{"id":33805,"date":"2021-01-25T00:00:53","date_gmt":"2021-01-25T00:00:53","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33805"},"modified":"2025-07-31T01:25:49","modified_gmt":"2025-07-31T01:25:49","slug":"systemata-deductiva-formalia-definitiones-et-exempla","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/systemata-deductiva-formalia-definitiones-et-exempla\/","title":{"rendered":"Systemata Deductiva Formalia: Definitiones et Exempla"},"content":{"rendered":"<p><center><\/p>\n<h1 style=\"text-align:center;\">Systemata Deductiva Formalia in Logica Propositionali<\/h1>\n<p style=\"text-align:center;\"><em><strong>Summarium:<\/strong><\/br>In hac lectione recensentur systemata deductiva formalia. Explicatur quomodo haec systemata adhibentur ad enodandas relationes quae inter diversas expressiones logicas exsistere possunt, atque elementa fundamentalia quibus hae demonstrationes construuntur: lingua, axiomatibus, regulisque inferendi. Memorantur axiomata \u0141ukasiewicz et explicatur modus ponens ut motor deductionis calculi propositionis. Praeterea tractantur ratiocinia, theorema et praemissae, atque exponitur quomodo deductiones intra systemata deductiva perficiantur.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Proposita Discendi:<\/strong><\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Intelligere<\/strong> notionem systematum deductivorum formalium in logica propositionali.<\/li>\n<li><strong>Agoscere<\/strong> partes elementares systematum deductivorum formalium.<\/li>\n<li><strong>Noscere<\/strong> axiomata \u0141ukasiewicz in calculo propositionis.<\/li>\n<li><strong>Intellegere<\/strong> modum ponentem ut machinam deductionis calculi propositionis.<\/li>\n<li><strong>Intelligere<\/strong> quomodo deductiones in systematibus deductivis fiant atque differentiam inter praemissas, ratiocinia et theoremata.<\/li>\n<li><strong>Intelligere<\/strong> quomodo deductiones generentur per schemata axiomatum et regulas inferendi.<\/li>\n<li><strong>Cognoscere<\/strong> potentiam logicae ad conectendas expressiones easque pro verbis linguae communis substituendas.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>INDEX CONTENTORUM<\/u>:<\/strong><br \/>\n<a href=\"#1\">QUID EST SYSTEMA DEDUCTIVUM FORMALE?<\/a><br \/>\n<a href=\"#2\">AXIOMATA \u0141UKASIEWICZ LOGICAE PROPOSITIONIS<\/a><br \/>\n<a href=\"#3\">MODUS PONENS: MACHINA DEDUCTIONIS CALCULI PROPOSITIONIS<\/a><br \/>\n<a href=\"#4\">RATIONES, THEOREMATA ET PRAEMISSAE<\/a><br \/>\n<a href=\"#5\">QUOMODO DEMONSTRATIO IN LOGICA PROPOSITIONALI PERFICITUR?<\/a><br \/>\n<a href=\"#6\">NOTIO AEQUIVALENTIAE PROBATAE<\/a><br \/>\n<a href=\"#7\">(META)THEOREMA DEDUCTIONIS<\/a><br \/>\n<a href=\"#8\">CONVERSUM THEOREMATIS DEDUCTIONIS<\/a><br \/>\n<a href=\"#9\">DEDUCTIONES DE EXPRESSIONIBUS ET DEDUCTIONES DE DEDUCTIONIBUS<\/a><br \/>\n<a href=\"#10\">REGULA MONOTONIAE<\/a><br \/>\n<a href=\"#11\">SYNTESIS ET REFLECTIONES DE SYSTEMATIBUS DEDUCTIVIS ET LOGICA PROPOSITIONALI<\/a>\n<\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/OvoEDefcSZg\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/center><\/p>\n<p style=\"text-align: justify;\">Pervenimus, in nostro studio logicae, ad punctum conversionis, nam hic initium sumimus recensendi Systemata Deductiva Logicae Propositionalis. Hic est locus ubi omnia quae antea vidimus effici incipiunt et lucem accipit verus spiritus logicae, quia studebimus essentiam demonstrationum. Hoc loco praesumitur te iam didicisse quomodo expressiones scribantur ac quid sit logica propositionalis intellegas; quodsi tibi nondum plene liquet, commendatur ut priores huius lectionis classes recognoscas.<\/p>\n<p style=\"text-align: justify;\">His expletis, sequitur nunc ut perscrutemur modum quo expressiones logicae propositionis inter se connectuntur ad deductionem formandam. Mechanismus per quem istae relationes construuntur est <strong>systema deductivum formale.<\/strong><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Quid est Systema Deductivum Formale?<\/h2>\n<p style=\"text-align: justify;\">Systemata deductiva formalia, sive systemata calculi deductivi, tria elementa fundamentalia continent:<\/p>\n<ol style=\"color: #000000; text-align: justify;\">\n<li><strong>Lingua Formali.<\/strong><\/li>\n<li><strong>Schema Axiomaticum.<\/strong><\/li>\n<li><strong>Regulae Inferendi Elementariae.<\/strong><\/li>\n<\/ol>\n<p style=\"text-align: justify;\">Iam omnia quae ad linguas formales spectant recognovimus. Nunc nobis introducenda sunt schemata axiomatum et regulae inferendi elementariae.<\/p>\n<p style=\"text-align: justify;\">Ad construendum systema deductivum calculi propositionis incipiemus componendo systema deductivum ex <strong>Axiomatibus <a href=\"https:\/\/es.wikipedia.org\/wiki\/Jan_%C5%81ukasiewicz\" rel=\"noopener\" target=\"_blank\">\u0141ukasiewicz<\/a><\/strong>, et ut regula inferendi elementaria adhibebitur <strong>Modus Ponens.<\/strong><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Axiomata \u0141ukasiewicz ad Logicam Propositionalem<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=206s\" target=\"_blank\" rel=\"noopener\"><strong>Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta<\/span><\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> sunt expressiones calculi propositionis,<\/strong><\/a> tum sequentia sunt axiomata calculi propositionis:<\/p>\n<table>\n<tbody>\n<tr>\n<td>[A1]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow (\\beta \\rightarrow \\alpha))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>[A2]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((\\alpha \\rightarrow (\\beta \\rightarrow \\gamma))\\rightarrow ((\\alpha\\rightarrow \\beta)\\rightarrow(\\alpha \\rightarrow \\gamma)))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>[A3]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((\\neg\\beta \\rightarrow \\neg\\alpha)\\rightarrow(\\alpha\\rightarrow \\beta))<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Modus Ponens: Machina Deductionis Calculi Propositionis<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=392s\" target=\"_blank\" rel=\"noopener\"><strong>Si <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> sunt expressiones validae calculi propositionis, <\/strong><\/a>tunc modus ponens statuit ex <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span> deduci <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span>. In forma ratiocinii hoc scribitur hoc modo:<\/p>\n<table style=\"text-align: justify;\">\n<caption>Structura Modi Ponentis<\/caption>\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/td>\n<td>; Praemissa<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span><\/td>\n<td>; Praemissa<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/td>\n<td>; MP(1,2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Hic abbreviate repraesentatur Modus Ponens inter gradus (1) et (2) per scripturam \u00abMP(1,2)\u00bb, et huius totius compendium exprimitur per notationem:<\/p>\n<p style=\"text-align: center;\">Ergo<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha, (\\alpha \\rightarrow \\beta)\\}\\vdash \\beta <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Mox videbimus ex axiomatis \u0141ukasiewicz et Modo Ponente omnes technicas deductionis calculi propositionis extrahi posse, quae regulas fundamentales ratiocinii communis compendiunt atque fundamentum praebent <strong>logicae classicae.<\/strong><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Ratiocinia, Theoremata et Praemissae<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=506s\" target=\"_blank\" rel=\"noopener\"><strong>In systematibus deductivis logicae propositionalis ratiocinia<\/strong><\/a> (sive deductiones) perficiuntur, quae sunt quaelibet series expressionum in qua unaquaeque est vel praemissa vel expressio ex praemissis orta utens solum axiomatibus \u0141ukasiewicz et modo ponente. Theorema est exitus deductionis sine ullis praemissis. Praemissa esse potest quaelibet expressio quae neque est axioma neque ex axiomatibus deducitur. Generaliter, cum habeamus collectionem praemissarum <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> et expressionem <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> quae ex aliquo elemento <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span>, axiomatibus et modo ponente obtinetur, scribitur \u00ab<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma \\vdash \\alpha<\/span><\/span>\u00bb et dicimus:<\/p>\n<p style=\"text-align: center;\"><em>ex <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> deducitur <\/em><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span>\n<p style=\"text-align: justify;\">Si <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> est copia vacua, tum loco scripturae \u00ab<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset\\vdash \\alpha<\/span><\/span>\u00bb scribitur \u00ab<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash \\alpha <\/span><\/span>\u00ab. Hoc legitur \u00ab<span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> est theorema\u00bb. Haec forma repraesentationis theorematum extendi potest ad repraesentationem axiomatum, ita ut si <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> sint expressiones, tunc axiomata \u0141ukasiewicz scribi possint hoc modo:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>[A1]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow (\\beta \\rightarrow \\alpha))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>[A2]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash((\\alpha \\rightarrow (\\beta \\rightarrow \\gamma))\\rightarrow ((\\alpha\\rightarrow \\beta)\\rightarrow(\\alpha \\rightarrow \\gamma)))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>[A3]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash((\\neg\\beta \\rightarrow \\neg\\alpha)\\rightarrow(\\alpha\\rightarrow \\alpha))<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Ex hoc dicitur axiomata esse enuntiata per se manifesta, vel theoremata esse expressiones quae ex vacuo inferuntur, vel axiomata et theoremata esse proprietates calculi propositionis.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Quomodo Demonstratio in Logica Propositionali Perficitur?<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=783s\" target=\"_blank\" rel=\"noopener\"><strong>Nunc a tractatione theoretica ad praxim procedemus.<\/strong><\/a> Et quidem de exsecutione demonstrationis multa dici possunt; sed quamvis multa praeclara dicantur de systematibus deductivis et logica propositionali, et omnia intelligantur, hoc non necessario significabit competentiis ad demonstrationem perficiendam iam excultis esse. Quam ob rem, ut modum demonstrationum faciendarum doceamus, unam demonstrationem theorematis simplicis examinabimus.<\/p>\n<p style=\"text-align: justify; color: #880000;\"><strong>Theorema<\/strong><\/p>\n<p style=\"text-align: justify;\">Si <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> est expressio logicae propositionis, tunc valet quod<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha\\rightarrow \\alpha)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000088;\"><strong>Demonstratio<\/strong><\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> (\\alpha\\rightarrow ( \\alpha \\rightarrow \\alpha)) <\/span><\/span><\/td>\n<td>; A1<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> (\\alpha\\rightarrow ((\\alpha\\rightarrow \\alpha)\\rightarrow\\alpha)) <\/span><\/span><\/td>\n<td>; A1<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ( (\\alpha\\rightarrow((\\alpha\\rightarrow\\alpha)\\rightarrow\\alpha)) \\rightarrow ((\\alpha\\rightarrow (\\alpha\\rightarrow\\alpha))\\rightarrow( \\alpha\\rightarrow \\alpha))) <\/span><\/span><\/td>\n<td>; A2<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ((\\alpha\\rightarrow (\\alpha\\rightarrow\\alpha))\\rightarrow( \\alpha\\rightarrow \\alpha)) <\/span><\/span><\/td>\n<td>; MP(2,3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ( \\alpha\\rightarrow \\alpha) <\/span><\/span><\/td>\n<td>; MP(1,5)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Ergo<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\alpha\\rightarrow\\alpha)<\/span><\/span><\/p>\n<p>Finis demonstrationis.<\/p>\n<p style=\"text-align: justify;\">Ut videre licet, in systematibus deductivis et logica propositionali demonstrationes neutiquam triviales sunt, sed semel constructae facile replicari possunt.<\/p>\n<p style=\"text-align: justify;\">Nunc autem, antequam nos totos in deductiones his technicis fiendas proiciamus, prius aliquas proprietates et definitiones evolvemus quae ad hanc rem perficiendam valde utiles erunt; nam si solis hisce rationibus utamur, gravissimos errores offendamus.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Notio aequivalentiae probatae<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1191s\" target=\"_blank\" rel=\"noopener\"><strong>Si <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> sunt expressiones quaelibet et simul verum est <\/strong><\/a>quod <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash \\beta<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\beta\\} \\vdash \\alpha<\/span><\/span>, tunc dicitur <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> aequivalere probatae, et scribitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha \\dashv \\vdash \\beta<\/span><\/span>. Hoc symbolice compendiose exprimitur sic:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\{\\alpha\\}\\vdash\\beta \\wedge \\{\\beta\\}\\vdash\\alpha \\right) \\Leftrightarrow \\left(\\alpha\\dashv\\vdash\\beta\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Hoc legitur: ex <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> infertur <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span>, et ex <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> infertur <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> si et tantum si <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> sunt aequivalentes probatae.<\/p>\n<p style=\"text-align: justify;\">Haec est meta-proprietas logicae propositionalis.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h2>(Meta)Theorema Deductionis<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1355s\" target=\"_blank\" rel=\"noopener\"><strong>Si <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> sunt expressiones calculi propositionis,<\/strong><\/a> et <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> est collectio praemissarum; tunc habetur: si ex <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma \\cup \\{\\alpha\\}<\/span><\/span> deducitur <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span>, tum ex <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> deducitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span>. Symbolice hoc exprimitur sic:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\Gamma \\cup \\{\\alpha\\}\\vdash \\beta \\right) \\Rightarrow \\left( \\Gamma\\vdash(\\alpha\\rightarrow\\beta)\\right)<\/span>\n<p style=\"text-align: justify; color: #880000;\"><strong>Demonstratio:<\/strong><\/p>\n<p style=\"text-align: justify;\">Ut verificetur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma \\cup \\{\\alpha\\}\\vdash \\beta<\/span><\/span>, necesse est habere deductionem huiusmodi:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_1<\/span><\/span><\/td>\n<td>; Prima praemissa e <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(n)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_n<\/span><\/span><\/td>\n<td>; Praemissa n ex <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td>(n+1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\gamma}_1<\/span><\/span><\/td>\n<td>; Modus Ponens inter aliquas lineas superiores<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(n+m)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\gamma}_m<\/span><\/span><\/td>\n<td>; Modus Ponens inter aliquas lineas superiores<\/td>\n<\/tr>\n<tr>\n<td>(n+m+1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/td>\n<td>; Praemissa<\/td>\n<\/tr>\n<tr>\n<td>(n+m+2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/td>\n<td>; Modus Ponens (n+m+1, una ex lineis superioribus, praeter (n+m+1))<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Ergo<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\Gamma\\cup\\{\\alpha\\} \\vdash \\beta <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ut hoc fieri possit, necesse est ut saltem una ex expressionibus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_1, \\cdots \\gamma_n,\\overline{\\gamma_1},\\cdots,\\overline{\\gamma_m}<\/span><\/span> sit formae <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\rightarrow \\beta)<\/span><\/span>; at omnes illae lineae solummodo involvunt elementa <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> et axiomata \u0141ukasiewicz in deductione sua, unde sequitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma\\vdash (\\alpha \\rightarrow \\beta)<\/span><\/span>. Quare theorema demonstratum est.<\/p>\n<p>Finis demonstrationis.<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<h2>Reciproca Theorematis Deductionis<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1668s\" target=\"_blank\" rel=\"noopener\"><strong>Iisdem condicionibus quibus theorema deductionis statuitur, sequitur<\/strong><\/a><\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\left(\\Gamma\\vdash(\\alpha \\rightarrow \\beta)\\right) \\Rightarrow \\left( \\Gamma \\cup \\{\\alpha\\}\\vdash \\beta \\right)\n\n<\/span>\n<p style=\"text-align: justify; color: #880000;\"><strong>Demonstratio:<\/strong><\/p>\n<p style=\"text-align: justify;\">Si ponitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma\\vdash (\\alpha\\rightarrow \\beta)<\/span><\/span>, tum habetur deductio huiusmodi formae<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_1<\/span><\/span><\/td>\n<td>; Prima praemissa e <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(n)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_n<\/span><\/span><\/td>\n<td>; Praemissa n e <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td>(n+1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span><\/td>\n<td>; Modus Ponens (inter quasdam lineas superiores)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Iam si <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> addatur ut praemissa ad hoc ratiocinium, tunc habebuntur lineae sequentis formae<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(n+2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/td>\n<td>; Praemissa addita<\/td>\n<\/tr>\n<tr>\n<td>(n+3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/td>\n<td>; MP(n+1, n+2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Ergo<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\Gamma \\cup \\{\\alpha\\} \\vdash \\beta<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Quod erat demonstrandum.<\/p>\n<p>Finis demonstrationis.<br \/>\n<a name=\"9\"><\/a><\/p>\n<h2>Deductiones de Expressionibus et Deductiones de Deductionibus<\/h2>\n<p style=\"text-align: justify;\">Demonstrationes sicut illa quae antea facta est ad obtinendum propositum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha\\rightarrow \\alpha)<\/span><\/span> sunt casus deductionum ex expressionibus, quia quisque gradus continet expressionem concretam. Similiter fieri potest deductiones ex aliis deductionibus conficere, ubi quisque gradus est ipsemet deductio. In praxi, utrumque modo simili efficitur, sed alterum nobis facultatem dat utamur theorema deductionis eiusque reciproco, quod magnam flexibilitatem rationandi tribuit. Ut hoc videamus, iterum demonstrabimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow \\alpha)<\/span><\/span>, sed nunc utens deductionibus loco simplicium expressionum. Una ex possibilibus viis sequens est:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow (\\alpha \\rightarrow \\alpha))<\/span><\/span><\/td>\n<td>; A1<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash ( \\alpha \\rightarrow \\alpha)<\/span><\/span><\/td>\n<td>; RTD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\cup \\{\\alpha\\}\\vdash \\alpha<\/span><\/span><\/td>\n<td>; RTD(2)<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash \\alpha<\/span><\/span><\/td>\n<td>; Animadvertendum est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\cup\\{\\alpha\\}=\\{\\alpha\\}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha\\rightarrow \\alpha)<\/span><\/span><\/td>\n<td>; TD(4)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Animadvertendum est hanc ratiocinationem non esse breviorem quam priorem, sed multo simpliciorem ad efficiendum, cum solummodo theorema deductionis, eius reciprocam et axioma A1 adhibuerimus ad structuram demonstrationis perficiendam.<\/p>\n<p style=\"text-align: justify;\">Apparet quidem in hoc processu nos usum fecisse tantum unius axioma \u0141ukasiewicz et neglexisse tam ceteros axiomata quam modum ponentem. Significatne hoc quod hac ratione ratiocinandi caetera axiomata et modum ponentem omittamus? Responsio est simul &#8216;ita&#8217; et &#8216;non&#8217;. Ex una parte, videmur eos ignorare, cum non explicite utamur; ex altera tamen, memorari oportet theorema deductionis et eius reciprocam ipsa ex axiomatibus \u0141ukasiewicz et modo ponente derivari, unde fit ut eorum usu etiam implicite istis elementis innitamur.<\/p>\n<p><a name=\"10\"><\/a><\/p>\n<h2>Regula Monotoniae<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1972s\" target=\"_blank\" rel=\"noopener\"><strong>Si <span class=\"katex-eq\" data-katex-display=\"false\">\\tau<\/span> est theorema,<\/strong><\/a> tunc, data quaelibet expressio <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span>, eveniet quod:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\beta\\}\\vdash\\tau<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Haec est revera regula facillime demonstranda, cum si <span class=\"katex-eq\" data-katex-display=\"false\">\\tau<\/span> est theorema, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash \\tau<\/span><\/span>. Id est, exsistit ratiocinatio quae, sine ulla praemissarum additione, ducit ad expressionem <span class=\"katex-eq\" data-katex-display=\"false\">\\tau<\/span>; ergo additio alicuius expressi ad praemissas (vacuam copiam) nullam mutationem efficit.<\/p>\n<p style=\"text-align: justify;\">Simili ratione sequens propositio proponi potest: si ex collectione praemissarum <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> infertur <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span>, tunc eveniet quod:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma\\cup\\{\\alpha\\}\\vdash\\gamma<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ubi <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> est quaelibet expressio.<\/p>\n<p><a name=\"11\"><\/a><\/p>\n<h2>Synthesis et Meditationes de Systematibus Deductivis et Logica Propositionali<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1933s\" target=\"_blank\" rel=\"noopener\"><strong>Cum linguam logicae propositionalis regula inferendi et expressionibus fundamentalibus instruximus:<\/strong><\/a> scilicet Modo Ponente et Axiomatis \u0141ukasiewicz, id quod efficimus est simile machinamentum deductionis constituere et \u00abmachinam motoriam\u00bb quae eam in motum inducit. Ex hoc momento incipiunt naturaliter emergere omnes regulae fundamentales deductionis quas in lectionibus sequentibus examinabimus.<\/p>\n<p style=\"text-align: justify;\">Adhuc unum observandum. Expressiones logicae propositionalis revera sunt meta-expressiones linguae binorum symbolorum quam antea inspeximus. Meminerimus gratiam harum meta-expressionum esse quod nobis permittunt metasymbolos eorum per quaslibet expressiones linguae substituere ad novam structuram eandem exprimendam obtinendam. Cum systemati deductivo schemata axiomatum et regulas inferendi addimus, systema logicum construitur quod deductiones inter expressiones connectendas gignit. Inde emergit structura deductiva quae infinitas deductiones complecti potest: omnes quae effici possunt per substitutionem meta-variabilium per quaslibet expressiones. Potentia logicae revera tunc explicatur, cum intellegimus nos posse, loco expressionum binorum symbolorum originum, uti expressionibus linguae nostrae usitatae, et spectare quid eveniat.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Systemata Deductiva Formalia in Logica Propositionali Summarium:In hac lectione recensentur systemata deductiva formalia. Explicatur quomodo haec systemata adhibentur ad enodandas relationes quae inter diversas expressiones logicas exsistere possunt, atque elementa fundamentalia quibus hae demonstrationes construuntur: lingua, axiomatibus, regulisque inferendi. Memorantur axiomata \u0141ukasiewicz et explicatur modus ponens ut motor deductionis calculi propositionis. Praeterea tractantur ratiocinia, theorema [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":33789,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[1358,1298],"tags":[],"class_list":["post-33805","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","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>Systemata Deductiva Formalia: Definitiones et Exempla - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Investiga fundamenta systematum deductivorum formalium in logica propositionali, axiomatibus, inferentiis et theorematibus inclusis.\" \/>\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\/systemata-deductiva-formalia-definitiones-et-exempla\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Systemata Deductiva Formalia: Definitiones et Exempla\" \/>\n<meta property=\"og:description\" content=\"Investiga fundamenta systematum deductivorum formalium in logica propositionali, axiomatibus, inferentiis et theorematibus inclusis.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/systemata-deductiva-formalia-definitiones-et-exempla\/\" \/>\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=\"2021-01-25T00:00:53+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-31T01:25:49+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/01\/sitemasdeductivos-1024x465.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Systemata Deductiva Formalia: Definitiones et Exempla\" \/>\n<meta name=\"twitter:description\" content=\"Investiga fundamenta systematum deductivorum formalium in logica propositionali, axiomatibus, inferentiis et theorematibus inclusis.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/01\/sitemasdeductivos.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\\\/systemata-deductiva-formalia-definitiones-et-exempla\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/systemata-deductiva-formalia-definitiones-et-exempla\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Systemata Deductiva Formalia: Definitiones et Exempla\",\"datePublished\":\"2021-01-25T00:00:53+00:00\",\"dateModified\":\"2025-07-31T01:25:49+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/systemata-deductiva-formalia-definitiones-et-exempla\\\/\"},\"wordCount\":1889,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/systemata-deductiva-formalia-definitiones-et-exempla\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/01\\\/sitemasdeductivos.jpg\",\"articleSection\":[\"Logica Propositionalis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/systemata-deductiva-formalia-definitiones-et-exempla\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/systemata-deductiva-formalia-definitiones-et-exempla\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/systemata-deductiva-formalia-definitiones-et-exempla\\\/\",\"name\":\"Systemata Deductiva Formalia: Definitiones et Exempla - 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