Lógica proposicional - Wikipedia, la enciclopedia libre

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="es" dir="ltr"> <head> <meta charset="UTF-8"> <title>Lógica proposicional - Wikipedia, la enciclopedia libre</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )eswikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t."," \t,"],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","enero","febrero","marzo","abril","mayo","junio","julio","agosto","septiembre","octubre","noviembre","diciembre"],"wgRequestId":"2d39f45c-6c20-4b4a-9fed-e9f0ce97f35c","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Lógica_proposicional","wgTitle":"Lógica proposicional","wgCurRevisionId":165114985,"wgRevisionId":165114985,"wgArticleId":258939,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Wikipedia:Artículos con pasajes que requieren referencias","Wikipedia:Artículos con identificadores GND","Wikipedia:Páginas que utilizan un formato obsoleto en la etiqueta math","Sistemas lógicos","Lógica proposicional"],"wgPageViewLanguage":"es","wgPageContentLanguage":"es","wgPageContentModel":"wikitext","wgRelevantPageName":"Lógica_proposicional","wgRelevantArticleId":258939,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true, "wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"es","pageLanguageDir":"ltr","pageVariantFallbacks":"es"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":40000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q200694","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":true,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={ "ext.gadget.imagenesinfobox":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.a-commons-directo","ext.gadget.ReferenceTooltips","ext.gadget.refToolbar","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth", "ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=es&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=es&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=es&modules=ext.gadget.imagenesinfobox&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=es&modules=site.styles&only=styles&skin=vector-2022"> <noscript><link rel="stylesheet" href="/w/load.php?lang=es&modules=noscript&only=styles&skin=vector-2022"></noscript> <meta name="generator" content="MediaWiki 1.44.0-wmf.16"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Lógica proposicional - Wikipedia, la enciclopedia libre"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//es.m.wikipedia.org/wiki/L%C3%B3gica_proposicional"> <link rel="alternate" type="application/x-wiki" title="Editar" href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (es)"> <link rel="EditURI" type="application/rsd+xml" href="//es.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://es.wikipedia.org/wiki/L%C3%B3gica_proposicional"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.es"> <link rel="alternate" type="application/atom+xml" title="Canal Atom de Wikipedia" href="/w/index.php?title=Especial:CambiosRecientes&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Lógica_proposicional rootpage-Lógica_proposicional skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Ir al contenido</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Sitio"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Menú principal" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Menú principal" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Menú principal</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Menú principal</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">mover a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">ocultar</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navegación </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Wikipedia:Portada" title="Visitar la página principal [z]" accesskey="z"><span>Portada</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Portal:Comunidad" title="Acerca del proyecto, lo que puedes hacer, dónde encontrar información"><span>Portal de la comunidad</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Actualidad" title="Encuentra información de contexto sobre acontecimientos actuales"><span>Actualidad</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Especial:CambiosRecientes" title="Lista de cambios recientes en la wiki [r]" accesskey="r"><span>Cambios recientes</span></a></li><li id="n-newpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginasNuevas"><span>Páginas nuevas</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Especial:Aleatoria" title="Cargar una página al azar [x]" accesskey="x"><span>Página aleatoria</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Ayuda:Contenidos" title="El lugar para aprender"><span>Ayuda</span></a></li><li id="n-bug_in_article" class="mw-list-item"><a href="/wiki/Wikipedia:Informes_de_error"><span>Notificar un error</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginasEspeciales"><span>Páginas especiales</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Wikipedia:Portada" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="La enciclopedia libre" src="/static/images/mobile/copyright/wikipedia-tagline-es.svg" width="120" height="13" style="width: 7.5em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Especial:Buscar" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Buscar en este wiki [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Buscar</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Buscar en Wikipedia" aria-label="Buscar en Wikipedia" autocapitalize="sentences" title="Buscar en este wiki [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Especial:Buscar"> </div> <button class="cdx-button cdx-search-input__end-button">Buscar</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Herramientas personales"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Apariencia"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Apariencia" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Apariencia</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=es.wikipedia.org&uselang=es" class=""><span>Donaciones</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Crear_una_cuenta&returnto=L%C3%B3gica+proposicional" title="Te recomendamos crear una cuenta e iniciar sesión; sin embargo, no es obligatorio" class=""><span>Crear una cuenta</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Entrar&returnto=L%C3%B3gica+proposicional" title="Te recomendamos iniciar sesión, aunque no es obligatorio [o]" accesskey="o" class=""><span>Acceder</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Más opciones" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Herramientas personales" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Herramientas personales</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menú de usuario" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=es.wikipedia.org&uselang=es"><span>Donaciones</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Crear_una_cuenta&returnto=L%C3%B3gica+proposicional" title="Te recomendamos crear una cuenta e iniciar sesión; sin embargo, no es obligatorio"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Crear una cuenta</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Entrar&returnto=L%C3%B3gica+proposicional" title="Te recomendamos iniciar sesión, aunque no es obligatorio [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Acceder</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Páginas para editores desconectados <a href="/wiki/Ayuda:Introducci%C3%B3n" aria-label="Obtenga más información sobre editar"><span>más información</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Especial:MisContribuciones" title="Una lista de modificaciones hechas desde esta dirección IP [y]" accesskey="y"><span>Contribuciones</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:MiDiscusi%C3%B3n" title="Discusión sobre ediciones hechas desde esta dirección IP [n]" accesskey="n"><span>Discusión</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Sitio"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contenidos" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contenidos</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ocultar</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li 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">Inicio</div> </a> </li> <li id="toc-Introducción" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Introducción"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Introducción</span> </div> </a> <button aria-controls="toc-Introducción-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 subsección Introducción</span> </button> <ul id="toc-Introducción-sublist" class="vector-toc-list"> <li id="toc-Conectivas_lógicas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conectivas_lógicas"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Conectivas lógicas</span> </div> </a> <ul id="toc-Conectivas_lógicas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Leyes_notables_en_lógica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Leyes_notables_en_lógica"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Leyes notables en lógica</span> </div> </a> <ul id="toc-Leyes_notables_en_lógica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Límites_de_la_lógica_proposicional" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Límites_de_la_lógica_proposicional"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Límites de la lógica proposicional</span> </div> </a> <ul id="toc-Límites_de_la_lógica_proposicional-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dos_sistemas_formales_de_lógica_proposicional" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dos_sistemas_formales_de_lógica_proposicional"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Dos sistemas formales de lógica proposicional</span> </div> </a> <button aria-controls="toc-Dos_sistemas_formales_de_lógica_proposicional-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 subsección Dos sistemas formales de lógica proposicional</span> </button> <ul id="toc-Dos_sistemas_formales_de_lógica_proposicional-sublist" class="vector-toc-list"> <li id="toc-Sistema_axiomático" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sistema_axiomático"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Sistema axiomático</span> </div> </a> <ul id="toc-Sistema_axiomático-sublist" class="vector-toc-list"> <li id="toc-Alfabeto" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Alfabeto"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Alfabeto</span> </div> </a> <ul id="toc-Alfabeto-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gramática" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Gramática"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Gramática</span> </div> </a> <ul id="toc-Gramática-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Axiomas" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Axiomas"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Axiomas</span> </div> </a> <ul id="toc-Axiomas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reglas_de_inferencia" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Reglas_de_inferencia"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.4</span> <span>Reglas de inferencia</span> </div> </a> <ul id="toc-Reglas_de_inferencia-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Deducción_natural" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Deducción_natural"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Deducción natural</span> </div> </a> <ul id="toc-Deducción_natural-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Formas_de_argumentos_básicas_y_derivadas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formas_de_argumentos_básicas_y_derivadas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Formas de argumentos básicas y derivadas</span> </div> </a> <button aria-controls="toc-Formas_de_argumentos_básicas_y_derivadas-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 subsección Formas de argumentos básicas y derivadas</span> </button> <ul id="toc-Formas_de_argumentos_básicas_y_derivadas-sublist" class="vector-toc-list"> <li id="toc-Ejemplo_de_una_demostración" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ejemplo_de_una_demostración"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Ejemplo de una demostración</span> </div> </a> <ul id="toc-Ejemplo_de_una_demostración-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lenguaje_formal_en_la_notación_BNF" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lenguaje_formal_en_la_notación_BNF"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Lenguaje formal en la notación BNF</span> </div> </a> <ul id="toc-Lenguaje_formal_en_la_notación_BNF-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semántica" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Semántica"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Semántica</span> </div> </a> <button aria-controls="toc-Semántica-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 subsección Semántica</span> </button> <ul id="toc-Semántica-sublist" class="vector-toc-list"> <li id="toc-Tablas_de_verdad" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tablas_de_verdad"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Tablas de verdad</span> </div> </a> <ul id="toc-Tablas_de_verdad-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Formas_normales" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formas_normales"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Formas normales</span> </div> </a> <ul id="toc-Formas_normales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historia"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Historia</span> </div> </a> <ul id="toc-Historia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Véase_también" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Véase_también"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Véase también</span> </div> </a> <ul id="toc-Véase_también-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referencias" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referencias"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Referencias</span> </div> </a> <ul id="toc-Referencias-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografía" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografía"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Bibliografía</span> </div> </a> <ul id="toc-Bibliografía-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enlaces_externos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Enlaces_externos"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Enlaces externos</span> </div> </a> <ul id="toc-Enlaces_externos-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="Contenidos" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Tabla de contenidos" > <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="Cambiar a la tabla de contenidos" > <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">Cambiar a la tabla de contenidos</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">Lógica proposicional</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 a un artículo en otro idioma. Disponible en 51 idiomas" > <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-51" 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">51 idiomas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Proposisionele_logika" title="Proposisionele logika – afrikáans" lang="af" hreflang="af" data-title="Proposisionele logika" data-language-autonym="Afrikaans" data-language-local-name="afrikáans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8_%D8%A7%D9%84%D9%82%D8%B6%D8%A7%D9%8A%D8%A7" title="حساب القضايا – árabe" lang="ar" hreflang="ar" data-title="حساب القضايا" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/L%C3%B3xica_proposicional" title="Lóxica proposicional – asturiano" lang="ast" hreflang="ast" data-title="Lóxica proposicional" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D1%96%D0%BA%D0%B0_%D0%B2%D1%8B%D0%BA%D0%B0%D0%B7%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F%D1%9E" title="Логіка выказванняў – bielorruso" lang="be" hreflang="be" data-title="Логіка выказванняў" data-language-autonym="Беларуская" data-language-local-name="bielorruso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%97%D1%8C%D0%BB%D1%96%D1%87%D1%8D%D0%BD%D1%8C%D0%BD%D0%B5_%D0%B2%D1%8B%D0%BA%D0%B0%D0%B7%D0%B2%D0%B0%D0%BD%D1%8C%D0%BD%D1%8F%D1%9E" title="Зьлічэньне выказваньняў – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Зьлічэньне выказваньняў" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%BF%D0%BE%D0%B7%D0%B8%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%B0_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Пропозиционална логика – búlgaro" lang="bg" hreflang="bg" data-title="Пропозиционална логика" 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/L%C3%B2gica_proposicional" title="Lògica proposicional – catalán" lang="ca" hreflang="ca" data-title="Lògica proposicional" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/V%C3%BDrokov%C3%A1_logika" title="Výroková logika – checo" lang="cs" hreflang="cs" data-title="Výroková logika" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BB%D0%B0%D0%BD%C4%83%D0%BB%C4%83%D1%85%D1%81%D0%B5%D0%BD_%D1%88%D1%83%D1%82%D0%BB%D0%B0%D0%B2%C4%95" title="Каланăлăхсен шутлавĕ – chuvasio" lang="cv" hreflang="cv" data-title="Каланăлăхсен шутлавĕ" data-language-autonym="Чӑвашла" data-language-local-name="chuvasio" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhesymeg_osodiadol" title="Rhesymeg osodiadol – galés" lang="cy" hreflang="cy" data-title="Rhesymeg osodiadol" data-language-autonym="Cymraeg" data-language-local-name="galés" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Aussagenlogik" title="Aussagenlogik – alemán" lang="de" hreflang="de" data-title="Aussagenlogik" data-language-autonym="Deutsch" data-language-local-name="alemán" 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%A0%CF%81%CE%BF%CF%84%CE%B1%CF%83%CE%B9%CE%B1%CE%BA%CF%8C%CF%82_%CE%BB%CE%BF%CE%B3%CE%B9%CF%83%CE%BC%CF%8C%CF%82" title="Προτασιακός λογισμός – griego" lang="el" hreflang="el" data-title="Προτασιακός λογισμός" data-language-autonym="Ελληνικά" data-language-local-name="griego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Propositional_calculus" title="Propositional calculus – inglés" lang="en" hreflang="en" data-title="Propositional calculus" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Asertologiko" title="Asertologiko – esperanto" lang="eo" hreflang="eo" data-title="Asertologiko" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lauseloogika" title="Lauseloogika – estonio" lang="et" hreflang="et" data-title="Lauseloogika" data-language-autonym="Eesti" data-language-local-name="estonio" 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/Logika_proposizional" title="Logika proposizional – euskera" lang="eu" hreflang="eu" data-title="Logika proposizional" data-language-autonym="Euskara" data-language-local-name="euskera" 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%AD%D8%B3%D8%A7%D8%A8_%DA%AF%D8%B2%D8%A7%D8%B1%D9%87%E2%80%8C%D8%A7%DB%8C" title="حساب گزاره‌ای – persa" lang="fa" hreflang="fa" data-title="حساب گزاره‌ای" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Propositiologiikka" title="Propositiologiikka – finés" lang="fi" hreflang="fi" data-title="Propositiologiikka" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Calcul_des_propositions" title="Calcul des propositions – francés" lang="fr" hreflang="fr" data-title="Calcul des propositions" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/%C3%9Ctjsaagenloogik" title="Ütjsaagenloogik – frisón septentrional" lang="frr" hreflang="frr" data-title="Ütjsaagenloogik" data-language-autonym="Nordfriisk" data-language-local-name="frisón septentrional" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/L%C3%B3xica_proposicional" title="Lóxica proposicional – gallego" lang="gl" hreflang="gl" data-title="Lóxica proposicional" data-language-autonym="Galego" data-language-local-name="gallego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%97%D7%A9%D7%99%D7%91_%D7%94%D7%A4%D7%A1%D7%95%D7%A7%D7%99%D7%9D" title="תחשיב הפסוקים – hebreo" lang="he" hreflang="he" data-title="תחשיב הפסוקים" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A4%BF%E0%A4%9C%E0%A5%8D%E0%A4%9E%E0%A4%AA%E0%A5%8D%E0%A4%A4%E0%A4%BF%E0%A4%95_%E0%A4%95%E0%A4%B2%E0%A4%A8" title="प्रतिज्ञप्तिक कलन – hindi" lang="hi" hreflang="hi" data-title="प्रतिज्ञप्तिक कलन" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/%C3%8Dt%C3%A9letlogika" title="Ítéletlogika – húngaro" lang="hu" hreflang="hu" data-title="Ítéletlogika" 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/%D4%B1%D5%BD%D5%B8%D6%82%D5%B5%D5%A9%D5%B6%D5%A5%D6%80%D5%AB_%D5%BF%D6%80%D5%A1%D5%B4%D5%A1%D5%A2%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Ասույթների տրամաբանություն – armenio" lang="hy" hreflang="hy" data-title="Ասույթների տրամաբանություն" data-language-autonym="Հայերեն" data-language-local-name="armenio" 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/Kalkulus_proposisional" title="Kalkulus proposisional – indonesio" lang="id" hreflang="id" data-title="Kalkulus proposisional" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Propozicionala_logiko" title="Propozicionala logiko – ido" lang="io" hreflang="io" data-title="Propozicionala logiko" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Logica_proposizionale" title="Logica proposizionale – italiano" lang="it" hreflang="it" data-title="Logica proposizionale" 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/%E5%91%BD%E9%A1%8C%E8%AB%96%E7%90%86" title="命題論理 – japonés" lang="ja" hreflang="ja" data-title="命題論理" data-language-autonym="日本語" data-language-local-name="japonés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AA%85%EC%A0%9C_%EB%85%BC%EB%A6%AC" title="명제 논리 – coreano" lang="ko" hreflang="ko" data-title="명제 논리" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D1%81%D2%AF%D0%B9%D0%BB%D3%A9%D3%A9" title="Логикалык сүйлөө – kirguís" lang="ky" hreflang="ky" data-title="Логикалык сүйлөө" data-language-autonym="Кыргызча" data-language-local-name="kirguís" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Logica_propositionalis" title="Logica propositionalis – latín" lang="la" hreflang="la" data-title="Logica propositionalis" data-language-autonym="Latina" data-language-local-name="latín" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Teigini%C5%B3_logika" title="Teiginių logika – lituano" lang="lt" hreflang="lt" data-title="Teiginių logika" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Propositielogica" title="Propositielogica – neerlandés" lang="nl" hreflang="nl" data-title="Propositielogica" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Utsegnslogikk" title="Utsegnslogikk – noruego nynorsk" lang="nn" hreflang="nn" data-title="Utsegnslogikk" data-language-autonym="Norsk nynorsk" data-language-local-name="noruego nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Setningslogikk" title="Setningslogikk – noruego bokmal" lang="nb" hreflang="nb" data-title="Setningslogikk" data-language-autonym="Norsk bokmål" data-language-local-name="noruego bokmal" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rachunek_zda%C5%84" title="Rachunek zdań – polaco" lang="pl" hreflang="pl" data-title="Rachunek zdań" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%88%D9%8A_%D8%AD%D8%B3%D8%A7%D8%A8_(%D9%85%D9%86%D8%B7%D9%82)" title="قضیوي حساب (منطق) – pastún" lang="ps" hreflang="ps" data-title="قضیوي حساب (منطق)" data-language-autonym="پښتو" data-language-local-name="pastún" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/L%C3%B3gica_proposicional" title="Lógica proposicional – portugués" lang="pt" hreflang="pt" data-title="Lógica proposicional" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0_%D0%B2%D1%8B%D1%81%D0%BA%D0%B0%D0%B7%D1%8B%D0%B2%D0%B0%D0%BD%D0%B8%D0%B9" title="Логика высказываний – ruso" lang="ru" hreflang="ru" data-title="Логика высказываний" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Propositional_logic" title="Propositional logic – Simple English" lang="en-simple" hreflang="en-simple" data-title="Propositional logic" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/V%C3%BDrokov%C3%A1_logika" title="Výroková logika – eslovaco" lang="sk" hreflang="sk" data-title="Výroková logika" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Propozicijska_logika" title="Propozicijska logika – esloveno" lang="sl" hreflang="sl" data-title="Propozicijska logika" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D1%81%D0%BA%D0%B0%D0%B7%D0%BD%D0%B8_%D1%80%D0%B0%D1%87%D1%83%D0%BD" title="Исказни рачун – serbio" lang="sr" hreflang="sr" data-title="Исказни рачун" data-language-autonym="Српски / srpski" data-language-local-name="serbio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Satslogik" title="Satslogik – sueco" lang="sv" hreflang="sv" data-title="Satslogik" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%84%E0%B8%A5%E0%B8%84%E0%B8%B9%E0%B8%A5%E0%B8%B1%E0%B8%AA%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%9B%E0%B8%A3%E0%B8%B0%E0%B8%9E%E0%B8%88%E0%B8%99%E0%B9%8C" title="แคลคูลัสเชิงประพจน์ – tailandés" lang="th" hreflang="th" data-title="แคลคูลัสเชิงประพจน์" data-language-autonym="ไทย" data-language-local-name="tailandés" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%96nermeler_mant%C4%B1%C4%9F%C4%B1" title="Önermeler mantığı – turco" lang="tr" hreflang="tr" data-title="Önermeler mantığı" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F_%D0%B2%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D0%BB%D0%B5%D0%BD%D1%8C" title="Числення висловлень – ucraniano" lang="uk" hreflang="uk" data-title="Числення висловлень" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%E1%BB%87nh_%C4%91%E1%BB%81_to%C3%A1n_h%E1%BB%8Dc" title="Mệnh đề toán học – vietnamita" lang="vi" hreflang="vi" data-title="Mệnh đề toán học" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%91%BD%E9%A2%98%E9%80%BB%E8%BE%91" title="命题逻辑 – chino" lang="zh" hreflang="zh" data-title="命题逻辑" data-language-autonym="中文" data-language-local-name="chino" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%91%BD%E9%A1%8C%E9%82%8F%E8%BC%AF" title="命題邏輯 – cantonés" lang="yue" hreflang="yue" data-title="命題邏輯" data-language-autonym="粵語" data-language-local-name="cantonés" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q200694#sitelinks-wikipedia" title="Editar enlaces interlingüísticos" class="wbc-editpage">Editar enlaces</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espacios de nombres"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/L%C3%B3gica_proposicional" title="Ver la página de contenido [c]" accesskey="c"><span>Artículo</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discusi%C3%B3n:L%C3%B3gica_proposicional" rel="discussion" title="Discusión acerca de la página [t]" accesskey="t"><span>Discusión</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Cambiar variante de idioma" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">español</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistas"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/L%C3%B3gica_proposicional"><span>Leer</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit" title="Editar esta página [e]" accesskey="e"><span>Editar</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=history" title="Versiones anteriores de esta página [h]" accesskey="h"><span>Ver historial</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Página de herramientas"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Herramientas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Herramientas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Herramientas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mover a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ocultar</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Más opciones" > <div class="vector-menu-heading"> Acciones </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/L%C3%B3gica_proposicional"><span>Leer</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit" title="Editar esta página [e]" accesskey="e"><span>Editar</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=history"><span>Ver historial</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:LoQueEnlazaAqu%C3%AD/L%C3%B3gica_proposicional" title="Lista de todas las páginas de la wiki que enlazan aquí [j]" accesskey="j"><span>Lo que enlaza aquí</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:CambiosEnEnlazadas/L%C3%B3gica_proposicional" rel="nofollow" title="Cambios recientes en las páginas que enlazan con esta [k]" accesskey="k"><span>Cambios en enlazadas</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=es" title="Subir archivos [u]" accesskey="u"><span>Subir archivo</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=L%C3%B3gica_proposicional&oldid=165114985" title="Enlace permanente a esta versión de la página"><span>Enlace permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=info" title="Más información sobre esta página"><span>Información de la página</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citar&page=L%C3%B3gica_proposicional&id=165114985&wpFormIdentifier=titleform" title="Información sobre cómo citar esta página"><span>Citar esta página</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:Acortador_de_URL&url=https%3A%2F%2Fes.wikipedia.org%2Fwiki%2FL%25C3%25B3gica_proposicional"><span>Obtener URL acortado</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fes.wikipedia.org%2Fwiki%2FL%25C3%25B3gica_proposicional"><span>Descargar código QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimir/exportar </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Libro&bookcmd=book_creator&referer=L%C3%B3gica+proposicional"><span>Crear un libro</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=L%C3%B3gica_proposicional&action=show-download-screen"><span>Descargar como PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=L%C3%B3gica_proposicional&printable=yes" title="Versión imprimible de esta página [p]" accesskey="p"><span>Versión para imprimir</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> En otros proyectos </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Propositional_logic" hreflang="en"><span>Wikimedia Commons</span></a></li><li class="wb-otherproject-link wb-otherproject-wikiversity mw-list-item"><a href="https://es.wikiversity.org/wiki/L%C3%B3gica_proposicional" hreflang="es"><span>Wikiversidad</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q200694" title="Enlace al elemento conectado del repositorio de datos [g]" accesskey="g"><span>Elemento de Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Página de herramientas"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apariencia"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ocultar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">De Wikipedia, la enciclopedia libre</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="es" dir="ltr"><p>La <b>lógica proposicional</b>, también llamada <b>lógica de enunciados</b>, <b>lógica de orden cero</b> o <b>cálculo proposicional</b>, es un <a href="/wiki/Sistema_formal" title="Sistema formal">sistema formal</a> cuyos elementos más simples representan proposiciones o enunciados, y cuyas <a href="/wiki/Constante_l%C3%B3gica" title="Constante lógica">constantes lógicas</a>, llamadas <a href="/wiki/Conectivas_l%C3%B3gicas" class="mw-redirect" title="Conectivas lógicas">conectivas lógicas</a>, representan <a href="/wiki/Operaci%C3%B3n_matem%C3%A1tica" class="mw-redirect" title="Operación matemática">operaciones</a> sobre proposiciones, capaces de formar otras proposiciones de mayor complejidad.<sup id="cite_ref-1" class="reference separada"><a href="#cite_note-1"><span class="corchete-llamada">[</span>1<span class="corchete-llamada">]</span></a></sup> </p><p>Las lógicas proposicionales carecen de cuantificadores o variables de individuo, pero tienen <a href="/wiki/Variable_proposicional" title="Variable proposicional">variables proposicionales</a> (es decir, que se pueden interpretar como proposiciones con un valor de verdad definido), de ahí el nombre proposicional. Los sistemas de lógica proposicional incluyen además <a href="/wiki/Conectiva_l%C3%B3gica" title="Conectiva lógica">conectivas lógicas</a>, por lo que dentro de este tipo de lógica se puede analizar la <a href="/wiki/Inferencia" title="Inferencia">inferencia lógica</a> de proposiciones a partir de proposiciones, pero sin tener en cuenta la estructura interna de las proposiciones más simples.<sup id="cite_ref-iep_2-0" class="reference separada"><a href="#cite_note-iep-2"><span class="corchete-llamada">[</span>2<span class="corchete-llamada">]</span></a></sup> </p><p>Como las lógicas proposicionales no tienen cuantificadores o variables de individuo, cualquier secuencia de signos que constituya una <a href="/wiki/F%C3%B3rmula_bien_formada" title="Fórmula bien formada">fórmula bien formada</a> admite una valoración en la proposición es verdadera o falsa dependiendo del valor de verdad asignado a las proposiciones que la compongan. Esto implica que cualquier fórmula bien formada define una función proposicional. Por tanto, cualquier sistema lógico basado en la lógica proposicional es <a href="/wiki/Decidibilidad" title="Decidibilidad">decidible</a> y en un número finito de pasos se puede determinar la verdad o falsedad semántica de una proposición. Esto hace que la lógica proposicional sea <a href="/wiki/Completitud_(l%C3%B3gica)" title="Completitud (lógica)">completa</a> y con una semántica muy sencilla. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introducción"><span id="Introducci.C3.B3n"></span>Introducción</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=1" title="Editar sección: Introducción"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Considérese el siguiente <a href="/wiki/Argumento" title="Argumento">argumento</a>: </p> <ol><li>Mañana es miércoles <b>o</b> mañana es jueves.</li> <li>Mañana <b>no</b> es jueves.</li> <li><b>Por lo tanto</b>, mañana es miércoles.</li></ol> <p>Es un argumento <a href="/wiki/Validez_l%C3%B3gica" class="mw-redirect" title="Validez lógica">válido</a>. Quiere decir que es imposible que las <a href="/wiki/Premisa" title="Premisa">premisas</a> (1) y (2) sean verdaderas y la <a href="/wiki/Conclusi%C3%B3n" title="Conclusión">conclusión</a> (3) falsa. </p><p>Sin embargo, a pesar de que el argumento sea válido, esto no quiere decir que la conclusión sea verdadera. En otras palabras, si las premisas son falsas, entonces la conclusión también podría serlo. Pero si las premisas son verdaderas, entonces la conclusión también lo es. La validez del argumento no depende del significado de las expresiones «mañana es miércoles» ni «mañana es jueves», sino de la estructura misma del argumento. Estas premisas podrían cambiarse por otras y el argumento permanecería válido. Por ejemplo: </p> <ol><li>Hoy está soleado <b>o</b> está nublado.</li> <li>Hoy <b>no</b> está nublado.</li> <li><b>Por lo tanto</b>, hoy está soleado.</li></ol> <p>La validez de los dos argumentos anteriores depende del significado de las expresiones «o» y «no». Si alguna de estas expresiones se cambia por otra, entonces los argumentos podrían dejar de ser válidos. Por ejemplo, considérese el siguiente argumento inválido: </p> <ol><li><b>Ni</b> está soleado <b>ni</b> está nublado.</li> <li><b>No</b> está nublado.</li> <li><b>Por lo tanto</b>, está soleado.</li></ol> <p>Estas expresiones como «o» y «no», de las que depende la validez de los argumentos, se llaman <a href="/wiki/Conectiva_l%C3%B3gica" title="Conectiva lógica">conectivas lógicas</a>. En cuanto a expresiones como «está nublado» y «mañana es jueves», lo único que importa de ellas es que tengan un <a href="/wiki/Valor_de_verdad" title="Valor de verdad">valor de verdad</a>. Es por esto que se las reemplaza por simples letras, cuya intención es simbolizar una expresión con valor de verdad cualquiera. A estas letras se las llama <a href="/wiki/Variables_proposicionales" class="mw-redirect" title="Variables proposicionales">variables proposicionales</a>, y en general se toman del alfabeto latino, empezando por la letra <i>p</i> (de «proposición») luego <i>q</i>, <i>r</i>, <i>s</i>, etc. Es así que los dos primeros argumentos de esta sección se podrían reescribir así: </p> <ol><li><i>p</i> <b>o</b> <i>q</i></li> <li><b>No</b> <i>q</i></li> <li><b>Por lo tanto</b>, <i>p</i></li></ol> <p>Y el tercer argumento, a pesar de no ser válido, se puede reescribir así: </p> <ol><li><b>Ni</b> <i>p</i> <b>ni</b> <i>q</i></li> <li><b>No</b> <i>q</i></li> <li><b>Por lo tanto</b>, <i>p</i></li></ol> <div class="mw-heading mw-heading3"><h3 id="Conectivas_lógicas"><span id="Conectivas_l.C3.B3gicas"></span>Conectivas lógicas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=2" title="Editar sección: Conectivas lógicas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículo principal:</span> <i><a href="/wiki/Conectiva_l%C3%B3gica" title="Conectiva lógica"> Conectiva lógica</a></i></div> <table class="vertical-navbox nowraplinks plainlist" style="float:right;clear:right;width:22.0em;margin:0 0 1.0em 1.0em;background:#f9f9f9;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%"><tbody><tr><th style="padding:0.2em 0.4em 0.2em;font-size:145%;line-height:1.2em"><a href="/wiki/Conectiva_l%C3%B3gica" title="Conectiva lógica">Conectivas lógicas</a></th></tr><tr><td style="padding:0.2em 0 0.4em"><span typeof="mw:File"><a href="/wiki/Archivo:Logical_connectives_Hasse_diagram.svg" class="mw-file-description"><img alt="Diagrama de Hasse de las 16 conectivas lógicas" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/200px-Logical_connectives_Hasse_diagram.svg.png" decoding="async" width="200" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/300px-Logical_connectives_Hasse_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/400px-Logical_connectives_Hasse_diagram.svg.png 2x" data-file-width="744" data-file-height="1052" /></a></span></td></tr><tr><td style="padding:0 0.1em 0.4em"> <ul><li><a href="/wiki/Tautolog%C3%ADa" title="Tautología">Tautología</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }"></span></li> <li><a href="/wiki/Conjunci%C3%B3n_opuesta" title="Conjunción opuesta">Conjunción opuesta</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span></li> <li><a href="/wiki/Implicaci%C3%B3n_opuesta" title="Implicación opuesta">Implicación opuesta</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">←</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0fb4bce772117bbaf55b7ca1539ceff9ae218c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftarrow }"></span></li> <li><a href="/wiki/Condicional_material" title="Condicional material">Condicional material</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }"></span></li> <li><a href="/wiki/Disyunci%C3%B3n_l%C3%B3gica" title="Disyunción lógica">Disyunción lógica</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"></span></li> <li><a href="/wiki/Negaci%C3%B3n_l%C3%B3gica" title="Negación lógica">Negación lógica</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/099107443792f5fec9bebe39b919a690db7198c1" 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 \lnot }"></span></li> <li><a href="/wiki/Disyunci%C3%B3n_exclusiva" title="Disyunción exclusiva">Disyunción exclusiva</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nleftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nleftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce85ed756bc5a6cdf0f62892f57a6a1f96803ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nleftrightarrow }"></span></li> <li><a href="/wiki/Bicondicional" title="Bicondicional">Bicondicional</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></span></li> <li><a href="/wiki/Afirmaci%C3%B3n_l%C3%B3gica" title="Afirmación lógica">Afirmación lógica</a></li> <li><a href="/wiki/Disyunci%C3%B3n_opuesta" title="Disyunción opuesta">Disyunción opuesta</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4618f22b0f780805eb94bb407578d9bc9487947a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \downarrow }"></span></li> <li><a href="/wiki/Adjunci%C3%B3n_l%C3%B3gica" title="Adjunción lógica">Adjunción lógica</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↛</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c458d67617e028ed10948d2dbcfef80e9e060a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nrightarrow }"></span></li> <li><a href="/wiki/Adjunci%C3%B3n_opuesta" title="Adjunción opuesta">Adjunción opuesta</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nleftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nleftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7694c9fc8eebe8a57c8156dd3c2caf022a619439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nleftarrow }"></span></li> <li><a href="/wiki/Conjunci%C3%B3n_l%C3%B3gica" title="Conjunción lógica">Conjunción lógica</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></span></li> <li><a href="/wiki/Contradicci%C3%B3n" title="Contradicción">Contradicción</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }"></span></li></ul></td> </tr><tr><td style="text-align:right;font-size:115%"><style data-mw-deduplicate="TemplateStyles:r149274968">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}.mw-parser-output .infobox .navbar{font-size:100%}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-ver"><a href="/wiki/Plantilla:Conectivas_l%C3%B3gicas" title="Plantilla:Conectivas lógicas"><abbr title="Ver esta plantilla">v</abbr></a></li><li class="nv-discusión"><a href="/w/index.php?title=Plantilla_discusi%C3%B3n:Conectivas_l%C3%B3gicas&action=edit&redlink=1" class="new" title="Plantilla discusión:Conectivas lógicas (aún no redactado)"><abbr title="Conversar sobre esta plantilla">t</abbr></a></li><li class="nv-editar"><a class="external text" href="https://es.wikipedia.org/w/index.php?title=Plantilla:Conectivas_l%C3%B3gicas&action=edit"><abbr title="Editar esta plantilla">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>A continuación hay una tabla que despliega todas las conectivas lógicas que ocupan a la lógica proposicional, incluyendo ejemplos de su uso en el <a href="/wiki/Lenguaje_natural" class="mw-redirect" title="Lenguaje natural">lenguaje natural</a> y los símbolos que se utilizan para representarlas en <a href="/wiki/Lenguaje_formal" title="Lenguaje formal">lenguaje formal</a>. </p> <table class="wikitable"> <tbody><tr> <th>Conectiva </th> <th>Expresión en el<br />lenguaje natural </th> <th>Ejemplo </th> <th>Símbolo en<br />este artículo </th> <th>Símbolos<br />alternativos </th></tr> <tr align="center"> <td align="left"><a href="/wiki/Negaci%C3%B3n_l%C3%B3gica" title="Negación lógica">Negación</a> </td> <td>no </td> <td align="left"><b>No</b> está lloviendo. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mspace width="thinmathspace" /> </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/e696c9f6fbea13e9bc4e7cbb549287558d8d0a94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.937ex; height:1.176ex;" alt="{\displaystyle \neg \,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce5528a8c86c5b0121f9448aa9a117429f5b9c88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.195ex; height:1.343ex;" alt="{\displaystyle \sim \,}"></span> </td></tr> <tr align="center"> <td align="left"><a href="/wiki/Conjunci%C3%B3n_l%C3%B3gica" title="Conjunción lógica">Conjunción</a> </td> <td>y </td> <td align="left">Está lloviendo <b>y</b> está nublado. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></span> </td> <td>& </td></tr> <tr align="center"> <td align="left"><a href="/wiki/Disyunci%C3%B3n_l%C3%B3gica" title="Disyunción lógica">Disyunción</a> </td> <td>o </td> <td align="left">Está lloviendo <b>o</b> está soleado. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"></span> </td> <td>| </td></tr> <tr align="center"> <td align="left"><a href="/wiki/Condicional_material" title="Condicional material">Condicional material</a> </td> <td>si... entonces </td> <td align="left"><b>Si</b> está soleado, <b>entonces</b> es de día. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe19c397bbb5552edae0800c4089d7e9161dfeec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.711ex; height:1.843ex;" alt="{\displaystyle \to \,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \supset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊃</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27bfe0828a2ed4c9c6b70987a85c02a1f005843c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \supset }"></span> </td></tr> <tr align="center"> <td align="left"><a href="/wiki/Bicondicional" title="Bicondicional">Bicondicional</a> </td> <td>si y solo si </td> <td align="left">Está nublado <b>si y solo si</b> hay nubes visibles. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8bb536993d997af505a4152639b2c09b89a96d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.195ex; height:1.509ex;" alt="{\displaystyle \equiv \,}"></span> </td></tr> <tr align="center"> <td align="left"><a href="/wiki/Disyunci%C3%B3n_opuesta" title="Disyunción opuesta">Disyunción opuesta</a> </td> <td>ni... ni </td> <td align="left"><b>Ni</b> está soleado <b>ni</b> está nublado. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \downarrow \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e366ca00f78638a4b58c0d4abf0c3498c8f9dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.55ex; height:2.509ex;" alt="{\displaystyle \downarrow \,}"></span> </td> <td> </td></tr> <tr align="center"> <td align="left"><a href="/wiki/Disyunci%C3%B3n_exclusiva" title="Disyunción exclusiva">Disyunción exclusiva</a> </td> <td>o bien... o bien </td> <td align="left"><b>O bien</b> está soleado, <b>o bien</b> está nublado. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nleftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nleftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce85ed756bc5a6cdf0f62892f57a6a1f96803ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nleftrightarrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus ,\not \equiv ,W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> <mo>,</mo> <mo>≢</mo> <mo>,</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus ,\not \equiv ,W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2a422603388a83cb3fd9dc0509130f6d7dfa22a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.121ex; height:2.676ex;" alt="{\displaystyle \oplus ,\not \equiv ,W}"></span> </td></tr></tbody></table> <p>En la lógica proposicional, las conectivas lógicas se tratan como <a href="/wiki/Funci%C3%B3n_de_verdad" title="Función de verdad">funciones de verdad</a>. Es decir, como <a href="/wiki/Funci%C3%B3n_matem%C3%A1tica" class="mw-redirect" title="Función matemática">funciones</a> que toman conjuntos de valores de verdad y devuelven valores de verdad. Por ejemplo, la conectiva lógica «no» es una función que si toma el valor de verdad V, devuelve F, y si toma el valor de verdad F, devuelve V. Por lo tanto, si se aplica la función «no» a una letra que represente una proposición falsa, el resultado será algo verdadero. Si es falso que «está lloviendo», entonces será verdadero que «no está lloviendo». </p><p>El significado de las conectivas lógicas no es nada más que su comportamiento como funciones de verdad. Cada conectiva lógica se distingue de las otras por los valores de verdad que devuelve frente a las distintas combinaciones de valores de verdad que puede recibir. Esto quiere decir que el significado de cada conectiva lógica puede ilustrarse mediante una tabla que despliegue los valores de verdad que la función devuelve frente a todas las combinaciones posibles de valores de verdad que puede recibir. </p> <table width="90%"> <tbody><tr> <th>Negación </th> <th>Conjunción </th> <th>Disyunción </th> <th>Condicional </th> <th>Bicondicional </th> <th>Disyunción exclusiva </th></tr> <tr valign="top" align="center"> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c||c}\phi &\neg \phi \\\hline F&V\\V&F\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid"> <mtr> <mtd> <mi>ϕ</mi> </mtd> <mtd> <mi mathvariant="normal">¬</mi> <mi>ϕ</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c||c}\phi &\neg \phi \\\hline F&V\\V&F\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ebe3cc67ad891d2c2a181adb5d26dd0d3d6599a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:9.655ex; height:10.509ex;" alt="{\displaystyle {\begin{array}{c||c}\phi &\neg \phi \\\hline F&V\\V&F\\\end{array}}}"></span> </p> </td> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \land \psi \\\hline V&V&V\\F&V&F\\V&F&F\\F&F&F\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid solid"> <mtr> <mtd> <mi>ϕ</mi> </mtd> <mtd> <mi>ψ</mi> </mtd> <mtd> <mi>ϕ</mi> <mo>∧</mo> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \land \psi \\\hline V&V&V\\F&V&F\\V&F&F\\F&F&F\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df160afa3165947916bfd9a8d8834014ba87db7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:16.31ex; height:16.843ex;" alt="{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \land \psi \\\hline V&V&V\\F&V&F\\V&F&F\\F&F&F\\\end{array}}}"></span> </p> </td> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \lor \psi \\\hline V&V&V\\F&V&V\\V&F&V\\F&F&F\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid solid"> <mtr> <mtd> <mi>ϕ</mi> </mtd> <mtd> <mi>ψ</mi> </mtd> <mtd> <mi>ϕ</mi> <mo>∨</mo> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \lor \psi \\\hline V&V&V\\F&V&V\\V&F&V\\F&F&F\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38cff03b746d6d4eb93c037d9431dc0e8028d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:16.31ex; height:16.843ex;" alt="{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \lor \psi \\\hline V&V&V\\F&V&V\\V&F&V\\F&F&F\\\end{array}}}"></span> </p> </td> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \to \psi \\\hline V&V&V\\F&V&V\\V&F&F\\F&F&V\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid solid"> <mtr> <mtd> <mi>ϕ</mi> </mtd> <mtd> <mi>ψ</mi> </mtd> <mtd> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \to \psi \\\hline V&V&V\\F&V&V\\V&F&F\\F&F&V\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06c5c2db1621c6d38c6cea0984cfccdf36d5606a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:17.342ex; height:16.843ex;" alt="{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \to \psi \\\hline V&V&V\\F&V&V\\V&F&F\\F&F&V\\\end{array}}}"></span> </p> </td> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \leftrightarrow \psi \\\hline V&V&V\\F&V&F\\V&F&F\\F&F&V\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid solid"> <mtr> <mtd> <mi>ϕ</mi> </mtd> <mtd> <mi>ψ</mi> </mtd> <mtd> <mi>ϕ</mi> <mo stretchy="false">↔</mo> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \leftrightarrow \psi \\\hline V&V&V\\F&V&F\\V&F&F\\F&F&V\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1895fb32ec99f9944df99c600e501f622c3d1755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:17.342ex; height:16.843ex;" alt="{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \leftrightarrow \psi \\\hline V&V&V\\F&V&F\\V&F&F\\F&F&V\\\end{array}}}"></span> </p> </td> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \nleftrightarrow \psi \\\hline V&V&F\\F&V&V\\V&F&V\\F&F&F\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid solid"> <mtr> <mtd> <mi>ϕ</mi> </mtd> <mtd> <mi>ψ</mi> </mtd> <mtd> <mi>ϕ</mi> <mo>↮</mo> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \nleftrightarrow \psi \\\hline V&V&F\\F&V&V\\V&F&V\\F&F&F\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6048e11e216a8cc4e9776bb015ec0d9b837bcba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:17.342ex; height:16.843ex;" alt="{\displaystyle {\begin{array}{c|c||c}\phi &\psi &\phi \nleftrightarrow \psi \\\hline V&V&F\\F&V&V\\V&F&V\\F&F&F\\\end{array}}}"></span> </p> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Leyes_notables_en_lógica"><span id="Leyes_notables_en_l.C3.B3gica"></span>Leyes notables en lógica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=3" title="Editar sección: Leyes notables en lógica"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Entre las reglas de la lógica proposicional clásica algunas de la más notables son listadas a continuación: </p> <ul><li><a href="/wiki/Ley_de_doble_negaci%C3%B3n" class="mw-redirect" title="Ley de doble negación">Ley de doble negación</a></li> <li>Leyes de <a href="/wiki/Idempotencia" title="Idempotencia">idempotencia</a></li> <li>Leyes <a href="/wiki/Asociatividad_(%C3%A1lgebra)" title="Asociatividad (álgebra)">Asociatividad (Buscar 'Lógica Preposicional' dentro del artículo)</a></li> <li><a href="/w/index.php?title=Leyes_conmutativas&action=edit&redlink=1" class="new" title="Leyes conmutativas (aún no redactado)">Leyes conmutativas</a></li> <li><a href="/w/index.php?title=Leyes_distributivas&action=edit&redlink=1" class="new" title="Leyes distributivas (aún no redactado)">Leyes distributivas</a></li> <li><a href="/wiki/Leyes_de_De_Morgan" title="Leyes de De Morgan">Leyes de De Morgan</a></li></ul> <p>Otras leyes como el <a href="/wiki/Principio_del_tercero_excluido" title="Principio del tercero excluido">principio del tercero excluido</a> son admisibles en lógica clásica, pero en <a href="/wiki/L%C3%B3gica_intuicionista" title="Lógica intuicionista">lógica intuicionista</a> y con fines a sus aplicaciones matemáticas no existe un equivalente del tercero excluido. </p> <div class="mw-heading mw-heading3"><h3 id="Límites_de_la_lógica_proposicional"><span id="L.C3.ADmites_de_la_l.C3.B3gica_proposicional"></span>Límites de la lógica proposicional</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=4" title="Editar sección: Límites de la lógica proposicional"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La maquinaria de la lógica proposicional permite formalizar y teorizar sobre la validez de una gran cantidad de argumentos. Sin embargo, también existen argumentos que son intuitivamente válidos, pero cuya validez no se puede probar por la lógica proposicional. Por ejemplo, considérese el siguiente argumento: </p> <ol><li>Todos los hombres son mortales.</li> <li>Sócrates es un hombre.</li> <li>Por lo tanto, Sócrates es mortal.</li></ol> <p>Como este argumento no contiene ninguna de las conectivas «no», «y», «o», etc., según la lógica proposicional, su formalización será la siguiente: </p> <ol><li><i>p</i></li> <li><i>q</i></li> <li>Por lo tanto, <i>r</i></li></ol> <p>Pero esta es una forma de argumento inválida, y eso contradice nuestra intuición de que el argumento es válido. Para teorizar sobre la validez de este tipo de argumentos, se necesita investigar la estructura interna de las variables proposicionales. De esto se ocupa la <a href="/wiki/L%C3%B3gica_de_primer_orden" title="Lógica de primer orden">lógica de primer orden</a>. Otros sistemas formales permiten teorizar sobre otros tipos de argumentos. Por ejemplo la <a href="/wiki/L%C3%B3gica_de_segundo_orden" title="Lógica de segundo orden">lógica de segundo orden</a>, la <a href="/wiki/L%C3%B3gica_modal" title="Lógica modal">lógica modal</a> y la <a href="/wiki/L%C3%B3gica_temporal" title="Lógica temporal">lógica temporal</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Dos_sistemas_formales_de_lógica_proposicional"><span id="Dos_sistemas_formales_de_l.C3.B3gica_proposicional"></span>Dos sistemas formales de lógica proposicional</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=5" title="Editar sección: Dos sistemas formales de lógica proposicional"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A continuación se presentan dos <a href="/wiki/Sistema_formal" title="Sistema formal">sistemas formales</a> estándar para la lógica proposicional. El primero es un sistema axiomático simple, y el segundo es un sistema sin axiomas, de <a href="/wiki/Deducci%C3%B3n_natural" title="Deducción natural">deducción natural</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Sistema_axiomático"><span id="Sistema_axiom.C3.A1tico"></span>Sistema axiomático</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=6" title="Editar sección: Sistema axiomático"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Alfabeto">Alfabeto</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=7" title="Editar sección: Alfabeto"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El alfabeto de un sistema formal es el conjunto de símbolos que pertenecen al lenguaje del sistema. Si L es el nombre de este sistema axiomático de lógica proposicional, entonces el alfabeto de L consiste en: </p> <ul><li>Una cantidad finita pero arbitrariamente grande de variables proposicionales. En general se las toma del alfabeto latino, empezando por la letra <i>p</i>, luego <i>q</i>, <i>r</i>, etc., y utilizando subíndices cuando es necesario o conveniente. Las variables proposicionales representan <a href="/wiki/Proposici%C3%B3n" title="Proposición">proposiciones</a> como "está lloviendo" o "los metales se expanden con el calor".</li> <li>Un conjunto de <a href="/wiki/Operador" title="Operador">operadores</a> lógicos: <span class="mwe-math-element"><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 ,\land ,\lor ,\to ,\leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo>,</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg ,\land ,\lor ,\to ,\leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c7da5209d8f06eb0a2364e2b229eec604b54c83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.434ex; height:2.343ex;" alt="{\displaystyle \neg ,\land ,\lor ,\to ,\leftrightarrow }"></span></li> <li>Dos signos de puntuación: los <a href="/wiki/Par%C3%A9ntesis" title="Paréntesis">paréntesis</a> izquierdo y derecho. Su única función es desambiguar ciertas expresiones ambiguas, en exactamente el mismo sentido en que desambiguan la expresión 2 + 2 ÷ 2, que puede significar tanto (2 + 2) ÷ 2, como 2 + (2 ÷ 2).</li></ul> <div class="mw-heading mw-heading4"><h4 id="Gramática"><span id="Gram.C3.A1tica"></span>Gramática</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=8" title="Editar sección: Gramática"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una vez definido el alfabeto, el siguiente paso es determinar qué combinaciones de símbolos pertenecen al lenguaje del sistema. Esto se logra mediante una <a href="/wiki/Gram%C3%A1tica_formal" title="Gramática formal">gramática formal</a>. La misma consiste en un conjunto de reglas que definen <a href="/wiki/Recursi%C3%B3n" title="Recursión">recursivamente</a> las <a href="/wiki/Cadena_de_caracteres" title="Cadena de caracteres">cadenas de caracteres</a> que pertenecen al lenguaje. A las cadenas de caracteres construidas según estas reglas se las llama <a href="/wiki/F%C3%B3rmula_bien_formada" title="Fórmula bien formada">fórmulas bien formadas</a>. Las reglas del sistema L son: </p> <ol><li>Las variables proposicionales del alfabeto de L son fórmulas bien formadas.</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c69f1c4a95b2d750b30fa4cf5d5d068a573ac0d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.773ex; height:2.509ex;" alt="{\displaystyle \phi \,}"></span> es una fórmula bien formada de L, entonces <span class="mwe-math-element"><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 \phi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>ϕ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \phi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d07ee06088be318ca569ee0a01e12f9a4ccf6bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.323ex; height:2.509ex;" alt="{\displaystyle \neg \phi \,}"></span> también lo es.</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c69f1c4a95b2d750b30fa4cf5d5d068a573ac0d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.773ex; height:2.509ex;" alt="{\displaystyle \phi \,}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0246f660f8317d29b9b2f21c339b3fe1171740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.9ex; height:2.509ex;" alt="{\displaystyle \psi \,}"></span> son fórmulas bien formadas de L, entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\phi \land \psi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>∧</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\phi \land \psi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d1def2daaadfa3b46a885a87ff1483cdf4f084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.29ex; height:2.843ex;" alt="{\displaystyle (\phi \land \psi )}"></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 (\phi \lor \psi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>∨</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\phi \lor \psi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9d0435f657d3bc697ac42734f2289781203420a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.29ex; height:2.843ex;" alt="{\displaystyle (\phi \lor \psi )}"></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 (\phi \to \psi )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\phi \to \psi )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2071ceffb8148b26081547a1d64ad8000d51f82f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.709ex; height:2.843ex;" alt="{\displaystyle (\phi \to \psi )\,}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\phi \leftrightarrow \psi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">↔</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\phi \leftrightarrow \psi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e5a2ab20370ef36d2da98d097412c88e470220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.322ex; height:2.843ex;" alt="{\displaystyle (\phi \leftrightarrow \psi )}"></span> también lo son.</li> <li>Solo las expresiones que pueden ser generadas mediante las cláusulas 1 a 3 en un número finito de pasos son fórmulas bien formadas de L.</li></ol> <p>Según estas reglas, las siguientes cadenas de caracteres son ejemplos de fórmulas bien formadas: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4fa5f88a712eb9b03398066a0577fdcf33e02c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.646ex; height:2.009ex;" alt="{\displaystyle p\,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \neg \neg q\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \neg \neg q\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a27c6ba3d6cb1441b0047a559e8ca9b953cd3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.108ex; height:2.009ex;" alt="{\displaystyle \neg \neg \neg q\,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c6e5ae2dd581f95af0ba40f3d09b0d7d9f5e497" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (p\land q)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (p\land q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (p\land q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7043948245d4012329d1739ae7ffc26c13798005" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.181ex; height:2.843ex;" alt="{\displaystyle \neg (p\land q)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\leftrightarrow \neg p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">↔</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\leftrightarrow \neg p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec2ff20835bf5e1a8b4056259f3a2b2d3fdbb3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.313ex; height:2.843ex;" alt="{\displaystyle (p\leftrightarrow \neg p)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((p\to q)\land p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((p\to q)\land p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f68ef9a3caca2e326f01c3c23b4d3e500bbdb581" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.224ex; height:2.843ex;" alt="{\displaystyle ((p\to q)\land p)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\neg (p\land (q\lor r))\lor s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∨</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\neg (p\land (q\lor r))\lor s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482543afdab7833f461629c3580a9b2f4c12b01b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.104ex; height:2.843ex;" alt="{\displaystyle (\neg (p\land (q\lor r))\lor s)}"></span></dd></dl> <p>Y los siguientes son ejemplos de fórmulas mal formadas<sup>[<i><a href="/wiki/Wikipedia:Verificabilidad" title="Wikipedia:Verificabilidad">cita requerida</a></i>]</sup>: </p> <dl><dd><table width="600px"> <tbody><tr> <th align="left">Fórmula </th> <th align="left">Error </th> <th align="left">Corrección </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9945593820cbeba6e869af1d247a9bfe2c33ecc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.366ex; height:2.843ex;" alt="{\displaystyle (p)\,}"></span> </td> <td>Sobran paréntesis </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4fa5f88a712eb9b03398066a0577fdcf33e02c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.646ex; height:2.009ex;" alt="{\displaystyle p\,}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (p)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (p)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d4cd0dcb2f047c8352909daedb256d89985255a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.916ex; height:2.843ex;" alt="{\displaystyle \neg (p)\,}"></span> </td> <td>Sobran paréntesis </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg p\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg p\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/857d8885633f874a8e20069e764f3694cfd33257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.107ex; height:2.009ex;" alt="{\displaystyle \neg p\,}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\neg p)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\neg p)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ebd3364fc86c83488199664f600bbf135c0ed4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.916ex; height:2.843ex;" alt="{\displaystyle (\neg p)\,}"></span> </td> <td>Sobran paréntesis </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg p\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg p\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/857d8885633f874a8e20069e764f3694cfd33257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.107ex; height:2.009ex;" alt="{\displaystyle \neg p\,}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\to q\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\to q\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c35f9558d53cd7b11c233f26c1a9182a409a8a22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.33ex; height:2.176ex;" alt="{\displaystyle p\to q\,}"></span> </td> <td>Faltan paréntesis </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to q)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to q)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be5341a146ba083773be498e982de426935737c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.049ex; height:2.843ex;" alt="{\displaystyle (p\to q)\,}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land q\to r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q\to r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a80df6366458db3d751eaaa1525e0901cebd8100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.294ex; height:2.843ex;" alt="{\displaystyle (p\land q\to r)}"></span> </td> <td>Faltan paréntesis </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((p\land q)\to r)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((p\land q)\to r)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b285e2f32b13b1fe00b88123643235749677de0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.49ex; height:2.843ex;" alt="{\displaystyle ((p\land q)\to r)\,}"></span> </td></tr></tbody></table></dd></dl> <p>Por otra parte, dado que la única función de los paréntesis es desambiguar las fórmulas, en general se acostumbra omitir los paréntesis <i>externos</i> de cada fórmula, ya que estos no cumplen ninguna función. Así por ejemplo, las siguientes fórmulas generalmente se consideran bien formadas: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\land q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∧</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\land q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4904e68d8180ec4c12e20fabc15017b60b098b9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\land q}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg p\to q\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg p\to q\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0ae316bfe19bd185f8a18830e7f5628f953a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.79ex; height:2.176ex;" alt="{\displaystyle \neg p\to q\,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land q)\lor \neg q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q)\lor \neg q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0607caaf0c772195eb540e0a1329f9b0977de36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.833ex; height:2.843ex;" alt="{\displaystyle (p\land q)\lor \neg q}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\leftrightarrow q)\leftrightarrow (q\leftrightarrow p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">↔</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\leftrightarrow q)\leftrightarrow (q\leftrightarrow p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69fe3a313510d9e71cc58e629876ae68d5ddc60c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.939ex; height:2.843ex;" alt="{\displaystyle (p\leftrightarrow q)\leftrightarrow (q\leftrightarrow p)}"></span></dd></dl> <p>Otra convención acerca del uso de los paréntesis es que las conjunciones y las disyunciones tienen «menor jerarquía» que los condicionales materiales y los bicondicionales. Esto significa que dada una fórmula sin paréntesis, las conjunciones y las disyunciones deben agruparse <i>antes</i> que los condicionales materiales y los bicondicionales. Por ejemplo: </p> <dl><dd><table width="600px"> <tbody><tr> <th align="left">Fórmula </th> <th align="left">Lectura correcta </th> <th align="left">Lectura incorrecta </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\land q\to r\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\land q\to r\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8a8a8f8ad5e60ca2734975fce5e4bd5e03d254" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.961ex; height:2.343ex;" alt="{\displaystyle p\land q\to r\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land q)\to r\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>r</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q)\to r\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27720240430d7ea061e601f8dcfe73762ccd19a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.681ex; height:2.843ex;" alt="{\displaystyle (p\land q)\to r\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\land (q\to r)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\land (q\to r)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a76f5d3e96d5f8c6aac69c46e4e348b2a55ede" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.77ex; height:2.843ex;" alt="{\displaystyle p\land (q\to r)\,}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg p\leftrightarrow q\lor r\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo>∨</mo> <mi>r</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg p\leftrightarrow q\lor r\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/021a8e42a3de4d5522756245c756cd0356ca2820" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.422ex; height:2.343ex;" alt="{\displaystyle \neg p\leftrightarrow q\lor r\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg p\leftrightarrow (q\lor r)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∨</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg p\leftrightarrow (q\lor r)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d008de51a241a7d4c525a74abd4440f921d68f96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.231ex; height:2.843ex;" alt="{\displaystyle \neg p\leftrightarrow (q\lor r)\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\neg p\leftrightarrow q)\lor r\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>r</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\neg p\leftrightarrow q)\lor r\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3466843590428eea19f388b13fc2800f9b71aedc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.231ex; height:2.843ex;" alt="{\displaystyle (\neg p\leftrightarrow q)\lor r\,}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\land q\leftrightarrow r\lor s\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">↔</mo> <mi>r</mi> <mo>∨</mo> <mi>s</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\land q\leftrightarrow r\lor s\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1015ba6a27f192d4e1fb7ad643769c10a1ea9b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.634ex; height:2.343ex;" alt="{\displaystyle p\land q\leftrightarrow r\lor s\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land q)\leftrightarrow (r\lor s)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>∨</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q)\leftrightarrow (r\lor s)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00e8c89a5bf8418b827f8ba1f1555b62f9c2c5f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.163ex; height:2.843ex;" alt="{\displaystyle (p\land q)\leftrightarrow (r\lor s)\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land (q\leftrightarrow r))\lor s\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">↔</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>s</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land (q\leftrightarrow r))\lor s\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41a326889b05a99b2a308df9947098b7b8eeb1d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.163ex; height:2.843ex;" alt="{\displaystyle (p\land (q\leftrightarrow r))\lor s\,}"></span> </td></tr></tbody></table></dd></dl> <p>Estas convenciones son análogas a las que existen en el <a href="/wiki/%C3%81lgebra_elemental" title="Álgebra elemental">álgebra elemental</a>, donde la multiplicación y la división siempre deben resolverse antes que la suma y la resta. Así por ejemplo, la ecuación 2 + 2 × 2 podría interpretarse como (2 + 2) × 2 o como 2 + (2 × 2). En el primer caso el resultado sería 8, y en el segundo caso sería 6. Pero como la multiplicación siempre debe resolverse antes que la suma, el resultado correcto en este caso es 6, no 8. </p> <div class="mw-heading mw-heading4"><h4 id="Axiomas">Axiomas</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=9" title="Editar sección: Axiomas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Los <a href="/wiki/Axioma" title="Axioma">axiomas</a> de un sistema formal son un conjunto de fórmulas bien formadas que se toman como punto de partida para demostraciones ulteriores. Un conjunto de axiomas estándar es el que descubrió <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Jan Łukasiewicz</a>: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\phi \to (\psi \to \phi ))\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">→</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\phi \to (\psi \to \phi ))\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27679853a3ac5048e4186124ec41eb4c0053bf29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.518ex; height:2.843ex;" alt="{\displaystyle (\phi \to (\psi \to \phi ))\,}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((\phi \to (\psi \to \chi ))\to ((\phi \to \psi )\to (\phi \to \chi )))\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">→</mo> <mi>χ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mi>χ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((\phi \to (\psi \to \chi ))\to ((\phi \to \psi )\to (\phi \to \chi )))\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29e9e63092b426253dd8502ce7220a27edde288c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.02ex; height:2.843ex;" alt="{\displaystyle ((\phi \to (\psi \to \chi ))\to ((\phi \to \psi )\to (\phi \to \chi )))\,}"></span></li> <li><span class="mwe-math-element"><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 \phi \to \neg \psi )\to (\psi \to \phi ))\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">→</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((\neg \phi \to \neg \psi )\to (\psi \to \phi ))\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb7ebed8f54ee241a9c6610c579992147eebad0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.555ex; height:2.843ex;" alt="{\displaystyle ((\neg \phi \to \neg \psi )\to (\psi \to \phi ))\,}"></span></li></ul> <div class="mw-heading mw-heading4"><h4 id="Reglas_de_inferencia">Reglas de inferencia</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=10" title="Editar sección: Reglas de inferencia"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="vertical-navbox nowraplinks plainlist" style="float:right;clear:right;width:22.0em;margin:0 0 1.0em 1.0em;background:#f9f9f9;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%"><tbody><tr><th style="padding:0.2em 0.4em 0.2em;font-size:145%;line-height:1.2em"><a href="/wiki/Reglas_de_inferencia" class="mw-redirect" title="Reglas de inferencia">Reglas de transformación</a></th></tr><tr><th style="padding:0.1em;background:#eaeaff;;background:#ddf;font-size:110%; border-bottom:1px #fefefe solid;"> <a class="mw-selflink selflink">Lógica proposicional</a></th></tr><tr><th style="padding:0.1em;background:#eaeaff;"> <a href="/wiki/Reglas_de_inferencia" class="mw-redirect" title="Reglas de inferencia">Reglas de inferencia</a></th></tr><tr><td style="padding:0 0.1em 0.4em;padding-top:0.15em;"> <ul><li><a href="/wiki/Modus_tollendo_tollens" title="Modus tollendo tollens"><span title="A→B, A ⊢ B"><i>Modus tollendo tollens</i></span></a> / <a href="/wiki/Modus_tollendo_ponens" title="Modus tollendo ponens"><span title="A→B, ¬B ⊢ ¬A"><i>ponens</i></span></a></li> <li><a href="/wiki/Modus_ponendo_ponens" title="Modus ponendo ponens"><span title="A→B, A ⊢ B"><i>Modus ponendo ponens</i></span></a> / <i><a href="/wiki/Modus_ponendo_tollens" title="Modus ponendo tollens">tollens</a></i></li> <li><a href="/wiki/Introducci%C3%B3n_del_bicondicional" title="Introducción del bicondicional"><span title="A→B, B→A ⊢ A↔B">Introducción del bicondicional </span></a> / <a href="/wiki/Eliminaci%C3%B3n_del_bicondicional" title="Eliminación del bicondicional">eliminación</a></li> <li><a href="/wiki/Introducci%C3%B3n_de_la_conjunci%C3%B3n" title="Introducción de la conjunción"><span title="A, B ⊢ A∧B">Introducción de la conjunción</span></a> / <a href="/wiki/Simplificaci%C3%B3n" title="Simplificación"><span title="A∧B ⊢ A">eliminación</span></a></li> <li><a href="/wiki/Introducci%C3%B3n_de_la_disyunci%C3%B3n" title="Introducción de la disyunción"><span title="A ⊢ A∨B">Introducción de la disyunción</span></a> / <a href="/wiki/Eliminaci%C3%B3n_de_la_disyunci%C3%B3n" title="Eliminación de la disyunción"><span title="A∨B, A→C, B→C ⊢ C">eliminación</span></a></li> <li><a href="/wiki/Modus_tollendo_ponens" title="Modus tollendo ponens"><span title="A∨B, ¬A ⊢ B">Silogismo disyuntivo</span></a> / <a href="/wiki/Silogismo_hipot%C3%A9tico" title="Silogismo hipotético"><span title="A→B, B→C ⊢ A→C"> hipotético</span></a></li> <li><a href="/wiki/Dilema_constructivo" title="Dilema constructivo"><span title="A→P, B→Q, A∨B ⊢ P∨Q">Dilema constructivo</span></a> / <a href="/wiki/Dilema_destructivo" title="Dilema destructivo"><span title="A→P, B→Q, ¬P∨¬Q ⊢ ¬A∨¬B">destructivo</span></a></li> <li><a href="/wiki/Absorci%C3%B3n_(l%C3%B3gica)" title="Absorción (lógica)"><span title="A→B ⊢ A→A∧B">Absorción</span></a></li></ul></td> </tr><tr><th style="padding:0.1em;background:#eaeaff;"> <a href="/wiki/Reglas_de_reemplazo" title="Reglas de reemplazo">Reglas de reemplazo</a></th></tr><tr><td style="padding:0 0.1em 0.4em;padding-top:0.15em;"> <div class="hlist hlist-separated" style="margin-left: 0em;"> <ul><li><a href="/wiki/Asociatividad_(%C3%A1lgebra)" title="Asociatividad (álgebra)"><span title="A∨(B∨C) = (A∨B)∨C">Asociatividad</span></a></li> <li><a href="/wiki/Conmutatividad" title="Conmutatividad">Conmutatividad</a></li> <li><a href="/wiki/Distributividad" title="Distributividad"><span title="A∧(B∨C) = (A∧B)∨(A∧C)">Distributividad</span></a></li> <li><a href="/wiki/Doble_negaci%C3%B3n_(l%C3%B3gica)" title="Doble negación (lógica)"><span title="¬¬A = A">Doble negación</span></a></li> <li><a href="/wiki/Leyes_de_De_Morgan" title="Leyes de De Morgan">Leyes de De Morgan</a></li> <li><a href="/wiki/Transposici%C3%B3n_(l%C3%B3gica)" title="Transposición (lógica)">Transposición</a></li> <li><a href="/wiki/Implicaci%C3%B3n_material_(regla_de_inferencia)" class="mw-redirect" title="Implicación material (regla de inferencia)"><span title="A→B ⊢ ¬A∨B">Implicación material</span></a></li> <li><a href="/wiki/Exportaci%C3%B3n_(l%C3%B3gica)" title="Exportación (lógica)"><span title="(A∧B)→C ⊢ A→(B→C)">Exportación</span></a></li> <li><a href="/wiki/Tautolog%C3%ADa_(regla_de_inferencia)" title="Tautología (regla de inferencia)">Tautología</a></li> <li><a href="/wiki/Introducci%C3%B3n_de_la_negaci%C3%B3n" title="Introducción de la negación">Introducción de la negación</a></li></ul></div></td> </tr><tr><th style="padding:0.1em;background:#eaeaff;;background:#ddf;font-size:110%;"> <a href="/wiki/L%C3%B3gica_de_primer_orden" title="Lógica de primer orden">Lógica predicativa</a></th></tr><tr><td style="padding:0 0.1em 0.4em;padding-top:0.15em;"> <ul><li><a href="/wiki/Generalizaci%C3%B3n_universal" title="Generalización universal">Generalización</a> / <a href="/wiki/Instanciaci%C3%B3n_universal" title="Instanciación universal">instanciación universal</a></li> <li><a href="/wiki/Generalizaci%C3%B3n_existencial" title="Generalización existencial">Generalización</a> / <a href="/wiki/Instanciaci%C3%B3n_existencial" title="Instanciación existencial">instanciación existencial</a></li></ul></td> </tr><tr><th style="padding:0.1em;background:#eaeaff;;background:#ddf;font-size:110%;"> <a href="/wiki/L%C3%B3gica_modal" title="Lógica modal">Lógica modal</a></th></tr><tr><td style="text-align:right;font-size:115%"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r149274968"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-ver"><a href="/wiki/Plantilla:Reglas_de_transformaci%C3%B3n" title="Plantilla:Reglas de transformación"><abbr title="Ver esta plantilla">v</abbr></a></li><li class="nv-discusión"><a href="/w/index.php?title=Plantilla_discusi%C3%B3n:Reglas_de_transformaci%C3%B3n&action=edit&redlink=1" class="new" title="Plantilla discusión:Reglas de transformación (aún no redactado)"><abbr title="Conversar sobre esta plantilla">t</abbr></a></li><li class="nv-editar"><a class="external text" href="https://es.wikipedia.org/w/index.php?title=Plantilla:Reglas_de_transformaci%C3%B3n&action=edit"><abbr title="Editar esta plantilla">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>Una <a href="/wiki/Regla_de_inferencia" title="Regla de inferencia">regla de inferencia</a> es una <a href="/wiki/Funci%C3%B3n_matem%C3%A1tica" class="mw-redirect" title="Función matemática">función</a> que va de conjuntos de fórmulas a fórmulas. Al conjunto de fórmulas que la función toma como argumento se lo llama <i>premisas</i>, mientras que a la fórmula que devuelve como valor se la llama <i>conclusión</i>. En general se busca que las reglas de inferencia transmitan la verdad de las premisas a la conclusión. Es decir, que sea imposible que las premisas sean verdaderas y la conclusión falsa. En el caso de L, la única regla de inferencia es el <a href="/wiki/Modus_ponens" class="mw-redirect" title="Modus ponens">modus ponens</a>, el cual dice: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\phi \to \psi ),\phi \vdash \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ϕ</mi> <mo>⊢</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\phi \to \psi ),\phi \vdash \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a346a991b49cf47efd2a8a7441f93ae240c65442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.965ex; height:2.843ex;" alt="{\displaystyle (\phi \to \psi ),\phi \vdash \psi }"></span></dd></dl> <p>Recordando que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c69f1c4a95b2d750b30fa4cf5d5d068a573ac0d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.773ex; height:2.509ex;" alt="{\displaystyle \phi \,}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0246f660f8317d29b9b2f21c339b3fe1171740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.9ex; height:2.509ex;" alt="{\displaystyle \psi \,}"></span> no son fórmulas, sino metavariables que pueden ser reemplazadas por cualquier fórmula bien formada. </p> <div class="mw-heading mw-heading3"><h3 id="Deducción_natural"><span id="Deducci.C3.B3n_natural"></span>Deducción natural</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=11" title="Editar sección: Deducción natural"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículo principal:</span> <i><a href="/wiki/Deducci%C3%B3n_natural" title="Deducción natural"> Deducción natural</a></i></div> <p>Sea <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{2}={\mathcal {L}}(\mathrm {A} ,\Omega ,\mathrm {Z} ,\mathrm {I} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>,</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{2}={\mathcal {L}}(\mathrm {A} ,\Omega ,\mathrm {Z} ,\mathrm {I} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/735cecc77a6dbc4e9bfe0a867072e0efdc8dd918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.952ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}_{2}={\mathcal {L}}(\mathrm {A} ,\Omega ,\mathrm {Z} ,\mathrm {I} )}"></span>, donde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6366939c4ebbd4e8494d0dedc54c4b8dd7135a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {A} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></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 \mathrm {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faf96082c4b2e79d67626f995ae571403f51b5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathrm {Z} }"></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 \mathrm {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {I} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe7a69180f25bbb4c73e091f97c7c5f9941ed17b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.176ex;" alt="{\displaystyle \mathrm {I} }"></span>, se define como: </p> <ul><li>El conjunto Alfa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6366939c4ebbd4e8494d0dedc54c4b8dd7135a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {A} }"></span> es un <a href="/wiki/Conjunto_finito" title="Conjunto finito">conjunto finito</a> de símbolos que es lo suficientemente grande como para satisfacer las necesidades de una discusión dada, por ejemplo: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} =\{p,q,r,s,t,u\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} =\{p,q,r,s,t,u\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e5174b14ad727030cd30bee57a14fb609ec2906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.53ex; height:2.843ex;" alt="{\displaystyle \mathrm {A} =\{p,q,r,s,t,u\}.}"></span></dd></dl></li> <li>El conjunto Omega <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =\Omega _{1}\cup \Omega _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∪</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =\Omega _{1}\cup \Omega _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea62d7043f5caf4662322271d72fb79fb048fcc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.824ex; height:2.509ex;" alt="{\displaystyle \Omega =\Omega _{1}\cup \Omega _{2}}"></span> como partición de : <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{1}=\{\lnot \},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{1}=\{\lnot \},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a10c1cdba7209c59d10c8bcb597904f287ffe8e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.353ex; height:2.843ex;" alt="{\displaystyle \Omega _{1}=\{\lnot \},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{2}=\{\land ,\lor ,\to ,\leftrightarrow \}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{2}=\{\land ,\lor ,\to ,\leftrightarrow \}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30aad3ad9f24ec95550e00b11b2f625007cf2b2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.653ex; height:2.843ex;" alt="{\displaystyle \Omega _{2}=\{\land ,\lor ,\to ,\leftrightarrow \}.}"></span></dd></dl></li></ul> <p>En el siguiente ejemplo es de un cálculo proposicional, las reglas presentadas de transformación tienen que ser interpretadas como reglas de inferencia de un sistema de deducción natural. El sistema particular aquí presentado no tiene puntos iniciales, lo que significa que su interpretación para las aplicaciones lógicas deriva de un teorema de conjuntos de axiomas vacíos. </p><p>*El conjunto de puntos iniciales está vacío, este es <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {I} =\varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> <mo>=</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {I} =\varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4afb7696f5bff103ae12e86de6fa45440dbaba76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.746ex; height:2.176ex;" alt="{\displaystyle \mathrm {I} =\varnothing }"></span>. </p><p>*El conjunto de reglas de transformación <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faf96082c4b2e79d67626f995ae571403f51b5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathrm {Z} }"></span> se describe como : </p><p>Nuestro cálculo proposicional tiene diez reglas de inferencia. Estas reglas nos permiten derivar otras fórmulas verdaderas dado un conjunto de fórmulas que se supone que son verdaderas. Las primeros nueve simplemente declaran que podemos inferir ciertas fórmulas bien formadas de otras fórmulas bien formadas; y la última regla utiliza el razonamiento hipotético en el sentido de que la premisa de la regla asuma temporalmente una hipótesis( no probada) para formar parte del conjunto de fórmulas deducidas para ver si podemos inferir alguna otra fórmula. Dado que las primeras nueve reglas no son hipotéticas , usualmente se describirían como reglas no hipotéticas, y la última regla como una regla hipotética. </p><p>Al describir las reglas de transformación, podemos introducir un símbolo de metalenguaje <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c0d30cf8cb7dba179e317fcde9583d842e80f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \vdash }"></span>. Es básicamente una taquigrafía conveniente para decir " inferir que ". El formato es <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \vdash \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊢</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \vdash \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27a6c9717bd96aeec6ebe23cf9d76a1a3d82971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.677ex; height:2.509ex;" alt="{\displaystyle \Gamma \vdash \psi }"></span>, en el cual <span class="texhtml">Γ</span> es un conjunto de fórmulas llamadas premisas, y <span class="texhtml mvar" style="font-style:italic;">ψ</span> es una fórmula para hallar la conclusión. La regla de tranformacíon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \vdash \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊢</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \vdash \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27a6c9717bd96aeec6ebe23cf9d76a1a3d82971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.677ex; height:2.509ex;" alt="{\displaystyle \Gamma \vdash \psi }"></span> significa que si toda proposición en<span class="texhtml">Γ</span> es un teorema ( o tiene el mismo valor de verdad que los axiomas ), entonces <span class="texhtml mvar" style="font-style:italic;">ψ</span> es también un teorema. Tenga en cuenta que teniendo en cuenta la siguiente regla la introducción de conjunción <span class="texhtml">Γ</span> tiene más de una fórmula, siempre podemos reducirla con seguridad en una fórmula usando una conjunción. Así que para abreviar, a partir de ese momento podemos representar <span class="texhtml">Γ</span> como una fórmula en lugar de un conjunto. Otra omisión por conveniencia es cuando <span class="texhtml">Γ</span> es un <a href="/wiki/Conjunto_vac%C3%ADo" title="Conjunto vacío">conjunto vacío</a>, en cuyo caso <span class="texhtml">Γ</span> puede no aparecer. </p><p>Un sistema de lógica proposicional también puede construirse a partir de un <a href="/wiki/Conjunto" title="Conjunto">conjunto</a> vacío de <a href="/wiki/Axioma" title="Axioma">axiomas</a>. Para ello se especifican una serie de reglas de <a href="/wiki/Inferencia" title="Inferencia">inferencia</a> que intentan capturar el modo en que naturalmente razonamos acerca de las conectivas lógicas. </p> <dl><dt><a href="/wiki/Introducci%C3%B3n_de_la_negaci%C3%B3n" title="Introducción de la negación">Introducción de la negación</a></dt> <dd>De <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b805a61fbd9b41cd2976ebec792d73a0ebea0e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.662ex; height:2.843ex;" alt="{\displaystyle (p\to q)}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to \neg q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to \neg q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72c74d14db648d3eb4c441b8071741aaef126ebd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.213ex; height:2.843ex;" alt="{\displaystyle (p\to \neg q)}"></span>, se infiere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b198c79234d926cbee42c0f271d903ea55dc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.72ex; height:2.009ex;" alt="{\displaystyle \neg p}"></span>.</dd> <dd>Esto es, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(p\to q),(p\to \neg q)\}\vdash \neg p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>⊢</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(p\to q),(p\to \neg q)\}\vdash \neg p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/720d0c37392c616df777c7dde978496d645a2a91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.664ex; height:2.843ex;" alt="{\displaystyle \{(p\to q),(p\to \neg q)\}\vdash \neg p}"></span>.</dd> <dt><a href="/wiki/Eliminaci%C3%B3n_de_la_negaci%C3%B3n" class="mw-redirect" title="Eliminación de la negación">Eliminación de la negación</a></dt> <dd>De <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b198c79234d926cbee42c0f271d903ea55dc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.72ex; height:2.009ex;" alt="{\displaystyle \neg p}"></span>, se infiere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b70bebff1ed61e4f65a512ea61f2a8849d044d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.641ex; height:2.843ex;" alt="{\displaystyle (p\to r)}"></span>.</dd> <dd>Esto es, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\neg p\}\vdash (p\to r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo fence="false" stretchy="false">}</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\neg p\}\vdash (p\to r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e7588327b13f33b0e1a9d6d57029624b9b6250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.397ex; height:2.843ex;" alt="{\displaystyle \{\neg p\}\vdash (p\to r)}"></span>.</dd> <dt><a href="/wiki/Eliminaci%C3%B3n_de_la_doble_negaci%C3%B3n" class="mw-redirect" title="Eliminación de la doble negación">Eliminación de la doble negación</a></dt> <dd>De <span class="mwe-math-element"><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 \neg p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \neg p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61fb44bd43a9328064cdf2b5d70b91f0843ae54d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.27ex; height:2.009ex;" alt="{\displaystyle \neg \neg p}"></span>, se infiere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>.</dd> <dd>Esto es, <span class="mwe-math-element"><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 \neg p\vdash p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>⊢</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \neg p\vdash p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a4dc1d21a9ff681325eb6fdc0bc8950a1f3c24f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.15ex; height:2.509ex;" alt="{\displaystyle \neg \neg p\vdash p}"></span>.</dd> <dt><a href="/wiki/Introducci%C3%B3n_de_la_conjunci%C3%B3n" title="Introducción de la conjunción">Introducción de la conjunción</a></dt> <dd>De <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>, se infiere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c6e5ae2dd581f95af0ba40f3d09b0d7d9f5e497" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (p\land q)}"></span>.</dd> <dd>Esto es, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{p,q\}\vdash (p\land q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo fence="false" stretchy="false">}</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{p,q\}\vdash (p\land q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffc965c92212f65946c1624275321d5f7769edba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.939ex; height:2.843ex;" alt="{\displaystyle \{p,q\}\vdash (p\land q)}"></span>.</dd> <dt><a href="/wiki/Simplificaci%C3%B3n" title="Simplificación">Eliminación de la conjunción</a></dt> <dd>De <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c6e5ae2dd581f95af0ba40f3d09b0d7d9f5e497" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (p\land q)}"></span>, se infiere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>.</dd> <dd>De <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c6e5ae2dd581f95af0ba40f3d09b0d7d9f5e497" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (p\land q)}"></span>, se infiere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>.</dd> <dd>Esto es, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land q)\vdash p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q)\vdash p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0665b0a9606a1ced373dd66abd55e3cc078e3c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.511ex; height:2.843ex;" alt="{\displaystyle (p\land q)\vdash p}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land q)\vdash q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q)\vdash q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c10f27d977c39a95211031a47e6401864ed818f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.411ex; height:2.843ex;" alt="{\displaystyle (p\land q)\vdash q}"></span>.</dd> <dt><a href="/wiki/Introducci%C3%B3n_de_la_disyunci%C3%B3n" title="Introducción de la disyunción">Introducción de la disyunción</a></dt> <dd>De <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, se infiere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\lor q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\lor q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60f80b50074862b4c84850b692661ba67f49d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (p\lor q)}"></span>.</dd> <dd>Esto es, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\vdash (p\lor q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\vdash (p\lor q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54768af02f843eaea9e5675bedfca7072011200e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.6ex; height:2.843ex;" alt="{\displaystyle p\vdash (p\lor q)}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\vdash (p\lor q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\vdash (p\lor q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98e16538ec3d6d314ea8f35aa026b245356a2c07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.411ex; height:2.843ex;" alt="{\displaystyle q\vdash (p\lor q)}"></span>.</dd> <dt><a href="/wiki/Eliminaci%C3%B3n_de_la_disyunci%C3%B3n" title="Eliminación de la disyunción">Eliminación de la disyunción</a></dt> <dd>De <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\lor q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\lor q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60f80b50074862b4c84850b692661ba67f49d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (p\lor q)}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b70bebff1ed61e4f65a512ea61f2a8849d044d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.641ex; height:2.843ex;" alt="{\displaystyle (p\to r)}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q\to r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (q\to r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b915a57512bbcb7bace630a8aa2a3207e84e06e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.542ex; height:2.843ex;" alt="{\displaystyle (q\to r)}"></span>, se infiere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>.</dd> <dd>Esto es, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{p\lor q,p\to r,q\to r\}\vdash r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> <mo>⊢</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{p\lor q,p\to r,q\to r\}\vdash r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc0eb4f08c68e043385a53c668aab1100dfa93d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.538ex; height:2.843ex;" alt="{\displaystyle \{p\lor q,p\to r,q\to r\}\vdash r}"></span>.</dd> <dt><a href="/wiki/Introducci%C3%B3n_del_bicondicional" title="Introducción del bicondicional">Introducción del bicondicional</a></dt> <dd>De <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b805a61fbd9b41cd2976ebec792d73a0ebea0e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.662ex; height:2.843ex;" alt="{\displaystyle (p\to q)}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q\to p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (q\to p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36e5438d9c5ff7177a6db191b9494388a519a01e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.662ex; height:2.843ex;" alt="{\displaystyle (q\to p)}"></span>, se infiere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\leftrightarrow q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\leftrightarrow q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0529d17f35a432e0622afbfd52a6a7a1fb9098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.662ex; height:2.843ex;" alt="{\displaystyle (p\leftrightarrow q)}"></span>.</dd> <dd>Esto es, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{p\to q,q\to p\}\vdash (p\leftrightarrow q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>p</mi> <mo fence="false" stretchy="false">}</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{p\to q,q\to p\}\vdash (p\leftrightarrow q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f600279af526594bd33de4b6b1c564a93486bfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.438ex; height:2.843ex;" alt="{\displaystyle \{p\to q,q\to p\}\vdash (p\leftrightarrow q)}"></span>.</dd> <dt><a href="/wiki/Eliminaci%C3%B3n_del_bicondicional" title="Eliminación del bicondicional">Eliminación del bicondicional</a></dt> <dd>De <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\leftrightarrow q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\leftrightarrow q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0529d17f35a432e0622afbfd52a6a7a1fb9098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.662ex; height:2.843ex;" alt="{\displaystyle (p\leftrightarrow q)}"></span>, se infiere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b805a61fbd9b41cd2976ebec792d73a0ebea0e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.662ex; height:2.843ex;" alt="{\displaystyle (p\to q)}"></span>.</dd> <dd>De <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\leftrightarrow q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\leftrightarrow q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0529d17f35a432e0622afbfd52a6a7a1fb9098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.662ex; height:2.843ex;" alt="{\displaystyle (p\leftrightarrow q)}"></span>, se infiere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q\to p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (q\to p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36e5438d9c5ff7177a6db191b9494388a519a01e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.662ex; height:2.843ex;" alt="{\displaystyle (q\to p)}"></span>.</dd> <dd>Esto es, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\leftrightarrow q)\vdash (p\to q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\leftrightarrow q)\vdash (p\to q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbc95b4dfa2bdd93efec984213bca64926988202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.035ex; height:2.843ex;" alt="{\displaystyle (p\leftrightarrow q)\vdash (p\to q)}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\leftrightarrow q)\vdash (q\to p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\leftrightarrow q)\vdash (q\to p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10ed0fa571fd3a22a1e69d79680587399becf7d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.035ex; height:2.843ex;" alt="{\displaystyle (p\leftrightarrow q)\vdash (q\to p)}"></span>.</dd> <dt><a href="/wiki/Modus_ponens" class="mw-redirect" title="Modus ponens">Modus ponens</a> (eliminación del condicional)</dt> <dd>De <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b805a61fbd9b41cd2976ebec792d73a0ebea0e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.662ex; height:2.843ex;" alt="{\displaystyle (p\to q)}"></span>, se infiere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>.</dd> <dd>Esto es, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{p,p\to q\}\vdash q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>p</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo fence="false" stretchy="false">}</mo> <mo>⊢</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{p,p\to q\}\vdash q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92352e6898d1af1b88dfa4a2cd670e78e0af9bb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.161ex; height:2.843ex;" alt="{\displaystyle \{p,p\to q\}\vdash q}"></span>.</dd> <dt><a href="/wiki/Prueba_condicional" title="Prueba condicional">Prueba condicional</a> (introducción del condicional)</dt> <dd>De [aceptando que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> permite una prueba de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>], se infiere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b805a61fbd9b41cd2976ebec792d73a0ebea0e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.662ex; height:2.843ex;" alt="{\displaystyle (p\to q)}"></span>.</dd> <dd>Esto es, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\vdash q)\vdash (p\to q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>⊢</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\vdash q)\vdash (p\to q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2174ef026b2a4db9a80c8d0c5f535ae8c0e21e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.132ex; height:2.843ex;" alt="{\displaystyle (p\vdash q)\vdash (p\to q)}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Formas_de_argumentos_básicas_y_derivadas"><span id="Formas_de_argumentos_b.C3.A1sicas_y_derivadas"></span>Formas de argumentos básicas y derivadas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=12" title="Editar sección: Formas de argumentos básicas y derivadas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <th>Nombre </th> <th>Consecuente </th> <th>Descripción </th></tr> <tr> <td><a href="/wiki/Modus_ponens" class="mw-redirect" title="Modus ponens">Modus ponens</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((p\to q)\land p)\vdash q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((p\to q)\land p)\vdash q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e718ce4ec77f4e099094d7e89bacc112c6754294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.004ex; height:2.843ex;" alt="{\displaystyle ((p\to q)\land p)\vdash q}"></span> </td> <td>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>; y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>; por lo tanto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> </td></tr> <tr> <td><a href="/wiki/Modus_tollens" class="mw-redirect" title="Modus tollens">Modus tollens</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((p\to q)\land \neg q)\vdash \neg p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((p\to q)\land \neg q)\vdash \neg p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ae31de8933f036da05b3e54fce242079989424" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.104ex; height:2.843ex;" alt="{\displaystyle ((p\to q)\land \neg q)\vdash \neg p}"></span> </td> <td>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>; y no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>; por lo tanto no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> </td></tr> <tr> <td><a href="/wiki/Silogismo_hipot%C3%A9tico" title="Silogismo hipotético">Silogismo hipotético</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((p\to q)\land (q\to r))\vdash (p\to r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((p\to q)\land (q\to r))\vdash (p\to r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6419a065bbb8e2b35b636e171a6891e3b3ce541e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.948ex; height:2.843ex;" alt="{\displaystyle ((p\to q)\land (q\to r))\vdash (p\to r)}"></span> </td> <td>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>; y si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>; por lo tanto, si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> </td></tr> <tr> <td><a href="/wiki/Silogismo_disyuntivo" class="mw-redirect" title="Silogismo disyuntivo">Silogismo disyuntivo</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((p\lor q)\land \neg p)\vdash q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((p\lor q)\land \neg p)\vdash q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b568f7af03c0c71a3f12c5b8dd1a80359e019c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.523ex; height:2.843ex;" alt="{\displaystyle ((p\lor q)\land \neg p)\vdash q}"></span> </td> <td>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>; y no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>; por lo tanto, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> </td></tr> <tr> <td><a href="/wiki/Dilema_constructivo" title="Dilema constructivo">Dilema constructivo</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((p\to q)\land (r\to s)\land (p\lor r))\vdash (q\lor s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">→</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∨</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((p\to q)\land (r\to s)\land (p\lor r))\vdash (q\lor s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f10f552191f535c0912419950c4b2d9efc2f5ba2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.072ex; height:2.843ex;" alt="{\displaystyle ((p\to q)\land (r\to s)\land (p\lor r))\vdash (q\lor s)}"></span> </td> <td>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>; y si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>; pero <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>; por lo tanto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> </td></tr> <tr> <td><a href="/wiki/Dilema_destructivo" title="Dilema destructivo">Dilema destructivo</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((p\to q)\land (r\to s)\land (\neg q\lor \neg s))\vdash (\neg p\lor \neg r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">→</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((p\to q)\land (r\to s)\land (\neg q\lor \neg s))\vdash (\neg p\lor \neg r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/082ed3c4f255aa4af20a4bd7eb4233c8f7a1071b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.273ex; height:2.843ex;" alt="{\displaystyle ((p\to q)\land (r\to s)\land (\neg q\lor \neg s))\vdash (\neg p\lor \neg r)}"></span> </td> <td>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>; y si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>; pero no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> o no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>; por lo tanto no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> </td></tr> <tr> <td>Dilema bidireccional </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((p\to q)\land (r\to s)\land (p\lor \neg s))\vdash (q\lor \neg r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">→</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((p\to q)\land (r\to s)\land (p\lor \neg s))\vdash (q\lor \neg r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4af3f354fe025bd0bb2d0c74d19ea7f218a0f8c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.172ex; height:2.843ex;" alt="{\displaystyle ((p\to q)\land (r\to s)\land (p\lor \neg s))\vdash (q\lor \neg r)}"></span> </td> <td>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>; y si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>; pero <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>; por lo tanto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> o no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> </td></tr> <tr> <td>Simplificación </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land q)\vdash p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q)\vdash p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0665b0a9606a1ced373dd66abd55e3cc078e3c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.511ex; height:2.843ex;" alt="{\displaystyle (p\land q)\vdash p}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> son verdaderos; por lo tanto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es verdadero </td></tr> <tr> <td>Conjunción </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q\vdash (p\land q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,q\vdash (p\land q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/709afc672607de0171c41eb25c5ce256d138e282" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.704ex; height:2.843ex;" alt="{\displaystyle p,q\vdash (p\land q)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> son verdaderos separadamente; entonces son verdaderos conjuntamente. </td></tr> <tr> <td>Adición </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\vdash (p\lor q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\vdash (p\lor q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54768af02f843eaea9e5675bedfca7072011200e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.6ex; height:2.843ex;" alt="{\displaystyle p\vdash (p\lor q)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es verdadero; por lo tanto la disyunción (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) es verdadera </td></tr> <tr> <td>Composición </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((p\to q)\land (p\to r))\vdash (p\to (q\land r))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∧</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((p\to q)\land (p\to r))\vdash (p\to (q\land r))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b615022bda1dcf1227a1c08c628aada076144d42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.509ex; height:2.843ex;" alt="{\displaystyle ((p\to q)\land (p\to r))\vdash (p\to (q\land r))}"></span> </td> <td>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>; y si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>; por lo tanto si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es verdadero entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> son verdaderos </td></tr> <tr> <td><a href="/wiki/Ley_de_De_Morgan" class="mw-redirect" title="Ley de De Morgan">Ley de De Morgan</a> (1) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (p\land q)\vdash (\neg p\lor \neg q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (p\land q)\vdash (\neg p\lor \neg q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5ba4031d1b54ccbae36739f8e061d7409da4731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.623ex; height:2.843ex;" alt="{\displaystyle \neg (p\land q)\vdash (\neg p\lor \neg q)}"></span> </td> <td>La negación de (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) es equivalente a (no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) </td></tr> <tr> <td>Ley de De Morgan (2) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (p\lor q)\vdash (\neg p\land \neg q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (p\lor q)\vdash (\neg p\land \neg q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b11c95b4b53fbf10b0b5da5de9d9b4842aebd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.623ex; height:2.843ex;" alt="{\displaystyle \neg (p\lor q)\vdash (\neg p\land \neg q)}"></span> </td> <td>La negación de (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) es equivalente a (no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) </td></tr> <tr> <td>Conmutación (1) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\lor q)\vdash (q\lor p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∨</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\lor q)\vdash (q\lor p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca93348719e95e47d204c10aa0efd75d14b4cec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.972ex; height:2.843ex;" alt="{\displaystyle (p\lor q)\vdash (q\lor p)}"></span> </td> <td>(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) es equivalente a (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>) </td></tr> <tr> <td>Conmutación (2) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land q)\vdash (q\land p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∧</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q)\vdash (q\land p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92f08307b3fd630f8b09e3f863af5835f41282d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.972ex; height:2.843ex;" alt="{\displaystyle (p\land q)\vdash (q\land p)}"></span> </td> <td>(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) es equivalente a (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>) </td></tr> <tr> <td>Conmutación (3) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\leftrightarrow q)\vdash (q\leftrightarrow p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">↔</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\leftrightarrow q)\vdash (q\leftrightarrow p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/589981935ee28fdb99010ab82b7e4d6c54620f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.035ex; height:2.843ex;" alt="{\displaystyle (p\leftrightarrow q)\vdash (q\leftrightarrow p)}"></span> </td> <td>(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es equivalente a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) es equivalente a (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> es equivalente a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>) </td></tr> <tr> <td>Asociación (1) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\lor (q\lor r))\vdash ((p\lor q)\lor r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∨</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\lor (q\lor r))\vdash ((p\lor q)\lor r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24af9aee97f5bae3e9d873654bd994da2100ea96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.853ex; height:2.843ex;" alt="{\displaystyle (p\lor (q\lor r))\vdash ((p\lor q)\lor r)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>) es equivalente a (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> </td></tr> <tr> <td>Asociación (2) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land (q\land r))\vdash ((p\land q)\land r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∧</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land (q\land r))\vdash ((p\land q)\land r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3e01f4bb1e4c2b228ad0865181eb3aa82304f40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.853ex; height:2.843ex;" alt="{\displaystyle (p\land (q\land r))\vdash ((p\land q)\land r)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>) es equivalente a (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> </td></tr> <tr> <td>Distribución (1) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\land (q\lor r))\vdash ((p\land q)\lor (p\land r))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∨</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land (q\lor r))\vdash ((p\land q)\lor (p\land r))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4166fc9632d94252ace796e96e41c4ebd8cc83fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.415ex; height:2.843ex;" alt="{\displaystyle (p\land (q\lor r))\vdash ((p\land q)\lor (p\land r))}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>) es equivalente a (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) o (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>) </td></tr> <tr> <td>Distribución (2) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\lor (q\land r))\vdash ((p\lor q)\land (p\lor r))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∧</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\lor (q\land r))\vdash ((p\lor q)\land (p\lor r))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d3e9f3e87a4e9ba3db7af4f221b3ecde1ac20df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.415ex; height:2.843ex;" alt="{\displaystyle (p\lor (q\land r))\vdash ((p\lor q)\land (p\lor r))}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>) es equivalente a (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) y (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>) </td></tr> <tr> <td><a href="/wiki/Doble_negaci%C3%B3n_(l%C3%B3gica)" title="Doble negación (lógica)">Doble negación</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\vdash \neg \neg p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⊢</mo> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\vdash \neg \neg p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0878ca469a135d36959823281cfc87aa75d6b1eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.24ex; height:2.509ex;" alt="{\displaystyle p\vdash \neg \neg p}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es equivalente a la negación de no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> </td></tr> <tr> <td><a href="/wiki/Transposici%C3%B3n_(l%C3%B3gica)" title="Transposición (lógica)">Transposición</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to q)\vdash (\neg q\to \neg p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to q)\vdash (\neg q\to \neg p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e555816ccbdde01ff5d0a920603827e878238582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.136ex; height:2.843ex;" alt="{\displaystyle (p\to q)\vdash (\neg q\to \neg p)}"></span> </td> <td>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> es equivalente a si no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> entonces no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> </td></tr> <tr> <td><a href="/wiki/Implicaci%C3%B3n_material" title="Implicación material">Implicación material</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to q)\vdash (\neg p\lor q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to q)\vdash (\neg p\lor q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be71a7c4930ab4e646485d7bc479d0dc666fc4a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.554ex; height:2.843ex;" alt="{\displaystyle (p\to q)\vdash (\neg p\lor q)}"></span> </td> <td>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> es equivalente a no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> </td></tr> <tr> <td><a href="/wiki/Equivalencia_material" class="mw-redirect" title="Equivalencia material">Equivalencia material</a> (1) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\leftrightarrow q)\vdash ((p\to q)\land (q\to p))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\leftrightarrow q)\vdash ((p\to q)\land (q\to p))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1df89540f366fd02056dcab93751ff7e05317028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.09ex; height:2.843ex;" alt="{\displaystyle (p\leftrightarrow q)\vdash ((p\to q)\land (q\to p))}"></span> </td> <td>(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> si y solo si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) es equivalente a (si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es verdadero entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> es verdadero) y (si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> es verdadero entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es verdadero) </td></tr> <tr> <td>Equivalencia material (2) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\leftrightarrow q)\vdash ((p\land q)\lor (\neg p\land \neg q))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\leftrightarrow q)\vdash ((p\land q)\lor (\neg p\land \neg q))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/870257b25f89e8b9e2eb7ea3262f6ef7fc95c48b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.127ex; height:2.843ex;" alt="{\displaystyle (p\leftrightarrow q)\vdash ((p\land q)\lor (\neg p\land \neg q))}"></span> </td> <td>(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) es equivalente a cualquiera de los dos (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> son verdaderos) o (tanto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> son falsos) </td></tr> <tr> <td>Equivalencia material (3) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\leftrightarrow q)\vdash ((p\lor \neg q)\land (\neg p\lor q))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\leftrightarrow q)\vdash ((p\lor \neg q)\land (\neg p\lor q))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8641bba33fff164062c71f4e9b6f29951ab130f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.127ex; height:2.843ex;" alt="{\displaystyle (p\leftrightarrow q)\vdash ((p\lor \neg q)\land (\neg p\lor q))}"></span> </td> <td>(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>) es equivalente a: tanto (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> como no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> son verdaderos) y (no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> es verdadero) </td></tr> <tr> <td>Exportación<sup id="cite_ref-3" class="reference separada"><a href="#cite_note-3"><span class="corchete-llamada">[</span>3<span class="corchete-llamada">]</span></a></sup> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((p\land q)\to r)\vdash (p\to (q\to r))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((p\land q)\to r)\vdash (p\to (q\to r))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb60d0eb18ee7f17a0708421b59fc94be91eb57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.948ex; height:2.843ex;" alt="{\displaystyle ((p\land q)\to r)\vdash (p\to (q\to r))}"></span> </td> <td>desde (si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> son verdaderos, entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> es verdadero) se puede probar que (si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> es verdadero entonces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> es verdadero, si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es verdadero) </td></tr> <tr> <td>Importación </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to (q\to r))\vdash ((p\land q)\to r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to (q\to r))\vdash ((p\land q)\to r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3582deb17f8485a5fd69a68f24f57275f2b49141" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.948ex; height:2.843ex;" alt="{\displaystyle (p\to (q\to r))\vdash ((p\land q)\to r)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> implica que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> implica <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> es equivalente a que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> implican <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> </td></tr> <tr> <td><a href="/wiki/Tautolog%C3%ADa" title="Tautología">Tautología</a> (1) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\vdash (p\lor p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\vdash (p\lor p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2e65de3b6b30964d06b6ebb5531559e27a0111" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.7ex; height:2.843ex;" alt="{\displaystyle p\vdash (p\lor p)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es verdadero es equivalente a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es verdadero o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es verdadero </td></tr> <tr> <td>Tautología (2) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\vdash (p\land p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\vdash (p\land p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9321b67f5e270a2ac4968376e585924f709316b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.7ex; height:2.843ex;" alt="{\displaystyle p\vdash (p\land p)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es verdadero es equivalente a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es verdadero y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es verdadero </td></tr> <tr> <td><a href="/wiki/Principio_del_tercero_excluido" title="Principio del tercero excluido">Principio del tercero excluido</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash (p\lor \neg p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash (p\lor \neg p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34acb6a72aa2d8bd27a9f3e7a2b33c7a30b8b937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.346ex; height:2.843ex;" alt="{\displaystyle \vdash (p\lor \neg p)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es verdadero </td></tr> <tr> <td><a href="/wiki/Principio_de_no_contradicci%C3%B3n" title="Principio de no contradicción">Principio de no contradicción</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash \neg (p\land \neg p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash \neg (p\land \neg p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d7ab87b3558f3091c8e4547f54ddebf36b5d04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.897ex; height:2.843ex;" alt="{\displaystyle \vdash \neg (p\land \neg p)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> y no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> es falso </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Ejemplo_de_una_demostración"><span id="Ejemplo_de_una_demostraci.C3.B3n"></span>Ejemplo de una demostración</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=13" title="Editar sección: Ejemplo de una demostración"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Demostrar: <span class="mwe-math-element"><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\to A\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>A</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to A\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/486cf2087a0a1876d1bee0240e7cd864141823e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.487ex; height:2.176ex;" alt="{\displaystyle A\to A\,}"></span> </p><p>Una posible prueba de esto (que, aunque válida, pasa a contener más pasos de los necesarios) se puede disponer de la siguiente manera: </p> <center> <table class="wikitable" style="background:white"> <tbody><tr> <th align="left">Paso </th> <th>Fórmula </th> <th>Razón </th></tr> <tr> <td>1</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></td> <td>Premisa. </td></tr> <tr> <td>2</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31940c78d50bd53d7a2a8e89351754385ec6daeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.069ex; height:2.176ex;" alt="{\displaystyle A\lor A}"></span></td> <td>Desde (1) por introducción de la disyunción. </td></tr> <tr> <td>3</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\lor A)\land A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\lor A)\land A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1774dc1f83c8fd656bd1c5bf1d8d4bf4c77354" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.204ex; height:2.843ex;" alt="{\displaystyle (A\lor A)\land A}"></span></td> <td>Desde (1) y (2) por introducción de la conjunción. </td></tr> <tr> <td>4</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></td> <td>Desde (3) por eliminación de la conjunción. </td></tr> <tr> <td>5</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\vdash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊢</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\vdash A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6627a49a818281679e6699d646d0fd6a2ddd34fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle A\vdash A}"></span></td> <td>Resumen de (1) hasta (4). </td></tr> <tr> <td>6</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash A\to A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash A\to A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a20720173dd81a25520d06a3d798d0eb0689a582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.166ex; height:2.176ex;" alt="{\displaystyle \vdash A\to A}"></span></td> <td>Desde (5) por introducción del condicional. <a href="/wiki/Quod_erat_demonstrandum" title="Quod erat demonstrandum">QED</a> </td></tr></tbody></table> </center> <p>Interpretar <span class="mwe-math-element"><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\vdash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊢</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\vdash A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6627a49a818281679e6699d646d0fd6a2ddd34fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle A\vdash A}"></span> como: "Asumiendo que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, inferire <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>". Leer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c0d30cf8cb7dba179e317fcde9583d842e80f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \vdash }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fbf720da5a9387e23c628079fbc3e021399c911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.1ex; height:2.176ex;" alt="{\displaystyle A\to A}"></span> como "Suponiendo nada, inferir que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> implica <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>", o "Es una tautología que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> implica <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>", o "Siempre es cierto que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> implica <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>". </p> <div class="mw-heading mw-heading2"><h2 id="Lenguaje_formal_en_la_notación_BNF"><span id="Lenguaje_formal_en_la_notaci.C3.B3n_BNF"></span>Lenguaje formal en la notación BNF</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=14" title="Editar sección: Lenguaje formal en la notación BNF"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El <a href="/wiki/Lenguaje_formal" title="Lenguaje formal">lenguaje formal</a> de la lógica proposicional se puede generar con la <a href="/wiki/Gram%C3%A1tica_formal" title="Gramática formal">gramática formal</a> descrita usando la <a href="/wiki/Backus-Naur_form" class="mw-redirect" title="Backus-Naur form">notación BNF</a> como sigue: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcl}{\rm {\langle Bicondicional\rangle }}&::=&{\rm {\langle Condicional\rangle \leftrightarrow \langle Bicondicional\rangle \mid \langle Condicional\rangle }}\\{\rm {\langle Condicional\rangle }}&::=&{\rm {\langle Conjunci{\acute {o}}n\rangle \rightarrow \langle Condicional\rangle \mid \langle Conjunci{\acute {o}}n\rangle }}\\{\rm {\langle Conjunci{\acute {o}}n\rangle }}&::=&{\rm {\langle Disyunci{\acute {o}}n\rangle \vee \langle Conjunci{\acute {o}}n\rangle \mid \langle Disyunci{\acute {o}}n\rangle }}\\{\rm {\langle Disyunci{\acute {o}}n\rangle }}&::=&{\rm {\langle Literal\rangle \wedge \langle Disyunci{\acute {o}}n\rangle \mid \langle Literal\rangle }}\\{\rm {\langle Literal\rangle }}&::=&{\rm {\langle {\acute {A}}tomo\rangle \mid \neg \langle {\acute {A}}tomo\rangle }}\\{\rm {\langle {\acute {A}}tomo\rangle }}&::=&{\rm {\top \mid \bot \mid \langle Letra\rangle \mid \langle Agrupaci{\acute {o}}n\rangle }}\\{\rm {\langle Agrupaci{\acute {o}}n\rangle }}&::=&{\rm {(\langle Bicondicional\rangle )\mid [\langle Bicondicional\rangle ]\mid \{\langle Bicondicional\rangle \}}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mrow> </mtd> <mtd> <mo>::=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">↔</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∣</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mrow> </mtd> <mtd> <mo>::=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">j</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">o</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">→</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∣</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">j</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">o</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">n</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">j</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">o</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">n</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mrow> </mtd> <mtd> <mo>::=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">o</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∨</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">j</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">o</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∣</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">o</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">n</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">o</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">n</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mrow> </mtd> <mtd> <mo>::=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∧</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">o</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∣</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mrow> </mtd> <mtd> <mo>::=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∣</mo> <mi mathvariant="normal">¬</mi> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mrow> </mtd> <mtd> <mo>::=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> <mo>∣</mo> <mi mathvariant="normal">⊥</mi> <mo>∣</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∣</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">o</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">n</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">o</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">n</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mrow> </mtd> <mtd> <mo>::=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> <mo>∣</mo> <mo stretchy="false">[</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">]</mo> <mo>∣</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">}</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}{\rm {\langle Bicondicional\rangle }}&::=&{\rm {\langle Condicional\rangle \leftrightarrow \langle Bicondicional\rangle \mid \langle Condicional\rangle }}\\{\rm {\langle Condicional\rangle }}&::=&{\rm {\langle Conjunci{\acute {o}}n\rangle \rightarrow \langle Condicional\rangle \mid \langle Conjunci{\acute {o}}n\rangle }}\\{\rm {\langle Conjunci{\acute {o}}n\rangle }}&::=&{\rm {\langle Disyunci{\acute {o}}n\rangle \vee \langle Conjunci{\acute {o}}n\rangle \mid \langle Disyunci{\acute {o}}n\rangle }}\\{\rm {\langle Disyunci{\acute {o}}n\rangle }}&::=&{\rm {\langle Literal\rangle \wedge \langle Disyunci{\acute {o}}n\rangle \mid \langle Literal\rangle }}\\{\rm {\langle Literal\rangle }}&::=&{\rm {\langle {\acute {A}}tomo\rangle \mid \neg \langle {\acute {A}}tomo\rangle }}\\{\rm {\langle {\acute {A}}tomo\rangle }}&::=&{\rm {\top \mid \bot \mid \langle Letra\rangle \mid \langle Agrupaci{\acute {o}}n\rangle }}\\{\rm {\langle Agrupaci{\acute {o}}n\rangle }}&::=&{\rm {(\langle Bicondicional\rangle )\mid [\langle Bicondicional\rangle ]\mid \{\langle Bicondicional\rangle \}}}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3315572c1df8ee6a36ecf27f96e18b269f322ce6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.505ex; width:79.688ex; height:24.176ex;" alt="{\displaystyle {\begin{array}{rcl}{\rm {\langle Bicondicional\rangle }}&::=&{\rm {\langle Condicional\rangle \leftrightarrow \langle Bicondicional\rangle \mid \langle Condicional\rangle }}\\{\rm {\langle Condicional\rangle }}&::=&{\rm {\langle Conjunci{\acute {o}}n\rangle \rightarrow \langle Condicional\rangle \mid \langle Conjunci{\acute {o}}n\rangle }}\\{\rm {\langle Conjunci{\acute {o}}n\rangle }}&::=&{\rm {\langle Disyunci{\acute {o}}n\rangle \vee \langle Conjunci{\acute {o}}n\rangle \mid \langle Disyunci{\acute {o}}n\rangle }}\\{\rm {\langle Disyunci{\acute {o}}n\rangle }}&::=&{\rm {\langle Literal\rangle \wedge \langle Disyunci{\acute {o}}n\rangle \mid \langle Literal\rangle }}\\{\rm {\langle Literal\rangle }}&::=&{\rm {\langle {\acute {A}}tomo\rangle \mid \neg \langle {\acute {A}}tomo\rangle }}\\{\rm {\langle {\acute {A}}tomo\rangle }}&::=&{\rm {\top \mid \bot \mid \langle Letra\rangle \mid \langle Agrupaci{\acute {o}}n\rangle }}\\{\rm {\langle Agrupaci{\acute {o}}n\rangle }}&::=&{\rm {(\langle Bicondicional\rangle )\mid [\langle Bicondicional\rangle ]\mid \{\langle Bicondicional\rangle \}}}\end{array}}}"></span></dd></dl> <p>La gramática anterior define la precedencia de <a href="/wiki/Operador" title="Operador">operadores</a> de la siguiente manera: </p> <ol><li>Negación (<span class="mwe-math-element"><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> <mspace width="thinmathspace" /> </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/e696c9f6fbea13e9bc4e7cbb549287558d8d0a94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.937ex; height:1.176ex;" alt="{\displaystyle \neg \,}"></span>)</li> <li>Conjunción (<span class="mwe-math-element"><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> <mspace width="thinmathspace" /> </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/99963a571d61ee9f766194e3b0d287aa773e3ea2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.937ex; height:2.009ex;" alt="{\displaystyle \land \,}"></span>)</li> <li>Disyunción (<span class="mwe-math-element"><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> <mspace width="thinmathspace" /> </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/f6bb1610877969d50f381dea831cb6106fce9f8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.937ex; height:2.009ex;" alt="{\displaystyle \lor \,}"></span>)</li> <li>Condicional material (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe19c397bbb5552edae0800c4089d7e9161dfeec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.711ex; height:1.843ex;" alt="{\displaystyle \to \,}"></span>)</li> <li>Bicondicional (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></span>)</li></ol> <div class="mw-heading mw-heading2"><h2 id="Semántica"><span id="Sem.C3.A1ntica"></span>Semántica</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=15" title="Editar sección: Semántica"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una <a href="/wiki/Interpretaci%C3%B3n_(l%C3%B3gica)" title="Interpretación (lógica)">interpretación</a> para un sistema de lógica proposicional es una asignación de <a href="/wiki/Valor_de_verdad" title="Valor de verdad">valores de verdad</a> para cada variable proposicional, sumada a la asignación usual de significados para los <a href="/wiki/Operador" title="Operador">operadores</a> lógicos. A cada variable proposicional se le asigna uno de dos posibles valores de verdad: o V (verdadero) o F (falso). Esto quiere decir que si hay <i>n</i> variables proposicionales en el sistema, el número de interpretaciones distintas es de 2<sup><i>n</i></sup>. </p><p>Partiendo de esto es posible definir una cantidad de nociones semánticas. Si A y B son fórmulas cualquiera de un lenguaje L, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span> es un conjunto de fórmulas de L, y M es una interpretación de L, entonces: </p> <ul><li>A es verdadera bajo la interpretación M si y solo si M asigna el valor de verdad V a A.</li> <li>A es falsa bajo la interpretación M si y solo si M asigna el valor de verdad F a A.</li> <li>A es una <a href="/wiki/Tautolog%C3%ADa" title="Tautología">tautología</a> (o una <a href="/wiki/Verdad_l%C3%B3gica" title="Verdad lógica">verdad lógica</a>) si y solo si para toda interpretación M, M asigna el valor de verdad V a A.</li> <li>A es una <a href="/wiki/Contradicci%C3%B3n" title="Contradicción">contradicción</a> si y solo si para toda interpretación M, M asigna el valor de verdad F a A.</li> <li>A es satisfacible (o consistente) si y solo si existe al menos una interpretación M que asigne el valor de verdad V a A.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span> es <a href="/wiki/Consistencia_(l%C3%B3gica)" title="Consistencia (lógica)">consistente</a> si y solo si existe al menos una interpretación que haga verdaderas a todas las fórmulas en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span>.</li> <li>A es una <a href="/wiki/Consecuencia_sem%C3%A1ntica" class="mw-redirect" title="Consecuencia semántica">consecuencia semántica</a> de un conjunto de fórmulas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span> si y solo si <b>no existe</b> interpretación en la que todas las fórmulas que pertenecen a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span> sean verdaderas y A sea falsa. Cuando A es una consecuencia semántica de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span> en un lenguaje L, se escribe: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \models _{L}A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <msub> <mo>⊨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \models _{L}A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9357127be67af33c3e2598f996404142bed926f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.853ex; height:2.843ex;" alt="{\displaystyle \Gamma \models _{L}A}"></span>.</li> <li>A es una <a href="/wiki/Verdad_l%C3%B3gica" title="Verdad lógica">verdad lógica</a> si y solo si A es una consecuencia semántica del conjunto vacío. Cuando A es una verdad lógica de un lenguaje L, se escribe: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models _{L}A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>⊨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models _{L}A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4ca95d86ab35f6b1c930f10ff69b2adfea26b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.755ex; height:2.843ex;" alt="{\displaystyle \models _{L}A}"></span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Tablas_de_verdad">Tablas de verdad</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=16" title="Editar sección: Tablas de verdad"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículo principal:</span> <i><a href="/wiki/Tablas_de_verdad" class="mw-redirect" title="Tablas de verdad"> Tablas de verdad</a></i></div> <p>La tabla de verdad de una fórmula es una tabla en la que se presentan todas las posibles interpretaciones de las variables proposicionales que constituye la fórmula y el valor de verdad de la fórmula completa para cada interpretación. Por ejemplo, la tabla de verdad para la fórmula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (p\lor q)\to (p\to r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (p\lor q)\to (p\to r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12e39becdb0e23943c4da8061b0fc8f0f5bcb7b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.437ex; height:2.843ex;" alt="{\displaystyle \neg (p\lor q)\to (p\to r)}"></span> es: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c|c|c||c|c|c|c}p&q&r&(p\lor q)&\neg (p\lor q)&(p\to r)&\neg (p\lor q)\to (p\to r)\\\hline V&V&V&V&F&V&V\\V&V&F&V&F&F&V\\V&F&V&V&F&V&V\\V&F&F&V&F&F&V\\F&V&V&V&F&V&V\\F&V&F&V&F&V&V\\F&F&V&F&V&V&V\\F&F&F&F&V&V&V\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid solid solid solid solid solid"> <mtr> <mtd> <mi>p</mi> </mtd> <mtd> <mi>q</mi> </mtd> <mtd> <mi>r</mi> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>F</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c|c|c||c|c|c|c}p&q&r&(p\lor q)&\neg (p\lor q)&(p\to r)&\neg (p\lor q)\to (p\to r)\\\hline V&V&V&V&F&V&V\\V&V&F&V&F&F&V\\V&F&V&V&F&V&V\\V&F&F&V&F&F&V\\F&V&V&V&F&V&V\\F&V&F&V&F&V&V\\F&F&V&F&V&V&V\\F&F&F&F&V&V&V\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64c5085948723cb50e25cf399d7ef87647d7f0fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.338ex; width:63.797ex; height:29.843ex;" alt="{\displaystyle {\begin{array}{c|c|c||c|c|c|c}p&q&r&(p\lor q)&\neg (p\lor q)&(p\to r)&\neg (p\lor q)\to (p\to r)\\\hline V&V&V&V&F&V&V\\V&V&F&V&F&F&V\\V&F&V&V&F&V&V\\V&F&F&V&F&F&V\\F&V&V&V&F&V&V\\F&V&F&V&F&V&V\\F&F&V&F&V&V&V\\F&F&F&F&V&V&V\\\end{array}}}"></span></dd></dl> <p>Como se ve, esta fórmula tiene 2<sup><i>n</i></sup> interpretaciones posibles —una por cada línea de la tabla— donde <i>n</i> es el número de variables proposicionales (en este caso 3, es decir p, q, r) y resulta ser una <a href="/wiki/Tautolog%C3%ADa" title="Tautología">tautología</a>, es decir que bajo todas las interpretaciones posibles de las variables proposicionales, el valor de verdad de la fórmula completa termina siendo V. </p> <div class="mw-heading mw-heading2"><h2 id="Formas_normales">Formas normales</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=17" title="Editar sección: Formas normales"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A menudo es necesario transformar una fórmula en otra, sobre todo transformar una fórmula a su forma normal. Esto se consigue transformando la fórmula en otra equivalente y repitiendo el proceso hasta conseguir una fórmula que solo use los conectivos básicos (<span class="mwe-math-element"><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 ,\lor ,\neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land ,\lor ,\neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d544adbcc6fc37ba97898a965258ee41f600a54e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.719ex; height:2.343ex;" alt="{\displaystyle \land ,\lor ,\neg }"></span>). Para lograr esto se utilizan las equivalencias lógicas: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to q)\leftrightarrow (\neg p\lor q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to q)\leftrightarrow (\neg p\lor q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae13902d84ed2b2033e30b039fbee929e5bd658" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.458ex; height:2.843ex;" alt="{\displaystyle (p\to q)\leftrightarrow (\neg p\lor q)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\leftrightarrow q)\leftrightarrow [(\neg p\lor q)\land (\neg q\lor p)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">↔</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo>∨</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\leftrightarrow q)\leftrightarrow [(\neg p\lor q)\land (\neg q\lor p)]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44fe9ad9f8e43f9c36d854bbfb90ab477b573b10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.515ex; height:2.843ex;" alt="{\displaystyle (p\leftrightarrow q)\leftrightarrow [(\neg p\lor q)\land (\neg q\lor p)]}"></span></dd></dl> <p>Por ejemplo, considérese la siguiente fórmula: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\to q)\land (\neg q\leftrightarrow p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">↔</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to q)\land (\neg q\leftrightarrow p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2067e24c8504103343d211dbd236c707172952a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.458ex; height:2.843ex;" alt="{\displaystyle (p\to q)\land (\neg q\leftrightarrow p)}"></span></dd></dl> <p>La misma puede desarrollarse así: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\neg p\lor q)\land (q\lor p)\land (\neg p\lor \neg q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∨</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\neg p\lor q)\land (q\lor p)\land (\neg p\lor \neg q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eed42a7e691e76f081b1873d4bc3e4b885f0c346" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.709ex; height:2.843ex;" alt="{\displaystyle (\neg p\lor q)\land (q\lor p)\land (\neg p\lor \neg q)}"></span></dd></dl> <p>Se dice que una fórmula está en <i>forma normal disyuntiva</i> (FND) si y solo si tiene la siguiente forma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}\lor A_{2}\lor ...\lor A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∨</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>∨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}\lor A_{2}\lor ...\lor A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f33c4db29ae2b37ab203491fb7aa3208031e1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.341ex; height:2.509ex;" alt="{\displaystyle A_{1}\lor A_{2}\lor ...\lor A_{n}}"></span></dd></dl> <p>donde cada A es una conjunción de fórmulas. Por ejemplo, la siguiente fórmula está en forma normal disyuntiva: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\lor (q\land s)\lor (\neg q\land p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∧</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo>∧</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\lor (q\land s)\lor (\neg q\land p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d959d4577af20fcfe282abd8aa3c12fbc0a8f640" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:21.157ex; height:2.843ex;" alt="{\displaystyle p\lor (q\land s)\lor (\neg q\land p)}"></span></dd></dl> <p>Se dice que una fórmula está en <i>forma normal conjuntiva</i> (FNC) si y solo si tiene la siguiente forma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}\land A_{2}\land ...\land A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∧</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>∧</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}\land A_{2}\land ...\land A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05f312d135d2884867a0183d649aa320327aba04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.341ex; height:2.509ex;" alt="{\displaystyle A_{1}\land A_{2}\land ...\land A_{n}}"></span></dd></dl> <p>donde cada A es una disyunción de fórmulas. Por ejemplo, la siguiente fórmula está en forma normal conjuntiva: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\land (q\lor s)\land (\neg q\lor p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>∨</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo>∨</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\land (q\lor s)\land (\neg q\lor p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d1a9493f34635570d8c2281aadd9521337eda1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:21.157ex; height:2.843ex;" alt="{\displaystyle p\land (q\lor s)\land (\neg q\lor p)}"></span></dd></dl> <p>Por las <a href="/wiki/Leyes_de_De_Morgan" title="Leyes de De Morgan">leyes de De Morgan</a>, es posible pasar de una forma normal disyuntiva a una forma normal conjuntiva y viceversa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (A\lor B)\leftrightarrow (\neg A\land \neg B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (A\lor B)\leftrightarrow (\neg A\land \neg B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca399ec25b4b4e66fdcf9336d4cbdca7f1f186ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.063ex; height:2.843ex;" alt="{\displaystyle \neg (A\lor B)\leftrightarrow (\neg A\land \neg B)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (A\land B)\leftrightarrow (\neg A\lor \neg B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (A\land B)\leftrightarrow (\neg A\lor \neg B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4439b999ddd5f8f65ae62dbc69789a395e0f6fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.063ex; height:2.843ex;" alt="{\displaystyle \neg (A\land B)\leftrightarrow (\neg A\lor \neg B)}"></span></dd></dl> <p>Las FNC y FND son mutuamente duales. La demostración hace uso de las leyes de De Morgan y de la <a href="/wiki/Propiedad_distributiva" class="mw-redirect" title="Propiedad distributiva">propiedad distributiva</a> de la conjunción y la disyunción. Se debe cumplir que: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg [(A_{1}\land B_{1})\lor (A_{2}\land B_{2})\lor ...\lor (A_{n}\land B_{n})]\leftrightarrow [(\neg A_{1}\lor \neg B_{1})\land (\neg A_{2}\lor \neg B_{2})\land ...\land (\neg A_{n}\lor \neg B_{n})]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∨</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∧</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∧</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg [(A_{1}\land B_{1})\lor (A_{2}\land B_{2})\lor ...\lor (A_{n}\land B_{n})]\leftrightarrow [(\neg A_{1}\lor \neg B_{1})\land (\neg A_{2}\lor \neg B_{2})\land ...\land (\neg A_{n}\lor \neg B_{n})]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2c2cd871c3866d32e9d1b470c3b5acb81fe37f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:95.326ex; height:2.843ex;" alt="{\displaystyle \neg [(A_{1}\land B_{1})\lor (A_{2}\land B_{2})\lor ...\lor (A_{n}\land B_{n})]\leftrightarrow [(\neg A_{1}\lor \neg B_{1})\land (\neg A_{2}\lor \neg B_{2})\land ...\land (\neg A_{n}\lor \neg B_{n})]}"></span></dd></dl> <p>Y viceversa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg [(A_{1}\lor B_{1})\land (A_{2}\lor B_{2})\land ...\land (A_{n}\lor B_{n})]\leftrightarrow [(\neg A_{1}\land \neg B_{1})\lor (\neg A_{2}\land \neg B_{2})\lor ...\lor (\neg A_{n}\land \neg B_{n})]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∨</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∨</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∧</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∨</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∨</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg [(A_{1}\lor B_{1})\land (A_{2}\lor B_{2})\land ...\land (A_{n}\lor B_{n})]\leftrightarrow [(\neg A_{1}\land \neg B_{1})\lor (\neg A_{2}\land \neg B_{2})\lor ...\lor (\neg A_{n}\land \neg B_{n})]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/272c75419a5a854cf1b4dd75d9fa0efac2bd26a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:95.326ex; height:2.843ex;" alt="{\displaystyle \neg [(A_{1}\lor B_{1})\land (A_{2}\lor B_{2})\land ...\land (A_{n}\lor B_{n})]\leftrightarrow [(\neg A_{1}\land \neg B_{1})\lor (\neg A_{2}\land \neg B_{2})\lor ...\lor (\neg A_{n}\land \neg B_{n})]}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=18" title="Editar sección: Historia"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículo principal:</span> <i><a href="/wiki/Historia_de_la_l%C3%B3gica" title="Historia de la lógica"> Historia de la lógica</a></i></div> <p>La lógica es conocida como una de las ciencias más antiguas, tanto es así que se le atribuye a Aristóteles la paternidad de esta disciplina. Partiendo de que corresponde a Aristóteles haber sido el primero en tratar con todo detalle la lógica, se le considera su fundador. En un principio se llamó Analítica, en virtud del título de las obras en que trató los problemas lógicos. Más tarde los escritos de Aristóteles relativos a estos eventos fueron recopilados por sus discípulos con el título de Órganon, por considerar que la lógica era un instrumento para el conocimiento de la verdad. </p><p>Aristóteles se planteó cómo es posible probar y demostrar que un conocimiento es verdadero, es decir, que tiene una validez universal. Aristóteles encuentra el fundamento de la demostración en la deducción, procedimiento que consiste en derivar un hecho particular de algo universal. La forma en que se afecta esa derivación es el silogismo, por cuya razón la silogística llega a ser el centro de la lógica aristotélica. </p><p>Aunque la lógica proposicional (que es intercambiable con el cálculo proposicional) había sido insinuada por los filósofos anteriores, fue desarrollada como un sistema formal por el filósofo estoico <a href="/wiki/Crisipo_de_Solos" title="Crisipo de Solos">Crisipo</a> en el siglo <span style="font-variant:small-caps;text-transform:lowercase">III</span> a. C. y ampliada por sus sucesores de la misma escuela. La lógica proposicional se centró en proposiciones. Este avance fue diferente de la lógica silogística tradicional que se centró en los términos. Sin embargo, más tarde en la antigüedad, la lógica proposicional desarrollada por los estoicos no se comprendía. En consecuencia de ello, el sistema fue reinventado esencialmente por <a href="/wiki/Pedro_Abelardo" title="Pedro Abelardo">Pedro Abelardo</a> en el siglo <span style="font-variant:small-caps;text-transform:lowercase">XII</span>. </p><p>La lógica proposicional fue finalmente refinada usando la lógica simbólica, se acreditó ser el fundador de la lógica simbólica el matemático Gottfried Leibniz siglo <span style="font-variant:small-caps;text-transform:lowercase">XVII</span>/XVIII, por su trabajo ratiocinator del cálculo. Aunque su trabajo era unos de los primeros, era desconocido para la comunidad lógica más grande. En consecuencia, muchos de los avances logrados por Leibniz fueron recreados por lógicos como George Boole y Augustus De Morgan completamente independientes a Leibniz. </p><p>Así como la lógica proposicional puede considerarse un avance de la lógica silogísta anterior, la lógica de predicados de Gottlob Frege era un avance de la lógica proposicional anterior. Un autor describe esta lógica como la combinación de los rasgos distintivos de la lógica silogística y la lógica proposicional. Por lo tanto, la lógica predicativa marcó el comienzo de una nueva era en la historia de la lógica; sin embargo, los avances en la lógica proposicional se hicieron aún después de Frege, incluyendo Deducción Natural, Árboles de la Verdad y Tablas de Verdad. La deducción natural fue inventada por Gerhard Gentzen y Jan Lukasiewicz. Los árboles de la verdad fueron inventados por Evert Willem Beth. La invención de las tablas de la verdad, sin embargo, es de atribución controvertida. </p><p>Dentro de las obras de Frege y Bertrand Russell, hay ideas que influyen en la invención de las tablas de la verdad. La estructura tabular real se acredita generalmente a Ludwig Wittgenstein o a Emil Post ( o ambos independientemente). Adeám de Frege y Russell, otros acreditados con ideas anteriores a las tablas de la verdad incluyen a Philo, Boole, Charles Sanders Peirce. Otros acreditados de la estructura tabular incluyen Lukasiewicz, Alfred North Whitehead, Guillermo Stanley Jevons, John Venn, y Clarence Irving Lewis. En última instancia, algunos han llegado a la conclusión, como John Shosky, de que " está lejos de estar claro que a cualquier persona se le debe dar el título de 'inventor' de las tablas de la verdad". </p> <div class="mw-heading mw-heading2"><h2 id="Véase_también"><span id="V.C3.A9ase_tambi.C3.A9n"></span>Véase también</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=19" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%C3%81lgebra_de_Boole" title="Álgebra de Boole">Álgebra de Boole</a></li> <li><a href="/wiki/C%C3%A1lculo_l%C3%B3gico" title="Cálculo lógico">Cálculo lógico</a></li> <li><a href="/wiki/C%C3%A1lculo_proposicional_de_Frege" title="Cálculo proposicional de Frege">Cálculo proposicional de Frege</a></li> <li><a href="/wiki/Gr%C3%A1ficos_existenciales" title="Gráficos existenciales">Gráficos existenciales</a></li> <li><a href="/wiki/L%C3%B3gica_matem%C3%A1tica" title="Lógica matemática">Lógica matemática</a></li> <li><a href="/wiki/L%C3%B3gica_de_primer_orden" title="Lógica de primer orden">Lógica de primer orden</a></li> <li><a href="/wiki/L%C3%B3gica_modal" title="Lógica modal">Lógica modal</a></li> <li><a href="/wiki/Tabla_de_verdad" title="Tabla de verdad">Tabla de verdad</a></li> <li><a href="/wiki/Teor%C3%ADa_de_grafos" title="Teoría de grafos">Teoría de grafos</a></li> <li><a href="/wiki/Silogismo" title="Silogismo">Silogismo</a></li> <li><a href="/wiki/Valor_de_verdad" title="Valor de verdad">Valor de verdad</a></li> <li><a href="/wiki/Razonamiento_diagram%C3%A1tico" title="Razonamiento diagramático">Razonamiento diagramático</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencias">Referencias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=20" title="Editar sección: Referencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="listaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span id="CITAREFSimon_Blackburn" class="citation enciclopedia">Simon Blackburn (ed.). <a rel="nofollow" class="external text" href="http://www.oxfordreference.com/views/ENTRY.html?subview=Main&entry=t98.e2552">«propositional calculus»</a>. <i>Oxford Dictionary of Philosophy</i> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a><span class="reference-accessdate">. Consultado el 13 de agosto de 2009</span>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.atitle=Oxford+Dictionary+of+Philosophy&rft.btitle=propositional+calculus&rft.genre=bookitem&rft.pub=Oxford+University+Press&rft_id=http%3A%2F%2Fwww.oxfordreference.com%2Fviews%2FENTRY.html%3Fsubview%3DMain%26entry%3Dt98.e2552&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-iep-2"><span class="mw-cite-backlink"><a href="#cite_ref-iep_2-0">↑</a></span> <span class="reference-text"><span id="CITAREFKlement" class="citation enciclopedia">Klement, Kevin C. <a rel="nofollow" class="external text" href="http://www.iep.utm.edu/prop-log/">«Propositional Logic»</a>. <i>Internet Encyclopedia of Philosophy</i> <span style="color:var(--color-subtle, #555 );">(en inglés)</span><span class="reference-accessdate">. Consultado el 6 de febrero de 2012</span>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.atitle=Internet+Encyclopedia+of+Philosophy&rft.au=Klement%2C+Kevin+C.&rft.aufirst=Kevin+C.&rft.aulast=Klement&rft.btitle=Propositional+Logic&rft.genre=bookitem&rft_id=http%3A%2F%2Fwww.iep.utm.edu%2Fprop-log%2F&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span id="CITAREFToida2_de_agosto_de_2009" class="citation web">Toida, Shunichi (2 de agosto de 2009). <a rel="nofollow" class="external text" href="http://www.cs.odu.edu/~toida/nerzic/content/logic/prop_logic/implications/implication_proof.html">«Proof of Implications»</a>. <i>CS381 Discrete Structures/Discrete Mathematics Web Course Material</i> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. Department Of Computer Science, <a href="/wiki/Old_Dominion_University" class="mw-redirect" title="Old Dominion University">Old Dominion University</a><span class="reference-accessdate">. Consultado el 10 de marzo de 2010</span>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.atitle=Proof+of+Implications&rft.au=Toida%2C+Shunichi&rft.aufirst=Shunichi&rft.aulast=Toida&rft.date=2+de+agosto+de+2009&rft.genre=article&rft.jtitle=CS381+Discrete+Structures%2FDiscrete+Mathematics+Web+Course+Material&rft.pub=Department+Of+Computer+Science%2C+Old+Dominion+University&rft_id=http%3A%2F%2Fwww.cs.odu.edu%2F~toida%2Fnerzic%2Fcontent%2Flogic%2Fprop_logic%2Fimplications%2Fimplication_proof.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografía"><span id="Bibliograf.C3.ADa"></span>Bibliografía</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=21" title="Editar sección: Bibliografía"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span id="CITAREFEnderton1972" class="citation libro">Enderton, H. B. (1972). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalintr00ende"><i>A Mathematical Introduction to Logic</i></a>. Academic Press.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.au=Enderton%2C+H.+B.&rft.aufirst=H.+B.&rft.aulast=Enderton&rft.btitle=A+Mathematical+Introduction+to+Logic&rft.date=1972&rft.genre=book&rft.pub=Academic+Press&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalintr00ende&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFHamilton1981" class="citation libro">Hamilton, A. G. (1981). <i>Lógica para matemáticos</i>. Paraningo.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.au=Hamilton%2C+A.+G.&rft.aufirst=A.+G.&rft.aulast=Hamilton&rft.btitle=L%C3%B3gica+para+matem%C3%A1ticos&rft.date=1981&rft.genre=book&rft.pub=Paraningo&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFMendelson1997" class="citation libro">Mendelson, E. (1997). <i>Introduction to Mathematical Logic</i> (4ª edición). Chapman and May.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.au=Mendelson%2C+E.&rft.aufirst=E.&rft.aulast=Mendelson&rft.btitle=Introduction+to+Mathematical+Logic&rft.date=1997&rft.edition=4%C2%AA&rft.genre=book&rft.pub=Chapman+and+May&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFPla1991" class="citation libro">Pla, J. (1991). <i>Lliçons de lógica matemática</i>. P.P.U.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.au=Pla%2C+J.&rft.aufirst=J.&rft.aulast=Pla&rft.btitle=Lli%C3%A7ons+de+l%C3%B3gica+matem%C3%A1tica&rft.date=1991&rft.genre=book&rft.pub=P.P.U.&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFBadesaJanéJansana1998" class="citation libro">Badesa, C.; Jané, I.; Jansana, R. (1998). <i>Elementos de lógica formal</i>. Ariel.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.au=Badesa%2C+C.&rft.au=Jan%C3%A9%2C+I.&rft.au=Jansana%2C+R.&rft.aufirst=C.&rft.aulast=Badesa&rft.btitle=Elementos+de+l%C3%B3gica+formal&rft.date=1998&rft.genre=book&rft.pub=Ariel&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFBarnesMack1978" class="citation libro">Barnes, D. W.; Mack, J. M. (1978). <i>Una introducción algebraica a la lógica matemática</i>. Eunibar.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.au=Barnes%2C+D.+W.&rft.au=Mack%2C+J.+M.&rft.aufirst=D.+W.&rft.aulast=Barnes&rft.btitle=Una+introducci%C3%B3n+algebraica+a+la+l%C3%B3gica+matem%C3%A1tica&rft.date=1978&rft.genre=book&rft.pub=Eunibar&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFBridge1977" class="citation libro">Bridge, J. (1977). <a rel="nofollow" class="external text" href="https://archive.org/details/beginningmodelth0000brid"><i>Beginning Model Theory</i></a>. Oxford University Pres.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.au=Bridge%2C+J.&rft.aufirst=J.&rft.aulast=Bridge&rft.btitle=Beginning+Model+Theory&rft.date=1977&rft.genre=book&rft.pub=Oxford+University+Pres&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbeginningmodelth0000brid&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFErshovPaliutin1990" class="citation libro">Ershov, Y.; Paliutin, E. (1990). <i>Lógica matemática</i>. Mir.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.au=Ershov%2C+Y.&rft.au=Paliutin%2C+E.&rft.aufirst=Y.&rft.aulast=Ershov&rft.btitle=L%C3%B3gica+matem%C3%A1tica&rft.date=1990&rft.genre=book&rft.pub=Mir&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFHofstadter1987" class="citation libro">Hofstadter, D. (1987). <i>Gödel, Escher, Bach: un Eterno y Grácil Bucle</i>. Tusquets Editores.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.au=Hofstadter%2C+D.&rft.aufirst=D.&rft.aulast=Hofstadter&rft.btitle=G%C3%B6del%2C+Escher%2C+Bach%3A+un+Eterno+y+Gr%C3%A1cil+Bucle&rft.date=1987&rft.genre=book&rft.pub=Tusquets+Editores&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFJané1989" class="citation libro">Jané, I. (1989). <i>Álgebras de Boole y lógica</i>. Publicaciones U.B.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.au=Jan%C3%A9%2C+I.&rft.aufirst=I.&rft.aulast=Jan%C3%A9&rft.btitle=%C3%81lgebras+de+Boole+y+l%C3%B3gica&rft.date=1989&rft.genre=book&rft.pub=Publicaciones+U.B.&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFMonk1976" class="citation libro">Monk, J. D. (1976). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicallogi00jdon"><i>Mathematical Logic</i></a>. Springer-Verlag.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.au=Monk%2C+J.+D.&rft.aufirst=J.+D.&rft.aulast=Monk&rft.btitle=Mathematical+Logic&rft.date=1976&rft.genre=book&rft.pub=Springer-Verlag&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicallogi00jdon&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFNidditch1978" class="citation libro">Nidditch, P. H. (1978). <i>El desarrollo de la lógica matemática</i>. Cátedra.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.au=Nidditch%2C+P.+H.&rft.aufirst=P.+H.&rft.aulast=Nidditch&rft.btitle=El+desarrollo+de+la+l%C3%B3gica+matem%C3%A1tica&rft.date=1978&rft.genre=book&rft.pub=C%C3%A1tedra&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFVan_Dalen1983" class="citation libro">Van Dalen, D. (1983). <i>Logic and Structure</i> (2ª edición). Universitext, Springer-Verlag.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AL%C3%B3gica+proposicional&rft.au=Van+Dalen%2C+D.&rft.aufirst=D.&rft.aulast=Van+Dalen&rft.btitle=Logic+and+Structure&rft.date=1983&rft.edition=2%C2%AA&rft.genre=book&rft.pub=Universitext%2C+Springer-Verlag&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Enlaces_externos">Enlaces externos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=L%C3%B3gica_proposicional&action=edit&section=22" title="Editar sección: Enlaces externos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Wikiversity-logo-en.svg/20px-Wikiversity-logo-en.svg.png" decoding="async" width="20" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Wikiversity-logo-en.svg/30px-Wikiversity-logo-en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Wikiversity-logo-en.svg/40px-Wikiversity-logo-en.svg.png 2x" data-file-width="1000" data-file-height="900" /></span></span> <a href="/wiki/Wikiversidad" title="Wikiversidad">Wikiversidad</a> alberga proyectos de aprendizaje sobre <b><a href="https://es.wikiversity.org/wiki/L%C3%B3gica_proposicional" class="extiw" title="v:Lógica proposicional">Lógica proposicional</a></b>.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100612135253/http://portales.educared.net/wikiEducared/index.php?title=L%C3%B3gica_proposicional">Introducción a la lógica proposicional</a></li></ul> <p><br /> </p> <style data-mw-deduplicate="TemplateStyles:r161257576">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox table{margin:0}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral,#eaecf0);color:var(--color-base,#202122)}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}.mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .mw-authority-control .navbox-list{border-color:#f8f9fa}.mw-parser-output .mw-authority-control .navbox th{background-color:#eeeeff}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d);background-color:var(--background-color-neutral,#27292d);color:var(--color-base,#eaecf0)}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral,#27292d)!important;color:var(--color-base,#eaecf0)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}}</style><div class="mw-authority-control"><div role="navigation" class="navbox" aria-label="Navbox" style="width: inherit;padding:3px"><table class="hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width: 12%; text-align:center;"><a href="/wiki/Control_de_autoridades" title="Control de autoridades">Control de autoridades</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><b>Proyectos Wikimedia</b></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikidata" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span> Datos:</span> <span class="uid"><a href="https://www.wikidata.org/wiki/Q200694" class="extiw" title="wikidata:Q200694">Q200694</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Multimedia:</span> <span class="uid"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Propositional_logic">Propositional logic</a></span> / <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Special:MediaSearch?type=image&search=%22Q200694%22">Q200694</a></span></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikiversidad" title="Wikiversity"><img alt="Wikiversity" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/15px-Wikiversity-logo.svg.png" decoding="async" width="15" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/23px-Wikiversity-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/30px-Wikiversity-logo.svg.png 2x" data-file-width="1000" data-file-height="800" /></a></span> Recursos didácticos:</span> <span class="uid"><a href="https://es.wikiversity.org/wiki/L%C3%B3gica_proposicional" class="extiw" title="v:Lógica proposicional">Lógica proposicional</a></span></li></ul> <hr /> <ul><li><b>Identificadores</b></li> <li><span style="white-space:nowrap;"><a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4136098-9">4136098-9</a></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Biblioteca_Nacional_de_la_Rep%C3%BAblica_Checa" title="Biblioteca Nacional de la República Checa">NKC</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph127455">ph127455</a></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Biblioteca_Nacional_de_Israel" title="Biblioteca Nacional de Israel">NLI</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007541095305171">987007541095305171</a></span></li> <li><b>Diccionarios y enciclopedias</b></li> <li><span style="white-space:nowrap;"><a href="/wiki/Enciclopedia_Brit%C3%A1nica" title="Enciclopedia Británica">Britannica</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/logic-of-propositions">url</a></span></li></ul> </div></td></tr></tbody></table></div><div class="mw-mf-linked-projects hlist"> <ul><li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikidata" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span> Datos:</span> <span class="uid"><a href="https://www.wikidata.org/wiki/Q200694" class="extiw" title="wikidata:Q200694">Q200694</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Multimedia:</span> <span class="uid"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Propositional_logic">Propositional logic</a></span> / <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Special:MediaSearch?type=image&search=%22Q200694%22">Q200694</a></span></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikiversidad" title="Wikiversity"><img alt="Wikiversity" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/15px-Wikiversity-logo.svg.png" decoding="async" width="15" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/23px-Wikiversity-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/30px-Wikiversity-logo.svg.png 2x" data-file-width="1000" data-file-height="800" /></a></span> Recursos didácticos:</span> <span class="uid"><a href="https://es.wikiversity.org/wiki/L%C3%B3gica_proposicional" class="extiw" title="v:Lógica proposicional">Lógica proposicional</a></span></li></ul> </div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtenido de «<a dir="ltr" href="https://es.wikipedia.org/w/index.php?title=Lógica_proposicional&oldid=165114985">https://es.wikipedia.org/w/index.php?title=Lógica_proposicional&oldid=165114985</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categor%C3%ADas" title="Especial:Categorías">Categorías</a>: <ul><li><a href="/wiki/Categor%C3%ADa:Sistemas_l%C3%B3gicos" title="Categoría:Sistemas lógicos">Sistemas lógicos</a></li><li><a href="/wiki/Categor%C3%ADa:L%C3%B3gica_proposicional" title="Categoría:Lógica proposicional">Lógica proposicional</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorías ocultas: <ul><li><a href="/wiki/Categor%C3%ADa:Wikipedia:Art%C3%ADculos_con_pasajes_que_requieren_referencias" title="Categoría:Wikipedia:Artículos con pasajes que requieren referencias">Wikipedia:Artículos con pasajes que requieren referencias</a></li><li><a href="/wiki/Categor%C3%ADa:Wikipedia:Art%C3%ADculos_con_identificadores_GND" title="Categoría:Wikipedia:Artículos con identificadores GND">Wikipedia:Artículos con identificadores GND</a></li><li><a href="/wiki/Categor%C3%ADa:Wikipedia:P%C3%A1ginas_que_utilizan_un_formato_obsoleto_en_la_etiqueta_math" title="Categoría:Wikipedia:Páginas que utilizan un formato obsoleto en la etiqueta math">Wikipedia:Páginas que utilizan un formato obsoleto en la etiqueta math</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 se editó por última vez el 31 ene 2025 a las 19:11.</li> <li id="footer-info-copyright">El texto está disponible bajo la <a href="/wiki/Wikipedia:Texto_de_la_Licencia_Creative_Commons_Atribuci%C3%B3n-CompartirIgual_4.0_Internacional" title="Wikipedia:Texto de la Licencia Creative Commons Atribución-CompartirIgual 4.0 Internacional">Licencia Creative Commons Atribución-CompartirIgual 4.0</a>; pueden aplicarse cláusulas adicionales. Al usar este sitio aceptas nuestros <a class="external text" href="https://foundation.wikimedia.org/wiki/Policy:Terms_of_Use/es">términos de uso</a> y nuestra <a class="external text" href="https://foundation.wikimedia.org/wiki/Policy:Privacy_policy/es">política de privacidad</a>.<br />Wikipedia® es una marca registrada de la <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/es/">Fundación Wikimedia</a>, una organización sin ánimo de lucro.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy/es">Política de privacidad</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:Acerca_de">Acerca de Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:Limitaci%C3%B3n_general_de_responsabilidad">Limitación de responsabilidad</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Código de conducta</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Desarrolladores</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/es.wikipedia.org">Estadísticas</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement/es">Declaración de cookies</a></li> <li id="footer-places-mobileview"><a href="//es.m.wikipedia.org/w/index.php?title=L%C3%B3gica_proposicional&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Versión para móviles</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" lang="en" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Buscar</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Buscar en Wikipedia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Especial:Buscar"> </div> <button class="cdx-button cdx-search-input__end-button">Buscar</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contenidos" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Cambiar a la tabla de contenidos" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-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">Cambiar a la tabla de contenidos</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Lógica proposicional</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>51 idiomas</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Añadir tema</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-b766959bd-nsn2d","wgBackendResponseTime":144,"wgPageParseReport":{"limitreport":{"cputime":"0.504","walltime":"0.697","ppvisitednodes":{"value":3041,"limit":1000000},"postexpandincludesize":{"value":42279,"limit":2097152},"templateargumentsize":{"value":1715,"limit":2097152},"expansiondepth":{"value":7,"limit":100},"expensivefunctioncount":{"value":5,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":21903,"limit":5000000},"entityaccesscount":{"value":6,"limit":400},"timingprofile":["100.00% 264.985 1 -total"," 52.51% 139.147 1 Plantilla:Control_de_autoridades"," 13.64% 36.149 1 Plantilla:Conectivas_lógicas"," 13.01% 34.478 1 Plantilla:Listaref"," 10.23% 27.109 2 Plantilla:Cita_enciclopedia"," 8.78% 23.274 13 Plantilla:Cita_libro"," 2.56% 6.794 1 Plantilla:Reglas_de_transformación"," 1.92% 5.082 4 Plantilla:AP"," 1.15% 3.036 1 Plantilla:Cita_web"," 1.00% 2.658 1 Plantilla:Lista_plana"]},"scribunto":{"limitreport-timeusage":{"value":"0.156","limit":"10.000"},"limitreport-memusage":{"value":3390539,"limit":52428800}},"cachereport":{"origin":"mw-api-int.codfw.main-5b65fffc7d-bwqs9","timestamp":"20250215155241","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"L\u00f3gica proposicional","url":"https:\/\/es.wikipedia.org\/wiki\/L%C3%B3gica_proposicional","sameAs":"http:\/\/www.wikidata.org\/entity\/Q200694","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q200694","author":{"@type":"Organization","name":"Colaboradores de los proyectos Wikimedia"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2006-01-01T11:01:27Z","dateModified":"2025-01-31T19:11:06Z","headline":"sistema formal cuyos elementos m\u00e1s simples representan proposiciones o enunciados, y cuyas constantes l\u00f3gicas, llamadas conectivas l\u00f3gicas, representan operaciones sobre proposiciones, capaces de formar otras proposiciones de mayor complejidad"}</script> </body> </html>

CINXE.COM

Lógica proposicional - Wikipedia, la enciclopedia libre