{"id":33829,"date":"2021-01-28T13:00:28","date_gmt":"2021-01-28T13:00:28","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33829"},"modified":"2025-07-31T07:35:58","modified_gmt":"2025-07-31T07:35:58","slug":"quinque-symmetriae-logicae-propositionalis","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/","title":{"rendered":"Quinque Symmetriae Logicae Propositionalis"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>5 Symmetriae Logicae Propositionalis<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><em><strong>Summarium:<\/strong><br \/>Per totam hanc lectionem explorabimus quomodo duplex negatio, syllogismus hypotheticus, contrapositio implicationis, theorema deductionis et definitiones connectorum componantur ad symmetrias logicae propositionalis constituendas. Demonstrationibus claris ac simplicibus, disces quomodo aequivalentias intelligas easque ad tuas quaestiones logicas applices.<\/p>\n<p>Symmetriae in hac lectione tractatae sunt: <span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span>-Symmetria, <span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span>-Symmetria, <span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span>-Symmetria, <span class=\"katex-eq\" data-katex-display=\"false\">\\leftrightarrow<\/span>-Symmetria et <span class=\"katex-eq\" data-katex-display=\"false\">\\veebar<\/span>-Symmetria. Praeterea illustrantur interactiones inter demonstrationes et quomodo unaquaeque in praecedentibus nitatur ad deductiones futuras simpliciores reddendas. Haec lectio non solum cognitionem profundam Logicae Propositionalis tibi praebebit, sed etiam docebit quomodo demonstrationes priores ad discendi processum optimandum adhibeas.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong><u>Propositi Discendi<\/u>:<\/strong><br \/>Expleta hac lectione, discipulus erit capax<\/p>\n<ol>\n<li><strong>Memorandi<\/strong> notiones fundamentales logicae propositionalis, ut syllogismum hypotheticum et duplicem negationem.<\/li>\n<li><strong>Agnoscendi<\/strong> quinque Symmetrias Logicae Propositionalis.<\/li>\n<li><strong>Intelligendi<\/strong> processum demonstrationis aequivalentiarum symmetriarum.<\/li>\n<li><strong>Applicandi<\/strong> Praesumptionem, Theorema Deductionis eiusque Reciprocum in demonstrationibus.<\/li>\n<li><strong>Referendi<\/strong> definitiones connectorum logicorum ad symmetrias.<\/li>\n<li><strong>Aestimandi<\/strong> momentum unam demonstrationem semel perficiendi eamque in demonstrationibus futuris iterum adhibendi.<\/li>\n<li><strong>Excolendi<\/strong> facultates analyticas ac criticas in demonstrationibus logicis conficiendis.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX CONTENTORUM<\/strong><br \/>\n<a href=\"#1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span> &#8211; SYMMETRIA<\/a><br \/>\n<a href=\"#2\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span> &#8211; SYMMETRIA<\/a><br \/>\n<a href=\"#3\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span> &#8211; SYMMETRIA<\/a><br \/>\n<a href=\"#4\"><span class=\"katex-eq\" data-katex-display=\"false\">\\leftrightarrow<\/span> &#8211; SYMMETRIA<\/a><br \/>\n<a href=\"#5\"><span class=\"katex-eq\" data-katex-display=\"false\">\\veebar<\/span> &#8211; SYMMETRIA<\/a><br \/>\n<a href=\"#6\">OBSERVATIONES 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;\">Consequentia directa syllogismi hypothetici, duplex negationis et contrapositionis implicationis, theorematum deductionis atque definitionum connectorum sunt quinque symmetriae logicae propositionalis quas infra considerabimus.<\/p>\n<table style=\"text-align: justify;\">\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>-Symmetria<\/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>-Symmetria<\/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>-Symmetria<\/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>-Symmetria<\/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>-Symmetria<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Demonstrationes harum aequivalentiarum non sunt omnino triviales, sed, dissimiliter a quibusdam demonstrationibus quas iam vidimus, satis simplices sunt. Infra unaquaeque demonstratio in una directione ostenditur; demonstratio directionis inversae fere eadem est et lectori pro exercitio relinquitur.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-Symmetria<\/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;\">vide<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td> [\/latex]<\/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>; Praesumptio<\/td>\n<\/tr>\n<tr>\n<td> [\/latex]<\/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>; quia <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>(3)<\/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> [\/latex]<\/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>; quia <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;\">Ratiocinatio in sensu inverso parvis tantum mutationibus obtinetur incipiendo a praesumptione <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>-Symmetria<\/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;\">vide<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td> [\/latex]<\/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>; Praesumptio<\/td>\n<\/tr>\n<tr>\n<td> [\/latex]<\/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>; ex (1) quia <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> [\/latex]<\/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>-Symmetria<\/td>\n<\/tr>\n<tr>\n<td> [\/latex]<\/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> [\/latex]<\/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> [\/latex]<\/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>; ex (5) quia <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> [\/latex]<\/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;\">Denique, ratiocinando in sensu inverso, deducitur conclusio contraria incipiendo a praesumptione <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>-Symmetria<\/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;\">vide<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td> [\/latex]<\/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>; Praesumptio<\/td>\n<\/tr>\n<tr>\n<td> [\/latex]<\/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>; ex (1) quia <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> [\/latex]<\/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>-Symmetria (2)<\/td>\n<\/tr>\n<tr>\n<td> [\/latex]<\/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>; ex (3) quia <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;\">Sicut priore exemplo, si in sensu inverso ratiocinaris, deducitur cum minimis variationibus incipiendo a praesumptione <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>-Symmetria<\/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;\">vide<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td> [\/latex]<\/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>; Praesumptio<\/td>\n<\/tr>\n<tr>\n<td> [\/latex]<\/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>; ex (1) quia <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> [\/latex]<\/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>-Symmetria(2)<\/td>\n<\/tr>\n<tr>\n<td> [\/latex]<\/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>; ex (3) quia <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;\">Sicut in exemplo praecedenti, incipitur a praesumptione <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>-Symmetria<\/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;\">vide<\/span><\/strong><\/a><\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td> [\/latex]<\/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>; Praesumptio<\/td>\n<\/tr>\n<tr>\n<td> [\/latex]<\/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>-Symmetria(1)<\/td>\n<\/tr>\n<tr>\n<td> [\/latex]<\/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> [\/latex]<\/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> [\/latex]<\/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> [\/latex]<\/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>; ex (5) quia <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;\">Ut in omnibus casibus praecedentibus, sufficit praesumptionem in sensu inverso probare <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\beta \\leftrightarrow \\alpha)\\} \\vdash (\\beta \\leftrightarrow \\alpha)<\/span><\/span> ad deductionem in eodem sensu obtinendam.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Observationes Finales<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=XKY_nS5Mizk&#038;t=1145s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Res cui lector attendere debet<\/span><\/strong><\/a> est ordo in quo hae quinque symmetriae logicae propositionalis demonstrandae electae sunt. Animadvertendum est unamquamque ita factam esse ut aliqua ex demonstrationibus antea peractis utatur. Hoc declarat rationem qua demonstrationes faciendas est: semel perficiuntur (et numquam iterum!); postea, propositum tuum sit ut demonstrationes anteriores ad usum afferas ad futuras deductiones simpliciores reddendas.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>5 Symmetriae Logicae Propositionalis Summarium:Per totam hanc lectionem explorabimus quomodo duplex negatio, syllogismus hypotheticus, contrapositio implicationis, theorema deductionis et definitiones connectorum componantur ad symmetrias logicae propositionalis constituendas. Demonstrationibus claris ac simplicibus, disces quomodo aequivalentias intelligas easque ad tuas quaestiones logicas applices. Symmetriae in hac lectione tractatae sunt: -Symmetria, -Symmetria, -Symmetria, -Symmetria et -Symmetria. Praeterea illustrantur [&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":2,"footnotes":""},"categories":[1352,1358,1298],"tags":[],"class_list":["post-33829","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-logica-mathematica","category-logica-propositionalis","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Quinque Symmetriae Logicae Propositionalis - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Quinque Symmetrias Logicae Propositionalis per demonstrationes claras explora: negatio duplex, syllogismus hypotheticus, et alia.\" \/>\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\/la\/quinque-symmetriae-logicae-propositionalis\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Quinque Symmetriae Logicae Propositionalis\" \/>\n<meta property=\"og:description\" content=\"Quinque Symmetrias Logicae Propositionalis per demonstrationes claras explora: negatio duplex, syllogismus hypotheticus, et alia.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/\" \/>\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=\"2021-01-28T13:00:28+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-31T07:35:58+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/5simetrias.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Quinque Symmetriae Logicae Propositionalis\" \/>\n<meta name=\"twitter:description\" content=\"Quinque Symmetrias Logicae Propositionalis per demonstrationes claras explora: negatio duplex, syllogismus hypotheticus, et alia.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/5simetrias.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=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/quinque-symmetriae-logicae-propositionalis\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/quinque-symmetriae-logicae-propositionalis\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Quinque Symmetriae Logicae Propositionalis\",\"datePublished\":\"2021-01-28T13:00:28+00:00\",\"dateModified\":\"2025-07-31T07:35:58+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/quinque-symmetriae-logicae-propositionalis\\\/\"},\"wordCount\":948,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/quinque-symmetriae-logicae-propositionalis\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/5simetrias.jpg\",\"articleSection\":[\"Logica Mathematica\",\"Logica Propositionalis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/quinque-symmetriae-logicae-propositionalis\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/quinque-symmetriae-logicae-propositionalis\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/quinque-symmetriae-logicae-propositionalis\\\/\",\"name\":\"Quinque Symmetriae Logicae Propositionalis - toposuranos.com\\\/material\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/quinque-symmetriae-logicae-propositionalis\\\/#primaryimage\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/quinque-symmetriae-logicae-propositionalis\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/5simetrias.jpg\",\"datePublished\":\"2021-01-28T13:00:28+00:00\",\"dateModified\":\"2025-07-31T07:35:58+00:00\",\"description\":\"Quinque Symmetrias Logicae Propositionalis per demonstrationes claras explora: negatio duplex, syllogismus hypotheticus, et alia.\",\"breadcrumb\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/quinque-symmetriae-logicae-propositionalis\\\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/quinque-symmetriae-logicae-propositionalis\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/quinque-symmetriae-logicae-propositionalis\\\/#primaryimage\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/5simetrias.jpg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/5simetrias.jpg\",\"width\":1024,\"height\":356,\"caption\":\"Created with GIMP\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/quinque-symmetriae-logicae-propositionalis\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/cursos-de-matematica-y-fisica\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Quinque Symmetriae Logicae Propositionalis\"}]},{\"@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":"Quinque Symmetriae Logicae Propositionalis - toposuranos.com\/material","description":"Quinque Symmetrias Logicae Propositionalis per demonstrationes claras explora: negatio duplex, syllogismus hypotheticus, et alia.","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\/la\/quinque-symmetriae-logicae-propositionalis\/","og_locale":"es_ES","og_type":"article","og_title":"Quinque Symmetriae Logicae Propositionalis","og_description":"Quinque Symmetrias Logicae Propositionalis per demonstrationes claras explora: negatio duplex, syllogismus hypotheticus, et alia.","og_url":"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2021-01-28T13:00:28+00:00","article_modified_time":"2025-07-31T07:35:58+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/5simetrias.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"Quinque Symmetriae Logicae Propositionalis","twitter_description":"Quinque Symmetrias Logicae Propositionalis per demonstrationes claras explora: negatio duplex, syllogismus hypotheticus, et alia.","twitter_image":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/5simetrias.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"1 minuto"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/#article","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/"},"author":{"name":"giorgio.reveco","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"Quinque Symmetriae Logicae Propositionalis","datePublished":"2021-01-28T13:00:28+00:00","dateModified":"2025-07-31T07:35:58+00:00","mainEntityOfPage":{"@id":"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/"},"wordCount":948,"commentCount":0,"publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/5simetrias.jpg","articleSection":["Logica Mathematica","Logica Propositionalis","Mathematica"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/#respond"]}]},{"@type":"WebPage","@id":"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/","url":"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/","name":"Quinque Symmetriae Logicae Propositionalis - toposuranos.com\/material","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/#primaryimage"},"image":{"@id":"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/5simetrias.jpg","datePublished":"2021-01-28T13:00:28+00:00","dateModified":"2025-07-31T07:35:58+00:00","description":"Quinque Symmetrias Logicae Propositionalis per demonstrationes claras explora: negatio duplex, syllogismus hypotheticus, et alia.","breadcrumb":{"@id":"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/#primaryimage","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/5simetrias.jpg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/5simetrias.jpg","width":1024,"height":356,"caption":"Created with GIMP"},{"@type":"BreadcrumbList","@id":"http:\/\/toposuranos.com\/material\/la\/quinque-symmetriae-logicae-propositionalis\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"http:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"Quinque Symmetriae Logicae Propositionalis"}]},{"@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\/33829","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=33829"}],"version-history":[{"count":0,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/33829\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/27410"}],"wp:attachment":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=33829"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=33829"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=33829"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}