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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">Inhoud</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">naar zijbalk verplaatsen</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">verbergen</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">Top</div> </a> </li> <li id="toc-Informele_inleiding" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Informele_inleiding"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Informele inleiding</span> </div> </a> <ul id="toc-Informele_inleiding-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Syntaxis_en_semantiek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Syntaxis_en_semantiek"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Syntaxis en semantiek</span> </div> </a> <button aria-controls="toc-Syntaxis_en_semantiek-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>Syntaxis en semantiek-subkopje inklappen</span> </button> <ul id="toc-Syntaxis_en_semantiek-sublist" class="vector-toc-list"> <li id="toc-Formules" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formules"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Formules</span> </div> </a> <ul id="toc-Formules-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Valuaties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Valuaties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Valuaties</span> </div> </a> <ul id="toc-Valuaties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geldigheid_en_vervulbaarheid" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geldigheid_en_vervulbaarheid"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Geldigheid en vervulbaarheid</span> </div> </a> <ul id="toc-Geldigheid_en_vervulbaarheid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Identiteitswet" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Identiteitswet"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Identiteitswet</span> </div> </a> <ul id="toc-Identiteitswet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Waarheidstabellen" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Waarheidstabellen"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Waarheidstabellen</span> </div> </a> <ul id="toc-Waarheidstabellen-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bewijssystemen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bewijssystemen"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Bewijssystemen</span> </div> </a> <button aria-controls="toc-Bewijssystemen-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>Bewijssystemen-subkopje inklappen</span> </button> <ul id="toc-Bewijssystemen-sublist" class="vector-toc-list"> <li id="toc-Bewijssysteem_in_de_stijl_van_Hilbert" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bewijssysteem_in_de_stijl_van_Hilbert"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Bewijssysteem in de stijl van Hilbert</span> </div> </a> <ul id="toc-Bewijssysteem_in_de_stijl_van_Hilbert-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semantische_tableaus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semantische_tableaus"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Semantische tableaus</span> </div> </a> <ul id="toc-Semantische_tableaus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Natuurlijke_deductie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Natuurlijke_deductie"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Natuurlijke deductie</span> </div> </a> <ul id="toc-Natuurlijke_deductie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Resolutie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Resolutie"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Resolutie</span> </div> </a> <ul id="toc-Resolutie-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complexiteitstheorie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Complexiteitstheorie"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Complexiteitstheorie</span> </div> </a> <ul id="toc-Complexiteitstheorie-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="Inhoud" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Inhoudsopgave omschakelen" > <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">Inhoudsopgave omschakelen</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">Propositielogica</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="Ga naar een artikel in een andere taal. Beschikbaar in 49 talen" > <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-49" 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">49 talen</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 – Afrikaans" lang="af" hreflang="af" data-title="Proposisionele logika" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" 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 – Arabisch" lang="ar" hreflang="ar" data-title="حساب القضايا" data-language-autonym="العربية" data-language-local-name="Arabisch" 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 – Asturisch" lang="ast" hreflang="ast" data-title="Lóxica proposicional" data-language-autonym="Asturianu" data-language-local-name="Asturisch" 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 – Belarussisch" lang="be" hreflang="be" data-title="Логіка выказванняў" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" 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 – Bulgaars" lang="bg" hreflang="bg" data-title="Пропозиционална логика" data-language-autonym="Български" data-language-local-name="Bulgaars" 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 – Catalaans" lang="ca" hreflang="ca" data-title="Lògica proposicional" data-language-autonym="Català" data-language-local-name="Catalaans" 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 – Tsjechisch" lang="cs" hreflang="cs" data-title="Výroková logika" data-language-autonym="Čeština" data-language-local-name="Tsjechisch" 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 – Tsjoevasjisch" lang="cv" hreflang="cv" data-title="Каланăлăхсен шутлавĕ" data-language-autonym="Чӑвашла" data-language-local-name="Tsjoevasjisch" 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 – Welsh" lang="cy" hreflang="cy" data-title="Rhesymeg osodiadol" data-language-autonym="Cymraeg" data-language-local-name="Welsh" 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 – Duits" lang="de" hreflang="de" data-title="Aussagenlogik" data-language-autonym="Deutsch" data-language-local-name="Duits" 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 – Grieks" lang="el" hreflang="el" data-title="Προτασιακός λογισμός" data-language-autonym="Ελληνικά" data-language-local-name="Grieks" 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 – Engels" lang="en" hreflang="en" data-title="Propositional calculus" data-language-autonym="English" data-language-local-name="Engels" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/L%C3%B3gica_proposicional" title="Lógica proposicional – Spaans" lang="es" hreflang="es" data-title="Lógica proposicional" data-language-autonym="Español" data-language-local-name="Spaans" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lauseloogika" title="Lauseloogika – Estisch" lang="et" hreflang="et" data-title="Lauseloogika" data-language-autonym="Eesti" data-language-local-name="Estisch" 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 – Baskisch" lang="eu" hreflang="eu" data-title="Logika proposizional" data-language-autonym="Euskara" data-language-local-name="Baskisch" 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 – Perzisch" lang="fa" hreflang="fa" data-title="حساب گزاره‌ای" data-language-autonym="فارسی" data-language-local-name="Perzisch" 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 – Fins" lang="fi" hreflang="fi" data-title="Propositiologiikka" data-language-autonym="Suomi" data-language-local-name="Fins" 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 – Frans" lang="fr" hreflang="fr" data-title="Calcul des propositions" data-language-autonym="Français" data-language-local-name="Frans" 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 – Noord-Fries" lang="frr" hreflang="frr" data-title="Ütjsaagenloogik" data-language-autonym="Nordfriisk" data-language-local-name="Noord-Fries" 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 – Galicisch" lang="gl" hreflang="gl" data-title="Lóxica proposicional" data-language-autonym="Galego" data-language-local-name="Galicisch" 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 – Hebreeuws" lang="he" hreflang="he" data-title="תחשיב הפסוקים" data-language-autonym="עברית" data-language-local-name="Hebreeuws" 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 – Hongaars" lang="hu" hreflang="hu" data-title="Ítéletlogika" data-language-autonym="Magyar" data-language-local-name="Hongaars" 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 – Armeens" lang="hy" hreflang="hy" data-title="Ասույթների տրամաբանություն" data-language-autonym="Հայերեն" data-language-local-name="Armeens" 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 – Indonesisch" lang="id" hreflang="id" data-title="Kalkulus proposisional" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Logica_proposizionale" title="Logica proposizionale – Italiaans" lang="it" hreflang="it" data-title="Logica proposizionale" data-language-autonym="Italiano" data-language-local-name="Italiaans" 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 – Japans" lang="ja" hreflang="ja" data-title="命題論理" data-language-autonym="日本語" data-language-local-name="Japans" 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 – Koreaans" lang="ko" hreflang="ko" data-title="명제 논리" data-language-autonym="한국어" data-language-local-name="Koreaans" 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 – Kirgizisch" lang="ky" hreflang="ky" data-title="Логикалык сүйлөө" data-language-autonym="Кыргызча" data-language-local-name="Kirgizisch" 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 – Latijn" lang="la" hreflang="la" data-title="Logica propositionalis" data-language-autonym="Latina" data-language-local-name="Latijn" 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 – Litouws" lang="lt" hreflang="lt" data-title="Teiginių logika" data-language-autonym="Lietuvių" data-language-local-name="Litouws" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Utsegnslogikk" title="Utsegnslogikk – Noors - Nynorsk" lang="nn" hreflang="nn" data-title="Utsegnslogikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Noors - 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 – Noors - Bokmål" lang="nb" hreflang="nb" data-title="Setningslogikk" data-language-autonym="Norsk bokmål" data-language-local-name="Noors - Bokmål" 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 – Pools" lang="pl" hreflang="pl" data-title="Rachunek zdań" data-language-autonym="Polski" data-language-local-name="Pools" 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) – Pasjtoe" lang="ps" hreflang="ps" data-title="قضیوي حساب (منطق)" data-language-autonym="پښتو" data-language-local-name="Pasjtoe" 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 – Portugees" lang="pt" hreflang="pt" data-title="Lógica proposicional" data-language-autonym="Português" data-language-local-name="Portugees" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a 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<div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Paginahulpmiddelen"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Uiterlijk"> <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">Uiterlijk</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">naar zijbalk verplaatsen</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">verbergen</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">Uit Wikipedia, de vrije encyclopedie</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="nl" dir="ltr"><p>De <b>propositielogica</b> is een tak van <a href="/wiki/Logica" title="Logica">logica</a> die zich bezighoudt met het redeneren met <a href="/wiki/Propositie" title="Propositie">proposities</a>. Proposities zijn uitspraken of beweringen die ofwel waar, ofwel onwaar zijn. Voorbeelden hiervan zijn </p> <dl><dd><i>De Winkler Prins is een encyclopedie</i></dd></dl> <p>en </p> <dl><dd><i><a href="/wiki/Wickie_de_Viking_(tekenfilmserie)" title="Wickie de Viking (tekenfilmserie)">Wicky</a> heeft een noormannenhelm op</i>.</dd></dl> <p>In de propositielogica kunnen uitspraken alleen waar of onwaar zijn, dit in tegenstelling tot <a href="/wiki/Meerwaardige_logica" title="Meerwaardige logica">meerwaardige logica</a>'s waarbij uitspraken ook andere waarden kunnen hebben. </p><p>In vergelijking met andere types van logica is de propositielogica eenvoudig van opbouw (structuur, grammatica) maar beperkt in uitdrukkingsmogelijkheid. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Informele_inleiding">Informele inleiding</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositielogica&veaction=edit&section=1" title="Bewerk dit kopje: Informele inleiding" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositielogica&action=edit&section=1" title="De broncode bewerken van de sectie: Informele inleiding"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De propositielogica gaat over het redeneren met <i>proposities</i>. Een propositie is een uitspraak die waar of onwaar kan zijn. Proposities kunnen enkelvoudig zijn ("Morgen schijnt de zon" of "We gaan morgen picknicken") maar ook zijn samengesteld uit twee of meer andere proposities met behulp van (logische) voegwoorden, in deze context <i>connectieven</i> (soms ook <i>voegtekens</i>) genoemd ("Morgen schijnt de zon <i>en</i> we gaan morgen picknicken", "<i>Als</i> morgen de zon schijnt, <i>dan</i> gaan we morgen picknicken."). </p><p>In de propositielogica worden de volgende connectieven gebruikt om proposities samen te stellen: </p> <ul><li><a href="/wiki/Logische_negatie" title="Logische negatie">negatie</a> (ontkenning): <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"></span> of ~<span class="mwe-math-element"><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> (lees als: <b>niet</b> <span class="mwe-math-element"><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>).</li> <li><a href="/wiki/Logische_conjunctie" title="Logische conjunctie">conjunctie</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 A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></span> (lees als: <span class="mwe-math-element"><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> <b>en</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>).</li> <li><a href="/wiki/Logische_disjunctie" title="Logische disjunctie">disjunctie</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 A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"></span> (lees als: <span class="mwe-math-element"><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> <b>of</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>).</li> <li><a href="/wiki/Logische_implicatie" title="Logische implicatie">implicatie</a> (gevolgtrekking): <span class="mwe-math-element"><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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"></span> (lees als: <b>als</b> <span class="mwe-math-element"><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> <b>dan</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>).</li> <li><a href="/wiki/Dan_en_slechts_dan_als" title="Dan en slechts dan als">equivalentie</a> (gelijkwaardigheid): <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936ab098710910e69e56ec2734dd89063ce21efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftrightarrow B}"></span> (lees als: <span class="mwe-math-element"><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> <b>dan en slechts dan als</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>).</li></ul> <p>Verder is er één propositieconstante, genaamd <i>falsum</i> en aangeduid met <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \perp }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊥</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \perp }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb90d6db42aa12f9e2f31176a4ed4e741c69eca" 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 \perp }"></span>, die staat voor een <i>altijd onware</i> propositie. </p><p>In de propositielogica hangt de waarheid van samengestelde proposities alleen af van het gebruikte connectief en van de waarheid van de samenstellende delen. Zo is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></span> waar als <span class="mwe-math-element"><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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> beide waar zijn, terwijl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"></span> waar is als <span class="mwe-math-element"><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> waar is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> waar is of zowel <span class="mwe-math-element"><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> als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> waar zijn. </p><p>Stel bijvoorbeeld dat <span class="mwe-math-element"><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> voor de enkelvoudige propositie "Morgen schijnt de zon" staat 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> voor "We gaan morgen picknicken". Dan geldt: </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 A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"></span> is de uitspraak: <i>Morgen schijnt de zon <b>of</b> we gaan morgen picknicken</i>. Deze uitspraak is waar als morgen inderdaad de zon schijnt of als we gaan picknicken of allebei, en onwaar als morgen de zon niet schijnt en we ook niet gaan picknicken.</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 A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></span> is de uitspraak: <i>Morgen schijnt de zon <b>en</b> we gaan morgen picknicken</i>. Deze uitspraak is waar als morgen de zon schijnt en we ook gaan picknicken. Als de zon morgen niet schijnt of als we niet gaan picknicken of allebei (als de zon morgen niet schijnt en we niet gaan picknicken), dan is de uitspraak onwaar.</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 A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"></span> is de uitspraak: <i><b>Als</b> morgen de zon schijnt, <b>dan</b> gaan we morgen picknicken</i>. Aangezien de waarheid van deze uitspraak alleen mag afhangen van de waarheid van de samenstellende delen, vinden veel mensen de implicatie <i>tegenintuïtief</i>. Als de zon schijnt en we niet gaan picknicken, is de uitspraak uiteraard niet waar. In alle andere gevallen, als de zon niet schijnt of als de zon wel schijnt en we ook gaan picknicken, is de uitspraak waar. Zo komen we tot de conclusie dat de uitspraak hetzelfde betekent als <i>Morgen schijnt de zon niet <b>of</b> we gaan morgen picknicken</i>.</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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"></span> is de uitspraak: <i>Morgen schijnt de zon <b>niet</b></i>. Deze uitspraak is waar als de zon morgen niet schijnt en onwaar als de zon dan wel schijnt.</li></ul> <p>De bewering <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"></span> is waar als ten minste een van de twee beweringen waar is. De bewering is dus ook waar als beide beweringen waar zijn, iets wat in <a href="/wiki/Taal" title="Taal">natuurlijke taal</a> vaak niet zo bedoeld wordt. Dus er geldt: </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 (1=1)\lor (2=2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>=</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1=1)\lor (2=2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36ca70f903b70413236b581de002b4b113bd00d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.048ex; height:2.843ex;" alt="{\displaystyle (1=1)\lor (2=2)}"></span> is waar,</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 (1=1)\lor (1=2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>=</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1=1)\lor (1=2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51351f0de6749285e28e5a221fe0516417c821de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.048ex; height:2.843ex;" alt="{\displaystyle (1=1)\lor (1=2)}"></span> is waar, maar</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 (1=2)\lor (1>1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>=</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1=2)\lor (1>1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26f53519ec7e67ce4c7fa5be428b4b89765263f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.048ex; height:2.843ex;" alt="{\displaystyle (1=2)\lor (1>1)}"></span> is niet waar.</dd></dl> <p>Hoewel waarheid een basisbegrip van de propositielogica is, richt ze zich ook vooral op <i>geldigheid</i> en <i>onvervulbaarheid</i>. Een propositie is <i>geldig</i> (een <a href="/wiki/Tautologie_(logica)" title="Tautologie (logica)">tautologie</a>) als ze altijd waar is, onafhankelijk van welke waarheidswaarden aan de enkelvoudige proposities wordt toegekend. Zo is de propositie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land B)\leftrightarrow \neg (\neg A\lor \neg B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mi mathvariant="normal">¬</mi> <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 (A\land B)\leftrightarrow \neg (\neg A\lor \neg B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/945e0f6ed89e96e709ae77c66a884e440d9360a4" 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 (A\land B)\leftrightarrow \neg (\neg A\lor \neg B)}"></span> altijd waar. Een propositie is <i>onvervulbaar</i> als ze altijd onwaar is. </p><p>Ook onderzoekt de logica de geldigheid van redeneerstappen. Als <span class="mwe-math-element"><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> een ware propositie is 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 A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"></span> is dat ook, dan is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> eveneens een ware propositie. Dit geldt voor alle waarheidswaarden van <span class="mwe-math-element"><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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. Uit de <a href="/wiki/Premisse_(logica)" title="Premisse (logica)">premissen</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 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> 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 A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"></span> kan dus de conclusie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> worden afgeleid. Deze redeneerregel wordt <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a> genoemd. </p> <div class="mw-heading mw-heading2"><h2 id="Syntaxis_en_semantiek">Syntaxis en semantiek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositielogica&veaction=edit&section=2" title="Bewerk dit kopje: Syntaxis en semantiek" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositielogica&action=edit&section=2" title="De broncode bewerken van de sectie: Syntaxis en semantiek"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Als men over propositielogica spreekt, heeft men het meestal over de <a href="/wiki/Klassieke_logica" title="Klassieke logica">klassieke propositielogica</a>. Hieronder wordt de syntaxis en semantiek van deze logica gegeven. </p> <div class="mw-heading mw-heading3"><h3 id="Formules">Formules</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositielogica&veaction=edit&section=3" title="Bewerk dit kopje: Formules" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositielogica&action=edit&section=3" title="De broncode bewerken van de sectie: Formules"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Laat een aftelbaar oneindige verzameling van enkelvoudige proposities, ook propositievariabelen genoemd, gegeven zijn. Dan definiëren we de verzameling van formules als de kleinste verzameling waarvoor geldt: </p> <ul><li>propositievariabelen: als <span class="mwe-math-element"><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> een propositievariabele is, dan is <span class="mwe-math-element"><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> ook een formule;</li> <li>conjunctie: als <span class="mwe-math-element"><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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> formules zijn, dan is ook <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdcd2d1d13bc1f915aa141415965509a4e2b4f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.899ex; height:2.843ex;" alt="{\displaystyle (A\land B)}"></span> een formule;</li> <li>disjunctie: als <span class="mwe-math-element"><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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> formules zijn, dan is ook <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\lor B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\lor B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ccf97709eacad945b83238a85ec89cc220fed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.899ex; height:2.843ex;" alt="{\displaystyle (A\lor B)}"></span> een formule;</li> <li>negatie: als <span class="mwe-math-element"><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> een formule is, dan is ook <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"></span> een formule.</li></ul> <p>Bovendien definiëren we de volgende afkortingen: </p> <ul><li>implicatie: <span class="mwe-math-element"><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 B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\to B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2874d32e52c03fcb9112bab28893069fed6fc3e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.93ex; height:2.843ex;" alt="{\displaystyle (A\to B)}"></span> betekent: <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A\lor B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c13321f163366a0912c15b09d401b33e29a1ad46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.64ex; height:2.176ex;" alt="{\displaystyle \neg A\lor B}"></span>.</li> <li>equivalentie: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\leftrightarrow B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\leftrightarrow B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377a507f43f614500c9c326f3135fbb062cd5fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.93ex; height:2.843ex;" alt="{\displaystyle (A\leftrightarrow B)}"></span> betekent: <span class="mwe-math-element"><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 B)\land (B\to A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\to B)\land (B\to A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21459944695dadd2c3234fee78fa67e2f2752be2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.444ex; height:2.843ex;" alt="{\displaystyle (A\to B)\land (B\to A)}"></span>.</li></ul> <p><a href="/wiki/Bewerkingsvolgorde" title="Bewerkingsvolgorde">Bewerkingsvolgorde</a>: We spreken af dat negatie (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></span>) het sterkste bindt, daarna conjunctie (<span class="mwe-math-element"><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>), daarna disjunctie (<span class="mwe-math-element"><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>), en daarna implicatie (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"></span>) en bi-implicatie (<span class="mwe-math-element"><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>). Bovendien spreken we af dat de implicatie van rechts naar links associatief is. Dat betekent dat we <span class="mwe-math-element"><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\to p\land \neg q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">→</mo> <mi>p</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\to \neg q\to p\land \neg q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbd15adee583b04fe3de4d16a1497d71688e1e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:17.479ex; height:2.343ex;" alt="{\displaystyle p\to \neg q\to p\land \neg q}"></span> mogen schrijven in plaats van <span class="mwe-math-element"><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\to (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> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <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 stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to (\neg q\to (p\land \neg q)))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/827a6bc2b5362e5470bf44d425efbd0fc5041a1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.817ex; height:2.843ex;" alt="{\displaystyle (p\to (\neg q\to (p\land \neg q)))}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Valuaties">Valuaties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositielogica&veaction=edit&section=4" title="Bewerk dit kopje: Valuaties" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositielogica&action=edit&section=4" title="De broncode bewerken van de sectie: Valuaties"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Om de <a href="/wiki/Formele_semantiek" title="Formele semantiek">semantiek</a> (d.w.z. de betekenis) van een formule te definiëren, hebben we het begrip <i>valuatie</i> nodig. Een valuatie <i>v</i> is een <a href="/wiki/Afbeelding_(wiskunde)" title="Afbeelding (wiskunde)">functie</a> die aan (een deel van) de propositievariabelen een waarheidswaarde, waar (1) of onwaar (0), toekent. Op basis van de waarheidswaarden van de propositievariabelen die in een formule voorkomen, kunnen we dan de waarheidswaarde van de volledige formule bepalen. Daarvoor breiden we het begrip valuatie uit zodat ook aan formules waarheidswaarden worden toegekend, en wel als volgt: </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 v(A\land B)={\begin{cases}1&{\mbox{als }}v(A)=1{\mbox{ en }}v(B)=1\\0&{\mbox{anders}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> en </mtext> </mstyle> </mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>anders</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(A\land B)={\begin{cases}1&{\mbox{als }}v(A)=1{\mbox{ en }}v(B)=1\\0&{\mbox{anders}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/023408511f5c3b2fce9bc6415d9a9a88e65a26ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.8ex; height:6.176ex;" alt="{\displaystyle v(A\land B)={\begin{cases}1&{\mbox{als }}v(A)=1{\mbox{ en }}v(B)=1\\0&{\mbox{anders}}\end{cases}}}"></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 v(A\lor B)={\begin{cases}0&{\mbox{als }}v(A)=0{\mbox{ en }}v(B)=0\\1&{\mbox{anders}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> en </mtext> </mstyle> </mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>anders</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(A\lor B)={\begin{cases}0&{\mbox{als }}v(A)=0{\mbox{ en }}v(B)=0\\1&{\mbox{anders}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af5c0f56b3a2bb85b67295bfc64a3368552c356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.8ex; height:6.176ex;" alt="{\displaystyle v(A\lor B)={\begin{cases}0&{\mbox{als }}v(A)=0{\mbox{ en }}v(B)=0\\1&{\mbox{anders}}\end{cases}}}"></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 v(\neg A)={\begin{cases}0&{\mbox{als }}v(A)=1\\1&{\mbox{als }}v(A)=0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(\neg A)={\begin{cases}0&{\mbox{als }}v(A)=1\\1&{\mbox{als }}v(A)=0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ab0fad9346be454d6c35c08815ff194330dcff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.556ex; height:6.176ex;" alt="{\displaystyle v(\neg A)={\begin{cases}0&{\mbox{als }}v(A)=1\\1&{\mbox{als }}v(A)=0\end{cases}}}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Geldigheid_en_vervulbaarheid">Geldigheid en vervulbaarheid</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositielogica&veaction=edit&section=5" title="Bewerk dit kopje: Geldigheid en vervulbaarheid" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositielogica&action=edit&section=5" title="De broncode bewerken van de sectie: Geldigheid en vervulbaarheid"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Een formule <span class="mwe-math-element"><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> is <i>vervulbaar</i> als er ten minste één valuatie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> bestaat zodat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(A)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(A)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c5c4828d4486a6156432e4499e408bced6ab7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.941ex; height:2.843ex;" alt="{\displaystyle v(A)=1}"></span>. Een formule is <i>onvervulbaar</i> als ze niet vervulbaar is; met andere woorden, als er geen valuatie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> bestaat zodat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(A)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(A)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c5c4828d4486a6156432e4499e408bced6ab7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.941ex; height:2.843ex;" alt="{\displaystyle v(A)=1}"></span> is. </p><p>Een formule <span class="mwe-math-element"><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> is <i>geldig</i> (is een <a href="/wiki/Tautologie_(logica)" title="Tautologie (logica)"> tautologie</a>) als voor alle valuaties <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> geldt dat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(A)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(A)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c5c4828d4486a6156432e4499e408bced6ab7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.941ex; height:2.843ex;" alt="{\displaystyle v(A)=1}"></span>. Een geldige formule wordt ook wel een <i>logische wet</i> genoemd. Het feit dat een formule <span class="mwe-math-element"><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> geldig is, wordt genoteerd als <span class="mwe-math-element"><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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eaa8a353b3edc29e8597a301cf098812dfb2025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.403ex; height:2.843ex;" alt="{\displaystyle \models A}"></span>. Voorbeelden van geldige formules zijn: </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 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></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 (A\land B)\to (A\lor B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>A</mi> <mo>∨</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land B)\to (A\lor B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfcadb420b47913d08e1bfea93e7e599ed9ff0ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.412ex; height:2.843ex;" alt="{\displaystyle (A\land B)\to (A\lor B)}"></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 A\lor \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor \neg A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59bfe2f8010f7f758cae49d2b8fa7b2c4f812b63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.619ex; height:2.176ex;" alt="{\displaystyle A\lor \neg A}"></span> (<a href="/wiki/Wet_van_de_uitgesloten_derde" title="Wet van de uitgesloten derde">wet van de uitgesloten derde</a>; deze geldt bij de klassieke propositielogica, maar niet bij de <a href="/wiki/Intu%C3%AFtionisme" title="Intuïtionisme">intuitionistische propositielogica</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 \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> 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 \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> (de <a href="/wiki/Wetten_van_De_Morgan" title="Wetten van De Morgan">wetten van De Morgan</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 (A\to B)\leftrightarrow (\neg B\to \neg A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\to B)\leftrightarrow (\neg B\to \neg A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d142e036e0a935d52fb85f56267a3221cf69f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.576ex; height:2.843ex;" alt="{\displaystyle (A\to B)\leftrightarrow (\neg B\to \neg A)}"></span> (<a href="/wiki/Bewijs_door_contrapositie" title="Bewijs door contrapositie">contrapositie</a>)</li></ul> <p>Een gevolgtrekking heeft een aantal hypotheses en een conclusie. Een gevolgtrekking is geldig als in alle valuaties waarin de hypotheses waar zijn de conclusie ook waar is. De notatie <span class="mwe-math-element"><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},\ldots ,A_{n}\models B}"> <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> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⊨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},\ldots ,A_{n}\models B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7fd1922aa2f7e93859bead6c39ebc264c983fbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.006ex; height:2.843ex;" alt="{\displaystyle A_{1},\ldots ,A_{n}\models B}"></span> geeft aan dat de gevolgtrekking met hypotheses <span class="mwe-math-element"><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},\ldots ,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> <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},\ldots ,A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea095056a1ec9fe76f95b91813d03b69ffbf35a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.937ex; height:2.509ex;" alt="{\displaystyle A_{1},\ldots ,A_{n}}"></span> en conclusie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> geldig is. In de klassieke propositielogica geldt <span class="mwe-math-element"><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},\ldots ,A_{n}\models B}"> <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> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⊨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},\ldots ,A_{n}\models B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7fd1922aa2f7e93859bead6c39ebc264c983fbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.006ex; height:2.843ex;" alt="{\displaystyle A_{1},\ldots ,A_{n}\models B}"></span> dan en slechts dan als <span class="mwe-math-element"><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 \cdots \land A_{n})\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A_{1}\land \cdots \land A_{n})\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bdf3d0cb6718d45bc6ec8088002be71c2cde3a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.835ex; height:2.843ex;" alt="{\displaystyle (A_{1}\land \cdots \land A_{n})\to B}"></span> geldig is. </p><p>Twee formules <span class="mwe-math-element"><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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> zijn <i><a href="/wiki/Logische_equivalentie" title="Logische equivalentie">logisch equivalent</a></i> als <span class="mwe-math-element"><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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> waar zijn in precies dezelfde valuaties. Dat betekent dat voor alle valuaties <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> moet gelden dat als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(A)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(A)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c5c4828d4486a6156432e4499e408bced6ab7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.941ex; height:2.843ex;" alt="{\displaystyle v(A)=1}"></span> is, dan ook <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(B)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(B)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0828d612d2b2216754b7bef3768cdbc3d2155771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.962ex; height:2.843ex;" alt="{\displaystyle v(B)=1}"></span>, en dat als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(B)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(B)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0828d612d2b2216754b7bef3768cdbc3d2155771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.962ex; height:2.843ex;" alt="{\displaystyle v(B)=1}"></span> is, dan ook <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(A)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(A)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c5c4828d4486a6156432e4499e408bced6ab7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.941ex; height:2.843ex;" alt="{\displaystyle v(A)=1}"></span>. In de klassieke propositielogica zijn <span class="mwe-math-element"><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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> dan en slechts dan logisch equivalent, als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936ab098710910e69e56ec2734dd89063ce21efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftrightarrow B}"></span> een geldige formule is. </p> <div class="mw-heading mw-heading3"><h3 id="Identiteitswet">Identiteitswet</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositielogica&veaction=edit&section=6" title="Bewerk dit kopje: Identiteitswet" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositielogica&action=edit&section=6" title="De broncode bewerken van de sectie: Identiteitswet"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zijn <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> 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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> proposities met <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(F)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(F)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0a12e89d233c69a592e05e87d0354d6f5eeea7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.939ex; height:2.843ex;" alt="{\displaystyle v(F)=0}"></span> 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 v(T)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(T)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a0f035568d8097d62b81bb6eac1e245b6cbc3ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.834ex; height:2.843ex;" alt="{\displaystyle v(T)=1}"></span>, dan is: </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 (A\lor F)\leftrightarrow A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\lor F)\leftrightarrow A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c06bed7dc7742de5a30f78eacffc1d0b445a087" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.233ex; height:2.843ex;" alt="{\displaystyle (A\lor F)\leftrightarrow A}"></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 (A\land T)\leftrightarrow A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land T)\leftrightarrow A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/844df7c1ca6a8d38d9b8164cb11b6548c316d4b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.128ex; height:2.843ex;" alt="{\displaystyle (A\land T)\leftrightarrow A}"></span></li></ul> <p>Deze equivalenties vormen de <i>identiteitswet</i> van de propositielogica. </p> <div class="mw-heading mw-heading3"><h3 id="Waarheidstabellen">Waarheidstabellen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositielogica&veaction=edit&section=7" title="Bewerk dit kopje: Waarheidstabellen" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositielogica&action=edit&section=7" title="De broncode bewerken van de sectie: Waarheidstabellen"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote" style="margin-bottom:0.5em; padding:0.5em 0 0.5em 1.6em; font-size:95%;" role="note"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/15px-1rightarrow_blue.svg.png" decoding="async" width="15" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/23px-1rightarrow_blue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/30px-1rightarrow_blue.svg.png 2x" data-file-width="480" data-file-height="480" /></span></span> <i>Zie <a href="/wiki/Waarheidstabel" title="Waarheidstabel">Waarheidstabel</a> voor het hoofdartikel over dit onderwerp.</i></div> <p>Waarheidstabellen vormen een methode om te onderzoeken of een formule geldig of vervulbaar is. Bovendien kunnen waarheidstabellen worden gebruikt om te weten te komen of een gevolgtrekking geldig is, alsook om te achterhalen of twee formules logisch equivalent zijn. In een waarheidstabel worden alle mogelijke waarheidswaarden van propositievariabelen opgesomd. Elke rij in de tabel correspondeert met een valuatie, terwijl elke kolom correspondeert met een formule. In de tabel wordt dan opgenomen of de formule onder de gegeven valuatie waar is of niet. Een voorbeeld van een waarheidstabel met enkele eenvoudige formules: </p> <table class="wikitable"> <tbody><tr> <th><span class="mwe-math-element"><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></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"></span></th> <th><span class="mwe-math-element"><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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936ab098710910e69e56ec2734dd89063ce21efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftrightarrow B}"></span> </th></tr> <tr align="center"> <td>0</td> <td>0</td> <td>1</td> <td>0</td> <td>0</td> <td>1</td> <td>1 </td></tr> <tr align="center"> <td>0</td> <td>1</td> <td>1</td> <td>0</td> <td>1</td> <td>1</td> <td>0 </td></tr> <tr align="center"> <td>1</td> <td>0</td> <td>0</td> <td>0</td> <td>1</td> <td>0</td> <td>0 </td></tr> <tr align="center"> <td>1</td> <td>1</td> <td>0</td> <td>1</td> <td>1</td> <td>1</td> <td>1 </td></tr></tbody></table> <p>Een formule is geldig als er in de waarheidstabel alleen maar 1'en in de kolom eronder staan. De formule is vervulbaar als er minstens één 1 in de kolom staat. Zo blijkt uit de bovenstaande waarheidstabel dat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"></span> 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 A\leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936ab098710910e69e56ec2734dd89063ce21efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftrightarrow B}"></span> allemaal vervulbaar maar niet geldig zijn. </p><p>Bij het onderzoeken van een complexe logische stelling zoals A → (B ∧ ¬C) zal het aantal mogelijke combinaties van de verschillende variabelen (hier A, B en C) exponentieel toenemen met het aantal variabelen in de stelling. De waarheidstabel van een stelling met <i>n</i> variabelen telt immers 2<sup>n</sup> rijen. Voor een formule met 3 of zelfs 4 variabelen is dat met de hand nog wel te doen (de tabel telt 8 of 16 rijen), maar bij veel praktische toepassingen ontstaan formules met honderden of zelfs duizenden propositievariabelen. In zulke gevallen is het zelfs met de computer praktisch niet meer mogelijk de volledige waarheidstabel uit te rekenen en moeten andere methoden worden gevonden om te beslissen of een formule geldig of vervulbaar is. </p> <div class="mw-heading mw-heading2"><h2 id="Bewijssystemen">Bewijssystemen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositielogica&veaction=edit&section=8" title="Bewerk dit kopje: Bewijssystemen" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositielogica&action=edit&section=8" title="De broncode bewerken van de sectie: Bewijssystemen"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In tegenstelling tot waarheidstabellen, waar de betekenis van de logische connectieven een belangrijke rol speelt, wordt in een bewijssysteem slechts gewerkt met syntactische constructies. Formules worden aan de hand van regels omgeschreven in andere formules totdat een bepaalde voorwaarde is vervuld en men concludeert dat de formule geldig (of in sommige gevallen onvervulbaar) is. Belangrijke eigenschappen die een bepaald bewijssysteem moet hebben zijn <i>correctheid</i> (als een formule met het bewijssysteem bewezen kan worden is ze ook geldig) en <i>volledigheid</i> (als een formule geldig is kan ze ook bewezen worden). </p><p>Voor de propositielogica bestaan verschillende correcte en volledige bewijssystemen. </p> <div class="mw-heading mw-heading3"><h3 id="Bewijssysteem_in_de_stijl_van_Hilbert">Bewijssysteem in de stijl van Hilbert</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositielogica&veaction=edit&section=9" title="Bewerk dit kopje: Bewijssysteem in de stijl van Hilbert" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositielogica&action=edit&section=9" title="De broncode bewerken van de sectie: Bewijssysteem in de stijl van Hilbert"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Een bewijssysteem in de stijl van Hilbert bestaat uit een aantal axioma's en een aantal inferentieregels. Een bewijs in zo'n systeem is een eindig rijtje formules dat eindigt in de formule die bewezen moet worden, zodat voor elk element van het rijtje geldt dat het ofwel een axioma is, ofwel door een inferentieregel volgt uit formules die eerder in het rijtje voorkomen (en daardoor al bewezen zijn). Anders dan de meeste andere bewijssystemen die met inferentieregels werken, zoals natuurlijke deductie, introduceren inferentieregels in een Hilbert-systeem geen aannames. Omdat daardoor de structuur van bewijssystemen in de stijl van Hilbert heel eenvoudig is, worden zulke bewijssystemen vaak gebruikt in de <a href="/wiki/Bewijstheorie" title="Bewijstheorie">bewijstheorie</a>, waar formele bewijzen niet alleen een manier zijn om de waarheid van uitspraken te bepalen, maar zelf ook het onderwerp van studie zijn. (Dat de structuur van een bewijssysteem eenvoudig is, betekent overigens niet dat het ook eenvoudig is een bewijs erin op te stellen.) </p><p>Het bekendste Hilbert-bewijssysteem voor de propositielogica bevat de volgende inferentieregels: </p> <ul><li><i>modus ponens</i>: uit <span class="mwe-math-element"><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> 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 A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"></span> volgt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> (hierbij staan <span class="mwe-math-element"><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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> voor willekeurige propositielogische formules);</li> <li><i>uniforme substitutie</i>: uit <span class="mwe-math-element"><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> volgt <span class="mwe-math-element"><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[p/B]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">[</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A[p/B]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf28f8612ce66c9b0f4fb35d8a9908bb39483d2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.133ex; height:2.843ex;" alt="{\displaystyle A[p/B]}"></span>, de formule die ontstaat door in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> alle voorkomens van de propositie <span class="mwe-math-element"><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> te vervangen door <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>.</li></ul> <p>Bovendien bevat het de volgende axioma's: </p> <ul><li>P1: <span class="mwe-math-element"><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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\to p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edf3d9dfaa106174c31fe6f412ad65c67ea3afbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.042ex; height:2.176ex;" alt="{\displaystyle p\to p}"></span></li> <li>P2: <span class="mwe-math-element"><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 p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <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\to (q\to p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5d2b17824e9ea7e78ef194c3e2c806d0beb58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.535ex; height:2.843ex;" alt="{\displaystyle p\to (q\to p)}"></span></li> <li>P3: <span class="mwe-math-element"><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))\to ((p\to q)\to (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> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <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>p</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\to (q\to r))\to ((p\to q)\to (p\to r))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/032298b4ca9ff6f0c27a1378bc3fc2eff83fdeb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.476ex; height:2.843ex;" alt="{\displaystyle (p\to (q\to r))\to ((p\to q)\to (p\to r))}"></span></li> <li>P4: <span class="mwe-math-element"><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 \neg q)\to (q\to 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> <mi mathvariant="normal">¬</mi> <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 (\neg p\to \neg q)\to (q\to p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a3e269db70453359b0db5a71a861aca67164920" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.039ex; height:2.843ex;" alt="{\displaystyle (\neg p\to \neg q)\to (q\to p)}"></span></li></ul> <p>De bovenstaande axioma's zijn voldoende voor formules die alleen de logische connectieven <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"></span> 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 \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></span> bevatten. Voor formules die conjuncties (<span class="mwe-math-element"><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>) en disjuncties (<span class="mwe-math-element"><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>) bevatten, kunnen we de volgende axioma's toevoegen: </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 p\to q\to (p\land q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> <mo stretchy="false">→</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\to q\to (p\land q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d16ac08c06669cb77cc72730e6d84e34374ed77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:16.187ex; height:2.843ex;" alt="{\displaystyle p\to q\to (p\land q)}"></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 (p\land q)\to 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 stretchy="false">→</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q)\to p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd390432fdeb73e64dd58573cd20eca8f78dc6e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.414ex; height:2.843ex;" alt="{\displaystyle (p\land q)\to p}"></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 (p\land q)\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 stretchy="false">→</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\land q)\to q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11d45991b7ac4681d8d62dc8ad6bd6770ebd2f45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.314ex; height:2.843ex;" alt="{\displaystyle (p\land q)\to q}"></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 p\to (p\lor q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">→</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\to (p\lor q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e913aa6ef53e0310de47a0cf28f8e65ebea3ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.504ex; height:2.843ex;" alt="{\displaystyle p\to (p\lor q)}"></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 q\to (p\lor q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">→</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\to (p\lor q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df82af9bd40b34486b6d662523785d3603e7c546" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.314ex; height:2.843ex;" alt="{\displaystyle q\to (p\lor q)}"></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 (p\to r)\to (q\to r)\to (p\lor 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> <mi>r</mi> <mo stretchy="false">)</mo> <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 stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to r)\to (q\to r)\to (p\lor q)\to r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3df4a9e5833fab496ce6cda92c9beb22db73c3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.705ex; height:2.843ex;" alt="{\displaystyle (p\to r)\to (q\to r)\to (p\lor q)\to r}"></span></li></ul> <p>Het axioma P1 is eigenlijk niet nodig, omdat het op de volgende manier uit de axioma's P2 en P3 kan worden afgeleid: </p> <table class="wikitable"> <tbody><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 (p\to (q\to r))\to ((p\to q)\to (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> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">→</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <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>p</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\to (q\to r))\to ((p\to q)\to (p\to r))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/032298b4ca9ff6f0c27a1378bc3fc2eff83fdeb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.476ex; height:2.843ex;" alt="{\displaystyle (p\to (q\to r))\to ((p\to q)\to (p\to r))}"></span> </td> <td>(P3) </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 (p\to (q\to p))\to ((p\to q)\to (p\to p))}"> <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>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <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>p</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\to (q\to p))\to ((p\to q)\to (p\to p))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed50531ce2b736188969fa7963b0114afc90cbb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.717ex; height:2.843ex;" alt="{\displaystyle (p\to (q\to p))\to ((p\to q)\to (p\to p))}"></span> </td> <td>Uniforme substitutie: uit 1 volgt 1<span class="mwe-math-element"><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/p]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [r/p]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68d55737aceb7e0200b8ac60e1d33a1d4a5bb7f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.674ex; height:2.843ex;" alt="{\displaystyle [r/p]}"></span> </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 p\to (q\to p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <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\to (q\to p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5d2b17824e9ea7e78ef194c3e2c806d0beb58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.535ex; height:2.843ex;" alt="{\displaystyle p\to (q\to p)}"></span> </td> <td>(P2) </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 (p\to q)\to (p\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 stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to q)\to (p\to p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c016b015a9862f87ab78c8ed6d720c376128b354" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.039ex; height:2.843ex;" alt="{\displaystyle (p\to q)\to (p\to p)}"></span> </td> <td>Modus ponens uit 2, 3 </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 (p\to (q\to p))\to (p\to p)}"> <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>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">→</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\to (q\to p))\to (p\to p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f0428933b708c30e45a06e6a4d49f8a7c32d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.631ex; height:2.843ex;" alt="{\displaystyle (p\to (q\to p))\to (p\to p)}"></span> </td> <td>Uniforme substitutie uit 4, <span class="mwe-math-element"><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/(q\to p)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <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 [q/(q\to p)]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/276682d6d1e374d7bf056a039ab5e6dceba622e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.188ex; height:2.843ex;" alt="{\displaystyle [q/(q\to p)]}"></span> </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 p\to p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\to p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edf3d9dfaa106174c31fe6f412ad65c67ea3afbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.042ex; height:2.176ex;" alt="{\displaystyle p\to p}"></span> </td> <td>Modus ponens uit 3, 5 </td></tr></tbody></table> <p>Toch wordt P1 meestal wel in de axioma's van dit bewijssysteem opgenomen. </p> <div class="mw-heading mw-heading3"><h3 id="Semantische_tableaus">Semantische tableaus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositielogica&veaction=edit&section=10" title="Bewerk dit kopje: Semantische tableaus" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositielogica&action=edit&section=10" title="De broncode bewerken van de sectie: Semantische tableaus"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote" style="margin-bottom:0.5em; padding:0.5em 0 0.5em 1.6em; font-size:95%;" role="note"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/15px-1rightarrow_blue.svg.png" decoding="async" width="15" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/23px-1rightarrow_blue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/30px-1rightarrow_blue.svg.png 2x" data-file-width="480" data-file-height="480" /></span></span> <i>Zie <a href="/wiki/Semantisch_tableau" title="Semantisch tableau">Semantisch tableau</a> voor het hoofdartikel over dit onderwerp.</i></div> <p>Een semantisch tableau onderzoekt een stelling door de implicaties van een stelling te ontleden. Als een stelling waar is, wat betekent dat dan voor de waarheidswaarde van de variabelen in de stelling? Als A∧B waar is, dan moeten zowel A als B waar zijn. Als A→B waar is, dan is A onwaar en maakt de waarde van B niet uit of dan moeten A en B allebei waar zijn. Op deze manier kan een boom getekend worden die een complexe formule opsplitst in steeds kleinere delen, totdat duidelijk is welke waarden de variabelen aan moeten nemen om de stelling waar te maken. Semantische tableaus worden vaak gebruikt in combinatie met het <a href="/wiki/Reductio_ad_absurdum" class="mw-redirect" title="Reductio ad absurdum">bewijs uit het ongerijmde</a> om te bewijzen dat een stelling volgt uit een aantal hypothesen. </p> <div class="mw-heading mw-heading3"><h3 id="Natuurlijke_deductie">Natuurlijke deductie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositielogica&veaction=edit&section=11" title="Bewerk dit kopje: Natuurlijke deductie" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositielogica&action=edit&section=11" title="De broncode bewerken van de sectie: Natuurlijke deductie"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote" style="margin-bottom:0.5em; padding:0.5em 0 0.5em 1.6em; font-size:95%;" role="note"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/15px-1rightarrow_blue.svg.png" decoding="async" width="15" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/23px-1rightarrow_blue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/30px-1rightarrow_blue.svg.png 2x" data-file-width="480" data-file-height="480" /></span></span> <i>Zie <a href="/wiki/Natuurlijke_deductie" title="Natuurlijke deductie">Natuurlijke deductie</a> voor het hoofdartikel over dit onderwerp.</i></div> <p>Natuurlijke deductie is een door <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gerhard Gentzen</a> ontwikkeld bewijssysteem dat gebaseerd is op een natuurlijke manier van redeneren. Kenmerkend aan natuurlijke deductie is dat bewijzen een boomstructuur hebben en dat aannames ingevoerd en weer teruggetrokken kunnen worden. Daardoor ontstaat een natuurlijke notie van deelbewijs. Elk logisch connectief heeft een regel die vastlegt hoe gewenste conclusies die dit connectief als hoofdconnectief hebben, bewezen kunnen worden (de zogenaamde introductieregels), en een regel die vastlegt hoe aannames die dit connectief als hoofdconnectief hebben, kunnen worden gebruikt (de zogenaamde eliminatieregels). </p> <div class="mw-heading mw-heading3"><h3 id="Resolutie">Resolutie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositielogica&veaction=edit&section=12" title="Bewerk dit kopje: Resolutie" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositielogica&action=edit&section=12" title="De broncode bewerken van de sectie: Resolutie"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote" style="margin-bottom:0.5em; padding:0.5em 0 0.5em 1.6em; font-size:95%;" role="note"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/15px-1rightarrow_blue.svg.png" decoding="async" width="15" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/23px-1rightarrow_blue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/30px-1rightarrow_blue.svg.png 2x" data-file-width="480" data-file-height="480" /></span></span> <i>Zie <a href="/wiki/Resolutie_(logica)" title="Resolutie (logica)">Resolutie (logica)</a> voor het hoofdartikel over dit onderwerp.</i></div> <p>Met de bewijsmethode resolutie kan van formules in <a href="/wiki/Conjunctieve_normaalvorm" title="Conjunctieve normaalvorm">conjunctieve normaalvorm</a> de onvervulbaarheid worden aangetoond. Resolutie is gebaseerd op de geldige gevolgtrekking </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\lor B_{1}\lor \ldots \lor B_{n})\land (\neg A\lor C_{1}\lor \ldots \lor C_{m})\models B_{1}\lor \ldots \lor B_{n}\lor C_{1}\lor \ldots \lor C_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∨</mo> <mo>…</mo> <mo>∨</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>∨</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∨</mo> <mo>…</mo> <mo>∨</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊨</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∨</mo> <mo>…</mo> <mo>∨</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∨</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∨</mo> <mo>…</mo> <mo>∨</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\lor B_{1}\lor \ldots \lor B_{n})\land (\neg A\lor C_{1}\lor \ldots \lor C_{m})\models B_{1}\lor \ldots \lor B_{n}\lor C_{1}\lor \ldots \lor C_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffcb9909d5d201dc8a8d5883f45a50bcddeb98bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:77.552ex; height:2.843ex;" alt="{\displaystyle (A\lor B_{1}\lor \ldots \lor B_{n})\land (\neg A\lor C_{1}\lor \ldots \lor C_{m})\models B_{1}\lor \ldots \lor B_{n}\lor C_{1}\lor \ldots \lor C_{m}}"></span></dd></dl> <p>Er wordt geprobeerd door steeds deze gevolgtrekking toe te passen op de conjuncten van de formule de lege disjunctie af te leiden. Als dat is gelukt, is de formule niet vervulbaar. Resolutie wordt vaak toegepast in automatische stellingbewijzers. </p> <div class="mw-heading mw-heading2"><h2 id="Complexiteitstheorie">Complexiteitstheorie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositielogica&veaction=edit&section=13" title="Bewerk dit kopje: Complexiteitstheorie" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositielogica&action=edit&section=13" title="De broncode bewerken van de sectie: Complexiteitstheorie"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Het <a href="/wiki/Beslissingsprobleem" title="Beslissingsprobleem">probleem</a> om van een gegeven propositielogische formule na te gaan of het vervulbaar is, wordt het <a href="/wiki/Vervulbaarheidsprobleem" title="Vervulbaarheidsprobleem">vervulbaarheidsprobleem</a> genoemd (meestal afgekort tot <i>SAT</i>, van het Engelse <i>satisfiability</i>). SAT is <a href="/wiki/Berekenbaarheid" title="Berekenbaarheid">beslisbaar</a>. We kunnen namelijk voor elke formule automatisch de waarheidstabel opstellen en kijken of de formule in één rij daarvan waar is. Het aantal rijen in de waarheidstabel neemt echter exponentieel toe in verhouding met het aantal propositievariabelen dat in de formule voorkomt. Al voor een relatief klein aantal propositievariabelen is het in de praktijk niet meer mogelijk om de gehele waarheidstabel op te stellen. Aangezien SAT <a href="/wiki/NP-volledig" title="NP-volledig">NP-volledig</a> is, bestaat er op dit moment geen algoritme dat van <i>alle</i> formules efficiënt kan achterhalen of ze vervulbaar zijn of niet. Er bestaan echter algoritmes die in de praktijk voortreffelijke resultaten boeken. </p>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Overgenomen van "<a dir="ltr" href="https://nl.wikipedia.org/w/index.php?title=Propositielogica&oldid=67296975">https://nl.wikipedia.org/w/index.php?title=Propositielogica&oldid=67296975</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categorie:Alles" title="Categorie:Alles">Categorieën</a>: <ul><li><a href="/wiki/Categorie:Logica" title="Categorie:Logica">Logica</a></li><li><a href="/wiki/Categorie:Theoretische_informatica" title="Categorie:Theoretische informatica">Theoretische informatica</a></li><li><a href="/wiki/Categorie:Booleaanse_algebra" title="Categorie:Booleaanse algebra">Booleaanse algebra</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"> Deze pagina is voor het laatst bewerkt op 31 mrt 2024 om 20:02.</li> <li id="footer-info-copyright">De tekst is beschikbaar onder de licentie <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.nl">Creative Commons Naamsvermelding/Gelijk delen</a>, er kunnen aanvullende voorwaarden van toepassing zijn. 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