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Logique classique — Wikipédia
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class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Logique classique</span></h1> <div class="tagline"> première formalisation du langage et du raisonnement mathématique développée à partir de la fin du XIXe siècle en logique mathématique </div> </div> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"><a role="button" href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_classique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#p-lang" data-mw="interface" data-event-name="menu.languages" title="Langue" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> <span class="minerva-icon minerva-icon--language"></span> <span>Langue</span> </a></li> <li id="page-actions-watch" class="page-actions-menu__list-item"><a 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<span>Modifier</span> </a></li> </ul> </nav><!-- version 1.0.2 (change every time you update a partial) --> <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <div class="bandeau-container metadata bandeau-article bandeau-niveau-ebauche"> <div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"> <span class="noviewer" typeof="mw:File"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Fichier:Consistency.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Consistency.png/45px-Consistency.png" decoding="async" width="45" height="10" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Consistency.png/68px-Consistency.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/3/33/Consistency.png 2x" data-file-width="70" data-file-height="15"></a></span> </div> <div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p><strong class="bandeau-titre">Cet article est une <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Aide:%C3%89bauche?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Aide:Ébauche">ébauche</a> concernant la <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Logique">logique</a>.</strong></p> <p>Vous pouvez partager vos connaissances en l’améliorant (<b><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Aide:Comment_modifier_une_page?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Aide:Comment modifier une page">comment ?</a></b>) selon les recommandations des <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Projet:Accueil?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Projet:Accueil">projets correspondants</a>.</p> <p>Consultez la liste des <b>tâches à accomplir</b> en <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Discussion:Logique_classique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Discussion:Logique classique">page de discussion</a>.</p> </div> </div> <p>La <b>logique classique</b> est la première formalisation du langage et du raisonnement mathématique développée à partir de la fin du <a href="https://fr-m-wikipedia-org.translate.goog/wiki/XIXe_si%C3%A8cle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="XIXe siècle"><abbr class="abbr" title="19ᵉ siècle"><span class="romain">XIX</span><sup style="font-size:72%">e</sup></abbr> siècle</a> en <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_math%C3%A9matique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Logique mathématique">logique mathématique</a>. Appelée simplement <i>logique</i> à ses débuts, c'est l'apparition d'autres systèmes logiques formels, notamment de la <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_intuitionniste?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Logique intuitionniste">logique intuitionniste</a>, qui a suscité l'adjonction de l'adjectif <i>classique</i> au terme <i>logique</i>. À cette époque, le terme de <i>logique classique</i> fait référence à la <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_aristot%C3%A9licienne?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Logique aristotélicienne">logique aristotélicienne</a><sup id="cite_ref-1" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_classique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-2" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_classique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>.</p> <p>La logique classique est caractérisée par des postulats qui la fondent et la différencient de la logique intuitionniste, exprimés dans le formalisme du <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Calcul_des_propositions?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Calcul des propositions">calcul des propositions</a> ou du <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Calcul_des_pr%C3%A9dicats?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Calcul des prédicats">calcul des prédicats</a> :</p> <ul> <li>Le <i><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Principe_du_tiers_exclu?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Principe du tiers exclu">tiers exclu</a></i> énonce que pour toute <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Proposition_(math%C3%A9matiques)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect mw-disambig" title="Proposition (mathématiques)">proposition mathématique</a> considérée, elle-même ou sa négation est vraie : <span class="mwe-math-element"><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></li> </ul> <ul> <li>Le <i><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Raisonnement_par_l%27absurde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Raisonnement par l'absurde">raisonnement par l'absurde</a></i> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot \neg A\Rightarrow A}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ¬<!-- ¬ --> </mi> <mi mathvariant="normal"> ¬<!-- ¬ --> </mi> <mi> A </mi> <mo stretchy="false"> ⇒<!-- ⇒ --> </mo> <mi> A </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \lnot \neg A\Rightarrow A} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8adafbdf0d6d19a202c97501f2786a266bd28006" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.201ex; height:2.176ex;" alt="{\displaystyle \lnot \neg A\Rightarrow A}"></span></li> </ul> <ul> <li>La <i><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Contraposition?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Contraposition">contraposition</a></i> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big (}\neg B\Rightarrow \neg A{\big )}\Rightarrow {\big (}A\Rightarrow B{\big )}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mi mathvariant="normal"> ¬<!-- ¬ --> </mi> <mi> B </mi> <mo stretchy="false"> ⇒<!-- ⇒ --> </mo> <mi mathvariant="normal"> ¬<!-- ¬ --> </mi> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> <mo stretchy="false"> ⇒<!-- ⇒ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mi> A </mi> <mo stretchy="false"> ⇒<!-- ⇒ --> </mo> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\big (}\neg B\Rightarrow \neg A{\big )}\Rightarrow {\big (}A\Rightarrow B{\big )}} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5608ee76b0dfe912dc7d97bfe51eef29da3525ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.217ex; height:3.176ex;" alt="{\displaystyle {\big (}\neg B\Rightarrow \neg A{\big )}\Rightarrow {\big (}A\Rightarrow B{\big )}}"></span></li> </ul> <ul> <li>L'<i><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Implication_(logique)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Implication (logique)">implication matérielle</a></i> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\Rightarrow B)\Leftrightarrow (\neg A\lor 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> <mo stretchy="false"> ⇔<!-- ⇔ --> </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ¬<!-- ¬ --> </mi> <mi> A </mi> <mo> ∨<!-- ∨ --> </mo> <mi> B </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (A\Rightarrow B)\Leftrightarrow (\neg A\lor B)} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bc002c2947a77531cd2be6c39e90da9a12e0886" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.994ex; height:2.843ex;" alt="{\displaystyle (A\Rightarrow B)\Leftrightarrow (\neg A\lor B)}"></span></li> </ul> <p>Ces principes sont équivalents par raisonnement intuitionniste, c’est-à-dire que l'on peut montrer que n'importe lequel d'entre eux permet de déduire les autres en utilisant les règles intuitionnistes.</p> <p>On y ajoute généralement l'une des <i><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Lois_de_De_Morgan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lois de De Morgan">lois de De Morgan</a></i> : <span class="mwe-math-element"><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)\Rightarrow (\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)\Rightarrow (\neg A\lor \neg B)} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/515b296575804313123d4b2751966f1332a5221d" 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)\Rightarrow (\neg A\lor \neg B)}"></span></p> <p>Ces principes contribuent au fait que les <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Correspondance_de_Curry-Howard?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Correspondance de Curry-Howard">modèles calculatoires de la logique classique</a> sont beaucoup plus complexes que ceux de la logique intuitionniste.</p> <p>Le principe</p> <center> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow 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\Rightarrow B)\vee (B\Rightarrow A)} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" 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\Rightarrow B)\vee (B\Rightarrow A)}"></span> </center> <p>est valide en logique classique, et n'est pas démontrable en logique intuitionniste<sup id="cite_ref-3" class="reference"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_classique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup>, mais son adjonction à la logique intuitionniste n'engendre pas la logique classique.</p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="fr" dir="ltr"> <h2 id="mw-toc-heading">Sommaire</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_classique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Logique_classique_et_tables_de_v%C3%A9rit%C3%A9"><span class="tocnumber">1</span> <span class="toctext">Logique classique et tables de vérité</span></a></li> <li class="toclevel-1 tocsection-2"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_classique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#(A_%E2%87%92_B)_%E2%88%A8_(B_%E2%87%92_A),_la_logique_classique_et_la_logique_intuitionniste"><span class="tocnumber">2</span> <span class="toctext"><i>(A ⇒ B) ∨ (B ⇒ A)</i>, la logique classique et la logique intuitionniste</span></a></li> <li class="toclevel-1 tocsection-3"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_classique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#La_loi_de_Peirce"><span class="tocnumber">3</span> <span class="toctext">La loi de Peirce</span></a></li> <li class="toclevel-1 tocsection-4"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_classique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Notes_et_r%C3%A9f%C3%A9rences"><span class="tocnumber">4</span> <span class="toctext">Notes et références</span></a></li> <li class="toclevel-1 tocsection-5"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_classique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Lien_externe"><span class="tocnumber">5</span> <span class="toctext">Lien externe</span></a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Logique_classique_et_tables_de_vérité"><span id="Logique_classique_et_tables_de_v.C3.A9rit.C3.A9"></span>Logique classique et tables de vérité</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Logique_classique&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Logique classique et tables de vérité" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"> <div class="bandeau-cell bandeau-icone-css loupe"> Article détaillé : <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Table_de_v%C3%A9rit%C3%A9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Table de vérité">table de vérité</a>. </div> </div> <p>La <a href="https://fr-m-wikipedia-org.translate.goog/wiki/S%C3%A9mantique_formelle_(logique)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sémantique formelle (logique)">sémantique</a> (autrement dit la signification) de la logique classique se fait par des <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Table_de_v%C3%A9rit%C3%A9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Table de vérité">tables de vérité</a>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="(A_⇒_B)_∨_(B_⇒_A),_la_logique_classique_et_la_logique_intuitionniste"><span id=".28A_.E2.87.92_B.29_.E2.88.A8_.28B_.E2.87.92_A.29.2C_la_logique_classique_et_la_logique_intuitionniste"></span><i>(A ⇒ B) ∨ (B ⇒ A)</i>, la logique classique et la logique intuitionniste</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Logique_classique&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : (A ⇒ B) ∨ (B ⇒ A), la logique classique et la logique intuitionniste" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Fichier:Kripke_countermodel_of_A_-)_B_V_B_-)_A1.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Kripke_countermodel_of_A_-%29_B_V_B_-%29_A1.png/220px-Kripke_countermodel_of_A_-%29_B_V_B_-%29_A1.png" decoding="async" width="220" height="247" class="mw-file-element" data-file-width="227" data-file-height="255"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 247px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Kripke_countermodel_of_A_-%29_B_V_B_-%29_A1.png/220px-Kripke_countermodel_of_A_-%29_B_V_B_-%29_A1.png" data-width="220" data-height="247" data-srcset="//upload.wikimedia.org/wikipedia/commons/e/e5/Kripke_countermodel_of_A_-%29_B_V_B_-%29_A1.png 1.5x" data-class="mw-file-element"> </span></a> <figcaption> Figure 1: Contre-modèle de Kripke de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow 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\Rightarrow B)\vee (B\Rightarrow A)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" 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\Rightarrow B)\vee (B\Rightarrow A)}"> </noscript><span class="lazy-image-placeholder" style="width: 20.444ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" data-alt="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </figcaption> </figure> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Fichier:Kripke_countermodel_of_A_V_not_A.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/1/13/Kripke_countermodel_of_A_V_not_A.png" decoding="async" width="154" height="100" class="mw-file-element" data-file-width="154" data-file-height="100"> </noscript><span class="lazy-image-placeholder" style="width: 154px;height: 100px;" data-src="//upload.wikimedia.org/wikipedia/commons/1/13/Kripke_countermodel_of_A_V_not_A.png" data-width="154" data-height="100" data-class="mw-file-element"> </span></a> <figcaption> Figure 2: Contre-modèle de Kripke de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\vee \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\vee \neg A} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/323f5c2a8924ef707dcc24cee3bce1b148ec1367" 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\vee \neg A}"> </noscript><span class="lazy-image-placeholder" style="width: 7.619ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/323f5c2a8924ef707dcc24cee3bce1b148ec1367" data-alt="{\displaystyle A\vee \neg A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </figcaption> </figure> <p>Si on ajoute la proposition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow 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\Rightarrow B)\vee (B\Rightarrow A)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" 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\Rightarrow B)\vee (B\Rightarrow A)}"> </noscript><span class="lazy-image-placeholder" style="width: 20.444ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" data-alt="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> à la logique intuitionniste, on obtient une logique qui n'est plus la logique intuitionniste, car <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow 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\Rightarrow B)\vee (B\Rightarrow A)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" 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\Rightarrow B)\vee (B\Rightarrow A)}"> </noscript><span class="lazy-image-placeholder" style="width: 20.444ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" data-alt="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> n'est pas une conséquence de la logique intuitionniste, et qui n'est pas encore la logique classique, car le tiers exclu n'est pas une conséquence de cette formule.</p> <p>Les <a href="https://fr-m-wikipedia-org.translate.goog/wiki/S%C3%A9mantique_de_Kripke?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sémantique de Kripke">modèles de Kripke</a> sont essentiels pour comprendre la différence entre logique classique et logique intuitionniste. Le présent paragraphe en est une illustration et explique l'affirmation de l'alinéa ci-dessus.</p> <p>La figure 1 montre un <a href="https://fr-m-wikipedia-org.translate.goog/wiki/S%C3%A9mantique_de_Kripke?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#S%C3%A9mantique_de_la_logique_intuitionniste" title="Sémantique de Kripke">contre modèle de Kripke</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 {\mathcal {M}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> M </mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathcal {M}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.791ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd" data-alt="{\displaystyle {\mathcal {M}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> de la proposition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow 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\Rightarrow B)\vee (B\Rightarrow A)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" 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\Rightarrow B)\vee (B\Rightarrow A)}"> </noscript><span class="lazy-image-placeholder" style="width: 20.444ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" data-alt="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. Ça n'est pas un modèle de Kripke de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow 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\Rightarrow B)\vee (B\Rightarrow A)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" 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\Rightarrow B)\vee (B\Rightarrow A)}"> </noscript><span class="lazy-image-placeholder" style="width: 20.444ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" data-alt="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> parce que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}},w_{1}\nvDash (A\Rightarrow B)\vee (B\Rightarrow A)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> M </mi> </mrow> </mrow> <mo> , </mo> <msub> <mi> w </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> ⊭<!-- ⊭ --> </mo> <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 {\mathcal {M}},w_{1}\nvDash (A\Rightarrow B)\vee (B\Rightarrow A)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb0a5e9f3c00c48911eb04472cd5570f98bf5199" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.697ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}},w_{1}\nvDash (A\Rightarrow B)\vee (B\Rightarrow A)}"> </noscript><span class="lazy-image-placeholder" style="width: 29.697ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb0a5e9f3c00c48911eb04472cd5570f98bf5199" data-alt="{\displaystyle {\mathcal {M}},w_{1}\nvDash (A\Rightarrow B)\vee (B\Rightarrow A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. En effet,</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 {\mathcal {M}},w_{2}\nvDash A\Rightarrow B}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> M </mi> </mrow> </mrow> <mo> , </mo> <msub> <mi> w </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> ⊭<!-- ⊭ --> </mo> <mi> A </mi> <mo stretchy="false"> ⇒<!-- ⇒ --> </mo> <mi> B </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathcal {M}},w_{2}\nvDash A\Rightarrow B} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d22ef6393a292340e0d5d0dae41238e57e3ecd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.375ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M}},w_{2}\nvDash A\Rightarrow B}"> </noscript><span class="lazy-image-placeholder" style="width: 16.375ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d22ef6393a292340e0d5d0dae41238e57e3ecd7" data-alt="{\displaystyle {\mathcal {M}},w_{2}\nvDash A\Rightarrow B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}},w_{3}\nvDash B\Rightarrow A}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> M </mi> </mrow> </mrow> <mo> , </mo> <msub> <mi> w </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> ⊭<!-- ⊭ --> </mo> <mi> B </mi> <mo stretchy="false"> ⇒<!-- ⇒ --> </mo> <mi> A </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathcal {M}},w_{3}\nvDash B\Rightarrow A} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5c746dc7de03fa96dad75a63fb426029f9b455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.375ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M}},w_{3}\nvDash B\Rightarrow A}"> </noscript><span class="lazy-image-placeholder" style="width: 16.375ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5c746dc7de03fa96dad75a63fb426029f9b455" data-alt="{\displaystyle {\mathcal {M}},w_{3}\nvDash B\Rightarrow A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>donc</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 {\mathcal {M}},w_{1}\nvDash (A\Rightarrow B)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> M </mi> </mrow> </mrow> <mo> , </mo> <msub> <mi> w </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> ⊭<!-- ⊭ --> </mo> <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 {\mathcal {M}},w_{1}\nvDash (A\Rightarrow B)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e77466e3158e43cd567d7b5614e707dd6ddfb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.184ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}},w_{1}\nvDash (A\Rightarrow B)}"> </noscript><span class="lazy-image-placeholder" style="width: 18.184ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e77466e3158e43cd567d7b5614e707dd6ddfb0" data-alt="{\displaystyle {\mathcal {M}},w_{1}\nvDash (A\Rightarrow B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}},w_{1}\nvDash (B\Rightarrow A)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> M </mi> </mrow> </mrow> <mo> , </mo> <msub> <mi> w </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <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 {\mathcal {M}},w_{1}\nvDash (B\Rightarrow A)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ade5bf3478ad2d418ce72bd7115f98ccba809492" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.184ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}},w_{1}\nvDash (B\Rightarrow A)}"> </noscript><span class="lazy-image-placeholder" style="width: 18.184ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ade5bf3478ad2d418ce72bd7115f98ccba809492" data-alt="{\displaystyle {\mathcal {M}},w_{1}\nvDash (B\Rightarrow A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>d'où l'affirmation ci-dessus. Par conséquent, la proposition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow 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\Rightarrow B)\vee (B\Rightarrow A)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" 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\Rightarrow B)\vee (B\Rightarrow A)}"> </noscript><span class="lazy-image-placeholder" style="width: 20.444ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" data-alt="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> n'est pas valide dans la logique intuitionniste.</p> <p>Tous les modèles de Kripke linéaires, c'est-à-dire tous les modèles dans lesquels chaque monde a un seul autre monde accessible, sont des modèles de Kripke de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow 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\Rightarrow B)\vee (B\Rightarrow A)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" 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\Rightarrow B)\vee (B\Rightarrow A)}"> </noscript><span class="lazy-image-placeholder" style="width: 20.444ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" data-alt="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. C'est le cas de la figure 2. En revanche ce modèle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}'}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> M </mi> </mrow> </mrow> <mo> ′ </mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathcal {M}}'} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7595952d5fa141cfbbfdc76e51c57bd52f5a5b4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.475ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M}}'}"> </noscript><span class="lazy-image-placeholder" style="width: 3.475ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7595952d5fa141cfbbfdc76e51c57bd52f5a5b4e" data-alt="{\displaystyle {\mathcal {M}}'}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> n'est pas un modèle de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\vee \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\vee \neg A} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/323f5c2a8924ef707dcc24cee3bce1b148ec1367" 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\vee \neg A}"> </noscript><span class="lazy-image-placeholder" style="width: 7.619ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/323f5c2a8924ef707dcc24cee3bce1b148ec1367" data-alt="{\displaystyle A\vee \neg A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, car</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 {\mathcal {M}}',w_{1}\nvDash A}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> M </mi> </mrow> </mrow> <mo> ′ </mo> </msup> <mo> , </mo> <msub> <mi> w </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> ⊭<!-- ⊭ --> </mo> <mi> A </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathcal {M}}',w_{1}\nvDash A} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9997387a397bfb7290d8494c04f9cf3d80ebdd26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.681ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}',w_{1}\nvDash A}"> </noscript><span class="lazy-image-placeholder" style="width: 11.681ex;height: 2.843ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9997387a397bfb7290d8494c04f9cf3d80ebdd26" data-alt="{\displaystyle {\mathcal {M}}',w_{1}\nvDash A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}',w_{1}\nvDash \neg A}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> M </mi> </mrow> </mrow> <mo> ′ </mo> </msup> <mo> , </mo> <msub> <mi> w </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> ⊭<!-- ⊭ --> </mo> <mi mathvariant="normal"> ¬<!-- ¬ --> </mi> <mi> A </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathcal {M}}',w_{1}\nvDash \neg A} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e98f3e0617c10d7dfe3299c7157ef7b610756a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.232ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}',w_{1}\nvDash \neg A}"> </noscript><span class="lazy-image-placeholder" style="width: 13.232ex;height: 2.843ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e98f3e0617c10d7dfe3299c7157ef7b610756a" data-alt="{\displaystyle {\mathcal {M}}',w_{1}\nvDash \neg A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Puisque <span class="mwe-math-element"><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\vee \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\vee \neg A} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/323f5c2a8924ef707dcc24cee3bce1b148ec1367" 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\vee \neg A}"> </noscript><span class="lazy-image-placeholder" style="width: 7.619ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/323f5c2a8924ef707dcc24cee3bce1b148ec1367" data-alt="{\displaystyle A\vee \neg A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> n'est pas valide dans tous les modèles de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow 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\Rightarrow B)\vee (B\Rightarrow A)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" 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\Rightarrow B)\vee (B\Rightarrow A)}"> </noscript><span class="lazy-image-placeholder" style="width: 20.444ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" data-alt="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></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\vee \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\vee \neg A} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/323f5c2a8924ef707dcc24cee3bce1b148ec1367" 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\vee \neg A}"> </noscript><span class="lazy-image-placeholder" style="width: 7.619ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/323f5c2a8924ef707dcc24cee3bce1b148ec1367" data-alt="{\displaystyle A\vee \neg A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> ne peut pas être une conséquence de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow 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\Rightarrow B)\vee (B\Rightarrow A)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" 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\Rightarrow B)\vee (B\Rightarrow A)}"> </noscript><span class="lazy-image-placeholder" style="width: 20.444ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" data-alt="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. Donc plus généralement la logique classique ne peut pas être engendrée par l'ajout de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow 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\Rightarrow B)\vee (B\Rightarrow A)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" 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\Rightarrow B)\vee (B\Rightarrow A)}"> </noscript><span class="lazy-image-placeholder" style="width: 20.444ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233d0f56e9683a62e69ac33e5fc28fb3a8129acd" data-alt="{\displaystyle (A\Rightarrow B)\vee (B\Rightarrow A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> à la logique intuitionniste.</p> <p>Notons en passant que nous avons aussi montré que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\vee \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\vee \neg A} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/323f5c2a8924ef707dcc24cee3bce1b148ec1367" 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\vee \neg A}"> </noscript><span class="lazy-image-placeholder" style="width: 7.619ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/323f5c2a8924ef707dcc24cee3bce1b148ec1367" data-alt="{\displaystyle A\vee \neg A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> n'est pas valide en logique intuitionniste.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="La_loi_de_Peirce">La <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Loi_de_Peirce?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Loi de Peirce">loi de Peirce</a></h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Logique_classique&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : La loi de Peirce" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <p>La <a href="https://fr-m-wikipedia-org.translate.goog/wiki/Loi_de_Peirce?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Loi de Peirce">loi de Peirce</a> est la proposition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((A\Rightarrow B)\Rightarrow A)\Rightarrow A}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo stretchy="false"> ⇒<!-- ⇒ --> </mo> <mi> B </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ⇒<!-- ⇒ --> </mo> <mi> A </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ⇒<!-- ⇒ --> </mo> <mi> A </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle ((A\Rightarrow B)\Rightarrow A)\Rightarrow A} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec97ef2578663f88d37fff5483d5a65a81904cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.454ex; height:2.843ex;" alt="{\displaystyle ((A\Rightarrow B)\Rightarrow A)\Rightarrow A}"> </noscript><span class="lazy-image-placeholder" style="width: 21.454ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec97ef2578663f88d37fff5483d5a65a81904cc5" data-alt="{\displaystyle ((A\Rightarrow B)\Rightarrow A)\Rightarrow A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. Elle n'est pas valide en logique intuitionniste et son ajout à celle-ci produit la logique classique. Elle a la particularité de ne contenir que des implications à la différence des quatre propositions citées plus haut qui contiennent toutes une négation.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Logique_classique&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Notes et références" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <div class="references-small decimal" style=""> <div class="mw-references-wrap"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_classique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-1">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Couturat1901"><span class="ouvrage" id="Louis_Couturat1901">Louis Couturat, <cite class="italique">La logique de Leibniz d'après des documents inédits</cite>, Félix Alcan éditeur, <time>1901</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gallica.bnf.fr/ark:/12148/bpt6k110843d/f1">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=La+logique+de+Leibniz+d%27apr%C3%A8s+des+documents+in%C3%A9dits&rft.pub=F%C3%A9lix+Alcan+%C3%A9diteur&rft.aulast=Couturat&rft.aufirst=Louis&rft.date=1901&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ALogique+classique"></span></span></span> lire en ligne sur <i><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Gallica?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gallica">Gallica</a></i>, notamment l'appendice intitulé <span class="ouvrage">« <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gallica.bnf.fr/ark:/12148/bpt6k110843d/f456"><cite style="font-style:normal;"><i>Précis de logique classique</i></cite></a> »</span></span></li> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_classique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-2">↑</a> </span><span class="reference-text"> <span class="ouvrage" id="Dufumier1911"><span class="ouvrage" id="H._Dufumier1911">H. Dufumier, « <cite style="font-style:normal">La généralisation mathématique</cite> », <i>Revue de métaphysique et de morale</i>, <time>1911</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gallica.bnf.fr/ark:/12148/bpt6k11123b/f70">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=La+g%C3%A9n%C3%A9ralisation+math%C3%A9matique&rft.jtitle=Revue+de+m%C3%A9taphysique+et+de+morale&rft.aulast=Dufumier&rft.aufirst=H.&rft.date=1911&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ALogique+classique"></span></span></span> lire en ligne sur <i><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Gallica?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gallica">Gallica</a></i>.</span></li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_classique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-3">↑</a> </span><span class="reference-text"> <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Dirk_van_Dalen&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Dirk van Dalen (page inexistante)">Dirk van Dalen</a> <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/wiki/Dirk_van_Dalen" class="extiw" title="de:Dirk van Dalen"><span class="indicateur-langue" title="Article en allemand : « Dirk van Dalen »">(de)</span></a>, <i>Logic and Structure</i>, chap. 5 « <i>Intuitionistic logic</i> », exercice 9. (a), Springer-Verlag, 1991.</span></li> </ol> </div> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Lien_externe">Lien externe</h2><span class="mw-editsection"> <a role="button" href="https://fr-m-wikipedia-org.translate.goog/w/index.php?title=Logique_classique&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Modifier la section : Lien externe" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>modifier</span> </a> </span> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <ul> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.isima.fr/~leborgne/IsimathLogique/logique.pdf">Rappel : Bases de : logique, ensembles, limites.</a></li> <li><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_intuitionniste?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Logique intuitionniste">Logique intuitionniste</a></li> <li><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Logique_minimale?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Logique minimale">Logique minimale</a></li> </ul> <div class="navbox-container" style="clear:both;"> </div> <ul id="bandeau-portail" class="bandeau-portail"> <li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="https://fr-m-wikipedia-org.translate.goog/wiki/Portail:Math%C3%A9matiques?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Portail des mathématiques"> <noscript> <img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/24px-Racine_carr%C3%A9e_bleue.svg.png" decoding="async" 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href="https://fr-m-wikipedia-org.translate.goog/wiki/Portail:Logique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Portail:Logique">Portail de la logique</a></span> </span></li> </ul><!-- NewPP limit report Parsed by mw‐web.eqiad.main‐565d46677b‐qjnxz Cached time: 20241128121335 Cache expiry: 2592000 Reduced expiry: false Complications: [show‐toc] CPU time usage: 0.223 seconds Real time usage: 0.388 seconds Preprocessor visited node count: 1284/1000000 Post‐expand include size: 52108/2097152 bytes Template argument size: 12074/2097152 bytes Highest expansion depth: 12/100 Expensive parser function count: 2/500 Unstrip recursion depth: 0/20 Unstrip post‐expand size: 4347/5000000 bytes Lua time usage: 0.100/10.000 seconds Lua memory usage: 5610963/52428800 bytes Number of Wikibase entities loaded: 1/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 244.542 1 -total 35.49% 86.779 1 Modèle:Références 28.06% 68.623 1 Modèle:Ébauche 22.00% 53.808 1 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interwiki-mk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mk.wikipedia.org/wiki/%25D0%259A%25D0%25BB%25D0%25B0%25D1%2581%25D0%25B8%25D1%2587%25D0%25BD%25D0%25B0_%25D0%25BB%25D0%25BE%25D0%25B3%25D0%25B8%25D0%25BA%25D0%25B0" title="Класична логика – macédonien" lang="mk" hreflang="mk" data-title="Класична логика" data-language-autonym="Македонски" data-language-local-name="macédonien" class="interlanguage-link-target"><span>Македонски</span></a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Klassieke_logica" title="Klassieke logica – néerlandais" lang="nl" hreflang="nl" data-title="Klassieke logica" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li> <li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/L%25C3%25B3gica_cl%25C3%25A1ssica" title="Lógica clássica – portugais" lang="pt" hreflang="pt" data-title="Lógica clássica" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li> <li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%259A%25D0%25BB%25D0%25B0%25D1%2581%25D1%2581%25D0%25B8%25D1%2587%25D0%25B5%25D1%2581%25D0%25BA%25D0%25B0%25D1%258F_%25D0%25BB%25D0%25BE%25D0%25B3%25D0%25B8%25D0%25BA%25D0%25B0" title="Классическая логика – russe" lang="ru" hreflang="ru" data-title="Классическая логика" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li> <li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sk.wikipedia.org/wiki/Klasick%25C3%25A1_logika" title="Klasická logika – slovaque" lang="sk" hreflang="sk" data-title="Klasická logika" data-language-autonym="Slovenčina" data-language-local-name="slovaque" class="interlanguage-link-target"><span>Slovenčina</span></a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Klassisk_logik" title="Klassisk logik – suédois" lang="sv" hreflang="sv" data-title="Klassisk logik" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target"><span>Svenska</span></a></li> <li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ta.wikipedia.org/wiki/%25E0%25AE%25AE%25E0%25AE%25B0%25E0%25AE%25AA%25E0%25AF%2581_%25E0%25AE%2585%25E0%25AE%25B3%25E0%25AE%25B5%25E0%25AF%2588%25E0%25AE%25AF%25E0%25AE%25BF%25E0%25AE%25AF%25E0%25AE%25B2%25E0%25AF%258D" title="மரபு அளவையியல் – tamoul" lang="ta" hreflang="ta" data-title="மரபு அளவையியல்" data-language-autonym="தமிழ்" data-language-local-name="tamoul" class="interlanguage-link-target"><span>தமிழ்</span></a></li> <li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tl.wikipedia.org/wiki/Klasikong_lohika" title="Klasikong lohika – tagalog" lang="tl" hreflang="tl" data-title="Klasikong lohika" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%259A%25D0%25BB%25D0%25B0%25D1%2581%25D0%25B8%25D1%2587%25D0%25BD%25D0%25B0_%25D0%25BB%25D0%25BE%25D0%25B3%25D1%2596%25D0%25BA%25D0%25B0" title="Класична логіка – ukrainien" lang="uk" hreflang="uk" data-title="Класична логіка" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li> <li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh.wikipedia.org/wiki/%25E7%25BB%258F%25E5%2585%25B8%25E9%2580%25BB%25E8%25BE%2591" title="经典逻辑 – chinois" lang="zh" hreflang="zh" data-title="经典逻辑" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li> <li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh-yue.wikipedia.org/wiki/%25E5%258F%25A4%25E5%2585%25B8%25E9%2582%258F%25E8%25BC%25AF" title="古典邏輯 – cantonais" lang="yue" hreflang="yue" data-title="古典邏輯" data-language-autonym="粵語" data-language-local-name="cantonais" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> </section> </div> <div class="minerva-footer-logo"> <img src="/static/images/mobile/copyright/wikipedia-wordmark-fr.svg" alt="Wikipédia" width="119" height="18" style="width: 7.4375em; height: 1.125em;"> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod">La dernière modification de cette page a été faite le 18 septembre 2023 à 17:53.</li> <li id="footer-info-copyright">Le contenu est disponible sous licence <a class="external" rel="nofollow" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://creativecommons.org/licenses/by-sa/4.0/deed.fr">CC BY-SA 4.0</a> 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