{"id":33877,"date":"2021-03-08T13:00:04","date_gmt":"2021-03-08T13:00:04","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33877"},"modified":"2025-08-01T01:49:08","modified_gmt":"2025-08-01T01:49:08","slug":"consequentia-et-aequivalentia-semantica","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/consequentia-et-aequivalentia-semantica\/","title":{"rendered":"Consequentia et Aequivalentia Semantica"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Consequentia et Aequivalentia Semantica<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>SUMMARIUM<\/strong><br \/><em>In hac lectione studebimus Consequentiam et Aequivalentiam Semanticam in logica propositionali, quod est continuatio naturalis eorum quae antea vidimus. Discemus quomodo notio consequentiae semanticae ex attributionibus valorum veritatis obtineatur et quomodo haec idea ad theorema deductionis pertineat. Praeterea, videbimus exempla practica usus tabularum veritatis ad proprietates utiles obtinendas, ut Eliminatio Coniunctionis et Introductio Disiunctionis. Explorabimus etiam notionem Aequivalentiae Semanticae et quomodo ea ad proprietates iam notas referatur. Denique ostendemus quomodo usus exemplorum (modelorum) et technicarum deductionis nos adiuvet in studio quaestionum de consequentia et aequivalentia semantica.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>PROPOSITA DISCENDI:<\/strong><br \/>\nHis peractis, discipulus poterit\n<\/p>\n<ol>\n<li><strong>Intelligere<\/strong> notionem consequentiae semanticae.<\/li>\n<li><strong>Intelligere<\/strong> varias interpretationes symboli \u22a8.<\/li>\n<li><strong>Intelligere<\/strong> demonstrationem theorematum deductionis in versione semantica eiusque usum in studio consequentiae et aequivalentiae semanticae.<\/li>\n<li><strong>Intelligere<\/strong> definitionem aequivalentiae semanticae eiusque relationem ad valores veritatis.<\/li>\n<li><strong>Applicare<\/strong> theorematis deductionis versionem semanticam ad quaestiones de consequentia in quaestiones de validitate transformandas.<\/li>\n<li><strong>Applicare<\/strong> proprietates utiles in usu tabularum veritatis ad aequivalentias semanticas demonstrandas.<\/li>\n<li><strong>Applicare<\/strong> leges absorptionis, distributionis et DeMorgan ad expressionum complexarum simplificationem.<\/li>\n<li><strong>Examinare<\/strong> relationem inter exempla (modela) et deductiones in studio logicae propositionis.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\">ATTRIBUTIONES ET MODELA<\/a><br \/>\n<a href=\"#2\">THEOREMA DEDUCTIONIS (VERSIO SEMANTICA)<\/a><br \/>\n<a href=\"#3\">USUS THEOREMATIS DEDUCTIONIS IN STUDIO CONSEQUENTIAE ET AEQUIVALENTIAE SEMANTICAE<\/a><br \/>\n<a href=\"#4\">AEQUIVALENTIA SEMANTICA ET PROPRIETATES<\/a><br \/>\n<a href=\"#5\">SYNTESIS<\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/vjkzDxbG8LY\" 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;\">Studium Consequentiae et Aequivalentiae Semanticae est continuatio naturalis eorum quae tractavimus cum <a href=\"https:\/\/toposuranos.com\/semantica-de-la-logica-proposicional\/\" rel=\"noopener\" target=\"_blank\">semanticam logicae propositionis<\/a> consideravimus. Nunc considerabimus quomodo ex attributionibus valorum veritatis notio consequentiae semanticae proveniat, et quomodo ex hoc naturaliter oriatur versio semantica <a href=\"https:\/\/toposuranos.com\/tecnicas-deduccion-logica-proposicional\/\" rel=\"noopener\" target=\"_blank\">theorematis deductionis<\/a>. Inde ostendentur exempla practica usus tabularum veritatis ad quasdam proprietates utiles obtinendas. Haec omnia videre etiam potes in <a href=\"https:\/\/www.youtube.com\/watch?v=vjkzDxbG8LY\" rel=\"noopener\" target=\"_blank\">canali YouTube<\/a>.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Attributiones et Modela<\/h2>\n<p style=\"text-align: justify;\">Primum incipiamus cum definitione quae est momenti maximi pro progressibus quos in hoc capite videbimus: definitio consequentiae semanticae.<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify;\"><span style=\"color: #880000;\"><strong>DEFINITIO:<\/strong><\/span> Expressio <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> est <strong>consequentia (semantica)<\/strong> alterius expressionis <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> si pro omni attributione <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}<\/span> valet<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}\\models F \\Rightarrow \\mathcal{A}\\models G<\/span>\n<p>Hoc repraesentatur scribendo <span class=\"katex-eq\" data-katex-display=\"false\">F\\models G<\/span> et legitur \u00abexpressio <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> modulat expressionem <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>\u00bb vel \u00ab<span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> est consequentia (semantica) expressionis <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span>.\u00bb<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Hac definitione data, animadvertendum est symbolum <span class=\"katex-eq\" data-katex-display=\"false\">\\models<\/span> diversas habere interpretationes secundum contextum:<\/p>\n<ul style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A} \\models F<\/span> significat <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}(F) = 1<\/span>; id est, \u00ab<span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}<\/span> modulat <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span>.\u00bb<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">G \\models F<\/span> significat si aliqua attributio <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> modulat, tunc etiam <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> modulat; hoc legitur ut \u00ab<span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> est consequentia <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>.\u00bb<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\models F<\/span> significat <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> omnibus attributionibus verum est; id est, <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> est tautologia.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Itaque, quamvis symbolum <span class=\"katex-eq\" data-katex-display=\"false\">\\models<\/span> varias habeat interpretationes, contextus eas clare definit.<\/p>\n<p style=\"text-align: justify;\">Notio consequentiae (semanticae) proxima est notioni \u00abimplicationis\u00bb quam antea tractavimus, cum <span class=\"katex-eq\" data-katex-display=\"false\">F\\models G<\/span> implicet <span class=\"katex-eq\" data-katex-display=\"false\">\\models (F\\rightarrow G)<\/span>. Re vera, hoc simile est theorema deductionis quod antea in lectionibus tractavimus.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Theorema Deductionis (Versio Semantica)<\/h2>\n<p><strong><span style=\"color: #000000;\">[<a href=\"https:\/\/www.youtube.com\/watch?v=vjkzDxbG8LY&amp;t=444s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">videre<\/span><\/a>]<\/span><\/strong><\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: justify;\"><span style=\"color: #aa0000;\"><strong>THEOREMA:<\/strong><\/span> Si <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> sunt quaelibet expressiones, tunc valet<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> F\\models G \\Leftrightarrow \\models (F\\rightarrow G) <\/span>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify;\"><span style=\"color: #0000aa;\"><strong>Demonstratio:<\/strong><\/span><\/p>\n<p style=\"text-align: justify;\">Demonstratio huius theorematum facile obtinetur observando tabulas veritatis<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #dddddd;\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddddd;\"><span class=\"katex-eq\" data-katex-display=\"false\">G<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddddd;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg F<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddddd;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\rightarrow G):=(\\neg F \\vee G)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Si attendamus ad significationem <span class=\"katex-eq\" data-katex-display=\"false\">F\\models G<\/span>, videbimus hoc aequivalere dicere <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}\\models F \\Rightarrow \\mathcal{A}\\models G<\/span>, quod rursus idem est ac dicere <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}\\not\\models F \\vee \\mathcal{A}\\models G<\/span>. Nunc, si advertamus <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}\\not\\models F<\/span> idem esse ac <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}\\models \\neg F<\/span>, tunc sequitur <span class=\"katex-eq\" data-katex-display=\"false\">F\\models G<\/span> aequivalere dicere <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A} \\models \\neg F \\vee \\mathcal{A}\\models G<\/span>. Si igitur tabulam veritatis pro <span class=\"katex-eq\" data-katex-display=\"false\">F \\rightarrow G<\/span> conficiamus et regione in qua <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A} \\models \\neg F \\vee \\mathcal{A}\\models G<\/span> valet <span style=\"color: #008800;\"><strong>viridi<\/strong><\/span> notemus, tum videbimus hoc:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #dddddd;\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddddd;\"><span class=\"katex-eq\" data-katex-display=\"false\">G<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddddd;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg F<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddddd;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\rightarrow G):=(\\neg F \\vee G)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center; background-color: #008800; color: #ffffff;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center; background-color: #008800; color: #ffffff;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center; background-color: #008800; color: #ffffff;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center; background-color: #008800; color: #ffffff;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center; background-color: #008800; color: #ffffff;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center; background-color: #008800; color: #ffffff;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Hinc habemus quod, cum <span class=\"katex-eq\" data-katex-display=\"false\">F\\models G<\/span>, semper evenit <span class=\"katex-eq\" data-katex-display=\"false\">\\models (F \\rightarrow G)<\/span> et vicissim; quod est nihil aliud quam theorema deductionis et eius reciproca in versione semantica.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Supponamus nos velle scire utrum expressio <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> sit consequentia alicuius alterius expressionis <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span>. Ad hoc referemur ut <strong>quaestio de consequentia.<\/strong> Utendo theorema superiore, haec quaestio transformari potest in <strong>quaestionem de validitate,<\/strong> quia \u00ab<span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> est consequentia <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> si et tantum si <span class=\"katex-eq\" data-katex-display=\"false\">(F\\rightarrow G)<\/span> est theorema.\u00bb<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Usus theorematum deductionis in studio consequentiae et aequivalentiae semanticae<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=vjkzDxbG8LY&amp;t=796s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Ex tabulis veritatis<\/span><\/strong><\/a> inferre possumus nonnullas proprietates quae alias iam visae sunt.<\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td colspan=\"2\" style=\"text-align: justify;\"><strong><span style=\"color: #000088;\">EXEMPLUM<\/span><\/strong>: Ostendere usus tabularum veritatis valere sequentis proprietates<\/td>\n<\/tr>\n<tr>\n<td>Eliminatio Coniunctionis:<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(F\\wedge G)\\models F<\/span><\/td>\n<\/tr>\n<tr>\n<td>Introductio Disiunctionis:<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">F\\models (F\\vee G)<\/span><\/td>\n<\/tr>\n<tr>\n<td>Contradictio:<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(F\\wedge\\neg F)\\models G<\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\"><span style=\"color: #008800;\"><strong>Solutio:<\/strong><\/span><span style=\"color: #000000;\"> Utendo theorema deductionis quod modo recensuimus, possumus quaestionem de consequentia in quaestionem de validitate convertere.<\/span><\/p>\n<p style=\"text-align: justify;\">Ad solvendum casum <strong>Eliminationis Coniunctionis,<\/strong> tabulam veritatis sequentem construere possumus<\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #bbbbbb;\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/td>\n<td style=\"text-align: center; background-color: #bbbbbb;\"><span class=\"katex-eq\" data-katex-display=\"false\">G<\/span><\/td>\n<td style=\"text-align: center; background-color: #bbbbbb;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\wedge G)<\/span><\/td>\n<td style=\"text-align: center; background-color: #bbbbbb;\"><span class=\"katex-eq\" data-katex-display=\"false\">((F\\wedge G) \\rightarrow F)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddd00;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddd00;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddd00;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddd00;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Hoc ostendit <span class=\"katex-eq\" data-katex-display=\"false\">((F\\wedge G)\\rightarrow F)<\/span> esse tautologiam et, propter reciprocum theorema deductionis, concluditur <span class=\"katex-eq\" data-katex-display=\"false\">(F\\wedge G) \\models F<\/span>.<\/p>\n<p style=\"text-align: justify;\"><strong>Introductio Disiunctionis<\/strong> analogice resolvitur per tabulam veritatis aptam construendam<\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #bbbbbb;\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/td>\n<td style=\"text-align: center; background-color: #bbbbbb;\"><span class=\"katex-eq\" data-katex-display=\"false\">G<\/span><\/td>\n<td style=\"text-align: center; background-color: #bbbbbb;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\vee G)<\/span><\/td>\n<td style=\"text-align: center; background-color: #bbbbbb;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\rightarrow(F\\vee G))<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddd00;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddd00;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddd00;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddd00;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Hic observamus <span class=\"katex-eq\" data-katex-display=\"false\">(F\\rightarrow (F\\vee G))<\/span> esse tautologiam et, igitur, per reciprocum theorema deductionis, habetur <span class=\"katex-eq\" data-katex-display=\"false\">F\\models (F\\vee G)<\/span>\n<p style=\"text-align: justify;\">Et denique, proprietas <strong>Contradictionis<\/strong> demonstratur utendo eadem methodo<\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #bbbbbb;\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/td>\n<td style=\"text-align: center; background-color: #bbbbbb;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg F<\/span><\/td>\n<td style=\"text-align: center; background-color: #bbbbbb;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\wedge \\neg F)<\/span><\/td>\n<td style=\"text-align: center; background-color: #bbbbbb;\"><span class=\"katex-eq\" data-katex-display=\"false\">G<\/span><\/td>\n<td style=\"text-align: center; background-color: #bbbbbb;\"><span class=\"katex-eq\" data-katex-display=\"false\">((F\\wedge \\neg F)\\rightarrow G)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddd00;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddd00;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddd00;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<td style=\"text-align: center; background-color: #dddd00;\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Hac tabula veritatis demonstravimus <span class=\"katex-eq\" data-katex-display=\"false\">((F\\wedge \\neg F)\\rightarrow G)<\/span> esse tautologiam et, igitur, per reciprocum theorema deductionis, habetur <span class=\"katex-eq\" data-katex-display=\"false\">(F\\wedge \\neg F)\\models G<\/span>.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Aequivalentia Semantica et Proprietates<\/h2>\n<p><span style=\"color: #000000;\"><strong>[<a href=\"https:\/\/www.youtube.com\/watch?v=vjkzDxbG8LY&amp;t=1058s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">videre<\/span><\/a>]<\/strong><\/span><\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: justify;\"><span style=\"color: #880000;\"><strong>DEFINITIO:<\/strong><\/span> Si simul occurrunt <span class=\"katex-eq\" data-katex-display=\"false\">F\\models G<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">G\\models F<\/span>, tunc dicitur <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> inter se esse <strong>semantice aequivalentes<\/strong>. Hoc repraesentatur scribendo <span class=\"katex-eq\" data-katex-display=\"false\">F\\equiv G<\/span>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\"><strong>Ex hac definitione sequitur duas expressiones semantice aequivalentes esse si et tantum si easdem habent valores veritatis.<\/strong><\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: justify;\"><strong><span style=\"color: #000088;\">EXEMPLUM:<\/span><\/strong> Ostendi potest per tabulas veritatis valere <strong>aequivalentias semanticas symmetriae.<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\downarrow G) \\equiv (G\\downarrow F)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\vee G) \\equiv (G\\vee F)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\wedge G) \\equiv (G\\wedge F)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\leftrightarrow G) \\equiv (G\\leftrightarrow F)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\underline{\\vee} G) \\equiv (G\\underline{\\vee} F)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: justify;\"><strong><span style=\"color: #000088;\">EXEMPLUM:<\/span><\/strong> Si <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> est quaelibet expressio, <span class=\"katex-eq\" data-katex-display=\"false\">\\top<\/span> est tautologia et <span class=\"katex-eq\" data-katex-display=\"false\">\\bot<\/span> est contradictio, tunc per tabulas veritatis demonstrari possunt sequentes aequivalentiae semanticae<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\wedge \\top) \\equiv F<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\vee \\top) \\equiv \\top<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\wedge \\bot) \\equiv \\bot<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\vee \\bot) \\equiv F<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify;\">Hae aequivalentiae cognoscuntur ut <strong>leges absorptionis.<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: justify;\"><strong><span style=\"color: #000088;\">EXEMPLUM:<\/span><\/strong> In semantica logicae propositionis valent etiam aequivalentiae distributionis inter coniunctionem et disiunctionem.<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\wedge (G\\vee H)) \\equiv ((F\\wedge G) \\vee (F\\wedge H))<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\vee (G\\wedge H)) \\equiv ((F\\vee G) \\wedge (F\\vee H))<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: justify;\"><strong><span style=\"color: #000088;\">EXEMPLUM:<\/span><\/strong> In semantica logicae propositionis valent etiam <strong>Leges DeMorgan.<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(F\\wedge G) \\equiv (\\neg F \\vee \\neg G)<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(F\\vee G) \\equiv (\\neg F \\wedge \\neg G)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: justify;\"><strong><span style=\"color: #880000;\">EXERCITATIO:<\/span><\/strong> Exercitium utile est demonstrare per tabulas veritatis valere revera aequivalentias semanticas Legum Absorptionis, Distributivitatis et Legum DeMorgan.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: justify;\"><strong><span style=\"color: #000088;\">EXEMPLUM:<\/span><\/strong> Demonstrandum est, utens aequivalentias semanticas, sequentem aequivalentiam:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">((C\\wedge D) \\vee A) \\wedge (C\\wedge D) \\vee B) \\wedge (E \\vee \\neg E))\\equiv ((A\\wedge B)\\vee(C\\wedge D))<\/span>.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"color: #000000;\"><span style=\"color: #008800;\"><strong>Solutio:<\/strong><\/span> Hanc aequivalentiam demonstrare possumus per tabulas veritatis, sed si hoc facimus, tractare debebimus expressionem cum 5 variabilibus propositionis, quod significat conficiendam esse tabulam veritatis cum <span class=\"katex-eq\" data-katex-display=\"false\">2^5 = 32<\/span> ordinibus; hoc igitur vitandum est. Ad hoc assequendum, utamur aequivalentias quas iam monstravimus.<\/p>\n<p style=\"text-align: justify;\">Primum animadvertamus <span class=\"katex-eq\" data-katex-display=\"false\">(E\\vee \\neg E)<\/span> esse tautologiam. Denotemus hanc tautologiam per <span class=\"katex-eq\" data-katex-display=\"false\">\\top<\/span>. Tunc, utentes legibus absorptionis, habebimus<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">((C\\wedge D) \\vee A) \\wedge (C\\wedge D) \\vee B) \\wedge (E \\vee \\neg E)) \\equiv ((C\\wedge D) \\vee A) \\wedge (C\\wedge D) \\vee B)) <\/span>\n<p style=\"text-align: justify;\">Utendo legibus distributionis obtinemus<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> ((C\\wedge D) \\vee A) \\wedge (C\\wedge D) \\vee B)) \\equiv ((C\\wedge D) \\vee (A\\wedge B))<\/span>\n<p style=\"text-align: justify;\">Denique, per symmetriam<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> ((C\\wedge D) \\vee (A\\wedge B)) \\equiv ((A\\wedge B) \\vee (C\\wedge D))<\/span>\n<p style=\"text-align: justify;\">Itaque, sequendo has aequivalentias, habetur aequivalentia<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">((C\\wedge D) \\vee A) \\wedge (C\\wedge D) \\vee B) \\wedge (E \\vee \\neg E)) \\equiv ((A\\wedge B) \\vee (C\\wedge D))<\/span>\n<p style=\"text-align: justify;\">quod erat demonstrandum.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Synthesis<\/h2>\n<p style=\"text-align: justify;\">Si spectemus expositionem huius ultimi exempli, videbimus quod, numero variabilium aucto, difficultas tractandi quaestiones de consequentia et aequivalentia semantica exponens crescit si ex solis tabulis veritatis pendemus. Tamen vidimus ex expositione notiones modelorum emergere aliquid simile technicis deductionis, quas iam satis diligenter tractavimus. Haec relatio inter modela et deductiones est quod mox videre poterimus, et coniunctio utriusque est quae tandem innumerabiles dolores capitis in studio logicae nobis evitabit.<a href=\"https:\/\/amzn.to\/3t6XASK\" target=\"_blank\" rel=\"noopener\"><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consequentia et Aequivalentia Semantica SUMMARIUMIn hac lectione studebimus Consequentiam et Aequivalentiam Semanticam in logica propositionali, quod est continuatio naturalis eorum quae antea vidimus. Discemus quomodo notio consequentiae semanticae ex attributionibus valorum veritatis obtineatur et quomodo haec idea ad theorema deductionis pertineat. Praeterea, videbimus exempla practica usus tabularum veritatis ad proprietates utiles obtinendas, ut Eliminatio Coniunctionis [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28055,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":13,"footnotes":""},"categories":[1352,1358,1298],"tags":[],"class_list":["post-33877","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 v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Consequentia et Aequivalentia Semantica - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce de Consequentia et Aequivalentia Semantica in logica propositionali per exempla practica. 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