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id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Início</div> </a> </li> <li id="toc-Linguagens" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Linguagens"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Linguagens</span> </div> </a> <button aria-controls="toc-Linguagens-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>Alternar a subsecção Linguagens</span> </button> <ul id="toc-Linguagens-sublist" class="vector-toc-list"> <li id="toc-Linguagem_natural" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linguagem_natural"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Linguagem natural</span> </div> </a> <ul id="toc-Linguagem_natural-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linguagens_formais" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linguagens_formais"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Linguagens formais</span> </div> </a> <ul id="toc-Linguagens_formais-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Conectivos_lógicos_comuns" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conectivos_lógicos_comuns"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Conectivos lógicos comuns</span> </div> </a> <button aria-controls="toc-Conectivos_lógicos_comuns-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>Alternar a subsecção Conectivos lógicos comuns</span> </button> <ul id="toc-Conectivos_lógicos_comuns-sublist" class="vector-toc-list"> <li id="toc-Lista_de_conectivos_lógicos_comuns" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lista_de_conectivos_lógicos_comuns"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Lista de conectivos lógicos comuns</span> </div> </a> <ul id="toc-Lista_de_conectivos_lógicos_comuns-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-História_das_notações" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#História_das_notações"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>História das notações</span> </div> </a> <ul id="toc-História_das_notações-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Redundância" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Redundância"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Redundância</span> </div> </a> <ul id="toc-Redundância-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Propriedades" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propriedades"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Propriedades</span> </div> </a> <ul id="toc-Propriedades-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-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="Conteúdo" 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="Alternar o índice" > <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">Alternar o índice</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">Conectivo lógico</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="Ir para um artigo noutra língua. Disponível em 39 línguas" > <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 línguas</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 — árabe" lang="ar" hreflang="ar" data-title="رابطة منطقية" data-language-autonym="العربية" data-language-local-name="árabe" 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 — azerbaijano" lang="az" hreflang="az" data-title="Məntiqi əməliyyat" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijano" 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 — búlgaro" lang="bg" hreflang="bg" data-title="Логическа операция" data-language-autonym="Български" data-language-local-name="búlgaro" 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 — catalão" lang="ca" hreflang="ca" data-title="Connectiva lògica" data-language-autonym="Català" data-language-local-name="catalão" 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_operator" title="Logisk operator — dinamarquês" lang="da" hreflang="da" data-title="Logisk operator" data-language-autonym="Dansk" data-language-local-name="dinamarquês" 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 — alemão" lang="de" hreflang="de" data-title="Logische Verknüpfung" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%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 — grego" lang="el" hreflang="el" data-title="Λογικές συναρτήσεις" data-language-autonym="Ελληνικά" data-language-local-name="grego" 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-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Logical_connective" title="Logical connective — inglês" lang="en" hreflang="en" data-title="Logical connective" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</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 — espanhol" lang="es" hreflang="es" data-title="Conectiva lógica" data-language-autonym="Español" data-language-local-name="espanhol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Konnektor_(keeleteadus)" title="Konnektor (keeleteadus) — estónio" lang="et" hreflang="et" data-title="Konnektor (keeleteadus)" data-language-autonym="Eesti" data-language-local-name="estónio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Lokailu_logiko" title="Lokailu logiko — basco" lang="eu" hreflang="eu" data-title="Lokailu logiko" data-language-autonym="Euskara" data-language-local-name="basco" 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 — persa" lang="fa" hreflang="fa" data-title="رابط منطقی" data-language-autonym="فارسی" data-language-local-name="persa" 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 — francês" lang="fr" hreflang="fr" data-title="Connecteur logique" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</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 — hebraico" lang="he" hreflang="he" data-title="קשר לוגי" data-language-autonym="עברית" data-language-local-name="hebraico" class="interlanguage-link-target"><span>עברית</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 — húngaro" lang="hu" hreflang="hu" data-title="Logikai művelet" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</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 — arménio" lang="hy" hreflang="hy" data-title="Տրամաբանական գործողություն" data-language-autonym="Հայերեն" data-language-local-name="arménio" 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 — indonésio" lang="id" hreflang="id" data-title="Operator logika" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" 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 — islandês" lang="is" hreflang="is" data-title="Rökaðgerð" data-language-autonym="Íslenska" data-language-local-name="islandês" 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 — italiano" lang="it" hreflang="it" data-title="Connettivo logico" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</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 — japonês" lang="ja" hreflang="ja" data-title="論理演算" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</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 — cazaque" lang="kk" hreflang="kk" data-title="Пропозициялық өзара қатынас" data-language-autonym="Қазақша" data-language-local-name="cazaque" class="interlanguage-link-target"><span>Қазақша</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 — coreano" lang="ko" hreflang="ko" data-title="논리 연산" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</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 — macedónio" lang="mk" hreflang="mk" data-title="Логичка операција" data-language-autonym="Македонски" data-language-local-name="macedónio" 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 — malaio" lang="ms" hreflang="ms" data-title="Pengoperasi logik" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Logisk_konstant" title="Logisk konstant — norueguês nynorsk" lang="nn" hreflang="nn" data-title="Logisk konstant" data-language-autonym="Norsk nynorsk" data-language-local-name="norueguês 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 — polaco" lang="pl" hreflang="pl" data-title="Funktor zdaniotwórczy" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Conector_logic" title="Conector logic — romeno" lang="ro" hreflang="ro" data-title="Conector logic" data-language-autonym="Română" data-language-local-name="romeno" 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 — russo" lang="ru" hreflang="ru" data-title="Логическая операция" data-language-autonym="Русский" data-language-local-name="russo" 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 — eslovaco" lang="sk" hreflang="sk" data-title="Výroková spojka" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" 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 — sueco" lang="sv" hreflang="sv" data-title="Logisk operator" data-language-autonym="Svenska" data-language-local-name="sueco" 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 — tâmil" lang="ta" hreflang="ta" data-title="ஏரண இணைப்பி" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target"><span>தமிழ்</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 — tajique" lang="tg" hreflang="tg" data-title="Амалҳои мантиқӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="tajique" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a 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Wikipédia, a enciclopédia livre.</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="pt" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro: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/Diagrama_de_Hasse" title="Diagrama de Hasse">Diagrama de Hasse</a> dos conectivos lógicos.</figcaption></figure> <p>Em <a href="/wiki/L%C3%B3gica" title="Lógica">lógica</a>, um <b>conectivo lógico</b> (também chamado de <b>operador lógico</b>) é um <a href="/wiki/S%C3%ADmbolo_(formal)" title="Símbolo (formal)">símbolo</a> ou <a href="/wiki/Palavra" title="Palavra">palavra</a> usado para conectar duas ou mais <a href="/wiki/Senten%C3%A7a" title="Sentença">sentenças</a> (tanto na <a href="/wiki/Linguagem_formal" title="Linguagem formal">linguagem formal</a> quanto na <a href="/wiki/Linguagem_natural" class="mw-redirect" title="Linguagem natural">linguagem natural</a>) de uma maneira gramaticalmente válida, de modo que o sentido da sentença composta produzida dependa apenas das senteças originais. </p><p>Os conectivos lógicos mais comuns são os <b>conectivos binários</b> (também chamados de <b>conectivos diádicos</b>), que juntam duas sentenças, que podem ser consideradas os <a href="/wiki/Operando" title="Operando">operandos</a> da função. É também comum considerar <a href="/wiki/Nega%C3%A7%C3%A3o" title="Negação">negação</a> como um <b>conectivo unário</b>. </p><p>Conectivos lógicos e quantificadores são os dois principais tipos de <a href="/wiki/Constante_l%C3%B3gica" class="mw-redirect" title="Constante lógica">constantes lógicas</a> usadas em <a href="/wiki/Sistema_formal" title="Sistema formal">sistemas formais</a> como a <a href="/wiki/L%C3%B3gica_proposicional" title="Lógica proposicional">lógica proposicional</a> e a <a href="/wiki/L%C3%B3gica_de_predicados" title="Lógica de predicados">lógica de predicados</a>. A semântica de um conectivo lógico é, muitas vezes, mas não sempre, apresentada como uma <a href="/wiki/Fun%C3%A7%C3%A3o_de_verdade" title="Função de verdade">função de verdade</a>. </p><p>Um conectivo lógico é similar, mas não equivalente, a um <a href="/w/index.php?title=Operador_condicional&action=edit&redlink=1" class="new" title="Operador condicional (página não existe)">operador condicional</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>1<span>]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Linguagens">Linguagens</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&veaction=edit&section=1" title="Editar secção: Linguagens" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&action=edit&section=1" title="Editar código-fonte da secção: Linguagens"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Linguagem_natural">Linguagem natural</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&veaction=edit&section=2" title="Editar secção: Linguagem natural" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&action=edit&section=2" title="Editar código-fonte da secção: Linguagem natural"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Na gramática das linguagens naturais, duas sentenças podem ser unidas por uma conjunção gramatical para formar uma sentença <i>gramaticalmente</i> composta. Algumas dessas conjunções gramaticais, mas não todas, são funções de verdade. Por exemplo, considere as seguintes sentenças: </p> <dl><dd>A: João subiu a montanha.</dd> <dd>B: Pedro subiu a montanha.</dd> <dd>C: João subiu a montanha <i>e</i> Pedro subiu a montanha.</dd> <dd>D: João subiu a montanha, <i>então</i> Pedro subiu a montanha.</dd></dl> <p>As palavras <i>e</i> e <i>então</i> são conjunções <i>gramaticais</i> unindo as sentenças (A) e (B) para formar as sentenças compostas (C) e (D). O <i>e</i> em (C) é um conectivo <i>lógico</i>, pois o valor verdade de (C) é completamente determinado por (A) e (B): não faria sentido afirmar (A) e (B) e negar (C). No entanto, <i>então</i> em (D) não é um conectivo lógico, pois seria bastante razoável afirmar (A) e (B) e negar (D): talvez Pedro subiu a montanha para buscar um balde d'água, e não porque João subiu a montanha. </p><p>Várias palavras e pares de palavras expressam conectivos lógicos, e algumas delas são sinônimos. Exemplos (com o nome da relação em parênteses): </p> <ul><li>"e" (conjunção)</li> <li>"ou" (disjunção)</li> <li>"ou...ou" (disjunção exclusiva)</li> <li>"implica" (condicional)</li> <li>"se...então" (condicional)</li> <li>"<a href="/wiki/Se_e_somente_se" title="Se e somente se">se e somente se</a>" (bicondicional)</li> <li>"somente se" (condicional)</li> <li>"apenas no caso" (bicondicional)</li> <li>"mas" (conjunção)</li> <li>"contudo" (conjunção)</li> <li>"não ambos" (NAND)</li> <li>"nem...nem" (NOR)</li></ul> <p>A palavra "não" (negação) e as frases "é falso que" (negação) e "não é o caso que" (negação) também expressam um conectivo lógico - mesmo que elas sejam aplicadas a uma única sentença, e não conectem duas sentenças. </p> <div class="mw-heading mw-heading3"><h3 id="Linguagens_formais">Linguagens formais</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&veaction=edit&section=3" title="Editar secção: Linguagens formais" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&action=edit&section=3" title="Editar código-fonte da secção: Linguagens formais"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Em linguagens formais, funções verdade são representadas por símbolos inequívocos. Esses símbolos são chamados "conectivos lógicos", "operadores lógicos", "operadores proposicionais", ou, na lógica clássica, "conectivos de funções de verdade". Veja fórmulas bem formadas para saber as regras que permitem que novas fórmulas bem formadas sejam construídas ao juntar outras fórmulas bem formadas usando conectivos de funções de verdade. </p><p>Conectivos lógicos podem ser usados para ligar mais de duas afirmações, então é comum falar sobre "conectivo lógico <i>n</i>-ário". </p> <div class="mw-heading mw-heading2"><h2 id="Conectivos_lógicos_comuns"><span id="Conectivos_l.C3.B3gicos_comuns"></span>Conectivos lógicos comuns</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&veaction=edit&section=4" title="Editar secção: Conectivos lógicos comuns" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&action=edit&section=4" title="Editar código-fonte da secção: Conectivos lógicos comuns"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <table align="right" style="margin-left:2em; margin-bottom:1ex"> <tbody><tr> <th rowspan="2" colspan="2">Nome / Símbolo </th> <th colspan="5">Valor verdade </th> <th rowspan="2">Venn<small><br />diagrama</small> </th></tr> <tr> <td bgcolor="#FFFF66"><span class="texhtml mvar" style="font-style:italic;">P</span> = </td> <td bgcolor="#FFFF66" colspan="2" align="center">0 </td> <td bgcolor="#FFFF66" colspan="2" align="center">1 </td></tr> <tr> <td><a href="/wiki/Verdade" title="Verdade">Verdade</a>/<a href="/wiki/Tautologia" title="Tautologia">Tautologia</a> </td> <td align="right">⊤</td> <td>  </td> <td colspan="2" align="center">1 </td> <td colspan="2" align="center">1 </td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro:Venn11.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Venn11.svg/32px-Venn11.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Venn11.svg/48px-Venn11.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Venn11.svg/64px-Venn11.svg.png 2x" data-file-width="280" data-file-height="280" /></a></span> </td></tr> <tr> <td>Proposição <span class="texhtml mvar" style="font-style:italic;">P</span></td> <td></td> <td>  </td> <td colspan="2" align="center">0 </td> <td colspan="2" align="center">1 </td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro: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><a href="/wiki/Falso" class="mw-redirect" title="Falso">Falso</a>/<a href="/wiki/Contradi%C3%A7%C3%A3o" title="Contradição">Contradição</a> </td> <td align="right">⊥</td> <td>  </td> <td colspan="2" align="center">0 </td> <td colspan="2" align="center">0 </td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro:Venn00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Venn00.svg/32px-Venn00.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Venn00.svg/48px-Venn00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Venn00.svg/64px-Venn00.svg.png 2x" data-file-width="280" data-file-height="280" /></a></span> </td></tr> <tr> <td><a href="/wiki/Nega%C3%A7%C3%A3o" title="Negação">Negação</a> </td> <td align="right">¬</td> <td>  </td> <td colspan="2" align="center">1 </td> <td colspan="2" align="center">0 </td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro: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="2">Conectivos binários </th> <td bgcolor="#FFFF66"><span class="texhtml mvar" style="font-style:italic;">Q</span> = </td> <td bgcolor="#FFFF66">0 </td> <td bgcolor="#FFFF66">1 </td> <td bgcolor="#FFFF66">0 </td> <td bgcolor="#FFFF66">1 </td></tr> <tr> <td><a href="/wiki/Conjun%C3%A7%C3%A3o" title="Conjunção">Conjunção</a> </td> <td align="right">∧</td> <td></td> <td>0</td> <td>0</td> <td>0</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro: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><a href="/wiki/Porta_NAND" title="Porta NAND">Negação disjunta</a> </td> <td align="right">↑</td> <td></td> <td>1</td> <td>1</td> <td>1</td> <td>0</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro: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><a href="/wiki/Disjun%C3%A7%C3%A3o" class="mw-redirect" title="Disjunção">Disjunção</a> </td> <td align="right">∨</td> <td></td> <td>0</td> <td>1</td> <td>1</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro: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><a href="/wiki/Porta_NOR" title="Porta NOR">Negação conjunta</a> </td> <td align="right">↓</td> <td></td> <td>1</td> <td>0</td> <td>0</td> <td>0</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro: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><a href="/wiki/Condicional_material" title="Condicional material">Condicional material</a> </td> <td align="right">→</td> <td></td> <td>1</td> <td>1</td> <td>0</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro: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><a href="/wiki/Disjun%C3%A7%C3%A3o_exclusiva" class="mw-redirect" title="Disjunção exclusiva">Ou exclusivo</a> </td> <td align="right"><span class="mwe-math-element"><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></td> <td></td> <td>0</td> <td>1</td> <td>1</td> <td>0</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro: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><a href="/wiki/Bicondicional" class="mw-redirect" title="Bicondicional">Bicondicional</a> </td> <td align="right">↔</td> <td></td> <td>1</td> <td>0</td> <td>0</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro: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><a href="/w/index.php?title=Implica%C3%A7%C3%A3o_inversa&action=edit&redlink=1" class="new" title="Implicação inversa (página não existe)">Implicação inversa</a> </td> <td align="right">←</td> <td></td> <td>1</td> <td>0</td> <td>1</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro: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>Proposição <span class="texhtml mvar" style="font-style:italic;">P</span></td> <td></td> <td></td> <td>0</td> <td>0</td> <td>1</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro: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>Proposição <span class="texhtml mvar" style="font-style:italic;">Q</span></td> <td></td> <td></td> <td>0</td> <td>1</td> <td>0</td> <td>1</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro: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 colspan="8" align="center" style="letter-spacing:0.4em"><a href="/wiki/Fun%C3%A7%C3%A3o_de_verdade#Tabela_de_funções_de_verdade_binárias" title="Função de verdade">Mais informações</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Lista_de_conectivos_lógicos_comuns"><span id="Lista_de_conectivos_l.C3.B3gicos_comuns"></span>Lista de conectivos lógicos comuns</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&veaction=edit&section=5" title="Editar secção: Lista de conectivos lógicos comuns" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&action=edit&section=5" title="Editar código-fonte da secção: Lista de conectivos lógicos comuns"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Conectivos lógicos comumente usados: Negação (não): ¬, ~ Conjunção (e): ∧, & , ∙ Disjunção (ou): ∨ Implicação material (se...então): → ,⇒,⊃ Bicondicional (se e somente se): ↔,≡ ,= </p><p>Nomes alternativos para bicondicional são "sse", "xnor" e "bi-implicação". </p><p>Por exemplo, o significado das afirmações <i>está chovendo</i> e <i>eu estou dentro de casa</i> é transformado quando as duas são combinadas com conectivos lógicos. Veja os exemplos a seguir, onde as afirmações equivalem a <i>P = Está chovendo</i> e <i>Q = Eu estou dentro de casa</i>: </p> <ul><li>Está chovendo <b>e</b> eu estou dentro de casa (P ∧ Q)</li> <li><b>Se</b> está chovendo, <b>então</b> eu estou dentro de casa. (P → Q)</li> <li><b>Se</b> eu estou dentro de casa, <b>então</b> está chovendo. (Q → P)</li> <li>Eu estou dentro de casa <b>se e somente se</b> está chovendo (Q ↔ P)</li> <li><b>Não</b> está chovendo (¬P)</li></ul> <p>É também comum considerar a fórmula <i>sempre verdadeira</i> e a fórmula <i>sempre falsa</i> como sendo conectivos: Verdadeiro (⊤, 1 or T) Falso (⊥, 0, or F) </p> <div class="mw-heading mw-heading3"><h3 id="História_das_notações"><span id="Hist.C3.B3ria_das_nota.C3.A7.C3.B5es"></span>História das notações</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&veaction=edit&section=6" title="Editar secção: História das notações" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&action=edit&section=6" title="Editar código-fonte da secção: História das notações"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Negação: o símbolo ¬ apareceu em <a href="/w/index.php?title=Aritim%C3%A9tica_de_Heyting&action=edit&redlink=1" class="new" title="Aritimética de Heyting (página não existe)">Heyting</a> em 1929.<sup id="cite_ref-autogenerated1929_2-0" class="reference"><a href="#cite_note-autogenerated1929-2"><span>[</span>2<span>]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>3<span>]</span></a></sup> (compare ao símbolo <span typeof="mw:File"><a href="/wiki/Ficheiro:Begriffsschrift_connective1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Begriffsschrift_connective1.svg/50px-Begriffsschrift_connective1.svg.png" decoding="async" width="50" height="21" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Begriffsschrift_connective1.svg/75px-Begriffsschrift_connective1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Begriffsschrift_connective1.svg/100px-Begriffsschrift_connective1.svg.png 2x" data-file-width="512" data-file-height="216" /></a></span> de <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege</a> em <a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a>); o símbolo ~ apareceu em Russell em 1908;<sup id="cite_ref-autogenerated222_4-0" class="reference"><a href="#cite_note-autogenerated222-4"><span>[</span>4<span>]</span></a></sup> uma notação alternativa é adicionar uma linha horizontal em cima da fórmula, como em <span class="mwe-math-element"><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/d5e1bed5bc42d4e46dd9e5c7d2fc327927b87169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.025ex; height:3.009ex;" alt="{\displaystyle {\overline {P}}}"></span>; outra notação alternativa é usar uma aspa simples como em P'.</li> <li>Conjunção: o símbolo ∧ apareceu em Heyting em 1929<sup id="cite_ref-autogenerated1929_2-1" class="reference"><a href="#cite_note-autogenerated1929-2"><span>[</span>2<span>]</span></a></sup> (compare ao uso de <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano</a> da notação de interseção ∩ em teoria dos conjuntos<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span>[</span>5<span>]</span></a></sup>); & apareceu pelo menos em <a href="/wiki/Moses_Sch%C3%B6nfinkel" title="Moses Schönfinkel">Schönfinkel</a> em 1924;<sup id="cite_ref-autogenerated1924_6-0" class="reference"><a href="#cite_note-autogenerated1924-6"><span>[</span>6<span>]</span></a></sup> <b>∙</b> veio da interpretação de <a href="/wiki/George_Boole" title="George Boole">Boole</a> da lógica como uma álgebra elementar.</li> <li>Disjunção: o símbolo ∨ apareceu em Russell em 1908 (compare ao uso de Peano da notação de união ∪ em <a href="/wiki/Teoria_dos_conjuntos" title="Teoria dos conjuntos">teoria dos conjuntos</a>); o símbolo + também é usado, apesar da ambiguidade decorrente de na álgebra elementar ordinária o + ser considerado um ou exclusivo quando interpretado logicamente em uma aliança de dois elementos; pontualmente na história um + junto com um ponto no canto inferior à direita foi usado por Peirce.</li> <li>Implicação: o símbolo → pode ser visto em Hilbert em 1917; ⊃ foi usado por Russell em 1908 (compare à notação de C invertido de Peano); ⇒ foi usado em Vax.</li> <li>Bicondicional: o símbolo ≡ foi usado ao menos por Russell em 1908; ↔ foi usado ao menos por Tarski em 1940; ⇔ foi usado em Vax; outros símbolos apareceram pontualmente na história como ⊃⊂ em Gentzen, ~ em Schönfinkel ou ⊂⊃ em Chazal.</li> <li>Verdadeiro: o símbolo 1 veio da interpretação de Boole da lógica como uma álgebra elementar como a álbegra booleana de dois elementos; outras notações inclusive <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b20eaee46377072a177e3577ea439d142574f6aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.582ex; height:3.843ex;" alt="{\displaystyle \bigwedge }"></span> foram encontradas em Peano.</li> <li>Falso: o símbolo 0 vem também da interpretação de Boole da lógica como um anel [?]; outras notações inclusive <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigvee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋁</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigvee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424eb787b9be7b652deb858148ac5412c317aebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.582ex; height:3.843ex;" alt="{\displaystyle \bigvee }"></span> foram encontradas em Peano.</li></ul> <p>Alguns autores usaram letras para conectivos em algum momento da história: <b>u.</b> para conjunção (do Alemão "und", significa "e") e <b>o.</b> para disjunção (do Alemão "oder", significa "ou") nos primeiros trabalhos de Hilbert (1904); <b>N</b> para negação, <b>K</b> para conjunção, <b>A</b> para disjunção, <b>C</b> para implicação, <b>E</b> para bicondicional em Łukasiewicz (1929). </p> <div class="mw-heading mw-heading3"><h3 id="Redundância"><span id="Redund.C3.A2ncia"></span>Redundância</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&veaction=edit&section=7" title="Editar secção: Redundância" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&action=edit&section=7" title="Editar código-fonte da secção: Redundância"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O conectivo lógico da implicação recíproca ← é, na verdade, o mesmo que o condicional material com as premissas trocadas. Sendo assim, o símbolo da implicação recíproca é redundante. Em alguns cálculos lógicos, notavelmente na lógica clássica, certas afirmações compostas essencialmente diferentes são logicamente equivalentes. Um exemplo menos trivial de uma redundância é a clássica equivalência entre <i>¬P ∨ Q</i> e <i>P → Q</i>. Portanto, um sistema lógico de base clássica não precisa do operador condicional "→" se "¬" (não) e "∨" (ou) já são usados. Pode-se usar o "→" somente como um açúcar sintático para uma composição que tenha uma negação e uma disjunção. </p><p>Existem 16 funções boolianas associando os valores verdade de entrada <i>P</i> e <i>Q</i> com saídas binárias de 4 dígitos. Estas correspondem às escolhas possíveis de conectivos lógicos binários para a lógica clássica. Uma implementação diferente da lógica clássica pode escolher diferentes subconjuntos funcionalmente completos de conectivos. </p><p>Uma abordagem é escolher um conjunto <i>mínimo</i>, e definir outros conectivos por alguma forma lógica, como no exemplo com condicional material acima. A seguir estão os conjuntos mínimos funcionalmente completos de operadores na lógica clássica, cujas aridades não excedem 2: </p> <dl><dt>Um elemento</dt> <dd>{↑}, {↓}.</dd> <dt>Dois elementos</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 }"> <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 \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 \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 \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 \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>}, {←, <span class="mwe-math-element"><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>}, {→, <span class="mwe-math-element"><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>}, {→, <span class="mwe-math-element"><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 \leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not \leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4acf12661d69611db01ff7f036bf8c5b5dfec902" 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 \leftarrow }"></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 \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>}, {←, <span class="mwe-math-element"><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 \leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not \leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4acf12661d69611db01ff7f036bf8c5b5dfec902" 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 \leftarrow }"></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 \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>, ¬}, {<span class="mwe-math-element"><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 \leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not \leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4acf12661d69611db01ff7f036bf8c5b5dfec902" 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 \leftarrow }"></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 \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>, <span class="mwe-math-element"><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 \not \leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not \leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4acf12661d69611db01ff7f036bf8c5b5dfec902" 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 \leftarrow }"></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 \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 \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>, <span class="mwe-math-element"><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 \not \leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not \leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4acf12661d69611db01ff7f036bf8c5b5dfec902" 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 \leftarrow }"></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/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>}.</dd> <dt>Três elementos</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 }"> <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>, <span class="mwe-math-element"><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 \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 \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>, <span class="mwe-math-element"><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 \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>}, {<span class="mwe-math-element"><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>, <span class="mwe-math-element"><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>, <span class="mwe-math-element"><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 \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>, <span class="mwe-math-element"><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 \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 \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>, <span class="mwe-math-element"><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 \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>}, {<span class="mwe-math-element"><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>, <span class="mwe-math-element"><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>, <span class="mwe-math-element"><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>}.</dd></dl> <p>Veja mais detalhes sobre <a href="/wiki/Completude_funcional" title="Completude funcional">completude funcional</a>. </p><p>Outra abordagem é usar conectivos em igualdade de direitos, de um certo conjunto conveniente e funcionalmente completo, mas <i>não mínimo</i>. Essa abordagem requer mais axiomas proposicionais e cada equivalência entre formas lógicas tem que ser ou um <a href="/wiki/Axioma" title="Axioma">axioma</a> ou provada como um teorema. </p><p>Mas a lógica intuicionista tem a situação mais complicada. De seus cinco conectivos {∧, ∨, →, ¬, ⊥} somente a negação ¬ tem como ser reduzida a outros conectivos (¬p ≡ (p → ⊥)). Nem conjunção, disjunção e condicional material tem uma forma equivalente construída dos outros quatro conectivos lógicos. </p> <div class="mw-heading mw-heading2"><h2 id="Propriedades">Propriedades</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&veaction=edit&section=8" title="Editar secção: Propriedades" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Conectivo_l%C3%B3gico&action=edit&section=8" title="Editar código-fonte da secção: Propriedades"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Alguns conectivos lógicos possuem propriedades que podem ser expressas nos teoremas contendo o conectivo. Algumas dessas propriedades que um conectivo lógico pode ter são: </p> <ul><li><b><a href="/wiki/Associatividade" title="Associatividade">Associatividade</a></b>: Em uma expressão contendo dois ou mais do mesmo conectivo associativo em uma linha, a ordem das operações não importa enquanto a sequência de operandos não mudar.</li> <li><b><a href="/wiki/Comutatividade" title="Comutatividade">Comutatividade</a></b>: Os operandos do conectivo podem ser trocados (um pelo outro) preservando a equivalência lógica da expressão original.</li> <li><b><a href="/wiki/Distributividade" title="Distributividade">Distributividade</a></b>: Um conectivo denotado por • distribui sobre outro conectivo denotado por +, se <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> para todos os operandos <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>.</li> <li><b><a href="/wiki/Idempot%C3%AAncia" title="Idempotência">Idempotência</a></b>: Sempre que os operandos de uma operação são iguais, o composto é logicamente equivalente ao operando.</li> <li><b>Absorção</b>: Um par de conectivos <span class="mwe-math-element"><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>, <span class="mwe-math-element"><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> satisfaz a lei da absorção se <span class="mwe-math-element"><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> para todos os operandos <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>.</li> <li><b><a href="/wiki/Fun%C3%A7%C3%A3o_mon%C3%B3tona" title="Função monótona">Monotonicidade</a></b>: Se <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>) para todo <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} tal que <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>. Ex., <span class="mwe-math-element"><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 \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 \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 \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> <li><b><a href="/wiki/Transforma%C3%A7%C3%A3o_afim" class="mw-redirect" title="Transformação afim">Afinidade</a></b>: Cada variável sempre faz uma diferença no valor verdade da operação ou então nunca faz uma diferença. Ex., <span class="mwe-math-element"><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 \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 \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>, <span class="mwe-math-element"><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 \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> <h2 id="Referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Refer.C3.AAncias"></span>Referências</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite class="citation web">Cogwheel. <a rel="nofollow" class="external text" href="http://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> (em inglês). Stack Overflow<span class="reference-accessdate">. Consultado em 9 de abril de 2015</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AConectivo+l%C3%B3gico&rft.au=Cogwheel&rft.btitle=What+is+the+difference+between+logical+and+conditional+%2Foperator%2F&rft.genre=unknown&rft.pub=Stack+Overflow&rft_id=http%3A%2F%2Fstackoverflow.com%2Fquestions%2F3154132%2Fwhat-is-the-difference-between-logical-and-conditional-and-or-in-c&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">A referência emprega parâmetros obsoletos <code style="color:inherit; border:inherit; padding:inherit;">|apellido#=</code> (<a href="/wiki/Ajuda:Erros_nas_refer%C3%AAncias#deprecated_params" title="Ajuda:Erros nas referências">ajuda</a>)</span></span> </li> <li id="cite_note-autogenerated1929-2"><span class="mw-cite-backlink">↑ <sup><i><b><a href="#cite_ref-autogenerated1929_2-0">a</a></b></i></sup> <sup><i><b><a href="#cite_ref-autogenerated1929_2-1">b</a></b></i></sup></span> <span class="reference-text"><a href="/w/index.php?title=Heyting&action=edit&redlink=1" class="new" title="Heyting (página não existe)">Heyting</a> (1929) <i>Die formalen Regeln der intuitionistischen Logik</i>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Denis Roegel (2002), <i><a rel="nofollow" class="external text" href="http://www.loria.fr/~roegel/cours/symboles-logiques.pdf">Petit panorama des notations logiques du 20e siècle</a></i> (see chart on page 2).</span> </li> <li id="cite_note-autogenerated222-4"><span class="mw-cite-backlink"><a href="#cite_ref-autogenerated222_4-0">↑</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-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Peano (1889) <i>Arithmetices principia, nova methodo exposita</i>.</span> </li> <li id="cite_note-autogenerated1924-6"><span class="mw-cite-backlink"><a href="#cite_ref-autogenerated1924_6-0">↑</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> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtida de "<a dir="ltr" href="https://pt.wikipedia.org/w/index.php?title=Conectivo_lógico&oldid=66297392">https://pt.wikipedia.org/w/index.php?title=Conectivo_lógico&oldid=66297392</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categoria</a>: <ul><li><a href="/wiki/Categoria:L%C3%B3gica" title="Categoria:Lógica">Lógica</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorias ocultas: <ul><li><a href="/wiki/Categoria:!P%C3%A1ginas_que_usam_refer%C3%AAncias_com_par%C3%A2metros_obsoletas" title="Categoria:!Páginas que usam referências com parâmetros obsoletas">!Páginas que usam referências com parâmetros obsoletas</a></li><li><a href="/wiki/Categoria:!CS1_ingl%C3%AAs-fontes_em_l%C3%ADngua_(en)" title="Categoria:!CS1 inglês-fontes em língua (en)">!CS1 inglês-fontes em língua (en)</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Esta página foi editada pela última vez às 18h18min de 22 de julho de 2023.</li> <li id="footer-info-copyright">Este texto é disponibilizado nos termos da licença <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pt">Atribuição-CompartilhaIgual 4.0 Internacional (CC BY-SA 4.0) da Creative Commons</a>; pode estar sujeito a condições adicionais. 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