural languages. In <a href="/wiki/English_language" title="English language">English</a>, as in many languages, such expressions are typically <a href="/wiki/Grammatical_conjunction" class="mw-redirect" title="Grammatical conjunction">grammatical conjunctions</a>. However, they can also take the form of <a href="/wiki/Complementizer" title="Complementizer">complementizers</a>, <a href="/wiki/Verb" title="Verb">verb</a> <a href="/wiki/Suffix" title="Suffix">suffixes</a>, and <a href="/wiki/Grammatical_particle" title="Grammatical particle">particles</a>. The <a href="/wiki/Denotation" title="Denotation">denotations</a> of natural language connectives is a major topic of research in <a href="/wiki/Formal_semantics_(natural_language)" title="Formal semantics (natural language)">formal semantics</a>, a field that studies the logical structure of natural languages. </p><p>The meanings of natural language connectives are not precisely identical to their nearest equivalents in classical logic. In particular, disjunction can receive an <a href="/wiki/Exclusive_disjunction" class="mw-redirect" title="Exclusive disjunction">exclusive interpretation</a> in many languages. Some researchers have taken this fact as evidence that natural language <a href="/wiki/Semantics_(natural_language)" class="mw-redirect" title="Semantics (natural language)">semantics</a> is <a href="/wiki/Nonclassical_logic" class="mw-redirect" title="Nonclassical logic">nonclassical</a>. However, others maintain classical semantics by positing <a href="/wiki/Pragmatics" title="Pragmatics">pragmatic</a> accounts of exclusivity which create the illusion of nonclassicality. In such accounts, exclusivity is typically treated as a <a href="/wiki/Scalar_implicature" title="Scalar implicature">scalar implicature</a>. Related puzzles involving disjunction include <a href="/wiki/Free_choice_inference" title="Free choice inference">free choice inferences</a>, <a href="/wiki/Hurford_disjunction" title="Hurford disjunction">Hurford's Constraint</a>, and the contribution of disjunction in <a href="/wiki/Alternative_question" class="mw-redirect" title="Alternative question">alternative questions</a>. </p><p>Other apparent discrepancies between natural language and classical logic include the <a href="/wiki/Paradoxes_of_material_implication" title="Paradoxes of material implication">paradoxes of material implication</a>, <a href="/wiki/Donkey_anaphora" class="mw-redirect" title="Donkey anaphora">donkey anaphora</a> and the problem of <a href="/wiki/Counterfactual_conditionals" class="mw-redirect" title="Counterfactual conditionals">counterfactual conditionals</a>. These phenomena have been taken as motivation for identifying the denotations of natural language conditionals with logical operators including the <a href="/wiki/Strict_conditional" title="Strict conditional">strict conditional</a>, the <a href="/wiki/Variably_strict_conditional" class="mw-redirect" title="Variably strict conditional">variably strict conditional</a>, as well as various <a href="/wiki/Dynamic_semantics" title="Dynamic semantics">dynamic</a> operators. </p><p>The following table shows the standard classically definable approximations for the English connectives. </p> <table class="wikitable sortable"> <tbody><tr> <th>English word</th> <th>Connective</th> <th>Symbol</th> <th>Logical gate </th></tr> <tr> <td>not</td> <td><a href="/wiki/Negation" title="Negation">negation</a></td> <td><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></span></td> <td><a href="/wiki/Inverter_(logic_gate)" title="Inverter (logic gate)">NOT</a> </td></tr> <tr> <td>and</td> <td><a href="/wiki/Logical_conjunction" title="Logical conjunction">conjunction</a></td> <td><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 \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span></td> <td><a href="/wiki/AND_gate" title="AND gate">AND</a> </td></tr> <tr> <td>or</td> <td><a href="/wiki/Logical_disjunction" title="Logical disjunction">disjunction</a></td> <td><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 \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span></td> <td><a href="/wiki/OR_gate" title="OR gate">OR</a> </td></tr> <tr> <td>if...then</td> <td><a href="/wiki/Material_conditional" title="Material conditional">material implication</a></td> <td><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 \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"></span></td> <td><a href="/wiki/IMPLY_gate" title="IMPLY gate">IMPLY</a> </td></tr> <tr> <td>...if</td> <td><a href="/wiki/Converse_implication" class="mw-redirect" title="Converse implication">converse implication</a></td> <td><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 \leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">←</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0fb4bce772117bbaf55b7ca1539ceff9ae218c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftarrow }"></span></td> <td> </td></tr> <tr> <td>either...or</td> <td><a href="/wiki/Exclusive_or" title="Exclusive or">exclusive disjunction</a></td> <td><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 \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></span></td> <td><a href="/wiki/XOR_gate" title="XOR gate">XOR</a> </td></tr> <tr> <td>if and only if</td> <td><a href="/wiki/Logical_biconditional" title="Logical biconditional">biconditional</a></td> <td><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 \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></span></td> <td><a href="/wiki/XNOR_gate" title="XNOR gate">XNOR</a> </td></tr> <tr> <td>not both</td> <td><a href="/wiki/Sheffer_stroke" title="Sheffer stroke">alternative denial</a></td> <td><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 \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span></td> <td><a href="/wiki/NAND_gate" title="NAND gate">NAND</a> </td></tr> <tr> <td>neither...nor</td> <td><a href="/wiki/Logical_NOR" title="Logical NOR">joint denial</a></td> <td><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 \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4618f22b0f780805eb94bb407578d9bc9487947a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \downarrow }"></span></td> <td><a href="/wiki/NOR_gate" title="NOR gate">NOR</a> </td></tr> <tr> <td>but not</td> <td><a href="/wiki/Material_nonimplication" title="Material nonimplication">material nonimplication</a></td> <td><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 \not \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↛</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28250ddf37312b4f73f5c969749c95f6ec275849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \not \to }"></span></td> <td><a href="/wiki/NIMPLY_gate" title="NIMPLY gate">NIMPLY</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_connective&action=edit&section=6" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some logical connectives possess properties that may be expressed in the theorems containing the connective. Some of those properties that a logical connective may have are: </p> <dl><dt><a href="/wiki/Associativity" class="mw-redirect" title="Associativity">Associativity</a></dt> <dd>Within an expression containing two or more of the same associative connectives in a row, the order of the operations does not matter as long as the sequence of the operands is not changed.</dd> <dt><a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">Commutativity</a></dt> <dd>The operands of the connective may be swapped, preserving logical equivalence to the original expression.</dd> <dt><a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">Distributivity</a></dt> <dd>A connective denoted by · distributes over another connective denoted by +, if <span class="texhtml"><i>a</i> · (<i>b</i> + <i>c</i>) = (<i>a</i> · <i>b</i>) + (<i>a</i> · <i>c</i>)</span> for all operands <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, <span class="texhtml mvar" style="font-style:italic;">c</span>.</dd> <dt><a href="/wiki/Idempotence" title="Idempotence">Idempotence</a></dt> <dd>Whenever the operands of the operation are the same, the compound is logically equivalent to the operand.</dd> <dt><a href="/wiki/Absorption_Law" class="mw-redirect" title="Absorption Law">Absorption</a></dt> <dd>A pair of connectives ∧, ∨ satisfies the absorption law 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 a\land (a\lor b)=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∨</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\land (a\lor b)=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aabf1bc4aa530b4243da83d1676e9c6fbb83b46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.76ex; height:2.843ex;" alt="{\displaystyle a\land (a\lor b)=a}"></span> for all operands <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>.</dd> <dt><a href="/wiki/Monotonicity" class="mw-redirect" title="Monotonicity">Monotonicity</a></dt> <dd>If <i>f</i>(<i>a</i><sub>1</sub>, ..., <i>a</i><sub><i>n</i></sub>) ≤ <i>f</i>(<i>b</i><sub>1</sub>, ..., <i>b</i><sub><i>n</i></sub>) for all <i>a</i><sub>1</sub>, ..., <i>a</i><sub><i>n</i></sub>, <i>b</i><sub>1</sub>, ..., <i>b</i><sub><i>n</i></sub> ∈ {0,1} such that <i>a</i><sub>1</sub> ≤ <i>b</i><sub>1</sub>, <i>a</i><sub>2</sub> ≤ <i>b</i><sub>2</sub>, ..., <i>a</i><sub><i>n</i></sub> ≤ <i>b</i><sub><i>n</i></sub>. E.g., ∨, ∧, ⊤, ⊥.</dd> <dt><a href="/wiki/Affine_transformation" title="Affine transformation">Affinity</a></dt> <dd>Each variable always makes a difference in the truth-value of the operation or it never makes a difference. E.g., ¬, ↔, <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 \nleftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nleftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce85ed756bc5a6cdf0f62892f57a6a1f96803ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nleftrightarrow }"></span>, ⊤, ⊥.</dd> <dt><a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">Duality</a></dt> <dd>To read the truth-value assignments for the operation from top to bottom on its <a href="/wiki/Truth_table" title="Truth table">truth table</a> is the same as taking the complement of reading the table of the same or another connective from bottom to top. Without resorting to truth tables it may be formulated as <span class="texhtml"><i>g̃</i>(¬<i>a</i><sub>1</sub>, ..., ¬<i>a</i><sub><i>n</i></sub>) = ¬<i>g</i>(<i>a</i><sub>1</sub>, ..., <i>a</i><sub><i>n</i></sub>)</span>. E.g., ¬.</dd> <dt>Truth-preserving</dt> <dd>The compound all those arguments are tautologies is a tautology itself. E.g., ∨, ∧, ⊤, →, ↔, ⊂ (see <a href="/wiki/Validity_(logic)" title="Validity (logic)">validity</a>).</dd> <dt>Falsehood-preserving</dt> <dd>The compound all those argument are <a href="/wiki/Contradiction" title="Contradiction">contradictions</a> is a contradiction itself. E.g., ∨, ∧, <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 \nleftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nleftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce85ed756bc5a6cdf0f62892f57a6a1f96803ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nleftrightarrow }"></span>, ⊥, ⊄, ⊅ (see <a href="/wiki/Validity_(logic)" title="Validity (logic)">validity</a>).</dd> <dt><a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">Involutivity</a> (for unary connectives)</dt> <dd><span class="texhtml"><i>f</i>(<i>f</i>(<i>a</i>)) = <i>a</i></span>. E.g. negation in classical logic.</dd></dl> <p>For classical and intuitionistic logic, the "=" symbol means that corresponding implications "...→..." and "...←..." for logical compounds can be both proved as theorems, and the "≤" symbol means that "...→..." for logical compounds is a consequence of corresponding "...→..." connectives for propositional variables. Some <a href="/wiki/Many-valued_logic" title="Many-valued logic">many-valued logics</a> may have incompatible definitions of equivalence and order (entailment). </p><p>Both conjunction and disjunction are associative, commutative and idempotent in classical logic, most varieties of many-valued logic and intuitionistic logic. The same is true about distributivity of conjunction over disjunction and disjunction over conjunction, as well as for the absorption law. </p><p>In classical logic and some varieties of many-valued logic, conjunction and disjunction are dual, and negation is self-dual, the latter is also self-dual in intuitionistic logic. </p> <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-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Logical_connective&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">March 2012</span>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Order_of_precedence">Order of precedence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_connective&action=edit&section=7" title="Edit section: Order of precedence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As a way of reducing the number of necessary parentheses, one may introduce <a href="/wiki/Precedence_rule" class="mw-redirect" title="Precedence rule">precedence rules</a>: ¬ has higher precedence than ∧, ∧ higher than ∨, and ∨ higher than →. So for example, <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\vee Q\wedge {\neg R}\rightarrow S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> <mi>R</mi> </mrow> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\vee Q\wedge {\neg R}\rightarrow S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1fc0c4ae9cfddb8e178c7f59f2e77bd549afec5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.177ex; height:2.509ex;" alt="{\displaystyle P\vee Q\wedge {\neg R}\rightarrow S}"></span> is short for <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\vee (Q\wedge (\neg R)))\rightarrow S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\vee (Q\wedge (\neg R)))\rightarrow S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1893f6c4477aa8eca67ce3e0704533309fb8d4f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.604ex; height:2.843ex;" alt="{\displaystyle (P\vee (Q\wedge (\neg R)))\rightarrow S}"></span>. </p><p>Here is a table that shows a commonly used precedence of logical operators.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:35_20-0" class="reference"><a href="#cite_note-:35-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable" style="text-align: center;"> <tbody><tr> <th>Operator</th> <th>Precedence </th></tr> <tr> <td><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></span></td> <td>1 </td></tr> <tr> <td><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 \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></span></td> <td>2 </td></tr> <tr> <td><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 \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"></span></td> <td>3 </td></tr> <tr> <td><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 \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"></span></td> <td>4 </td></tr> <tr> <td><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 \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></span></td> <td>5 </td></tr></tbody></table> <p>However, not all compilers use the same order; for instance, an ordering in which disjunction is lower precedence than implication or bi-implication has also been used.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Sometimes precedence between conjunction and disjunction is unspecified requiring to provide it explicitly in given formula with parentheses. The order of precedence determines which connective is the "main connective" when interpreting a non-atomic formula. </p> <div class="mw-heading mw-heading2"><h2 id="Table_and_Hasse_diagram">Table and Hasse diagram</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_connective&action=edit&section=8" title="Edit section: Table and Hasse diagram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The 16 logical connectives can be <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partially ordered</a> to produce the following <a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a>. The partial order is defined by declaring 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 x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span> if and only if whenever <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> holds then so does <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 y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f72471aff7c6fbb27df0f971283a068efe091f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.009ex;" alt="{\displaystyle y.}"></span> </p> <table style="margin: 0 auto;"> <tbody><tr> <td><figure class="noresize" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Logical_connectives_table.svg/380px-Logical_connectives_table.svg.png" decoding="async" width="380" height="493" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Logical_connectives_table.svg/570px-Logical_connectives_table.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Logical_connectives_table.svg/760px-Logical_connectives_table.svg.png 2x" data-file-width="547" data-file-height="710" usemap="#ImageMap_848e5c72e34c9a5c" /></span><map name="ImageMap_848e5c72e34c9a5c"><area 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</td> <td><figure class="noresize" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/350px-Logical_connectives_Hasse_diagram.svg.png" decoding="async" width="350" height="495" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/525px-Logical_connectives_Hasse_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/700px-Logical_connectives_Hasse_diagram.svg.png 2x" data-file-width="744" data-file-height="1052" usemap="#ImageMap_04fce252ad7c1369" /></span><map name="ImageMap_04fce252ad7c1369"><area href="/wiki/Tautology_(logic)" shape="rect" coords="153,13,196,94" alt="X or ¬X" title="X or ¬X" /><area href="/wiki/Sheffer_stroke" shape="rect" coords="38,110,78,192" alt="¬A or ¬B" title="¬A or ¬B" /><area href="/wiki/Converse_implication" shape="rect" coords="122,109,164,192" alt="A or ¬B" title="A or ¬B" /><area href="/wiki/Material_conditional" shape="rect" coords="185,108,226,192" alt="¬A or B" title="¬A or B" /><area href="/wiki/Logical_disjunction" shape="rect" coords="270,109,312,192" alt="A or B" title="A or B" /><area href="/wiki/Negation" shape="rect" coords="6,205,48,290" alt="¬B" title="¬B" /><area href="/wiki/Negation" shape="rect" coords="69,206,111,290" alt="¬A" title="¬A" /><area href="/wiki/Exclusive_or" shape="rect" coords="131,207,173,290" alt="A xor B" title="A xor B" /><area href="/wiki/Logical_biconditional" shape="rect" coords="176,207,218,290" alt="A xnor B" title="A xnor B" /><area href="/wiki/Proposition" shape="rect" coords="239,207,280,290" alt="A" title="A" /><area href="/wiki/Proposition" shape="rect" coords="301,206,344,290" alt="B" title="B" /><area href="/wiki/Logical_NOR" shape="rect" coords="37,304,79,389" alt="¬A and ¬B" title="¬A and ¬B" /><area href="/wiki/Material_nonimplication" shape="rect" coords="122,304,164,389" alt="A and ¬B" title="A and ¬B" /><area href="/wiki/Converse_nonimplication" shape="rect" coords="184,304,227,389" alt="¬A and B" title="¬A and B" /><area href="/wiki/Logical_conjunction" shape="rect" coords="270,304,312,389" alt="A and B" title="A and B" /><area href="/wiki/Contradiction" shape="rect" coords="154,401,196,487" alt="X and ¬X" title="X and ¬X" /></map><figcaption></figcaption></figure> </td></tr> <tr> <td style="text-align:left;"><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/File:Logical_connectives_table.svg" title="commons:File:Logical connectives table.svg"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Magnify-clip_%28blue%29.svg/20px-Magnify-clip_%28blue%29.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Magnify-clip_%28blue%29.svg/30px-Magnify-clip_%28blue%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Magnify-clip_%28blue%29.svg/40px-Magnify-clip_%28blue%29.svg.png 2x" data-file-width="14" data-file-height="10" /></a></span></td> <td></td> <td style="text-align:right;"><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/File:Logical_connectives_Hasse_diagram.svg" title="commons:File:Logical connectives Hasse diagram.svg"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Magnify-clip_%28blue%29.svg/20px-Magnify-clip_%28blue%29.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Magnify-clip_%28blue%29.svg/30px-Magnify-clip_%28blue%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Magnify-clip_%28blue%29.svg/40px-Magnify-clip_%28blue%29.svg.png 2x" data-file-width="14" data-file-height="10" /></a></span>    <span typeof="mw:File"><a href="http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/1000px-Logical_connectives_Hasse_diagram.svg.png"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Loupe_light.svg/20px-Loupe_light.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Loupe_light.svg/30px-Loupe_light.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Loupe_light.svg/40px-Loupe_light.svg.png 2x" data-file-width="58" data-file-height="57" /></a></span> </td></tr></tbody></table> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_connective&action=edit&section=9" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Logical connectives are used in <a href="/wiki/Computer_science" title="Computer science">computer science</a> and in <a href="/wiki/Set_theory" title="Set theory">set theory</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Computer_science">Computer science</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_connective&action=edit&section=10" title="Edit section: Computer science"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Logic_gate" title="Logic gate">Logic gate</a></div> <p>A truth-functional approach to logical operators is implemented as <a href="/wiki/Logic_gate" title="Logic gate">logic gates</a> in <a href="/wiki/Digital_circuit" class="mw-redirect" title="Digital circuit">digital circuits</a>. Practically all digital circuits (the major exception is <a href="/wiki/DRAM" class="mw-redirect" title="DRAM">DRAM</a>) are built up from <a href="/wiki/Logical_NAND" class="mw-redirect" title="Logical NAND">NAND</a>, <a href="/wiki/Logical_NOR" title="Logical NOR">NOR</a>, <a href="/wiki/Negation" title="Negation">NOT</a>, and <a href="/wiki/Logic_gate" title="Logic gate">transmission gates</a>; see more details in <a href="/wiki/Truth_function#Computer_science" title="Truth function">Truth function in computer science</a>. Logical operators over <a href="/wiki/Bit_array" title="Bit array">bit vectors</a> (corresponding to finite <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebras</a>) are <a href="/wiki/Bitwise_operation" title="Bitwise operation">bitwise operations</a>. </p><p>But not every usage of a logical connective in <a href="/wiki/Computer_programming" title="Computer programming">computer programming</a> has a Boolean semantic. For example, <a href="/wiki/Lazy_evaluation" title="Lazy evaluation">lazy evaluation</a> is sometimes implemented for <span class="texhtml"><i>P</i> ∧ <i>Q</i></span> and <span class="texhtml"><i>P</i> ∨ <i>Q</i></span>, so these connectives are not commutative if either or both of the expressions <span class="texhtml mvar" style="font-style:italic;">P</span>, <span class="texhtml mvar" style="font-style:italic;">Q</span> have <a href="/wiki/Side_effect_(computer_science)" title="Side effect (computer science)">side effects</a>. Also, a <a href="/wiki/Conditional_(programming)" class="mw-redirect" title="Conditional (programming)">conditional</a>, which in some sense corresponds to the <a href="/wiki/Material_conditional" title="Material conditional">material conditional</a> connective, is essentially non-Boolean because for <code>if (P) then Q;</code>, the consequent Q is not executed if the <a href="/wiki/Antecedent_(logic)" title="Antecedent (logic)">antecedent</a> P is false (although a compound as a whole is successful ≈ "true" in such case). This is closer to intuitionist and <a href="/wiki/Constructive_mathematics" class="mw-redirect" title="Constructive mathematics">constructivist</a> views on the material conditional— rather than to classical logic's views. </p> <div class="mw-heading mw-heading3"><h3 id="Set_theory">Set theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_connective&action=edit&section=11" title="Edit section: Set theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Set_theory" title="Set theory">Set theory</a> and <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">Axiomatic set theory</a></div> <p>Logical connectives are used to define the fundamental operations of <a href="/wiki/Set_theory" title="Set theory">set theory</a>,<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> as follows: </p> <table class="wikitable" style="margin:1em auto; text-align:left;"> <caption>Set theory operations and connectives </caption> <tbody><tr> <th>Set operation </th> <th>Connective </th> <th>Definition </th></tr> <tr> <td><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">Intersection</a> </td> <td><a href="/wiki/Logical_conjunction" title="Logical conjunction">Conjunction</a> </td> <td><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 A\cap B=\{x:x\in A\land x\in B\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>∧</mo> <mi>x</mi> <mo>∈</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cap B=\{x:x\in A\land x\in B\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6220ee69216a0da0aa47f891c89adf5813bc61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.21ex; height:2.843ex;" alt="{\displaystyle A\cap B=\{x:x\in A\land x\in B\}}"></span><sup id="cite_ref-:0_23-0" class="reference"><a href="#cite_note-:0-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_24-0" class="reference"><a href="#cite_note-:1-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Union_(set_theory)" title="Union (set theory)">Union</a> </td> <td><a href="/wiki/Logical_disjunction" title="Logical disjunction">Disjunction</a> </td> <td><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 A\cup B=\{x:x\in A\lor x\in B\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>∨</mo> <mi>x</mi> <mo>∈</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cup B=\{x:x\in A\lor x\in B\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a991148b0bf73fd5cca039289e367cd6882b14b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.21ex; height:2.843ex;" alt="{\displaystyle A\cup B=\{x:x\in A\lor x\in B\}}"></span><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_23-1" class="reference"><a href="#cite_note-:0-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_24-1" class="reference"><a href="#cite_note-:1-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">Complement</a> </td> <td><a href="/wiki/Negation" title="Negation">Negation</a> </td> <td><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 {\overline {A}}=\{x:x\notin A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>:</mo> <mi>x</mi> <mo>∉</mo> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A}}=\{x:x\notin A\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6647863e2ace802013b4fde8ca36e21f21d72c3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.462ex; height:3.509ex;" alt="{\displaystyle {\overline {A}}=\{x:x\notin A\}}"></span><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_24-2" class="reference"><a href="#cite_note-:1-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Subset" title="Subset">Subset</a> </td> <td><a href="/wiki/Material_conditional" title="Material conditional">Implication</a> </td> <td><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 A\subseteq B\leftrightarrow (x\in A\rightarrow x\in B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>x</mi> <mo>∈</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\leftrightarrow (x\in A\rightarrow x\in B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/056f98659d9ebf9e1d10b80e2f226f3452a45b91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.491ex; height:2.843ex;" alt="{\displaystyle A\subseteq B\leftrightarrow (x\in A\rightarrow x\in B)}"></span><sup id="cite_ref-:2_29-0" class="reference"><a href="#cite_note-:2-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_24-3" class="reference"><a href="#cite_note-:1-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">Equality</a> </td> <td><a href="/wiki/Logical_biconditional" title="Logical biconditional">Biconditional</a> </td> <td><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 A=B\leftrightarrow (\forall X)[A\in X\leftrightarrow B\in X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>B</mi> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>A</mi> <mo>∈</mo> <mi>X</mi> <mo stretchy="false">↔</mo> <mi>B</mi> <mo>∈</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=B\leftrightarrow (\forall X)[A\in X\leftrightarrow B\in X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b9c356d049a459b2bde935a70b26587ff211b98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.358ex; height:2.843ex;" alt="{\displaystyle A=B\leftrightarrow (\forall X)[A\in X\leftrightarrow B\in X]}"></span><sup id="cite_ref-:2_29-1" class="reference"><a href="#cite_note-:2-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_24-4" class="reference"><a href="#cite_note-:1-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </td></tr></tbody></table> <p>This definition of set equality is equivalent to the <a href="/wiki/Axiom_of_extensionality" title="Axiom of extensionality">axiom of extensionality</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=Logical_connective&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output 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topics">List of Boolean algebra topics</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">Logical conjunction</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">Logical constant</a></li> <li><a href="/wiki/Modal_operator" title="Modal operator">Modal operator</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Term_logic" title="Term logic">Term logic</a></li> <li><a href="/wiki/Tetralemma" title="Tetralemma">Tetralemma</a></li> <li><a href="/wiki/Truth_function" title="Truth function">Truth function</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth table</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth values</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_connective&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style 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 id="CITEREFCogwheel" class="citation web cs1">Cogwheel. <a rel="nofollow" class="external text" href="https://stackoverflow.com/questions/3154132/what-is-the-difference-between-logical-and-conditional-and-or-in-c">"What is the difference between logical and conditional /operator/"</a>. <i>Stack Overflow</i><span class="reference-accessdate">. Retrieved <span class="nowrap">9 April</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stack+Overflow&rft.atitle=What+is+the+difference+between+logical+and+conditional+%2Foperator%2F&rft.au=Cogwheel&rft_id=https%3A%2F%2Fstackoverflow.com%2Fquestions%2F3154132%2Fwhat-is-the-difference-between-logical-and-conditional-and-or-in-c&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-chao2023-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-chao2023_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChao2023" class="citation book cs1 cs1-prop-foreign-lang-source">Chao, C. (2023). <i>数理逻辑：形式化方法的应用</i> [<i>Mathematical Logic: Applications of the Formalization Method</i>] (in Chinese). Beijing: Preprint. pp. 15–28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91%EF%BC%9A%E5%BD%A2%E5%BC%8F%E5%8C%96%E6%96%B9%E6%B3%95%E7%9A%84%E5%BA%94%E7%94%A8&rft.place=Beijing&rft.pages=15-28&rft.pub=Preprint.&rft.date=2023&rft.aulast=Chao&rft.aufirst=C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-heyting1930-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-heyting1930_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-heyting1930_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeyting1930" class="citation journal cs1 cs1-prop-foreign-lang-source">Heyting, A. (1930). "Die formalen Regeln der intuitionistischen Logik". <i>Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse</i> (in German): 42–56.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sitzungsberichte+der+Preussischen+Akademie+der+Wissenschaften%2C+Physikalisch-mathematische+Klasse&rft.atitle=Die+formalen+Regeln+der+intuitionistischen+Logik&rft.pages=42-56&rft.date=1930&rft.aulast=Heyting&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Denis Roegel (2002), <i><a rel="nofollow" class="external text" href="https://members.loria.fr/Roegel/loc/symboles-logiques-eng.pdf">A brief survey of 20th century logical notations</a></i> (see chart on page 2).</span> </li> <li id="cite_note-frege1879a-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-frege1879a_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrege1879" class="citation book cs1">Frege, G. (1879). <i>Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens</i>. Halle a/S.: Verlag von Louis Nebert. p. 10.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Begriffsschrift%2C+eine+der+arithmetischen+nachgebildete+Formelsprache+des+reinen+Denkens&rft.place=Halle+a%2FS.&rft.pages=10&rft.pub=Verlag+von+Louis+Nebert&rft.date=1879&rft.aulast=Frege&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-autogenerated222-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-autogenerated222_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-autogenerated222_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-autogenerated222_6-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell</a> (1908) <i>Mathematical logic as based on the theory of types</i> (American Journal of Mathematics 30, p222–262, also in From Frege to Gödel edited by van Heijenoort).</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"><a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano</a> (1889) <i><a href="/wiki/Arithmetices_principia,_nova_methodo_exposita" title="Arithmetices principia, nova methodo exposita">Arithmetices principia, nova methodo exposita</a></i>.</span> </li> <li id="cite_note-autogenerated1924-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-autogenerated1924_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-autogenerated1924_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Moses_Sch%C3%B6nfinkel" title="Moses Schönfinkel">Schönfinkel</a> (1924) <i> Über die Bausteine der mathematischen Logik</i>, translated as <i>On the building blocks of mathematical logic</i> in From Frege to Gödel edited by van Heijenoort.</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"><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce</a> (1867) <i>On an improvement in Boole's calculus of logic.</i></span> </li> <li id="cite_note-hilbert1918-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-hilbert1918_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbert1918" class="citation book cs1">Hilbert, D. (1918). Bernays, P. (ed.). <i>Prinzipien der Mathematik</i>. Lecture notes at Universität Göttingen, Winter Semester, 1917-1918</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Prinzipien+der+Mathematik&rft.date=1918&rft.aulast=Hilbert&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span>; Reprinted as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbert2013" class="citation encyclopaedia cs1">Hilbert, D. (2013). "Prinzipien der Mathematik". In Ewald, W.; Sieg, W. (eds.). <i>David Hilbert's Lectures on the Foundations of Arithmetic and Logic 1917–1933</i>. Heidelberg, New York, Dordrecht and London: Springer. pp. 59–221.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Prinzipien+der+Mathematik&rft.btitle=David+Hilbert%27s+Lectures+on+the+Foundations+of+Arithmetic+and+Logic+1917%E2%80%931933&rft.place=Heidelberg%2C+New+York%2C+Dordrecht+and+London&rft.pages=59-221&rft.pub=Springer&rft.date=2013&rft.aulast=Hilbert&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-bourbaki1954a-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-bourbaki1954a_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1954" class="citation book cs1">Bourbaki, N. (1954). <i>Théorie des ensembles</i>. Paris: Hermann & Cie, Éditeurs. p. 14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+des+ensembles&rft.place=Paris&rft.pages=14&rft.pub=Hermann+%26+Cie%2C+%C3%89diteurs&rft.date=1954&rft.aulast=Bourbaki&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-frege1879b-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-frege1879b_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrege1879" class="citation book cs1 cs1-prop-foreign-lang-source">Frege, G. (1879). <i>Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens</i> (in German). Halle a/S.: Verlag von Louis Nebert. p. 15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Begriffsschrift%2C+eine+der+arithmetischen+nachgebildete+Formelsprache+des+reinen+Denkens&rft.place=Halle+a%2FS.&rft.pages=15&rft.pub=Verlag+von+Louis+Nebert&rft.date=1879&rft.aulast=Frege&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-becker1933-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-becker1933_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBecker1933" class="citation book cs1 cs1-prop-foreign-lang-source">Becker, A. (1933). <i>Die Aristotelische Theorie der Möglichkeitsschlösse: Eine logisch-philologische Untersuchung der Kapitel 13-22 von Aristoteles' Analytica priora I</i> (in German). Berlin: Junker und Dünnhaupt Verlag. p. 4.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Die+Aristotelische+Theorie+der+M%C3%B6glichkeitsschl%C3%B6sse%3A+Eine+logisch-philologische+Untersuchung+der+Kapitel+13-22+von+Aristoteles%27+Analytica+priora+I&rft.place=Berlin&rft.pages=4&rft.pub=Junker+und+D%C3%BCnnhaupt+Verlag&rft.date=1933&rft.aulast=Becker&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-bourbaki1954b-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-bourbaki1954b_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1954" class="citation book cs1 cs1-prop-foreign-lang-source">Bourbaki, N. (1954). <i>Théorie des ensembles</i> (in French). Paris: Hermann & Cie, Éditeurs. p. 32.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+des+ensembles&rft.place=Paris&rft.pages=32&rft.pub=Hermann+%26+Cie%2C+%C3%89diteurs&rft.date=1954&rft.aulast=Bourbaki&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gentzen</a> (1934) <i>Untersuchungen über das logische Schließen</i>.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Chazal (1996) : Éléments de logique formelle.</span> </li> <li id="cite_note-hilbert1904-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-hilbert1904_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbert1905" class="citation encyclopaedia cs1">Hilbert, D. (1905) [1904]. "Über die Grundlagen der Logik und der Arithmetik". In Krazer, K. (ed.). <i>Verhandlungen des Dritten Internationalen Mathematiker Kongresses in Heidelberg vom 8. bis 13. August 1904</i>. pp. 174–185.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C3%9Cber+die+Grundlagen+der+Logik+und+der+Arithmetik&rft.btitle=Verhandlungen+des+Dritten+Internationalen+Mathematiker+Kongresses+in+Heidelberg+vom+8.+bis+13.+August+1904&rft.pages=174-185&rft.date=1905&rft.aulast=Hilbert&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="/wiki/J%C3%B3zef_Maria_Boche%C5%84ski" title="Józef Maria Bocheński">Bocheński</a> (1959), <i>A Précis of Mathematical Logic</i>, passim.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'DonnellHallPage2007" class="citation cs2">O'Donnell, John; Hall, Cordelia; Page, Rex (2007), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KKxyQQWQam4C&pg=PA120"><i>Discrete Mathematics Using a Computer</i></a>, Springer, p. 120, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781846285981" title="Special:BookSources/9781846285981"><bdi>9781846285981</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics+Using+a+Computer&rft.pages=120&rft.pub=Springer&rft.date=2007&rft.isbn=9781846285981&rft.aulast=O%27Donnell&rft.aufirst=John&rft.au=Hall%2C+Cordelia&rft.au=Page%2C+Rex&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DKKxyQQWQam4C%26pg%3DPA120&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span>.</span> </li> <li id="cite_note-:35-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-:35_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAllenHand2022" class="citation book cs1">Allen, Colin; Hand, Michael (2022). <i>Logic primer</i> (3rd ed.). Cambridge, Massachusetts: The MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-54364-4" title="Special:BookSources/978-0-262-54364-4"><bdi>978-0-262-54364-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic+primer&rft.place=Cambridge%2C+Massachusetts&rft.edition=3rd&rft.pub=The+MIT+Press&rft.date=2022&rft.isbn=978-0-262-54364-4&rft.aulast=Allen&rft.aufirst=Colin&rft.au=Hand%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJackson2012" class="citation cs2">Jackson, Daniel (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DDv8Ie_jBUQC&pg=PA263"><i>Software Abstractions: Logic, Language, and Analysis</i></a>, MIT Press, p. 263, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780262017152" title="Special:BookSources/9780262017152"><bdi>9780262017152</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Software+Abstractions%3A+Logic%2C+Language%2C+and+Analysis&rft.pages=263&rft.pub=MIT+Press&rft.date=2012&rft.isbn=9780262017152&rft.aulast=Jackson&rft.aufirst=Daniel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDDv8Ie_jBUQC%26pg%3DPA263&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span>.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPinter2014" class="citation book cs1">Pinter, Charles C. (2014). <i>A book of set theory</i>. Mineola, New York: Dover Publications, Inc. pp. 26–29. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-49708-2" title="Special:BookSources/978-0-486-49708-2"><bdi>978-0-486-49708-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+book+of+set+theory&rft.place=Mineola%2C+New+York&rft.pages=26-29&rft.pub=Dover+Publications%2C+Inc&rft.date=2014&rft.isbn=978-0-486-49708-2&rft.aulast=Pinter&rft.aufirst=Charles+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-:0-23"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_23-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_23-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-operations.html">"Set operations"</a>. <i>www.siue.edu</i><span class="reference-accessdate">. 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Retrieved <span class="nowrap">2024-06-11</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=1.5+Logic+and+Sets&rft_id=https%3A%2F%2Fwww.whitman.edu%2Fmathematics%2Fhigher_math_online%2Fsection01.05.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</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://mirror.clarkson.edu/isabelle/dist/library/HOL/HOL/Set.html">"Theory Set"</a>. <i>mirror.clarkson.edu</i><span class="reference-accessdate">. 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Retrieved <span class="nowrap">2024-06-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=autry.sites.grinnell.edu&rft.atitle=Set+Inclusion+and+Relations&rft_id=https%3A%2F%2Fautry.sites.grinnell.edu%2Fcsc208%2Freadings%2Fset-inclusion.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</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://web.mnstate.edu/peil/MDEV102/U1/S6/Complement3.htm">"Complement and Set Difference"</a>. <i>web.mnstate.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-06-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=web.mnstate.edu&rft.atitle=Complement+and+Set+Difference&rft_id=https%3A%2F%2Fweb.mnstate.edu%2Fpeil%2FMDEV102%2FU1%2FS6%2FComplement3.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCooper" class="citation web cs1">Cooper, A. <a rel="nofollow" class="external text" href="https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/">"Set Operations and Subsets – Foundations of Mathematics"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-06-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Set+Operations+and+Subsets+%E2%80%93+Foundations+of+Mathematics&rft.aulast=Cooper&rft.aufirst=A.&rft_id=https%3A%2F%2Fma225.wordpress.ncsu.edu%2Fset-operations-and-subsets%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-:2-29"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_29-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_29-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-basics.html">"Basic concepts"</a>. <i>www.siue.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-06-11</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.siue.edu&rft.atitle=Basic+concepts&rft_id=https%3A%2F%2Fwww.siue.edu%2F~jloreau%2Fcourses%2Fmath-223%2Fnotes%2Fsec-set-basics.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCooper" class="citation web cs1">Cooper, A. <a rel="nofollow" class="external text" href="https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/">"Set Operations and Subsets – Foundations of Mathematics"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-06-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Set+Operations+and+Subsets+%E2%80%93+Foundations+of+Mathematics&rft.aulast=Cooper&rft.aufirst=A.&rft_id=https%3A%2F%2Fma225.wordpress.ncsu.edu%2Fset-operations-and-subsets%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCooper" class="citation web cs1">Cooper, A. <a rel="nofollow" class="external text" href="https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/">"Set Operations and Subsets – Foundations of Mathematics"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-06-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Set+Operations+and+Subsets+%E2%80%93+Foundations+of+Mathematics&rft.aulast=Cooper&rft.aufirst=A.&rft_id=https%3A%2F%2Fma225.wordpress.ncsu.edu%2Fset-operations-and-subsets%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_connective&action=edit&section=14" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/J%C3%B3zef_Maria_Boche%C5%84ski" title="Józef Maria Bocheński">Bocheński, Józef Maria</a> (1959), <i>A Précis of Mathematical Logic</i>, translated from the French and German editions by Otto Bird, D. Reidel, Dordrecht, South Holland.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChao2023" class="citation book cs1 cs1-prop-foreign-lang-source">Chao, C. (2023). <i>数理逻辑：形式化方法的应用</i> [<i>Mathematical Logic: Applications of the Formalization Method</i>] (in Chinese). Beijing: Preprint. pp. 15–28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91%EF%BC%9A%E5%BD%A2%E5%BC%8F%E5%8C%96%E6%96%B9%E6%B3%95%E7%9A%84%E5%BA%94%E7%94%A8&rft.place=Beijing&rft.pages=15-28&rft.pub=Preprint.&rft.date=2023&rft.aulast=Chao&rft.aufirst=C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEnderton2001" class="citation cs2"><a href="/wiki/Herbert_Enderton" title="Herbert Enderton">Enderton, Herbert</a> (2001), <i>A Mathematical Introduction to Logic</i> (2nd ed.), Boston, MA: Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-238452-3" title="Special:BookSources/978-0-12-238452-3"><bdi>978-0-12-238452-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Mathematical+Introduction+to+Logic&rft.place=Boston%2C+MA&rft.edition=2nd&rft.pub=Academic+Press&rft.date=2001&rft.isbn=978-0-12-238452-3&rft.aulast=Enderton&rft.aufirst=Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGamut1991" class="citation cs2"><a href="/wiki/L._T._F._Gamut" title="L. T. F. Gamut">Gamut, L.T.F</a> (1991), "Chapter 2", <i>Logic, Language and Meaning</i>, vol. 1, University of Chicago Press, pp. 54–64, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/21372380">21372380</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+2&rft.btitle=Logic%2C+Language+and+Meaning&rft.pages=54-64&rft.pub=University+of+Chicago+Press&rft.date=1991&rft_id=info%3Aoclcnum%2F21372380&rft.aulast=Gamut&rft.aufirst=L.T.F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRautenberg,_W.2010" class="citation cs2"><a href="/wiki/Wolfgang_Rautenberg" title="Wolfgang Rautenberg">Rautenberg, W.</a> (2010), <i>A Concise Introduction to Mathematical Logic</i> (3rd ed.), <a href="/wiki/New_York_City" title="New York City">New York</a>: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer Science+Business Media</a>, <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%2F978-1-4419-1221-3">10.1007/978-1-4419-1221-3</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-1220-6" title="Special:BookSources/978-1-4419-1220-6"><bdi>978-1-4419-1220-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Concise+Introduction+to+Mathematical+Logic&rft.place=New+York&rft.edition=3rd&rft.pub=Springer+Science%2BBusiness+Media&rft.date=2010&rft_id=info%3Adoi%2F10.1007%2F978-1-4419-1221-3&rft.isbn=978-1-4419-1220-6&rft.au=Rautenberg%2C+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHumberstone2011" class="citation book cs1">Humberstone, Lloyd (2011). <i>The Connectives</i>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-01654-4" title="Special:BookSources/978-0-262-01654-4"><bdi>978-0-262-01654-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Connectives&rft.pub=MIT+Press&rft.date=2011&rft.isbn=978-0-262-01654-4&rft.aulast=Humberstone&rft.aufirst=Lloyd&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></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=Logical_connective&action=edit&section=15" 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 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Propositional+connective&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DPropositional_connective&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+connective" class="Z3988"></span></li> <li>Lloyd Humberstone (2010), "<a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/connectives-logic/">Sentence Connectives in Formal Logic</a>", <a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a> (An <a href="/wiki/Abstract_algebraic_logic" title="Abstract algebraic logic">abstract algebraic logic</a> approach to connectives.)</li> <li>John MacFarlane (2005), "<a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logical-constants/">Logical constants</a>", <a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output 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dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Common_logical_connectives" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><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:Logical_connectives" title="Template:Logical connectives"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Logical_connectives" title="Template talk:Logical connectives"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Logical_connectives" title="Special:EditPage/Template:Logical connectives"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Common_logical_connectives" style="font-size:114%;margin:0 4em">Common <a class="mw-selflink selflink">logical connectives</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a>/<a href="/wiki/Logical_truth" title="Logical truth">True</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 \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }"></span></li></ul> </div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Logical_connectives_Hasse_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/80px-Logical_connectives_Hasse_diagram.svg.png" decoding="async" width="80" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/120px-Logical_connectives_Hasse_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/160px-Logical_connectives_Hasse_diagram.svg.png 2x" data-file-width="744" data-file-height="1052" /></a></span></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sheffer_stroke" title="Sheffer stroke">Alternative denial</a> (<a href="/wiki/NAND_gate" title="NAND gate">NAND gate</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 \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span></li> <li><a href="/wiki/Converse_(logic)" title="Converse (logic)">Converse implication</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 \leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">←</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0fb4bce772117bbaf55b7ca1539ceff9ae218c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftarrow }"></span></li> <li><a href="/wiki/Material_conditional" title="Material conditional">Implication</a> (<a href="/wiki/IMPLY_gate" title="IMPLY gate">IMPLY gate</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 \rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }"></span></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">Disjunction</a> (<a href="/wiki/OR_gate" title="OR gate">OR gate</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 \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"></span></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Negation" title="Negation">Negation</a> (<a href="/wiki/Inverter_(logic_gate)" title="Inverter (logic gate)">NOT gate</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 \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></span></li> <li><a href="/wiki/Exclusive_or" title="Exclusive or">Exclusive or</a> (<a href="/wiki/XOR_gate" title="XOR gate">XOR gate</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 \not \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/363ed81fd02da85c658dde9f17737c13b7263e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \not \leftrightarrow }"></span></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">Biconditional</a> (<a href="/wiki/XNOR_gate" title="XNOR gate">XNOR gate</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 \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></span></li> <li><a href="/wiki/Statement_(logic)" title="Statement (logic)">Statement</a> (<a href="/wiki/Digital_buffer" title="Digital buffer">Digital buffer</a>)</li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Logical_NOR" title="Logical NOR">Joint denial</a> (<a href="/wiki/NOR_gate" title="NOR gate">NOR gate</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 \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4618f22b0f780805eb94bb407578d9bc9487947a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \downarrow }"></span></li> <li><a href="/wiki/Material_nonimplication" title="Material nonimplication">Nonimplication</a> (<a href="/wiki/NIMPLY_gate" title="NIMPLY gate">NIMPLY gate</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 \nrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↛</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c458d67617e028ed10948d2dbcfef80e9e060a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nrightarrow }"></span></li> <li><a href="/wiki/Converse_nonimplication" title="Converse nonimplication">Converse nonimplication</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 \nleftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nleftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7694c9fc8eebe8a57c8156dd3c2caf022a619439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nleftarrow }"></span></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">Conjunction</a> (<a href="/wiki/AND_gate" title="AND gate">AND gate</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 \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></span></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contradiction" title="Contradiction">Contradiction</a>/<a href="/wiki/False_(logic)" title="False (logic)">False</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 \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }"></span></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><span class="nowrap"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/18px-Socrates.png" 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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_logic" 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="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a class="mw-selflink selflink">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a 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 class="mw-selflink selflink">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a 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 href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Formal_semantics_(natural_language)" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse 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:Formal_semantics" title="Template:Formal semantics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Formal_semantics" title="Template talk:Formal semantics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Formal_semantics" title="Special:EditPage/Template:Formal semantics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Formal_semantics_(natural_language)" style="font-size:114%;margin:0 4em"><a href="/wiki/Formal_semantics_(natural_language)" title="Formal semantics (natural language)">Formal semantics (natural language)</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Central concepts</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/Principle_of_compositionality" title="Principle of compositionality">Compositionality</a></li> <li><a href="/wiki/Denotation" title="Denotation">Denotation</a></li> <li><a href="/wiki/Entailment" class="mw-redirect" title="Entailment">Entailment</a></li> <li><a href="/wiki/Extension_(semantics)" title="Extension (semantics)">Extension</a></li> <li><a href="/wiki/Generalized_quantifier" title="Generalized quantifier">Generalized quantifier</a></li> <li><a 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 href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">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 href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li></ul> 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