{"id":33913,"date":"2021-04-17T13:00:24","date_gmt":"2021-04-17T13:00:24","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33913"},"modified":"2025-08-01T02:23:54","modified_gmt":"2025-08-01T02:23:54","slug":"inductio-de-complexitate-formularum","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/inductio-de-complexitate-formularum\/","title":{"rendered":"Inductio de Complexitate Formularum"},"content":{"rendered":"<p><center><\/p>\n<h1>Inductio de Complexitate Formularum<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>SUMMARIUM<\/strong><br \/><em>In hac lectione disces de variatione inductionis mathematicae, quae \u00abinductio de complexitate expressionum\u00bb appellatur, quae admodum utilis est ad proprietates in logica propositionali demonstrandas. Per exemplum simplex, theorema substitutionis, videbis quomodo haec technica applicetur et quomodo demonstrari possit proprietatem valere pro omnibus expressionibus logicae propositionalis. Praeterea explicabitur quomodo hypothesis inductionis et passus inductivus operentur, ut hanc technicam in tuis demonstrationibus adhibere possis.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>PROPOSITUM DISCENDI:<\/strong><br \/>\nPost hanc lectionem, discipulus poterit:\n<\/p>\n<ol>\n<li><strong>Intelligere<\/strong> notionem inductionis de complexitate expressionum.<\/li>\n<li><strong>Applicare<\/strong> inductionem mathematicam de complexitate formularum in logica propositionali.<\/li>\n<li><strong>Intelligere<\/strong> demonstrationem per inductionem de complexitate et eius applicationem in logica propositionali.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\">INDUCTIO DE COMPLEXITATE<\/a><br \/>\n<a href=\"#2\">EXEMPLUM SIMPLEX: THEOREMA SUBSTITUTIONIS<\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/wipHo3D2lQY\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Inductio de complexitate<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=wipHo3D2lQY&amp;t=407s\" target=\"_blank\" rel=\"noopener\"><strong>Supponamus nos velle demonstrare proprietatem<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{P}<\/span> valere pro qualibet expressione <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span>. Hoc demonstrari potest per variationem <strong>inductionis mathematicae<\/strong>, quae \u00abinductio de complexitate expressionum\u00bb vocatur. Haec per hos gradus exsequitur:<\/p>\n<ul style=\"text-align: justify;\">\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=wipHo3D2lQY&amp;t=432s\" target=\"_blank\" rel=\"noopener\"><strong>Primum ostendimus<\/strong><\/a> omnes expressiones atomicas hanc proprietatem implere (hoc respondet casui n=1 in inductione classica).<\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=wipHo3D2lQY&amp;t=461s\" target=\"_blank\" rel=\"noopener\"><strong>Deinde, assumendo<\/strong><\/a> eam valere pro quibuslibet expressionibus <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>, demonstramus ex hoc sequi eam quoque valere pro expressionibus formae <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\downarrow G<\/span><\/span>; seu equivalenter, pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg F<\/span><\/span> et aliqua ex sequentibus: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\wedge G,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\vee G,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\rightarrow G.<\/span><\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Si hoc perficere possumus, concludimus proprietatem <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{P}<\/span> valere pro omnibus expressionibus logicae propositionalis. Hoc est quod dicimus \u00abinductionem mathematicam de complexitate expressionum\u00bb.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Exemplum Simplex: Theorema Substitutionis<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=wipHo3D2lQY&amp;t=681s\" target=\"_blank\" rel=\"noopener\"><strong>Ut clarius intellegamus quomodo fiat inductio<\/strong><\/a> de complexitate formularum, consideremus (meta)theorema substitutionis.<\/p>\n<p style=\"text-align: justify;\">Supponamus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\equiv G.<\/span><\/span> Sit <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> expressio quae <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> tamquam subexpressionem continet, et sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime<\/span><\/span> expressio quae fit substituendo omnes occurrentias <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> per <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>; tum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv H^\\prime.<\/span><\/span><\/p>\n<h3>Demonstratio per Inductionem de Complexitate Formularum<\/h3>\n<p style=\"text-align: justify;\">Demonstratio per inductionem de complexitate constat ostensione duarum rerum: 1) casus initialis (pro formulis atomicis) et 2) passus inductivus (si valet pro quibuslibet expressionibus <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>, tum valet etiam pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\downarrow G<\/span><\/span>, vel simplicius: valet pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg F<\/span><\/span> et saltem uno ex sequentibus: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\vee G,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\wedge G,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\rightarrow G,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\leftrightarrow G.<\/span><\/span>)<\/p>\n<p style=\"text-align: justify;\">Supponamus <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> esse expressionem atomicam, <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> subexpressionem <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span>, et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\equiv G.<\/span><\/span> Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime<\/span><\/span> est expressio quae fit substituendo omnes subexpressiones <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> in <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span>, cum <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> sit atomica, erit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime \\equiv G<\/span><\/span>. Praeterea, cum <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> sit atomica et <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> eius subexpressio, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv F<\/span><\/span>. Denique sequetur:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv F \\equiv G \\equiv H^\\prime<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ita demonstratur casus initialis pro expressionibus atomicis.<\/p>\n<p style=\"text-align: justify;\">Videamus nunc casum inductivum.<\/p>\n<h4>Hypothesis Inductionis<\/h4>\n<p style=\"text-align: justify;\">Supponamus theorema valere pro duabus quibuslibet expressionibus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_2<\/span><\/span>, quarum utraque <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> tamquam subexpressionem continet et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F \\equiv G.<\/span><\/span> Tum si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1^\\prime<\/span><\/span> est expressio ex <span class=\"katex-eq\" data-katex-display=\"false\">H_1<\/span> orta per substitutionem omnium <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> per <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>, et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_2^\\prime<\/span><\/span> similter ex <span class=\"katex-eq\" data-katex-display=\"false\">H_2<\/span>, sequitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1\\equiv H_1^\\prime<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_2\\equiv H_2^\\prime<\/span><\/span>.<\/p>\n<h4>Passus Inductivus<\/h4>\n<p style=\"text-align: justify;\">Hic examinamus an, ex hypothesi inductionis, theorema quoque valeat pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg H_1<\/span><\/span> (vel <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg H_2,<\/span><\/span> alterutro) et saltem uno ex sequentibus: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1 \\wedge H_2,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1 \\vee H_2,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1 \\rightarrow H_2.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H:= \\neg H_1<\/span><\/span>, tum ex hypothesi inductionis sequetur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv \\neg H_1^\\prime=: H^\\prime <\/span><\/span>, ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime<\/span><\/span> est expressio quae fit ex <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> substituendo omnes occurrentias <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> per <span class=\"katex-eq\" data-katex-display=\"false\">G.<\/span> Ergo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv H^\\prime.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Similiter, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H:= H_1 \\wedge H_2<\/span><\/span>, tum ex hypothesi inductionis habebitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv H_1^\\prime \\wedge H_2 \\equiv H_1^\\prime \\wedge H_2^\\prime =: H^\\prime <\/span><\/span>, ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime<\/span><\/span> est expressio quae fit ex <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> substituendo omnes occurrentias <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> per <span class=\"katex-eq\" data-katex-display=\"false\">G.<\/span> Ergo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv H^\\prime.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ergo inductio completa est et theorema substitutionis valet pro omnibus expressionibus logicae propositionalis.<\/p>\n<p style=\"text-align: justify;\">\nApplicando hanc formam inductionis, potest demonstrari proprietatem manere pro omnibus expressionibus systematis logici, quod est praecipue utile in logica propositionali ad structuras demonstrationum rigorosas instituendas. Praeterea eius utilitas ad disciplinas extenditur sicut intelligentia artificialis et progressus programmatum computatoriorum, ubi verificatio systematum logicorum est fundamentalis. Per hanc technicam, demonstrationes automatice perfici possunt et consistentia expressionum complexarum corroborari, quod periculum errorum minuit et praecisionem auget in mediis quae ex firmitate mathematica et logica pendent.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Inductio de Complexitate Formularum SUMMARIUMIn hac lectione disces de variatione inductionis mathematicae, quae \u00abinductio de complexitate expressionum\u00bb appellatur, quae admodum utilis est ad proprietates in logica propositionali demonstrandas. Per exemplum simplex, theorema substitutionis, videbis quomodo haec technica applicetur et quomodo demonstrari possit proprietatem valere pro omnibus expressionibus logicae propositionalis. Praeterea explicabitur quomodo hypothesis inductionis et [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28280,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":3,"footnotes":""},"categories":[1352,1358,1298],"tags":[],"class_list":["post-33913","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>Inductio de Complexitate Formularum - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Inductio de complexitate est technica mathematica adhibita ad proprietates generales formularum et expressionum logicarum demonstrandas.\" \/>\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\/inductio-de-complexitate-formularum\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Inductio de Complexitate Formularum\" \/>\n<meta property=\"og:description\" content=\"Inductio de complexitate est technica mathematica adhibita ad proprietates generales formularum et expressionum logicarum demonstrandas.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/inductio-de-complexitate-formularum\/\" \/>\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=\"2021-04-17T13:00:24+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-08-01T02:23:54+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Inductio de Complexitate Formularum\" \/>\n<meta name=\"twitter:description\" content=\"Inductio de complexitate est technica mathematica adhibita ad proprietates generales formularum et expressionum logicarum demonstrandas.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/complejidad.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\\\/inductio-de-complexitate-formularum\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/inductio-de-complexitate-formularum\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Inductio de Complexitate Formularum\",\"datePublished\":\"2021-04-17T13:00:24+00:00\",\"dateModified\":\"2025-08-01T02:23:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/inductio-de-complexitate-formularum\\\/\"},\"wordCount\":790,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/inductio-de-complexitate-formularum\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/09\\\/complejidad.jpg\",\"articleSection\":[\"Logica Mathematica\",\"Logica Propositionalis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/inductio-de-complexitate-formularum\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/inductio-de-complexitate-formularum\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/inductio-de-complexitate-formularum\\\/\",\"name\":\"Inductio de Complexitate Formularum - 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