{"id":27356,"date":"2021-01-27T13:00:16","date_gmt":"2021-01-27T13:00:16","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27356"},"modified":"2024-07-07T23:49:47","modified_gmt":"2024-07-07T23:49:47","slug":"4-techniques-de-deduction-indispensables","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/","title":{"rendered":"4 techniques de d\u00e9duction indispensables"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Apprenez 4 techniques de d\u00e9duction indispensables<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><em><strong>R\u00e9sum\u00e9:<\/strong><\/br>Dans ce cours, 4 techniques de d\u00e9duction en logique propositionnelle sont d\u00e9crites pour enrichir le calcul propositionnel rudimentaire pr\u00e9sent\u00e9 jusqu&#8217;\u00e0 pr\u00e9sent. La r\u00e8gle de pr\u00e9somption et sa combinaison avec la r\u00e8gle de monotonie sont pr\u00e9sent\u00e9es, ainsi que le syllogisme hypoth\u00e9tique et deux fa\u00e7ons d&#8217;obtenir cette r\u00e8gle de d\u00e9duction. Les \u00e9quivalences de double n\u00e9gation et le contrapos\u00e9 de l&#8217;implication sont \u00e9galement expliqu\u00e9s.<\/em><\/p>\n<p style=\"text-align:center;\"><strong><u>Objectifs d&#8217;apprentissage<\/u>:<\/strong><br \/>\u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de<\/p>\n<ol>\n<li><strong>Se souvenir<\/strong> de la structure d&#8217;un raisonnement et des exemples simples.<\/li>\n<li><strong>Comprendre<\/strong> la r\u00e8gle de pr\u00e9somption et son rapport avec le th\u00e9or\u00e8me de d\u00e9duction.<\/li>\n<li><strong>Comprendre<\/strong> la r\u00e8gle du syllogisme hypoth\u00e9tique et son rapport avec le modus ponens.<\/li>\n<li><strong>Appliquer<\/strong> le th\u00e9or\u00e8me de d\u00e9duction en logique propositionnelle.<\/li>\n<li><strong>Appliquer<\/strong> la r\u00e8gle de monotonie dans la d\u00e9duction des expressions.<\/li>\n<li><strong>Comprendre<\/strong> les \u00e9quivalences de double n\u00e9gation et le contrapos\u00e9 de l&#8217;implication en logique propositionnelle.<\/li>\n<li><strong>Conna\u00eetre<\/strong> les d\u00e9monstrations des techniques de d\u00e9duction et \u00eatre capable de les appliquer dans la pratique.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>TABLE DES MATI\u00c8RES<\/strong><br \/>\n<a href=\"#1\">R\u00c8GLE DE PR\u00c9SUMPTION (PRE)<\/a><br \/>\n<a href=\"#2\">LE SYLLOGISME HYPOTH\u00c9TIQUE (SH)<\/a><br \/>\n<a href=\"#3\">\u00c9QUIVALENCES DE DOUBLE N\u00c9GATION (DN)<\/a><br \/>\n<a href=\"#4\">\u00c9QUIVALENCE DU CONTRAPOS\u00c9 DE L&#8217;IMPLICATION (CPI)<\/a>\n<\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/6f_aavuC4E0\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\">Nous avons d\u00e9j\u00e0 vu la structure d&#8217;un raisonnement et des exemples simples. Maintenant, nous mettrons \u00e0 l&#8217;\u00e9preuve cette connaissance <strong>en raisonnant avec 4 techniques de d\u00e9duction en logique propositionnelle.<\/strong> \u00c0 travers cela, nous verrons non seulement que ces choses fonctionnent, mais nous commencerons \u00e9galement \u00e0 enrichir les proc\u00e9dures qui sortiront l&#8217;\u00e9tat rudimentaire dans lequel se trouve le calcul propositionnel pr\u00e9sent\u00e9 jusqu&#8217;\u00e0 pr\u00e9sent.<\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>Si <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> sont des expressions du calcul propositionnel, alors il est possible d&#8217;inf\u00e9rer les techniques de d\u00e9duction suivantes depuis les fondements :<\/strong><\/p>\n<p><a name=\"1\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>R\u00e8gle de pr\u00e9somption (Pre)<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=6f_aavuC4E0&amp;t=168s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">La r\u00e8gle de d\u00e9duction la plus simple<\/span><\/strong><\/a> de toutes est celle de pr\u00e9somption. Elle est obtenue directement en appliquant le <strong>r\u00e9ciproque du th\u00e9or\u00e8me de d\u00e9duction<\/strong> sur le th\u00e9or\u00e8me <span class=\"katex-eq\" data-katex-display=\"false\">\\vdash(\\alpha\\rightarrow\\alpha)<\/span>. Si cela vous semble du langage archa\u00efque, tout ce que vous devez savoir se trouve <a href=\"http:\/\/toposuranos.com\/material\/fr\/systemes-deductifs-formels-definitions-et-exemples\/\" rel=\"noopener\" target=\"_blank\"><strong>ici<\/strong><\/a>.<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash \\alpha <\/span>\n<p style=\"text-align: justify; color: #000000;\">Combin\u00e9e avec la r\u00e8gle de monotonie, elle vous permettra d&#8217;ajouter des expressions pratiques dans vos d\u00e9ductions.<\/p>\n<p><a name=\"2\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Le Syllogisme Hypoth\u00e9tique (SH)<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=6f_aavuC4E0&amp;t=206s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Le syllogisme hypoth\u00e9tique<\/span><\/strong><\/a>, ou transitivit\u00e9 de l&#8217;implication, est une sorte d&#8217;\u00e9volution du modus ponens. Sa formulation est la suivante :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha\\rightarrow\\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash (\\alpha\\rightarrow\\gamma)<\/span>\n<p style=\"text-align: justify; color: #000000;\">Il existe plusieurs fa\u00e7ons d&#8217;obtenir cette r\u00e8gle de d\u00e9duction, nous en verrons quelques-unes sous peu.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Si nous raisonnons \u00e0 partir des expressions, il sera simple de construire le raisonnement suivant :<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/td>\n<td>; Pr\u00e9misse<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/td>\n<td>; Pr\u00e9misse<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(\\beta\\rightarrow \\gamma)<\/span><\/td>\n<td>; Pr\u00e9misse<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/td>\n<td>; MP(1,2)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/td>\n<td>; MP(4,3)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Donc<span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha,(\\alpha\\rightarrow\\beta),(\\beta\\rightarrow\\gamma)\\}\\vdash\\gamma<\/span>\n<p style=\"text-align: justify; color: #000000;\">Enfin, en appliquant le th\u00e9or\u00e8me de d\u00e9duction sur cette derni\u00e8re expression, on a :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{(\\alpha\\rightarrow\\beta),(\\beta\\rightarrow\\gamma)\\}\\vdash(\\alpha\\rightarrow \\gamma)<\/span>\n<p style=\"text-align: justify; color: #000000;\">Une autre fa\u00e7on d&#8217;obtenir la d\u00e9monstration de cette r\u00e8gle est de raisonner \u00e0 partir des d\u00e9ductions, en construisant \u00e0 travers la pr\u00e9somption et la monotonie. Observez le raisonnement suivant \u00e0 partir des d\u00e9ductions :<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash \\alpha <\/span><\/td>\n<td>; Pr\u00e9somption et Monotonie<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash (\\alpha\\rightarrow \\beta) <\/span><\/td>\n<td>; Pr\u00e9somption et Monotonie<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash (\\beta\\rightarrow\\gamma) <\/span><\/td>\n<td>; Pr\u00e9somption et Monotonie<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash \\beta <\/span><\/td>\n<td>; MP(1,2)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash \\gamma <\/span><\/td>\n<td>; MP(4,3)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash (\\alpha \\rightarrow \\gamma) <\/span><\/td>\n<td>; TD(5)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Vous devez noter ici que les deux d\u00e9monstrations sont identiques, seulement elles ont \u00e9t\u00e9 d\u00e9velopp\u00e9es avec des styles diff\u00e9rents. En pratique, vous pouvez alterner entre les deux styles selon ce qui vous convient le mieux.<\/p>\n<p><a name=\"3\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>\u00c9quivalences de Double N\u00e9gation (DN)<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=6f_aavuC4E0&amp;t=500s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Les \u00e9quivalences de double n\u00e9gation<\/span><\/strong><\/a> reproduisent la notion intuitive que la double n\u00e9gation d&#8217;une affirmation est \u00e9quivalente \u00e0 la m\u00eame affirmation. Cela, \u00e9crit symboliquement, sera de la forme<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha\\dashv\\vdash\\neg\\neg\\alpha<\/span>\n<p style=\"text-align: justify; color: #000000;\">Voyons maintenant une d\u00e9monstration :<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg\\neg \\alpha \\rightarrow (\\neg\\neg\\neg\\neg \\alpha \\rightarrow\\neg\\neg\\alpha))<\/span><\/td>\n<td>; A1<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash ((\\neg\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\alpha)\\rightarrow(\\neg\\alpha \\rightarrow \\neg\\neg\\neg\\alpha))<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash ((\\neg\\alpha \\rightarrow \\neg\\neg\\neg\\alpha)\\rightarrow(\\neg\\neg\\alpha \\rightarrow \\alpha))<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash ((\\neg\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\alpha)\\rightarrow(\\neg\\neg\\alpha \\rightarrow \\alpha))<\/span><\/td>\n<td>; SH(2,3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash (\\neg\\neg\\neg\\neg \\alpha \\rightarrow\\neg\\neg\\alpha)<\/span><\/td>\n<td>; RTD(1)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash ((\\neg\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\alpha)\\rightarrow(\\neg\\neg\\alpha \\rightarrow \\alpha))<\/span><\/td>\n<td>; Monotonie(4)<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash (\\neg\\neg\\alpha \\rightarrow \\alpha)<\/span><\/td>\n<td>; MP(5,6)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash \\alpha<\/span><\/td>\n<td>; RTD(7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Donc<span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash \\alpha <\/span>\n<p style=\"text-align: justify; color: #000000;\">Pour faire la d\u00e9monstration dans l&#8217;autre sens, nous pouvons utiliser celle que nous venons de faire en la r\u00e9adaptant par une simple substitution, obtenant ce qui suit :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\neg \\alpha \\} \\vdash \\neg \\alpha <\/span>\n<p style=\"text-align: justify; color: #000000;\">Et \u00e0 partir de cela, nous formons la d\u00e9monstration dans l&#8217;autre sens :<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\neg \\alpha \\} \\vdash \\neg \\alpha <\/span><\/td>\n<td>; Ce que nous venons de prouver<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash(\\neg\\neg \\neg \\alpha\\rightarrow \\neg \\alpha) <\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash((\\neg\\neg \\neg \\alpha\\rightarrow \\neg \\alpha) \\rightarrow(\\alpha \\rightarrow\\neg\\neg\\alpha)) <\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash(\\alpha \\rightarrow\\neg\\neg\\alpha) <\/span><\/td>\n<td>; MP(2,3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash\\neg\\neg\\alpha <\/span><\/td>\n<td>; RTD(4)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Donc <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha \\} \\vdash \\neg\\neg \\alpha <\/span>\n<p style=\"text-align: justify; color: #000000;\">Enfin, de ces deux d\u00e9monstrations, on a <span class=\"katex-eq\" data-katex-display=\"false\"> \\alpha \\dashv\\vdash \\neg\\neg \\alpha <\/span>.<\/p>\n<p><a name=\"4\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>\u00c9quivalence du Contrapos\u00e9 de l&#8217;Implication (CpI)<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=6f_aavuC4E0&amp;t=948s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Cela correspond<\/span><\/strong><\/a> aux \u00e9quivalences suivantes<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta) \\dashv\\vdash (\\neg\\beta \\rightarrow \\neg\\alpha)<\/span>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\neg\\alpha\\rightarrow\\beta)\\dashv\\vdash (\\neg\\beta\\rightarrow\\alpha)<\/span>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\rightarrow\\neg\\beta) \\dashv\\vdash (\\beta\\rightarrow\\neg\\alpha)<\/span>\n<p style=\"text-align: justify; color: #000000;\">La d\u00e9monstration de cette premi\u00e8re relation se fait de la mani\u00e8re suivante :<\/p>\n<p style=\"text-align: justify; color: #000000;\">D&#8217;un c\u00f4t\u00e9, elle est obtenue directement depuis le troisi\u00e8me axiome<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash ((\\neg\\beta\\rightarrow \\neg\\alpha) \\rightarrow (\\alpha \\rightarrow\\beta))<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow \\neg\\alpha)\\}\\vdash (\\alpha \\rightarrow \\beta)<\/span><\/td>\n<td>; RTD(1)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Donc <span class=\"katex-eq\" data-katex-display=\"false\"> \\{(\\neg\\beta\\rightarrow \\neg\\alpha)\\}\\vdash (\\alpha \\rightarrow \\beta)<\/span>\n<p style=\"text-align: justify; color: #000000;\">Et de l&#8217;autre c\u00f4t\u00e9, la d\u00e9monstration peut \u00eatre obtenue \u00e0 partir du raisonnement suivant :<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg\\alpha \\dashv\\vdash \\alpha<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\neg \\alpha \\rightarrow \\alpha)<\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg\\beta \\dashv\\vdash \\beta<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\beta \\rightarrow \\neg\\neg \\beta)<\/span><\/td>\n<td>; TD(3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg \\alpha \\rightarrow \\alpha)<\/span><\/td>\n<td>; Mon(2)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\alpha \\rightarrow \\beta)<\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg \\alpha \\rightarrow \\beta)<\/span><\/td>\n<td>; SH(5,6)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\{(\\alpha \\rightarrow \\beta)\\} \\vdash (\\beta \\rightarrow \\neg\\neg \\beta)<\/span><\/td>\n<td>; Mon(4)<\/td>\n<\/tr>\n<tr>\n<td>(9)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg \\alpha \\rightarrow \\neg\\neg \\beta)<\/span><\/td>\n<td>; SH(7,8)<\/td>\n<\/tr>\n<tr>\n<td>(10)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\neg \\alpha \\rightarrow \\neg\\neg \\beta) \\rightarrow (\\neg \\beta \\rightarrow \\neg \\alpha )<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(11)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash ((\\neg\\neg \\alpha \\rightarrow \\neg\\neg \\beta) \\rightarrow (\\neg \\beta \\rightarrow \\neg \\alpha ))<\/span><\/td>\n<td>; Mon(10)<\/td>\n<\/tr>\n<tr>\n<td>(11)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg \\beta \\rightarrow \\neg \\alpha )<\/span><\/td>\n<td>; SH(10;11)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Donc <span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg \\beta \\rightarrow \\neg \\alpha )<\/span>\n<p style=\"text-align: justify; color: #000000;\">Donc, des deux raisonnements pr\u00e9c\u00e9dents, on a<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\"> (\\alpha \\rightarrow \\beta) \\dashv\\vdash (\\neg \\beta \\rightarrow \\neg \\alpha ) <\/span>\n<p style=\"text-align: justify; color: #000000;\">Pour d\u00e9montrer la deuxi\u00e8me, nous pouvons faire les deux raisonnements suivants :<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta \\dashv\\vdash \\neg\\neg\\beta<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg\\neg\\alpha \\dashv\\vdash \\neg\\alpha<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\beta \\rightarrow \\neg\\neg\\beta)<\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\alpha)<\/span><\/td>\n<td>; TD(2)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\alpha \\rightarrow \\beta)<\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\beta \\rightarrow \\neg\\neg\\beta)<\/span><\/td>\n<td>; Mon(3)<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\alpha)<\/span><\/td>\n<td>; Mon(4)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\alpha \\rightarrow \\neg\\neg\\beta)<\/span><\/td>\n<td>; SH(5,6)<\/td>\n<\/tr>\n<tr>\n<td>(9)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\beta)<\/span><\/td>\n<td>; SH(7,8)<\/td>\n<\/tr>\n<tr>\n<td>(10)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\beta) \\rightarrow (\\neg\\beta \\rightarrow \\neg\\neg\\alpha)<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(11)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash ((\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\beta) \\rightarrow (\\neg\\beta \\rightarrow \\neg\\neg\\alpha))<\/span><\/td>\n<td>; Mon(10)<\/td>\n<\/tr>\n<tr>\n<td>(12)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\beta \\rightarrow \\neg\\neg\\alpha)<\/span><\/td>\n<td>; MP(9,11)<\/td>\n<\/tr>\n<tr>\n<td>(13)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg \\alpha \\dashv \\vdash \\alpha<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(14)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg\\neg \\alpha\\rightarrow \\alpha)<\/span><\/td>\n<td>; TD(13)<\/td>\n<\/tr>\n<tr>\n<td>(15)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\} \\vdash (\\neg\\neg \\alpha\\rightarrow \\alpha)<\/span><\/td>\n<td>; Mon(14)<\/td>\n<\/tr>\n<tr>\n<td>(16)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\} \\vdash(\\neg\\beta \\rightarrow \\alpha)<\/span><\/td>\n<td>; SH(12,15)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Donc <span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\} \\vdash(\\neg\\beta \\rightarrow \\alpha) <\/span>\n<p style=\"text-align: justify; color: #000000;\">Il reste maintenant \u00e0 faire la d\u00e9monstration dans le sens inverse. Nous pouvons le faire par le raisonnement suivant :<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha \\dashv \\vdash \\neg\\neg\\alpha<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow \\neg\\neg\\alpha)<\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\neg\\beta\\rightarrow\\alpha)<\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\alpha \\rightarrow \\neg\\neg\\alpha)<\/span><\/td>\n<td>; Mon(2)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\neg\\beta\\rightarrow\\neg\\neg\\alpha)<\/span><\/td>\n<td>; SH(3,4)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\beta\\rightarrow\\neg\\neg\\alpha)\\rightarrow (\\neg\\alpha \\rightarrow \\beta) <\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash ((\\neg\\beta\\rightarrow\\neg\\neg\\alpha)\\rightarrow (\\neg\\alpha \\rightarrow \\beta)) <\/span><\/td>\n<td>; Mon(6)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\neg\\alpha \\rightarrow \\beta) <\/span><\/td>\n<td>; MP(5,7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Donc <span class=\"katex-eq\" data-katex-display=\"false\"> \\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\neg\\alpha \\rightarrow \\beta) <\/span>\n<p style=\"text-align: justify; color: #000000;\">Enfin, de ces deux raisonnements, on conclut que <span class=\"katex-eq\" data-katex-display=\"false\"> (\\neg\\beta\\rightarrow\\alpha) \\dashv \\vdash (\\neg\\alpha \\rightarrow \\beta) <\/span>, ce qui \u00e9tait \u00e0 d\u00e9montrer.<\/p>\n<p style=\"text-align: justify; color: #000000;\">La derni\u00e8re \u00e9quivalence sera laiss\u00e9e comme exercice. Pour la d\u00e9montrer, vous pouvez vous guider avec les deux d\u00e9monstrations d\u00e9j\u00e0 donn\u00e9es. C&#8217;est la meilleure fa\u00e7on de ma\u00eetriser les techniques de d\u00e9duction.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Apprenez 4 techniques de d\u00e9duction indispensables R\u00e9sum\u00e9:Dans ce cours, 4 techniques de d\u00e9duction en logique propositionnelle sont d\u00e9crites pour enrichir le calcul propositionnel rudimentaire pr\u00e9sent\u00e9 jusqu&#8217;\u00e0 pr\u00e9sent. La r\u00e8gle de pr\u00e9somption et sa combinaison avec la r\u00e8gle de monotonie sont pr\u00e9sent\u00e9es, ainsi que le syllogisme hypoth\u00e9tique et deux fa\u00e7ons d&#8217;obtenir cette r\u00e8gle de d\u00e9duction. Les [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27340,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":0,"footnotes":""},"categories":[617,631,569],"tags":[],"class_list":["post-27356","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-logique-mathematique","category-logique-propositionnelle","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>4 techniques de d\u00e9duction indispensables - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Apprenez 4 techniques de d\u00e9duction en logique propositionnelle. R\u00e8gle de pr\u00e9somption, syllogisme hypoth\u00e9tique, \u00e9quivalences de double n\u00e9gation et le contrapos\u00e9 de l&#039;implication.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"4 techniques de d\u00e9duction indispensables\" \/>\n<meta property=\"og:description\" content=\"Apprenez 4 techniques de d\u00e9duction en logique propositionnelle. R\u00e8gle de pr\u00e9somption, syllogisme hypoth\u00e9tique, \u00e9quivalences de double n\u00e9gation et le contrapos\u00e9 de l&#039;implication.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-01-27T13:00:16+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-07-07T23:49:47+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/sistemasdeductivos.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"4 techniques de d\u00e9duction indispensables\" \/>\n<meta name=\"twitter:description\" content=\"Apprenez 4 techniques de d\u00e9duction en logique propositionnelle. R\u00e8gle de pr\u00e9somption, syllogisme hypoth\u00e9tique, \u00e9quivalences de double n\u00e9gation et le contrapos\u00e9 de l&#039;implication.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/sistemasdeductivos.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"7 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/4-techniques-de-deduction-indispensables\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/4-techniques-de-deduction-indispensables\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"4 techniques de d\u00e9duction indispensables\",\"datePublished\":\"2021-01-27T13:00:16+00:00\",\"dateModified\":\"2024-07-07T23:49:47+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/4-techniques-de-deduction-indispensables\\\/\"},\"wordCount\":1792,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/4-techniques-de-deduction-indispensables\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/sistemasdeductivos.jpg\",\"articleSection\":[\"Logique Math\u00e9matique\",\"Logique Propositionnelle\",\"Math\u00e9matiques\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/4-techniques-de-deduction-indispensables\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/4-techniques-de-deduction-indispensables\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/4-techniques-de-deduction-indispensables\\\/\",\"name\":\"4 techniques de d\u00e9duction indispensables - toposuranos.com\\\/material\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/4-techniques-de-deduction-indispensables\\\/#primaryimage\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/4-techniques-de-deduction-indispensables\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/sistemasdeductivos.jpg\",\"datePublished\":\"2021-01-27T13:00:16+00:00\",\"dateModified\":\"2024-07-07T23:49:47+00:00\",\"description\":\"Apprenez 4 techniques de d\u00e9duction en logique propositionnelle. R\u00e8gle de pr\u00e9somption, syllogisme hypoth\u00e9tique, \u00e9quivalences de double n\u00e9gation et le contrapos\u00e9 de l'implication.\",\"breadcrumb\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/4-techniques-de-deduction-indispensables\\\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/4-techniques-de-deduction-indispensables\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/4-techniques-de-deduction-indispensables\\\/#primaryimage\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/sistemasdeductivos.jpg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/sistemasdeductivos.jpg\",\"width\":1024,\"height\":356,\"caption\":\"Created with GIMP\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/4-techniques-de-deduction-indispensables\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/cursos-de-matematica-y-fisica\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"4 techniques de d\u00e9duction indispensables\"}]},{\"@type\":\"WebSite\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#website\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/\",\"name\":\"toposuranos.com\\\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\",\"name\":\"toposuranos.com\\\/material\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\\\/material\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\"},\"sameAs\":[\"https:\\\/\\\/www.facebook.com\\\/groups\\\/toposuranos\",\"https:\\\/\\\/x.com\\\/topuranos\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\\\/\\\/www.linkedin.com\\\/company\\\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\\\/\\\/toposuranos.com\\\/material\"],\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/author\\\/giorgio-reveco\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"4 techniques de d\u00e9duction indispensables - toposuranos.com\/material","description":"Apprenez 4 techniques de d\u00e9duction en logique propositionnelle. R\u00e8gle de pr\u00e9somption, syllogisme hypoth\u00e9tique, \u00e9quivalences de double n\u00e9gation et le contrapos\u00e9 de l'implication.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/","og_locale":"es_ES","og_type":"article","og_title":"4 techniques de d\u00e9duction indispensables","og_description":"Apprenez 4 techniques de d\u00e9duction en logique propositionnelle. R\u00e8gle de pr\u00e9somption, syllogisme hypoth\u00e9tique, \u00e9quivalences de double n\u00e9gation et le contrapos\u00e9 de l'implication.","og_url":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2021-01-27T13:00:16+00:00","article_modified_time":"2024-07-07T23:49:47+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/sistemasdeductivos.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"4 techniques de d\u00e9duction indispensables","twitter_description":"Apprenez 4 techniques de d\u00e9duction en logique propositionnelle. R\u00e8gle de pr\u00e9somption, syllogisme hypoth\u00e9tique, \u00e9quivalences de double n\u00e9gation et le contrapos\u00e9 de l'implication.","twitter_image":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/sistemasdeductivos.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"7 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/#article","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/"},"author":{"name":"giorgio.reveco","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"4 techniques de d\u00e9duction indispensables","datePublished":"2021-01-27T13:00:16+00:00","dateModified":"2024-07-07T23:49:47+00:00","mainEntityOfPage":{"@id":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/"},"wordCount":1792,"commentCount":0,"publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/sistemasdeductivos.jpg","articleSection":["Logique Math\u00e9matique","Logique Propositionnelle","Math\u00e9matiques"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/#respond"]}]},{"@type":"WebPage","@id":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/","url":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/","name":"4 techniques de d\u00e9duction indispensables - toposuranos.com\/material","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/#primaryimage"},"image":{"@id":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/sistemasdeductivos.jpg","datePublished":"2021-01-27T13:00:16+00:00","dateModified":"2024-07-07T23:49:47+00:00","description":"Apprenez 4 techniques de d\u00e9duction en logique propositionnelle. R\u00e8gle de pr\u00e9somption, syllogisme hypoth\u00e9tique, \u00e9quivalences de double n\u00e9gation et le contrapos\u00e9 de l'implication.","breadcrumb":{"@id":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/#primaryimage","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/sistemasdeductivos.jpg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/sistemasdeductivos.jpg","width":1024,"height":356,"caption":"Created with GIMP"},{"@type":"BreadcrumbList","@id":"http:\/\/toposuranos.com\/material\/fr\/4-techniques-de-deduction-indispensables\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"http:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"4 techniques de d\u00e9duction indispensables"}]},{"@type":"WebSite","@id":"http:\/\/toposuranos.com\/material\/#website","url":"http:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"http:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"http:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"http:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"http:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/27356","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=27356"}],"version-history":[{"count":0,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/27356\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/27340"}],"wp:attachment":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=27356"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=27356"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=27356"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}