{"id":33817,"date":"2021-01-27T13:00:01","date_gmt":"2021-01-27T13:00:01","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33817"},"modified":"2025-07-31T02:00:03","modified_gmt":"2025-07-31T02:00:03","slug":"iv-technicae-deductionis-necessariae","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/iv-technicae-deductionis-necessariae\/","title":{"rendered":"IV technicae deductionis necessariae"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Disce IV technicas deductionis necessarias<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><em><strong>Summarium:<\/strong><\/br>In hac lectione describuntur quattuor technicae deductionis logicae propositionis, ad calculum propositionis rudimentarium hactenus propositum locupletandum. Exponitur regula praesumptionis eiusque coniunctio cum regula monotoniae, item syllogismus hypotheticus et duae viae ad hanc regulam deductionis obtinendam. Explicantur quoque aequivalentiae negationis duplicatae et contrapositivum implicationis.<\/em><\/p>\n<p style=\"text-align:center;\"><strong><u>Propositi Discendi<\/u>:<\/strong><br \/>Expleta hac lectione discipulus poterit<\/p>\n<ol>\n<li><strong>Meminisse<\/strong> structuram ratiocinii et simplices exempla.<\/li>\n<li><strong>Intellegere<\/strong> regulam praesumptionis eiusque relationem ad theorema deductionis.<\/li>\n<li><strong>Intellegere<\/strong> regulam syllogismi hypothetici eiusque nexum cum modo ponente.<\/li>\n<li><strong>Adhibere<\/strong> theorema deductionis in logica propositionis.<\/li>\n<li><strong>Adhibere<\/strong> regulam monotoniae in deductione expressionum.<\/li>\n<li><strong>Intellegere<\/strong> aequivalentias negationis duplicatae et contrapositivum implicationis in logica propositionis.<\/li>\n<li><strong>Noscere<\/strong> demonstrationes technicarum deductionis easque in praxi adhibere posse.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX RERUM<\/strong><br \/>\n<a href=\"#1\">REGULA PRAESUMPTIONIS (PRE)<\/a><br \/>\n<a href=\"#2\">SYLLOGISMUS HYPOTHETICUS (SH)<\/a><br \/>\n<a href=\"#3\">AEQUIVALENTIAE NEGATIONIS DUPLICATAE (DN)<\/a><br \/>\n<a href=\"#4\">AEQUIVALENTIA CONTRAPOSITIVI IMPLICATIONIS (CPI)<\/a>\n<\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/6f_aavuC4E0\" 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;\">Iam vidimus qualis sit structura ratiocinii et exempla simplicia. Nunc scientiam istam experiemur <strong>ratione utens IV technicis deductionis logicae propositionis.<\/strong> Per hoc non solum ostendemus haec praecepta valere, sed etiam initium faciemus ditandi processum qui calculum propositionis, hactenus rudimentarium, ex eo statu eruat.<\/p>\n<p style=\"text-align: justify;\"><strong>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> sunt expressiones calculi propositionis, tunc fieri potest ut sequentia deductionis praecepta ex fundamentis concludantur:<\/strong><\/p>\n<p><a name=\"1\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Regula Praesumptionis (Pre)<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=6f_aavuC4E0&amp;t=168s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Facillima omnium regula deductionis<\/span><\/strong><\/a> est regula praesumptionis. Haec directe obtinetur applicando <strong>reciprocum theoremae deductionis<\/strong> ad theorema <span class=\"katex-eq\" data-katex-display=\"false\">\\vdash(\\alpha\\rightarrow\\alpha)<\/span>. Si hoc tibi arcano sermone videtur enuntiatum, omnia quae scire debes inveniuntur <a href=\"http:\/\/toposuranos.com\/material\/es\/sistemas-deductivos-formales-y-definiciones\/\" rel=\"noopener\" target=\"_blank\"><strong>hic<\/strong><\/a>.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash \\alpha <\/span>\n<p style=\"text-align: justify;\">Cum regula monotoniae coniuncta, haec tibi permittet expressiones utiles intra deductiones tuas inserere.<\/p>\n<p><a name=\"2\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Syllogismus Hypotheticus (SH)<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=6f_aavuC4E0&amp;t=206s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Syllogismus hypotheticus<\/span><\/strong><\/a>, sive transitivitas implicationis, species est evolutionis modi ponentis. Eius enuntiatio est haec:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha\\rightarrow\\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash (\\alpha\\rightarrow\\gamma)<\/span>\n<p style=\"text-align: justify;\">Multae sunt viae ad hanc regulam deductionis obtinendam, duas mox considerabimus.<\/p>\n<p style=\"text-align: justify;\">Si ex expressionibus ratiocinemur, facile erit ratiocinium sequentem construere:<\/p>\n<table style=\"text-align: justify;\">\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 class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/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\\rightarrow \\gamma)<\/span><\/td>\n<td>; Praemissa<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/td>\n<td>; MP(1,2)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/td>\n<td>; MP(4,3)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Ergo<span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha,(\\alpha\\rightarrow\\beta),(\\beta\\rightarrow\\gamma)\\}\\vdash\\gamma<\/span>\n<p style=\"text-align: justify;\">Postremo, applicando theorema deductionis ad hanc ultimam expressionem, habetur:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{(\\alpha\\rightarrow\\beta),(\\beta\\rightarrow\\gamma)\\}\\vdash(\\alpha\\rightarrow \\gamma)<\/span>\n<p style=\"text-align: justify;\">Alia via ad hanc regulam demonstrandam est ratiocinando ex deductionibus, construendo per praesumptionem et monotoniam. Considera sequentem ratiocinationem ex deductionibus ortam:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash \\alpha <\/span><\/td>\n<td>; Praesumptio et Monotonia<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash (\\alpha\\rightarrow \\beta) <\/span><\/td>\n<td>; Praesumptio et Monotonia<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash (\\beta\\rightarrow\\gamma) <\/span><\/td>\n<td>; Praesumptio et Monotonia<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash \\beta <\/span><\/td>\n<td>; MP(1,2)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash \\gamma <\/span><\/td>\n<td>; MP(4,3)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash (\\alpha \\rightarrow \\gamma) <\/span><\/td>\n<td>; TD(5)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Hic animadvertendum est utramque demonstrationem identicas esse, solummodo diversis modis expositas. In praxi, inter hos duos stylos alternare potes, secundum quod tibi commodius est.<\/p>\n<p><a name=\"3\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Aequivalentiae Negationis Duplicatae (DN)<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=6f_aavuC4E0&amp;t=500s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Aequivalentiae negationis duplicatae<\/span><\/strong><\/a> exprimunt notionem intuitivam secundum quam negatio duplicata enuntiationis aequivalet ipsi enuntiationi. Hoc, symbolice scriptum, sic se habet:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha\\dashv\\vdash\\neg\\neg\\alpha<\/span>\n<p style=\"text-align: justify;\">Videamus nunc demonstrationem:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg\\neg \\alpha \\rightarrow (\\neg\\neg\\neg\\neg \\alpha \\rightarrow\\neg\\neg\\alpha))<\/span><\/td>\n<td>; A1<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash ((\\neg\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\alpha)\\rightarrow(\\neg\\alpha \\rightarrow \\neg\\neg\\neg\\alpha))<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash ((\\neg\\alpha \\rightarrow \\neg\\neg\\neg\\alpha)\\rightarrow(\\neg\\neg\\alpha \\rightarrow \\alpha))<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash ((\\neg\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\alpha)\\rightarrow(\\neg\\neg\\alpha \\rightarrow \\alpha))<\/span><\/td>\n<td>; SH(2,3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash (\\neg\\neg\\neg\\neg \\alpha \\rightarrow\\neg\\neg\\alpha)<\/span><\/td>\n<td>; RTD(1)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash ((\\neg\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\alpha)\\rightarrow(\\neg\\neg\\alpha \\rightarrow \\alpha))<\/span><\/td>\n<td>; Monotonia(4)<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash (\\neg\\neg\\alpha \\rightarrow \\alpha)<\/span><\/td>\n<td>; MP(5,6)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash \\alpha<\/span><\/td>\n<td>; RTD(7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Ergo <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash \\alpha <\/span>\n<p style=\"text-align: justify;\">Ut demonstrationem in alteram partem conficiamus, hac ipsa demonstratione uti possumus, eam aptando per simplicem substitutionem, unde hoc obtinetur:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\neg \\alpha \\} \\vdash \\neg \\alpha <\/span>\n<p style=\"text-align: justify;\">Et ex hoc argumentum in alteram partem componimus:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\neg \\alpha \\} \\vdash \\neg \\alpha <\/span><\/td>\n<td>; Quod modo probatum est<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash(\\neg\\neg \\neg \\alpha\\rightarrow \\neg \\alpha) <\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash((\\neg\\neg \\neg \\alpha\\rightarrow \\neg \\alpha) \\rightarrow(\\alpha \\rightarrow\\neg\\neg\\alpha)) <\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash(\\alpha \\rightarrow\\neg\\neg\\alpha) <\/span><\/td>\n<td>; MP(2,3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash\\neg\\neg\\alpha <\/span><\/td>\n<td>; RTD(4)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Ergo <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha \\} \\vdash \\neg\\neg \\alpha <\/span>\n<p style=\"text-align: justify;\">Denique, ex his duabus demonstrationibus sequitur <span class=\"katex-eq\" data-katex-display=\"false\"> \\alpha \\dashv\\vdash \\neg\\neg \\alpha <\/span>.<\/p>\n<p><a name=\"4\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Aequivalentia Contrapositivi Implicationis (CpI)<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=6f_aavuC4E0&amp;t=948s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Hoc respondet<\/span><\/strong><\/a> sequentibus aequivalentibus enuntiationibus<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta) \\dashv\\vdash (\\neg\\beta \\rightarrow \\neg\\alpha)<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n(\\neg\\alpha\\rightarrow\\beta)\\dashv\\vdash (\\neg\\beta\\rightarrow\\alpha)<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\rightarrow\\neg\\beta) \\dashv\\vdash (\\beta\\rightarrow\\neg\\alpha)<\/span>\n<p style=\"text-align: justify;\">Demonstratio huius primae relationis hoc modo perficitur:<\/p>\n<p style=\"text-align: justify;\">Ex una parte, obtinetur directe ex tertio axiomatico:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash ((\\neg\\beta\\rightarrow \\neg\\alpha) \\rightarrow (\\alpha \\rightarrow\\beta))<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow \\neg\\alpha)\\}\\vdash (\\alpha \\rightarrow \\beta)<\/span><\/td>\n<td>; RTD(1)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Ergo <span class=\"katex-eq\" data-katex-display=\"false\"> \\{(\\neg\\beta\\rightarrow \\neg\\alpha)\\}\\vdash (\\alpha \\rightarrow \\beta)<\/span>\n<p style=\"text-align: justify;\">Et in alteram partem, demonstratio elici potest ex ratiocinatione sequenti:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg\\alpha \\dashv \\vdash \\alpha<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\neg \\alpha \\rightarrow \\alpha)<\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg\\beta \\dashv \\vdash \\beta<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\beta \\rightarrow \\neg\\neg \\beta)<\/span><\/td>\n<td>; TD(3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg \\alpha \\rightarrow \\alpha)<\/span><\/td>\n<td>; Mon(2)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\alpha \\rightarrow \\beta)<\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg \\alpha \\rightarrow\\beta)<\/span><\/td>\n<td>; SH(5,6)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\{(\\alpha \\rightarrow \\beta)\\} \\vdash (\\beta \\rightarrow \\neg\\neg \\beta)<\/span><\/td>\n<td>; Mon(4)<\/td>\n<\/tr>\n<tr>\n<td>(9)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg \\alpha \\rightarrow \\neg\\neg \\beta)<\/span><\/td>\n<td>; SH(7,8)<\/td>\n<\/tr>\n<tr>\n<td>(10)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\neg \\alpha \\rightarrow \\neg\\neg \\beta) \\rightarrow (\\neg \\beta \\rightarrow \\neg \\alpha )<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(11)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash ((\\neg\\neg \\alpha \\rightarrow \\neg\\neg \\beta) \\rightarrow (\\neg \\beta \\rightarrow \\neg \\alpha ))<\/span><\/td>\n<td>; Mon(10)<\/td>\n<\/tr>\n<tr>\n<td>(11)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg \\beta \\rightarrow \\neg \\alpha )<\/span><\/td>\n<td>; SH(10;11)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Ergo <span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg \\beta \\rightarrow \\neg \\alpha )<\/span>\n<p style=\"text-align: justify;\">Quapropter, ex duobus ratiociniis superioribus sequitur:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> (\\alpha \\rightarrow \\beta) \\dashv\\vdash (\\neg \\beta \\rightarrow \\neg \\alpha ) <\/span>\n<p style=\"text-align: justify;\">Ad demonstrandam secundam, fieri possunt haec duo ratiocinia:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta \\dashv\\vdash \\neg\\neg\\beta<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg\\neg\\alpha \\dashv\\vdash \\neg\\alpha<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\beta \\rightarrow \\neg\\neg\\beta)<\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\alpha)<\/span><\/td>\n<td>; TD(2)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\alpha \\rightarrow \\beta)<\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\beta \\rightarrow \\neg\\neg\\beta)<\/span><\/td>\n<td>; Mon(3)<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\alpha)<\/span><\/td>\n<td>; Mon(4)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\alpha \\rightarrow \\neg\\neg\\beta)<\/span><\/td>\n<td>; SH(5,6)<\/td>\n<\/tr>\n<tr>\n<td>(9)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\beta)<\/span><\/td>\n<td>; SH(7,8)<\/td>\n<\/tr>\n<tr>\n<td>(10)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\beta) \\rightarrow (\\neg\\beta \\rightarrow \\neg\\neg\\alpha)<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(11)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash ((\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\beta) \\rightarrow (\\neg\\beta \\rightarrow \\neg\\neg\\alpha))<\/span><\/td>\n<td>; Mon(10)<\/td>\n<\/tr>\n<tr>\n<td>(12)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\beta \\rightarrow \\neg\\neg\\alpha)<\/span><\/td>\n<td>; MP(9,11)<\/td>\n<\/tr>\n<tr>\n<td>(13)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg \\alpha \\dashv \\vdash \\alpha<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(14)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg\\neg \\alpha\\rightarrow \\alpha)<\/span><\/td>\n<td>; TD(13)<\/td>\n<\/tr>\n<tr>\n<td>(15)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\} \\vdash (\\neg\\neg \\alpha\\rightarrow \\alpha)<\/span><\/td>\n<td>; Mon(14)<\/td>\n<\/tr>\n<tr>\n<td>(16)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\} \\vdash(\\neg\\beta \\rightarrow \\alpha)<\/span><\/td>\n<td>; SH(12,15)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Ergo <span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\} \\vdash(\\neg\\beta \\rightarrow \\alpha) <\/span>\n<p style=\"text-align: justify;\">Nunc demonstratio in sensu inverso perficienda est. Id fieri potest per ratiocinationem sequentem:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha \\dashv \\vdash \\neg\\neg\\alpha<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow \\neg\\neg\\alpha)<\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\neg\\beta\\rightarrow\\alpha)<\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\alpha \\rightarrow \\neg\\neg\\alpha)<\/span><\/td>\n<td>; Mon(2)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\neg\\beta\\rightarrow\\neg\\neg\\alpha)<\/span><\/td>\n<td>; SH(3,4)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\beta\\rightarrow\\neg\\neg\\alpha)\\rightarrow (\\neg\\alpha \\rightarrow \\beta) <\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash ((\\neg\\beta\\rightarrow\\neg\\neg\\alpha)\\rightarrow (\\neg\\alpha \\rightarrow \\beta)) <\/span><\/td>\n<td>; Mon(6)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\neg\\alpha \\rightarrow \\beta) <\/span><\/td>\n<td>; MP(5,7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Ergo <span class=\"katex-eq\" data-katex-display=\"false\"> \\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\neg\\alpha \\rightarrow \\beta) <\/span>\n<p style=\"text-align: justify;\">Denique, ex his duabus ratiocinationibus concluditur <span class=\"katex-eq\" data-katex-display=\"false\"> (\\neg\\beta\\rightarrow\\alpha) \\dashv \\vdash (\\neg\\alpha \\rightarrow \\beta) <\/span>, quod erat demonstrandum.<\/p>\n<p style=\"text-align: justify;\">Ultima aequivalentia exercitio relinquetur. Ad eam demonstrandam, duabus iam datis demonstrationibus uti potes ut duce. Haec optima est via ad artem deductionis bene callendam.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Disce IV technicas deductionis necessarias Summarium:In hac lectione describuntur quattuor technicae deductionis logicae propositionis, ad calculum propositionis rudimentarium hactenus propositum locupletandum. Exponitur regula praesumptionis eiusque coniunctio cum regula monotoniae, item syllogismus hypotheticus et duae viae ad hanc regulam deductionis obtinendam. Explicantur quoque aequivalentiae negationis duplicatae et contrapositivum implicationis. Propositi Discendi:Expleta hac lectione discipulus poterit Meminisse [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27340,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":6,"footnotes":""},"categories":[1352,1358,1298],"tags":[],"class_list":["post-33817","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>IV technicae deductionis necessariae - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce IV technicas deductionis in logica propositionali: regulam praesumptionis, syllogismum hypotheticum, aequivalentias negationis duplicatae et contrapositivum implicationis.\" \/>\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\/iv-technicae-deductionis-necessariae\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"IV technicae deductionis necessariae\" \/>\n<meta property=\"og:description\" content=\"Disce IV technicas deductionis in logica propositionali: regulam praesumptionis, syllogismum hypotheticum, aequivalentias negationis duplicatae et contrapositivum implicationis.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/iv-technicae-deductionis-necessariae\/\" \/>\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=\"2021-01-27T13:00:01+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-31T02:00:03+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/sistemasdeductivos.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"IV technicae deductionis necessariae\" \/>\n<meta name=\"twitter:description\" content=\"Disce IV technicas deductionis in logica propositionali: regulam praesumptionis, syllogismum hypotheticum, aequivalentias negationis duplicatae et contrapositivum implicationis.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/sistemasdeductivos.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\\\/iv-technicae-deductionis-necessariae\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/iv-technicae-deductionis-necessariae\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"IV technicae deductionis necessariae\",\"datePublished\":\"2021-01-27T13:00:01+00:00\",\"dateModified\":\"2025-07-31T02:00:03+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/iv-technicae-deductionis-necessariae\\\/\"},\"wordCount\":1463,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/iv-technicae-deductionis-necessariae\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/sistemasdeductivos.jpg\",\"articleSection\":[\"Logica Mathematica\",\"Logica Propositionalis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/iv-technicae-deductionis-necessariae\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/iv-technicae-deductionis-necessariae\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/iv-technicae-deductionis-necessariae\\\/\",\"name\":\"IV technicae deductionis necessariae - 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