{"id":27450,"date":"2021-02-12T13:00:53","date_gmt":"2021-02-12T13:00:53","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27450"},"modified":"2024-07-09T04:23:10","modified_gmt":"2024-07-09T04:23:10","slug":"demostracion-de-las-tecnicas-de-la-logica-clasica","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/","title":{"rendered":"Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>RESUMEN<\/strong><br \/><em>En esta clase se presentan varias t\u00e9cnicas de la l\u00f3gica cl\u00e1sica para introducir y eliminar conjunciones y disyunciones, adem\u00e1s de la regla del tercero excluido y la regla de contradicci\u00f3n, tambi\u00e9n conocida como el principio de explosi\u00f3n. Adem\u00e1s, se explica la t\u00e9cnica de prueba por casos y la reducci\u00f3n al absurdo, ambas muy \u00fatiles en demostraciones matem\u00e1ticas y l\u00f3gicas en general. Cada t\u00e9cnica se presenta formalmente y se proporciona una demostraci\u00f3n paso a paso para su comprensi\u00f3n. Si deseas profundizar en la l\u00f3gica proposicional y mejorar tus habilidades en la demostraci\u00f3n de teoremas, esta clase te ser\u00e1 de gran utilidad.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p><strong>OBJETIVOS DE APRENDIZAJE:<\/strong><\/p>\n<ol>\n<li><strong>Comprender<\/strong> la justificaci\u00f3n detr\u00e1s de las t\u00e9cnicas de introducci\u00f3n y eliminaci\u00f3n de la conjunci\u00f3n y disyunci\u00f3n.<\/li>\n<li><strong>Comprender<\/strong> la propiedad del tercero excluido o tautolog\u00eda (TAU) en la l\u00f3gica cl\u00e1sica.<\/li>\n<li><strong>Comprender<\/strong> la regla de contradicci\u00f3n (CON) o principio de explosi\u00f3n en la l\u00f3gica cl\u00e1sica.<\/li>\n<li><strong>Comprender<\/strong> la t\u00e9cnica de eliminaci\u00f3n de disyuntos (\u2228-eliminaci\u00f3n3) en la l\u00f3gica cl\u00e1sica.<\/li>\n<li><strong>Comprender<\/strong> la t\u00e9cnica de pruebas por casos (CAS) en la l\u00f3gica cl\u00e1sica.<\/li>\n<li><strong>Comprender<\/strong> la t\u00e9cnica de reducci\u00f3n al absurdo (absurdo) en la l\u00f3gica cl\u00e1sica.<\/li>\n<li><strong>Aplicar<\/strong> los conocimientos de las diferentes t\u00e9cnicas de la l\u00f3gica cl\u00e1sica para resolver problemas y demostraciones complejas.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDICE<\/strong><br \/>\n<a href=\"#1\"><strong>INTRODUCCI\u00d3N Y ELIMINACI\u00d3N DE CONJUNCIONES Y DISYUNCIONES<\/strong><\/a><br \/>\n<a href=\"#2\">\u2228-INTRODUCCI\u00d3N<\/a><br \/>\n<a href=\"#3\">\u2228-ELIMINACI\u00d3N<\/a><br \/>\n<a href=\"#4\">\u2227-INTRODUCCI\u00d3N<\/a><br \/>\n<a href=\"#5\">\u2227-ELIMINACI\u00d3N<\/a><br \/>\n<a href=\"#6\"><strong>T\u00c9CNICAS DE CONTRADICCIONES Y TAUTOLOG\u00cdAS<\/strong><\/a><br \/>\n<a href=\"#7\">REGLA DEL TERCERO EXCLUIDO O TAUTOLOG\u00cdA (TAU)<\/a><br \/>\n<a href=\"#8\">REGLA DE CONTRADICCI\u00d3N O PRINCIPIO DE EXPLOSI\u00d3N<\/a><br \/>\n<a href=\"#9\">\u2228-ELIMINACI\u00d3N3<\/a><br \/>\n<a href=\"#10\">PRUEBAS POR CASOS (CAS)<\/a><br \/>\n<a href=\"#11\">REDUCCI\u00d3N AL ABSURDO (ABSURDO)<\/a><\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/qAJ_oaSfd9k\" 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><br \/>\n<\/br><\/br><\/p>\n<h2>Introducci\u00f3n y Eliminaci\u00f3n de Conjunciones y Disyunciones<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=qAJ_oaSfd9k&amp;t=439s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Una de las t\u00e9cnicas<\/span><\/strong><\/a> de la l\u00f3gica cl\u00e1sica consiste en la introducci\u00f3n y eliminaci\u00f3n de conectores y disyuntores. A pesar de que estas t\u00e9cnicas se ejecutan de un modo m\u00e1s o menos intuitivo, su justificaci\u00f3n no es del todo trivial, pero se pueden obtener a partir de las reglas de la l\u00f3gica proposicional que ya hemos demostrado en clases anteriores. Formalmente, las t\u00e9cnicas de introducci\u00f3n y eliminaci\u00f3n de conectores y disyuntores son las siguientes:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>\u2228-Introducci\u00f3n<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha \\} \\vdash (\\alpha \\vee \\beta)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u2228-Eliminaci\u00f3n<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha\\vee\\beta), \\neg\\alpha \\} \\vdash\\beta <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u2227-Intrducci\u00f3n<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha.\\beta \\} \\vdash(\\alpha \\wedge \\beta) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u2227-Eliminaci\u00f3n<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge \\beta) \\} \\vdash \\alpha <\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Y sus demostraciones desde la l\u00f3gica proposicional son las que se muestran a continuaci\u00f3n:<\/p>\n<p><a name=\"2\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>\u2228-Introducci\u00f3n<\/h3>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\} \\vdash \\alpha<\/span><\/span><\/td>\n<td>; Pre<\/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 (\\neg \\beta \\rightarrow \\alpha))<\/span><\/span><\/td>\n<td>; A1, Mon<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\} \\vdash (\\neg \\beta \\rightarrow \\alpha)<\/span><\/span><\/td>\n<td>; MP(1,2)<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{\\alpha\\} \\vdash (\\beta \\vee \\alpha)}<\/span><\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\rightarrow<\/span><\/span>-Definici\u00f3n(3)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>\u2228-Eliminaci\u00f3n<\/h3>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee \\beta), \\neg\\alpha\\}\\vdash (\\alpha \\vee\\beta)<\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee \\beta), \\neg\\alpha\\}\\vdash \\neg\\alpha <\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\vee \\beta), \\neg\\alpha\\}\\vdash (\\neg \\alpha \\rightarrow \\beta)<\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\rightarrow<\/span><\/span>-Definici\u00f3n (1)<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{(\\alpha \\vee \\beta), \\neg\\alpha\\}\\vdash \\beta}<\/span><\/span><\/span><\/td>\n<td>; MP(2,3)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"4\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>\u2227-Introducci\u00f3n<\/h3>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\vee \\neg \\beta), \\neg\\neg\\beta\\} \\vdash \\neg\\alpha<\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-Eliminaci\u00f3n<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg\\beta\\} \\vdash ((\\neg\\alpha \\vee \\neg \\beta) \\rightarrow \\neg\\alpha)<\/span><\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg\\beta\\} \\vdash (\\neg \\neg\\alpha \\rightarrow \\neg (\\neg\\alpha \\vee \\neg \\beta))<\/span><\/span><\/td>\n<td>; CPI(2))<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg\\neg\\beta \\rightarrow (\\neg \\neg\\alpha \\rightarrow \\neg (\\neg\\alpha \\vee \\neg \\beta)))<\/span><\/span><\/td>\n<td>; TD(3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha, \\beta \\} \\vdash (\\neg\\neg\\beta \\rightarrow (\\neg \\neg\\alpha \\rightarrow \\neg (\\neg\\alpha \\vee \\neg \\beta)))<\/span><\/span><\/td>\n<td>; Monoton\u00eda x2 (4)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha, \\beta \\} \\vdash \\beta<\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha, \\beta \\} \\vdash \\neg\\neg\\beta<\/span><\/span><\/td>\n<td>; DN(6)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha, \\beta \\} \\vdash (\\neg \\neg\\alpha \\rightarrow \\neg (\\neg\\alpha \\vee \\neg \\beta))<\/span><\/span><\/td>\n<td>; MP(7,5)<\/td>\n<\/tr>\n<tr>\n<td>(9)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha, \\beta \\} \\vdash \\alpha<\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(10)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha, \\beta \\} \\vdash \\neg\\neg\\alpha<\/span><\/span><\/td>\n<td>; DN(9)<\/td>\n<\/tr>\n<tr>\n<td>(11)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha, \\beta \\} \\vdash \\neg (\\neg\\alpha \\vee \\neg \\beta)<\/span><\/span><\/td>\n<td>; MP(10,8)<\/td>\n<\/tr>\n<tr>\n<td>(12)<\/td>\n<td><span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{\\alpha, \\beta \\} \\vdash (\\alpha \\wedge \\beta)}<\/span><\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>-Definici\u00f3n(11)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"5\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>\u2227-Eliminaci\u00f3n<\/h3>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\wedge \\beta)\\} \\vdash (\\alpha \\wedge \\beta)<\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg \\alpha\\} \\vdash (\\neg \\alpha \\vee \\neg\\beta)<\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-Introducci\u00f3n<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg \\alpha \\rightarrow (\\neg \\alpha \\vee \\neg\\beta))<\/span><\/span><\/td>\n<td>; TD(2)<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg(\\neg \\alpha \\vee \\neg\\beta) \\rightarrow \\alpha)<\/span><\/span><\/td>\n<td>; CPI(3))<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash ( ( \\alpha \\wedge \\beta) \\rightarrow \\alpha)<\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>-definici\u00f3n(4)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{(\\alpha \\wedge \\beta)\\} \\vdash ( ( \\alpha \\wedge \\beta) \\rightarrow \\alpha)<\/span><\/span><\/td>\n<td>; Monoton\u00eda(5)<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{(\\alpha \\wedge \\beta)\\} \\vdash \\alpha}<\/span><\/span><\/span><\/td>\n<td>; MP(1,6)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"6\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>T\u00e9cnicas de Contradicciones y Tautolog\u00edas<\/h2>\n<p><a name=\"7\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>Regla del Tercero Excluido o Tautolog\u00eda (tau)<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=qAJ_oaSfd9k&amp;t=1208s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Otra de las caracter\u00edsticas<\/span><\/strong><\/a> m\u00e1s notorias de la l\u00f3gica cl\u00e1sica es la propiedad del tercero excluido (tertium non datur). En esta se establece que si se tienen dos afirmaciones, donde una de ellas niegue a la otra, entonces por necesidad una de las dos debe ser verdadera; o dicho de otro modo, la conjunci\u00f3n de dos afirmaciones en que una niega a la otra forma una tautolog\u00eda por necesidad. Formalmente, esto se expresa escribiendo:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\alpha \\vee\\alpha)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Y su demostraci\u00f3n es f\u00e1cil de obtener.<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash \\alpha<\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow \\alpha)<\/span><\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\vdash (\\neg \\alpha \\vee \\alpha)}<\/span><\/span><\/span><\/td>\n<td>; de (2) porque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta) := (\\neg \\alpha \\vee \\beta)<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Otra forma de enunciar el principio del tercero excluido es a trav\u00e9s de la ley de <strong>no contradicci\u00f3n<\/strong>, que establece que una afirmaci\u00f3n no puede ser verdadera y falsa al mismo tiempo y que se enuncia formalmente a trav\u00e9s de:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash \\neg(\\neg\\alpha \\wedge \\alpha)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Esta propiedad no necesita demostraci\u00f3n, no porque sea autoevidente por s\u00ed misma, sino porque se obtiene directamente de aplicar la definici\u00f3n de la conjunci\u00f3n sobre el principio del tercero excluido.<\/p>\n<p><a name=\"8\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>Regla de Contradicci\u00f3n o Principio de Explosi\u00f3n<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=qAJ_oaSfd9k&amp;t=1410s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Otra propiedad conocida de la l\u00f3gica<\/span><\/strong><\/a> cl\u00e1sica es el principio de explosi\u00f3n, que usualmente se enuncia a trav\u00e9s de la frase \u00abde premisas contradictorias se puede concluir cualquier cosa\u00bb. Su formulaci\u00f3n se suele presentar de cualquiera de las siguientes dos maneras:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\wedge \\alpha)\\}\\vdash \\beta<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, \\neg\\alpha\\}\\vdash \\beta<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">La demostraci\u00f3n de esta regla es sencilla:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha ,\\neg\\alpha\\} \\vdash \\neg\\alpha <\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha ,\\neg\\alpha\\} \\vdash (\\neg\\alpha \\vee \\beta) <\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-introducci\u00f3n<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha ,\\neg\\alpha\\} \\vdash (\\alpha \\rightarrow \\beta) <\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\rightarrow<\/span><\/span>-definici\u00f3n(2)<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha ,\\neg\\alpha\\} \\vdash \\alpha <\/span><\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{\\alpha ,\\neg\\alpha\\} \\vdash \\beta}<\/span><\/span><\/span><\/td>\n<td>; MP(4,3)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"9\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>\u2228-Eliminaci\u00f3n3<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=qAJ_oaSfd9k&amp;t=1555s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">El modus ponens puede ser escrito<\/span><\/strong><\/a> de dos maneras distintas. Una de las formas que ya conocemos es <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha,(\\alpha \\rightarrow \\beta)\\}\\vdash \\beta<\/span><\/span>. La otra es un poco menos familiar:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash\\beta \\; \\wedge \\; \\vdash \\alpha \\; \\Longrightarrow \\; \\vdash \\beta<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Centr\u00e1ndonos en esta segunda forma, es posible visualizar una expansi\u00f3n para esta regla que llamamos <strong>\u2228-Eliminaci\u00f3n3<\/strong>, debido a que se asemeja a una simplificaci\u00f3n que se obtiene a partir de una disyunci\u00f3n. Esta nos dice que si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> puede ser inferido a partir de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> y a partir de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> (de ambas a la vez) y a su vez la disyunci\u00f3n entre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> y <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> es un teorema, entonces <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> es un teorema. Esto lo resumimos formalmente a trav\u00e9s de la siguiente escritura:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha\\}\\vdash\\gamma\\; \\wedge \\; \\{\\beta\\}\\vdash\\gamma \\; \\wedge \\; \\vdash (\\alpha \\vee \\beta)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Longrightarrow<\/span><\/span> <span style=\"color: #000088;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash \\gamma<\/span><\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">La demostraci\u00f3n de esta t\u00e9cnica de la l\u00f3gica cl\u00e1sica es la siguiente:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\alpha \\vdash \\gamma}<\/span><\/span><\/span><\/td>\n<td>; Premisa<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\beta \\vdash \\gamma}<\/span><\/span><\/td>\n<td>; Premisa<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\vdash (\\alpha \\vee \\beta)}<\/span><\/span><\/td>\n<td>; Premisa<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\alpha \\rightarrow \\gamma)<\/span><\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\beta \\rightarrow \\gamma)<\/span><\/span><\/td>\n<td>; TD(2)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg \\gamma \\rightarrow \\neg \\alpha)<\/span><\/span><\/td>\n<td>; CPI(4)<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg \\gamma \\rightarrow \\neg \\beta)<\/span><\/span><\/td>\n<td>; CPI(5)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\neg \\gamma \\}\\vdash \\neg \\alpha<\/span><\/span><\/td>\n<td>; RTD(6)<\/td>\n<\/tr>\n<tr>\n<td>(9)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\neg \\gamma\\}\\vdash \\neg \\beta<\/span><\/span><\/td>\n<td>; RTD(7)<\/td>\n<\/tr>\n<tr>\n<td>(10)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\neg \\gamma\\}\\vdash (\\neg \\alpha \\wedge \\neg \\beta)<\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>-Introducci\u00f3n(8,9)<\/td>\n<\/tr>\n<tr>\n<td>(11)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg \\gamma \\rightarrow (\\neg \\alpha \\wedge \\neg \\beta))<\/span><\/span><\/td>\n<td>; TD(10)<\/td>\n<\/tr>\n<tr>\n<td>(12)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg(\\neg \\alpha \\wedge \\neg \\beta)\\rightarrow \\gamma )<\/span><\/span><\/td>\n<td>; CPI(11)<\/td>\n<\/tr>\n<tr>\n<td>(13)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> (A \\wedge B) := \\neg(\\neg A \\vee \\neg B)<\/span><\/span><\/td>\n<td>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> &#8211; Definici\u00f3n<\/td>\n<\/tr>\n<tr>\n<td>(14)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\neg(A \\wedge B) := \\neg\\neg(\\neg A \\vee \\neg B)<\/span><\/span><\/td>\n<td>; Negando a ambos lados en (13)<\/td>\n<\/tr>\n<tr>\n<td>(15)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\neg(\\neg\\alpha \\wedge \\neg\\beta) := \\neg\\neg(\\neg\\neg\\alpha \\vee \\neg\\neg\\beta)<\/span><\/span><\/td>\n<td>; Remplazando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A:=\\neg\\alpha<\/span><\/span> y <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B:=\\neg\\beta<\/span><\/span> en (14)<\/td>\n<\/tr>\n<tr>\n<td>(16)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\neg(\\neg\\alpha \\wedge \\neg\\beta) \\dashv \\vdash (\\alpha \\vee \\beta)<\/span><\/span><\/td>\n<td>; DN(15)<\/td>\n<\/tr>\n<tr>\n<td>(17)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash ((\\alpha \\vee \\beta) \\rightarrow \\neg(\\neg\\alpha \\wedge \\neg\\beta) )<\/span><\/span><\/td>\n<td>; TD(16)<\/td>\n<\/tr>\n<tr>\n<td>(17)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash ((\\alpha \\vee \\beta) \\rightarrow \\gamma )<\/span><\/span><\/td>\n<td>; SH(17,12)<\/td>\n<\/tr>\n<tr>\n<td>(18)<\/td>\n<td><span style=\"color: #000088;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{ \\vdash \\gamma}<\/span><\/span><\/span><\/td>\n<td>; MP(3,17)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"10\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>Pruebas por casos (cas)<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=qAJ_oaSfd9k&amp;t=1957s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">Otra de las t\u00e9cnicas de la l\u00f3gica cl\u00e1sica<\/span><\/a> es la prueba por casos. Si una expresi\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> se puede inferir tanto desde otra expresi\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> como desde su negaci\u00f3n, entonces la expresi\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> es, por necesidad, un teorema. Esto se representa formalmente a trav\u00e9s de la escritura: <span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha \\vdash \\beta \\; \\wedge \\; \\neg\\alpha \\vdash \\beta <\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Longrightarrow<\/span><\/span> <span style=\"color: #000088;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash \\beta<\/span><\/span><\/span>. Su demostraci\u00f3n es la siguiente:<\/p>\n<p style=\"color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp; \\alpha \\vdash \\beta &amp;; Premisa\\\\\n\n(2) &amp; \\neg \\alpha \\vdash \\beta &amp;; Premisa \\\\\n\n(3) &amp; \\vdash \\alpha \\vee \\neg\\alpha &amp;; TAU \\\\\n\n(4) &amp; \\vdash \\beta &amp;; \\vee-Eliminacion3(1,2,3)\n\n\\end{array}\n\n<\/span><\/span><\/p>\n<p><a name=\"11\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>Reducci\u00f3n al Absurdo (absurdo)<\/h3>\n<p style=\"text-align: justify; color: #000000;\">Una de las t\u00e9cnicas de la l\u00f3gica cl\u00e1sica m\u00e1s utilizadas en las demostraciones, especialmente en las matem\u00e1ticas, es la de reducci\u00f3n al absurdo. Esta consiste en que si a partir de una expresi\u00f3n <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> se infiere una contradicci\u00f3n (una afirmaci\u00f3n y su negaci\u00f3n), entonces la negaci\u00f3n de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> es una tautolog\u00eda. Formalmente se expresa como: <span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash \\beta \\; \\wedge \\; \\{\\alpha\\}\\vdash \\neg\\beta<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Longrightarrow<\/span><\/span> <span style=\"color: #000088;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash \\neg\\alpha<\/span><\/span><\/span>. Y se puede demostrar a trav\u00e9s del siguiente razonamiento:<\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{\\alpha\\}\\vdash \\beta}<\/span><\/span><\/span><\/td>\n<td>; Premisa<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span style=\"color: #880000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\{\\alpha\\}\\vdash \\neg\\beta}<\/span><\/span><\/td>\n<td>; Premisa<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow \\beta)<\/span><\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow \\neg\\beta)<\/span><\/span><\/td>\n<td>; TD(2)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg \\beta \\rightarrow \\neg \\alpha)<\/span><\/span><\/td>\n<td>; CPI(3)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\beta \\rightarrow \\neg \\alpha)<\/span><\/span><\/td>\n<td>; CPI(4)<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg \\beta \\}\\vdash \\neg \\alpha<\/span><\/span><\/td>\n<td>; RTD(5)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\beta \\}\\vdash \\neg \\alpha<\/span><\/span><\/td>\n<td>; RTD(6)<\/td>\n<\/tr>\n<tr>\n<td>(9)<\/td>\n<td><span style=\"color: #000088;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\vdash \\neg \\alpha}<\/span><\/span><\/span><\/td>\n<td>; CAS(7,8)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica RESUMENEn esta clase se presentan varias t\u00e9cnicas de la l\u00f3gica cl\u00e1sica para introducir y eliminar conjunciones y disyunciones, adem\u00e1s de la regla del tercero excluido y la regla de contradicci\u00f3n, tambi\u00e9n conocida como el principio de explosi\u00f3n. Adem\u00e1s, se explica la t\u00e9cnica de prueba por casos y [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27451,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":17,"footnotes":""},"categories":[600,602,563],"tags":[],"class_list":["post-27450","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-logica-matematica","category-logica-proposicional","category-matematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"L\u00f3gica Cl\u00e1sica: introducci\u00f3n y eliminaci\u00f3n de conjunciones y disyunciones, tercero excluido, explosi\u00f3n, casos y reducci\u00f3n al absurdo.\" \/>\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\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica\" \/>\n<meta property=\"og:description\" content=\"L\u00f3gica Cl\u00e1sica: introducci\u00f3n y eliminaci\u00f3n de conjunciones y disyunciones, tercero excluido, explosi\u00f3n, casos y reducci\u00f3n al absurdo.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/\" \/>\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=\"2021-02-12T13:00:53+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-07-09T04:23:10+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/logicaclasica.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1024\" \/>\n\t<meta property=\"og:image:height\" content=\"356\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica\" \/>\n<meta name=\"twitter:description\" content=\"L\u00f3gica Cl\u00e1sica: introducci\u00f3n y eliminaci\u00f3n de conjunciones y disyunciones, tercero excluido, explosi\u00f3n, casos y reducci\u00f3n al absurdo.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/logicaclasica.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=\"7 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/demostracion-de-las-tecnicas-de-la-logica-clasica\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/demostracion-de-las-tecnicas-de-la-logica-clasica\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica\",\"datePublished\":\"2021-02-12T13:00:53+00:00\",\"dateModified\":\"2024-07-09T04:23:10+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/demostracion-de-las-tecnicas-de-la-logica-clasica\\\/\"},\"wordCount\":1869,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/demostracion-de-las-tecnicas-de-la-logica-clasica\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/logicaclasica.jpg\",\"articleSection\":[\"L\u00f3gica Matem\u00e1tica\",\"L\u00f3gica Proposicional\",\"Matem\u00e1tica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/demostracion-de-las-tecnicas-de-la-logica-clasica\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/demostracion-de-las-tecnicas-de-la-logica-clasica\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/demostracion-de-las-tecnicas-de-la-logica-clasica\\\/\",\"name\":\"Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica - toposuranos.com\\\/material\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/demostracion-de-las-tecnicas-de-la-logica-clasica\\\/#primaryimage\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/demostracion-de-las-tecnicas-de-la-logica-clasica\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/logicaclasica.jpg\",\"datePublished\":\"2021-02-12T13:00:53+00:00\",\"dateModified\":\"2024-07-09T04:23:10+00:00\",\"description\":\"L\u00f3gica Cl\u00e1sica: introducci\u00f3n y eliminaci\u00f3n de conjunciones y disyunciones, tercero excluido, explosi\u00f3n, casos y reducci\u00f3n al absurdo.\",\"breadcrumb\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/demostracion-de-las-tecnicas-de-la-logica-clasica\\\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/demostracion-de-las-tecnicas-de-la-logica-clasica\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/demostracion-de-las-tecnicas-de-la-logica-clasica\\\/#primaryimage\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/logicaclasica.jpg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/logicaclasica.jpg\",\"width\":1024,\"height\":356},{\"@type\":\"BreadcrumbList\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/demostracion-de-las-tecnicas-de-la-logica-clasica\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/cursos-de-matematica-y-fisica\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica\"}]},{\"@type\":\"WebSite\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#website\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/\",\"name\":\"toposuranos.com\\\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\",\"name\":\"toposuranos.com\\\/material\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\\\/material\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\"},\"sameAs\":[\"https:\\\/\\\/www.facebook.com\\\/groups\\\/toposuranos\",\"https:\\\/\\\/x.com\\\/topuranos\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\\\/\\\/www.linkedin.com\\\/company\\\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\\\/\\\/toposuranos.com\\\/material\"],\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/author\\\/giorgio-reveco\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica - toposuranos.com\/material","description":"L\u00f3gica Cl\u00e1sica: introducci\u00f3n y eliminaci\u00f3n de conjunciones y disyunciones, tercero excluido, explosi\u00f3n, casos y reducci\u00f3n al absurdo.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/","og_locale":"es_ES","og_type":"article","og_title":"Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica","og_description":"L\u00f3gica Cl\u00e1sica: introducci\u00f3n y eliminaci\u00f3n de conjunciones y disyunciones, tercero excluido, explosi\u00f3n, casos y reducci\u00f3n al absurdo.","og_url":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2021-02-12T13:00:53+00:00","article_modified_time":"2024-07-09T04:23:10+00:00","og_image":[{"width":1024,"height":356,"url":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/logicaclasica.jpg","type":"image\/jpeg"}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica","twitter_description":"L\u00f3gica Cl\u00e1sica: introducci\u00f3n y eliminaci\u00f3n de conjunciones y disyunciones, tercero excluido, explosi\u00f3n, casos y reducci\u00f3n al absurdo.","twitter_image":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/logicaclasica.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"7 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/#article","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/"},"author":{"name":"giorgio.reveco","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica","datePublished":"2021-02-12T13:00:53+00:00","dateModified":"2024-07-09T04:23:10+00:00","mainEntityOfPage":{"@id":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/"},"wordCount":1869,"commentCount":0,"publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/logicaclasica.jpg","articleSection":["L\u00f3gica Matem\u00e1tica","L\u00f3gica Proposicional","Matem\u00e1tica"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/#respond"]}]},{"@type":"WebPage","@id":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/","url":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/","name":"Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica - toposuranos.com\/material","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/#primaryimage"},"image":{"@id":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/logicaclasica.jpg","datePublished":"2021-02-12T13:00:53+00:00","dateModified":"2024-07-09T04:23:10+00:00","description":"L\u00f3gica Cl\u00e1sica: introducci\u00f3n y eliminaci\u00f3n de conjunciones y disyunciones, tercero excluido, explosi\u00f3n, casos y reducci\u00f3n al absurdo.","breadcrumb":{"@id":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/#primaryimage","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/logicaclasica.jpg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/logicaclasica.jpg","width":1024,"height":356},{"@type":"BreadcrumbList","@id":"http:\/\/toposuranos.com\/material\/es\/demostracion-de-las-tecnicas-de-la-logica-clasica\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"http:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"Demostraci\u00f3n de las T\u00e9cnicas de la L\u00f3gica Cl\u00e1sica"}]},{"@type":"WebSite","@id":"http:\/\/toposuranos.com\/material\/#website","url":"http:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"http:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"http:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"http:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"http:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/27450","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=27450"}],"version-history":[{"count":0,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/27450\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/27451"}],"wp:attachment":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=27450"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=27450"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=27450"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}