{"id":33889,"date":"2021-04-08T13:00:53","date_gmt":"2021-04-08T13:00:53","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33889"},"modified":"2025-08-01T02:03:08","modified_gmt":"2025-08-01T02:03:08","slug":"completitudo-et-solventia-in-logica-propositionali","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/completitudo-et-solventia-in-logica-propositionali\/","title":{"rendered":"Completitudo et Solventia in Logica Propositionali"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Completitudo et Solventia in Logica Propositionali<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>SUMMARIUM<\/strong><br \/><em>Hac lectione tractatur relatio inter completitudinem et solventiam in logica propositionali. Quamvis technicae deductionis et semanticae in logica propositionali late disceptatae sint, parum attentionis data est relationi inter utramque faciem. Solventia significat proprietatem systematis logici ex complexu expressionum \u0393 expressionem G inferendi. Ex altera parte, completitudo significat proprietatem systematis logici, in quo, si G est consequentia semantica complexus expressionum \u0393, tunc exsistit probatio formalis cum praemissis \u0393 ex quibus G inferri potest. Demonstratur logicam propositionalem et solventem et completam esse, atque praebetur explanatio accurata utriusque proprietatis. In specie ostenditur quomodo solventia ex constitutione systematis deductivi logicae propositionis derivetur, et quomodo completitudo simpliciter inferatur. Haec analysis magni momenti est ad intellegendum quomodo logica propositionalis operetur atque ut ea efficaciter in variis scientiae campis adhibeatur.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>PROPOSITA DISCENDI:<\/strong><br \/>\nPost hanc lectionem, discipulus poterit:\n<\/p>\n<ol>\n<li><strong>Distinguere<\/strong> inter solventiam et completitudinem in systemate logico.<\/li>\n<li><strong>Applicare<\/strong> tabulam veritatis ad axiomata \u0141ukasiewicz ad demonstrandam solventiam logicae propositionis.<\/li>\n<li><strong>Explicare<\/strong> modum quo modus ponens rescribi possit per versionem semanticam theorematos deductionis.<\/li>\n<li><strong>Intellegere<\/strong> solventiam et completitudinem inter se relatas esse atque unam ex altera inferri posse.<\/li>\n<li><strong>Analyzare<\/strong> notionem tautologiae eiusque relationem ad theoremata in logica propositionali.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\">COMPLETITUDO ET SOLVENTIA IN LOGICA PROPOSITIONALI<\/a><br \/>\n<a href=\"#2\">LOGICA PROPOSITIONALIS SOLVENS EST<\/a><br \/>\n<a href=\"#3\">LOGICA PROPOSITIONALIS COMPLETA EST<\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/dGrwsPMHa90\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Completitudo et Solventia in Logica Propositionali<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=dGrwsPMHa90&amp;t=181s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Ad hoc punctum pervenientes, loquendum est de completitudine et solventia logicae propositionis.<\/span><\/strong><\/a> Hactenus multum dictum est de technicis deductionis atque de semantica logicae propositionis, sed omnia ita tractata sunt quasi hae duae facies prorsus independentes essent et nullam inter se haberent relationem. Veritas autem est prorsus contraria.<\/p>\n<table>\n<tbody>\n<tr>\n<td>\n<p style=\"text-align: justify;\"><strong>SOLVENTIA:<\/strong> Ex una parte dicitur systema logicum esse solventem cum, quoties expressio G ex complexu expressionum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/span> inferri possit, tunc G sit etiam consequentia (semantica) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/span>.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: justify;\"><strong>COMPLETITUDO:<\/strong> Ex altera parte dicetur systema esse completum, cum, si G est consequentia semantica alicuius complexus expressionum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/span>, exsistat probatio formalis cum praemissis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/span>, ex quibus G inferri potest.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-VKgbesdSWbQ\/YG6s7C9kkdI\/AAAAAAAAE2g\/K5hUDwI6KkAs5FNmmkTZ1RRV_LqV_kxlQCLcBGAsYHQ\/s0\/completitud%2By%2Bsolvencia.PNG\" alt=\"Completitudo et Solventia Logicae Propositionalis\" class=\" aligncenter lazyload\" width=\"285\" height=\"220\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-VKgbesdSWbQ\/YG6s7C9kkdI\/AAAAAAAAE2g\/K5hUDwI6KkAs5FNmmkTZ1RRV_LqV_kxlQCLcBGAsYHQ\/s0\/completitud%2By%2Bsolvencia.PNG\" alt=\"Completitudo et Solventia Logicae Propositionalis\" class=\" aligncenter lazyload\" width=\"285\" height=\"220\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">His notionibus perspectis videbimus completitudinem et solventiam in logica propositionali satisfieri.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Logica Propositionalis Solvens Est<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=dGrwsPMHa90&amp;t=292s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Facile est solventiam logicae propositionis obtinere<\/span><\/strong><\/a> contemplando constitutionem eius systematis deductivi. Si tabulam veritatis axiomatum \u0141ukasiewicz conficiamus, videbimus hos talis structurae esse ut semper verum pro valore veritatis reddant, id est:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\models (\\alpha \\rightarrow (\\beta \\rightarrow \\alpha))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\models ((\\alpha \\rightarrow (\\beta \\rightarrow \\gamma))\\rightarrow ((\\alpha \\rightarrow \\beta) \\rightarrow (\\alpha \\rightarrow \\gamma)))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\models ((\\neg\\beta \\rightarrow \\neg\\alpha)\\rightarrow(\\alpha \\rightarrow \\beta))<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Similiter, modus ponens scribi potest hoc modo: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha,(\\alpha\\rightarrow \\beta)\\}\\models \\beta<\/span><\/span>, quod per versionem semanticam theorematos deductionis obtineri potest. Re quidem vera, per hunc modum habebimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha\\rightarrow \\beta)\\}\\models (\\alpha\\rightarrow \\beta),<\/span><\/span> et deinde <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\models ((\\alpha\\rightarrow \\beta)\\rightarrow (\\alpha\\rightarrow \\beta)),<\/span><\/span> quae profecto tautologia est manifestissima.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Logica Propositionalis Completa Est<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=dGrwsPMHa90&amp;t=487s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Completitudo logicae propositionis significat<\/span><\/strong><\/a> quod, si B est consequentia semantica A, tum ex A B inferri potest. Aliter dicendum: omnes expressiones verae habent demonstrationem. Hoc est quod vocamus completitudinem. Id simpliciter concludi potest.<\/p>\n<p style=\"text-align: justify;\">Id simpliciter concludi potest. Fingamus ex A B non posse inferri, sive <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(A\\vdash B)<\/span><\/span>; per theorema deductionis hoc est idem ac dicere: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg (\\vdash A\\rightarrow B)<\/span><\/span>; nunc, si solventiam adhibeamus, hoc ducit ad <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\models A \\rightarrow B)<\/span><\/span>, quod, per reciprocum theorema deductionis (versio semantica), est idem ac <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(A\\models B)<\/span><\/span>. Summatim igitur habemus:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(A\\vdash B) \\Rightarrow \\neg(A\\models B) <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Quod idem est ac dicere:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(A\\models B) \\Rightarrow (A\\vdash B) <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Hoc significat: si A B modelat, tum ex A B inferri potest. Et si theoremata deductionis respective adhibeantur, consequenter habebimus:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\models A\\rightarrow B) \\Rightarrow (\\vdash A \\rightarrow B) <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Id est: si expressio est tautologia, tum est theorema; et, ut vidimus, theoremata sunt effectus demonstrationis.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Completitudo et Solventia in Logica Propositionali SUMMARIUMHac lectione tractatur relatio inter completitudinem et solventiam in logica propositionali. Quamvis technicae deductionis et semanticae in logica propositionali late disceptatae sint, parum attentionis data est relationi inter utramque faciem. Solventia significat proprietatem systematis logici ex complexu expressionum \u0393 expressionem G inferendi. Ex altera parte, completitudo significat proprietatem systematis [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28119,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":5,"footnotes":""},"categories":[1352,1358,1298],"tags":[],"class_list":["post-33889","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>Completitudo et Solventia in Logica Propositionali - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Denique intellege relationem inter completitudinem et solventiam in logica propositionali, quomodo hae proprietates fundamentales firmitatem atque cohaerentiam in systematibus logicis praestent.\" \/>\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\/completitudo-et-solventia-in-logica-propositionali\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Completitudo et Solventia in Logica Propositionali\" \/>\n<meta property=\"og:description\" content=\"Denique intellege relationem inter completitudinem et solventiam in logica propositionali, quomodo hae proprietates fundamentales firmitatem atque cohaerentiam in systematibus logicis praestent.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/completitudo-et-solventia-in-logica-propositionali\/\" \/>\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=\"2021-04-08T13:00:53+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-08-01T02:03:08+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/semantic-1024x576.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Completitudo et Solventia in Logica Propositionali\" \/>\n<meta name=\"twitter:description\" content=\"Denique intellege relationem inter completitudinem et solventiam in logica propositionali, quomodo hae proprietates fundamentales firmitatem atque cohaerentiam in systematibus logicis praestent.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/semantic.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\/completitudo-et-solventia-in-logica-propositionali\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/completitudo-et-solventia-in-logica-propositionali\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Completitudo et Solventia in Logica Propositionali\",\"datePublished\":\"2021-04-08T13:00:53+00:00\",\"dateModified\":\"2025-08-01T02:03:08+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/completitudo-et-solventia-in-logica-propositionali\/\"},\"wordCount\":639,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/completitudo-et-solventia-in-logica-propositionali\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/semantic.jpg\",\"articleSection\":[\"Logica Mathematica\",\"Logica Propositionalis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/la\/completitudo-et-solventia-in-logica-propositionali\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/la\/completitudo-et-solventia-in-logica-propositionali\/\",\"url\":\"http:\/\/toposuranos.com\/material\/la\/completitudo-et-solventia-in-logica-propositionali\/\",\"name\":\"Completitudo et Solventia in Logica Propositionali - 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