{"id":27447,"date":"2021-01-28T13:00:39","date_gmt":"2021-01-28T13:00:39","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27447"},"modified":"2024-07-09T03:11:15","modified_gmt":"2024-07-09T03:11:15","slug":"5-symetries-de-la-logique-propositionnelle","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/5-symetries-de-la-logique-propositionnelle\/","title":{"rendered":"5 sym\u00e9tries de la logique propositionnelle"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>5 sym\u00e9tries de la logique propositionnelle<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><em><strong>R\u00e9sum\u00e9 :<\/strong><br \/>Au cours de cette le\u00e7on, nous explorerons comment la double n\u00e9gation, le syllogisme hypoth\u00e9tique, le contrapos\u00e9 de l&#8217;implication, les th\u00e9or\u00e8mes de d\u00e9duction et les d\u00e9finitions des connecteurs se combinent pour former les sym\u00e9tries de la logique propositionnelle. Gr\u00e2ce \u00e0 des d\u00e9monstrations claires et simples, vous apprendrez \u00e0 ma\u00eetriser les \u00e9quivalences et \u00e0 les appliquer \u00e0 vos propres d\u00e9fis logiques.<\/p>\n<p>Les sym\u00e9tries abord\u00e9es dans le cours incluent : <span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span>-sym\u00e9trie, <span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span>-sym\u00e9trie, <span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span>-sym\u00e9trie, <span class=\"katex-eq\" data-katex-display=\"false\">\\leftrightarrow<\/span>-sym\u00e9trie et <span class=\"katex-eq\" data-katex-display=\"false\">\\veebar<\/span>-sym\u00e9trie. De plus, les interactions entre les d\u00e9monstrations et la mani\u00e8re dont chacune s&#8217;appuie sur les pr\u00e9c\u00e9dentes pour simplifier les d\u00e9ductions futures sont mises en \u00e9vidence. Ce cours ne vous fournira pas seulement une connaissance approfondie de la logique propositionnelle, mais vous apprendra \u00e9galement \u00e0 utiliser les d\u00e9monstrations pr\u00e9c\u00e9dentes pour optimiser votre processus d&#8217;apprentissage.<br \/>\n<\/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> des concepts de base de la logique propositionnelle, tels que le syllogisme hypoth\u00e9tique et la double n\u00e9gation.<\/li>\n<li><strong>Reconna\u00eetre<\/strong> les 5 sym\u00e9tries de la logique propositionnelle.<\/li>\n<li><strong>Comprendre<\/strong> le processus de d\u00e9monstration des \u00e9quivalences de sym\u00e9tries.<\/li>\n<li><strong>Appliquer<\/strong> la pr\u00e9somption, le th\u00e9or\u00e8me de d\u00e9duction et son r\u00e9ciproque dans les d\u00e9monstrations.<\/li>\n<li><strong>Relier<\/strong> les d\u00e9finitions des connecteurs logiques aux sym\u00e9tries.<\/li>\n<li><strong>Appr\u00e9cier<\/strong> l&#8217;importance de r\u00e9aliser des d\u00e9monstrations une seule fois et de les r\u00e9utiliser dans des d\u00e9monstrations futures.<\/li>\n<li><strong>D\u00e9velopper<\/strong> des comp\u00e9tences analytiques et critiques lors de la r\u00e9alisation de d\u00e9monstrations logiques.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>TABLE DES MATI\u00c8RES<\/strong><br \/>\n<a href=\"#1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span> &#8211; SYM\u00c9TRIE<\/a><br \/>\n<a href=\"#2\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span> &#8211; SYM\u00c9TRIE<\/a><br \/>\n<a href=\"#3\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span> &#8211; SYM\u00c9TRIE<\/a><br \/>\n<a href=\"#4\"><span class=\"katex-eq\" data-katex-display=\"false\">\\leftrightarrow<\/span> &#8211; SYM\u00c9TRIE<\/a><br \/>\n<a href=\"#5\"><span class=\"katex-eq\" data-katex-display=\"false\">\\veebar<\/span> &#8211; SYM\u00c9TRIE<\/a><br \/>\n<a href=\"#6\">REMARQUES FINALES<\/a>\n<\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/XKY_nS5Mizk\" 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;\">Une cons\u00e9quence directe du syllogisme hypoth\u00e9tique, de la double n\u00e9gation et du contrapos\u00e9 de l&#8217;implication, des th\u00e9or\u00e8mes de d\u00e9duction et des d\u00e9finitions des connecteurs sont les 5 sym\u00e9tries de la logique propositionnelle que nous examinerons ci-dessous.<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\downarrow \\beta) \\dashv\\vdash (\\beta\\downarrow \\alpha)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span><\/span>-sym\u00e9trie<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\vee \\beta) \\dashv\\vdash (\\beta\\vee \\alpha)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-sym\u00e9trie<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\wedge \\beta) \\dashv\\vdash (\\beta\\wedge \\alpha)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>-sym\u00e9trie<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\leftrightarrow \\beta) \\dashv\\vdash (\\beta\\leftrightarrow \\alpha)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\leftrightarrow<\/span><\/span>-sym\u00e9trie<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\veebar \\beta) \\dashv\\vdash (\\beta\\veebar\\alpha)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\veebar<\/span><\/span>-sym\u00e9trie<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Les d\u00e9monstrations de ces \u00e9quivalences ne sont pas toutes triviales, mais contrairement \u00e0 certaines d\u00e9monstrations que nous avons d\u00e9j\u00e0 vues, elles sont assez simples. Ci-dessous est pr\u00e9sent\u00e9 la d\u00e9monstration de chacune dans un seul sens ; la d\u00e9monstration dans l&#8217;autre sens est pratiquement identique et est laiss\u00e9e comme exercice au lecteur.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-sym\u00e9trie<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=XKY_nS5Mizk&#038;t=203s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">voir<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee \\beta)\\}\\vdash (\\alpha \\vee\\beta)<\/span><\/span><\/td>\n<td>; Pr\u00e9<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee \\beta)\\}\\vdash (\\neg \\alpha \\rightarrow \\beta)<\/span><\/span><\/td>\n<td>; parce que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\vee \\beta) := (\\neg \\alpha \\rightarrow \\beta)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee \\beta)\\}\\vdash (\\neg \\beta \\rightarrow \\alpha)<\/span><\/span><\/td>\n<td>; CPI(2)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{(\\alpha \\vee \\beta)\\}\\vdash ( \\beta \\vee \\alpha)}<\/span><\/span><\/td>\n<td>; parce que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( \\beta \\vee \\alpha) := (\\neg\\beta\\rightarrow\\alpha)<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Le raisonnement inverse est obtenu avec tr\u00e8s peu de variations en commen\u00e7ant par la pr\u00e9somption <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\beta\\vee\\alpha)\\}\\vdash (\\beta\\vee\\alpha)<\/span><\/span><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span><\/span>-sym\u00e9trie<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=XKY_nS5Mizk&#038;t=383s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">voir<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg(\\alpha \\downarrow \\beta)\\}\\vdash \\neg(\\alpha \\downarrow \\beta) <\/span><\/span><\/td>\n<td>; Pr\u00e9<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\neg(\\alpha \\downarrow \\beta)\\}\\vdash (\\alpha \\vee \\beta) <\/span><\/span><\/td>\n<td>; de (1) parce que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> (\\alpha\\vee\\beta) := \\neg(\\alpha \\downarrow \\beta)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\neg(\\alpha \\downarrow \\beta)\\}\\vdash (\\beta \\vee \\alpha) <\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-sym\u00e9trie<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg(\\alpha \\downarrow \\beta) \\rightarrow (\\beta \\vee \\alpha)) <\/span><\/span><\/td>\n<td>; TD(3)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg(\\beta \\vee \\alpha) \\rightarrow (\\alpha \\downarrow \\beta)) <\/span><\/span><\/td>\n<td>; CPI(4)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash ((\\beta \\downarrow \\alpha) \\rightarrow (\\alpha \\downarrow \\beta)) <\/span><\/span><\/td>\n<td>; de (5) parce que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\beta\\vee\\alpha) := \\neg(\\beta \\downarrow \\alpha)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{(\\beta \\downarrow \\alpha) \\} \\vdash (\\alpha \\downarrow \\beta)}<\/span><\/span><\/td>\n<td>; RTD(6)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Enfin, le raisonnement inverse est obtenu en commen\u00e7ant par la pr\u00e9somption <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg(\\beta\\downarrow\\alpha)\\}\\vdash \\neg(\\beta\\downarrow\\alpha)<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>-sym\u00e9trie<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=XKY_nS5Mizk&#038;t=659s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">voir<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge \\beta)\\} \\vdash (\\alpha \\wedge \\beta)<\/span><\/span><\/td>\n<td>; Pr\u00e9<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge \\beta)\\} \\vdash (\\neg\\alpha \\downarrow \\neg\\beta)<\/span><\/span><\/td>\n<td>; de (1) parce que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\wedge \\beta) := (\\neg\\alpha \\downarrow \\neg\\beta)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge \\beta)\\} \\vdash (\\neg\\beta \\downarrow \\neg\\alpha)<\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span><\/span>-sym\u00e9trie (2)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{(\\alpha \\wedge \\beta)\\} \\vdash ( \\beta \\wedge \\alpha)}<\/span><\/span><\/td>\n<td>; de (3) parce que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\beta \\wedge \\alpha) := (\\neg\\beta \\downarrow \\neg\\alpha)<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Comme dans le cas pr\u00e9c\u00e9dent, le raisonnement inverse est obtenu avec tr\u00e8s peu de variations en commen\u00e7ant par la pr\u00e9somption <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{( \\beta \\wedge \\alpha)\\}\\vdash ( \\beta \\wedge \\alpha)<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\leftrightarrow<\/span><\/span>-sym\u00e9trie<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=XKY_nS5Mizk&#038;t=830s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">voir<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\leftrightarrow \\beta)\\} \\vdash (\\alpha \\leftrightarrow \\beta)<\/span><\/span><\/td>\n<td>; Pr\u00e9<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\leftrightarrow \\beta)\\} \\vdash ((\\alpha \\rightarrow \\beta) \\wedge (\\alpha \\rightarrow \\beta))<\/span><\/span><\/td>\n<td>; de (1) parce que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\leftrightarrow \\beta) := ((\\alpha \\rightarrow \\beta) \\wedge (\\beta \\rightarrow \\alpha)) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\leftrightarrow \\beta)\\} \\vdash ((\\beta \\rightarrow \\alpha) \\wedge (\\alpha \\rightarrow \\beta) )<\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>-sym\u00e9trie(2)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{(\\alpha \\leftrightarrow \\beta)\\} \\vdash (\\beta \\leftrightarrow \\alpha)}<\/span><\/span><\/td>\n<td>; de (3) parce que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\beta \\leftrightarrow \\alpha) := ((\\beta \\rightarrow \\alpha) \\wedge (\\alpha \\rightarrow \\beta)) <\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Comme dans le cas pr\u00e9c\u00e9dent, mais en commen\u00e7ant par la pr\u00e9somption <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{( \\beta \\leftrightarrow \\alpha)\\}\\vdash ( \\beta\\leftrightarrow \\alpha)<\/span><\/span><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\veebar<\/span><\/span>-sym\u00e9trie<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=XKY_nS5Mizk&#038;t=982s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">voir<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\leftrightarrow \\beta)\\} \\vdash (\\alpha \\leftrightarrow \\beta) <\/span><\/span><\/td>\n<td>; Pr\u00e9<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\leftrightarrow \\beta)\\} \\vdash ( \\beta \\leftrightarrow \\alpha) <\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\leftrightarrow<\/span><\/span>-sym\u00e9trie(1)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash ((\\alpha \\leftrightarrow \\beta) \\rightarrow ( \\beta \\leftrightarrow \\alpha)) <\/span><\/span><\/td>\n<td>; TD(2)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg ( \\beta \\leftrightarrow \\alpha) \\rightarrow \\neg (\\alpha \\leftrightarrow \\beta)) <\/span><\/span><\/td>\n<td>; CPI(3)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg ( \\beta \\leftrightarrow \\alpha)\\} \\vdash \\neg (\\alpha \\leftrightarrow \\beta) <\/span><\/span><\/td>\n<td>; RTD(4)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{ ( \\beta \\veebar \\alpha)\\} \\vdash (\\alpha \\veebar \\beta)} <\/span><\/span><\/td>\n<td>; de (5) parce que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( \\beta \\veebar \\alpha) := \\neg\\beta \\leftrightarrow \\alpha)<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\veebar \\beta) := \\neg (\\alpha \\leftrightarrow \\beta)<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Comme dans tous les autres cas, il suffit de prouver la pr\u00e9somption dans le sens inverse <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\beta \\leftrightarrow \\alpha)\\} \\vdash (\\beta \\leftrightarrow \\alpha)<\/span><\/span> pour obtenir la d\u00e9duction dans ce sens.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Remarques finales<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=XKY_nS5Mizk&#038;t=1145s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Un aspect auquel le lecteur doit pr\u00eater attention<\/span><\/strong><\/a> est l&#8217;ordre dans lequel ces 5 sym\u00e9tries de la logique propositionnelle ont \u00e9t\u00e9 choisies pour \u00eatre d\u00e9montr\u00e9es. Notez que chacune est faite de mani\u00e8re \u00e0 utiliser certaines des d\u00e9monstrations pr\u00e9c\u00e9demment r\u00e9alis\u00e9es. Cela refl\u00e8te l&#8217;approche \u00e0 suivre lors de la r\u00e9alisation de d\u00e9monstrations : elles sont effectu\u00e9es une seule fois (et plus jamais !); apr\u00e8s cela, votre objectif doit \u00eatre de vous concentrer sur l&#8217;utilisation des d\u00e9monstrations pr\u00e9c\u00e9dentes pour simplifier les d\u00e9ductions futures.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>5 sym\u00e9tries de la logique propositionnelle R\u00e9sum\u00e9 :Au cours de cette le\u00e7on, nous explorerons comment la double n\u00e9gation, le syllogisme hypoth\u00e9tique, le contrapos\u00e9 de l&#8217;implication, les th\u00e9or\u00e8mes de d\u00e9duction et les d\u00e9finitions des connecteurs se combinent pour former les sym\u00e9tries de la logique propositionnelle. Gr\u00e2ce \u00e0 des d\u00e9monstrations claires et simples, vous apprendrez \u00e0 ma\u00eetriser [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27410,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":8,"footnotes":""},"categories":[617,631,569],"tags":[],"class_list":["post-27447","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>5 sym\u00e9tries de la logique propositionnelle - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"D\u00e9couvrez les 5 sym\u00e9tries de la logique propositionnelle avec des d\u00e9monstrations claires : double n\u00e9gation, syllogisme hypoth\u00e9tique et plus encore.\" 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