{"id":27464,"date":"2021-02-12T13:00:22","date_gmt":"2021-02-12T13:00:22","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27464"},"modified":"2024-07-09T04:41:37","modified_gmt":"2024-07-09T04:41:37","slug":"demonstracao-das-tecnicas-da-logica-classica","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/","title":{"rendered":"Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>RESUMO<\/strong><br \/><em>Nesta aula, v\u00e1rias t\u00e9cnicas da l\u00f3gica cl\u00e1ssica para introduzir e eliminar conjun\u00e7\u00f5es e disjun\u00e7\u00f5es s\u00e3o apresentadas, junto com a lei do terceiro exclu\u00eddo e a lei da contradi\u00e7\u00e3o, tamb\u00e9m conhecida como o princ\u00edpio da explos\u00e3o. Al\u00e9m disso, a t\u00e9cnica de prova por casos e redu\u00e7\u00e3o ao absurdo s\u00e3o explicadas, ambas muito \u00fateis em demonstra\u00e7\u00f5es matem\u00e1ticas e l\u00f3gicas em geral. Cada t\u00e9cnica \u00e9 apresentada formalmente e uma demonstra\u00e7\u00e3o passo a passo \u00e9 fornecida para compreens\u00e3o. Se voc\u00ea deseja aprofundar seu entendimento da l\u00f3gica proposicional e melhorar suas habilidades de demonstra\u00e7\u00e3o de teoremas, esta aula ser\u00e1 muito \u00fatil para voc\u00ea.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p><strong>OBJETIVOS DE APRENDIZAGEM:<\/strong><\/p>\n<ol>\n<li><strong>Compreender<\/strong> a justificativa por tr\u00e1s das t\u00e9cnicas de introdu\u00e7\u00e3o e elimina\u00e7\u00e3o da conjun\u00e7\u00e3o e disjun\u00e7\u00e3o.<\/li>\n<li><strong>Compreender<\/strong> a propriedade do terceiro exclu\u00eddo ou tautologia (TAU) na l\u00f3gica cl\u00e1ssica.<\/li>\n<li><strong>Compreender<\/strong> a regra da contradi\u00e7\u00e3o (CON) ou princ\u00edpio da explos\u00e3o na l\u00f3gica cl\u00e1ssica.<\/li>\n<li><strong>Compreender<\/strong> a t\u00e9cnica de elimina\u00e7\u00e3o de disjuntos (\u2228-elimina\u00e7\u00e3o3) na l\u00f3gica cl\u00e1ssica.<\/li>\n<li><strong>Compreender<\/strong> a t\u00e9cnica de prova por casos (CAS) na l\u00f3gica cl\u00e1ssica.<\/li>\n<li><strong>Compreender<\/strong> a t\u00e9cnica de redu\u00e7\u00e3o ao absurdo (absurdo) na l\u00f3gica cl\u00e1ssica.<\/li>\n<li><strong>Aplicar<\/strong> o conhecimento das diferentes t\u00e9cnicas da l\u00f3gica cl\u00e1ssica para resolver problemas e demonstra\u00e7\u00f5es complexas.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>\u00cdNDICE<\/strong><br \/>\n<a href=\"#1\"><strong>INTRODU\u00c7\u00c3O E ELIMINA\u00c7\u00c3O DE CONJUN\u00c7\u00d5ES E DISJUN\u00c7\u00d5ES<\/strong><\/a><br \/>\n<a href=\"#2\">\u2228-INTRODU\u00c7\u00c3O<\/a><br \/>\n<a href=\"#3\">\u2228-ELIMINA\u00c7\u00c3O<\/a><br \/>\n<a href=\"#4\">\u2227-INTRODU\u00c7\u00c3O<\/a><br \/>\n<a href=\"#5\">\u2227-ELIMINA\u00c7\u00c3O<\/a><br \/>\n<a href=\"#6\"><strong>T\u00c9CNICAS DE CONTRADI\u00c7\u00d5ES E TAUTOLOGIAS<\/strong><\/a><br \/>\n<a href=\"#7\">REGRA DO TERCEIRO EXCLU\u00cdDO OU TAUTOLOGIA (TAU)<\/a><br \/>\n<a href=\"#8\">REGRA DA CONTRADI\u00c7\u00c3O OU PRINC\u00cdPIO DA EXPLOS\u00c3O<\/a><br \/>\n<a href=\"#9\">\u2228-ELIMINA\u00c7\u00c3O3<\/a><br \/>\n<a href=\"#10\">PROVA POR CASOS (CAS)<\/a><br \/>\n<a href=\"#11\">REDU\u00c7\u00c3O AO 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>Introdu\u00e7\u00e3o e Elimina\u00e7\u00e3o de Conjun\u00e7\u00f5es e Disjun\u00e7\u00f5es<\/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;\">Uma das t\u00e9cnicas<\/span><\/strong><\/a> da l\u00f3gica cl\u00e1ssica \u00e9 a introdu\u00e7\u00e3o e elimina\u00e7\u00e3o de conectores e disjuntos. Embora essas t\u00e9cnicas sejam executadas de maneira mais ou menos intuitiva, sua justificativa n\u00e3o \u00e9 inteiramente trivial, mas pode ser obtida a partir das regras da l\u00f3gica proposicional que j\u00e1 demonstramos em aulas anteriores. Formalmente, as t\u00e9cnicas de introdu\u00e7\u00e3o e elimina\u00e7\u00e3o de conectores e disjuntos s\u00e3o as seguintes:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>\u2228-Introdu\u00e7\u00e3o<\/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-Elimina\u00e7\u00e3o<\/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-Introdu\u00e7\u00e3o<\/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-Elimina\u00e7\u00e3o<\/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;\">E suas demonstra\u00e7\u00f5es a partir da l\u00f3gica proposicional s\u00e3o mostradas a seguir:<\/p>\n<p><a name=\"2\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>\u2228-Introdu\u00e7\u00e3o<\/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>-Defini\u00e7\u00e3o(3)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>\u2228-Elimina\u00e7\u00e3o<\/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>-Defini\u00e7\u00e3o (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-Introdu\u00e7\u00e3o<\/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>-Elimina\u00e7\u00e3o<\/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>; Monotonicidade 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>-Defini\u00e7\u00e3o(11)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"5\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>\u2227-Elimina\u00e7\u00e3o<\/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>-Introdu\u00e7\u00e3o<\/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>-defini\u00e7\u00e3o(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>; Monotonicidade(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 Contradi\u00e7\u00f5es e Tautologias<\/h2>\n<p><a name=\"7\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>Regra do Terceiro Exclu\u00eddo ou Tautologia (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;\">Outra caracter\u00edstica not\u00e1vel<\/span><\/strong><\/a> da l\u00f3gica cl\u00e1ssica \u00e9 a propriedade do terceiro exclu\u00eddo (tertium non datur). Estabelece que se existem duas afirma\u00e7\u00f5es onde uma nega a outra, ent\u00e3o necessariamente uma das duas deve ser verdadeira; em outras palavras, a conjun\u00e7\u00e3o de duas afirma\u00e7\u00f5es em que uma nega a outra necessariamente forma uma tautologia. Formalmente, isso \u00e9 expresso como:<\/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;\">E sua demonstra\u00e7\u00e3o \u00e9 f\u00e1cil de obter.<\/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;\">Outra maneira de enunciar o princ\u00edpio do terceiro exclu\u00eddo \u00e9 atrav\u00e9s da lei da <strong>n\u00e3o contradi\u00e7\u00e3o<\/strong>, que estabelece que uma afirma\u00e7\u00e3o n\u00e3o pode ser verdadeira e falsa ao mesmo tempo e que \u00e9 formalmente enunciada como:<\/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 propriedade n\u00e3o precisa de demonstra\u00e7\u00e3o, n\u00e3o porque seja autoevidente por si mesma, mas porque \u00e9 obtida diretamente aplicando a defini\u00e7\u00e3o de conjun\u00e7\u00e3o ao princ\u00edpio do terceiro exclu\u00eddo.<\/p>\n<p><a name=\"8\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h3>Regra da Contradi\u00e7\u00e3o ou Princ\u00edpio da Explos\u00e3o<\/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;\">Outra propriedade conhecida da l\u00f3gica<\/span><\/strong><\/a> cl\u00e1ssica \u00e9 o princ\u00edpio da explos\u00e3o, que geralmente \u00e9 enunciado atrav\u00e9s da frase \u00abde premissas contradit\u00f3rias, qualquer coisa pode ser conclu\u00edda\u00bb. Sua formula\u00e7\u00e3o \u00e9 frequentemente apresentada de uma das duas maneiras a seguir:<\/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;\">A demonstra\u00e7\u00e3o desta regra \u00e9 simples:<\/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>-introdu\u00e7\u00e3o<\/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>-defini\u00e7\u00e3o(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-Elimina\u00e7\u00e3o3<\/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;\">O modus ponens pode ser escrito<\/span><\/strong><\/a> de duas maneiras diferentes. Uma forma que j\u00e1 conhecemos \u00e9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha,(\\alpha \\rightarrow \\beta)\\}\\vdash \\beta<\/span><\/span>. A outra \u00e9 um pouco 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;\">Focando nesta segunda forma, \u00e9 poss\u00edvel visualizar uma expans\u00e3o para esta regra que chamamos de <strong>\u2228-Elimina\u00e7\u00e3o3<\/strong>, porque se assemelha a uma simplifica\u00e7\u00e3o obtida a partir de uma disjun\u00e7\u00e3o. Diz-nos que se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> pode ser inferido a partir de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> e de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> (ambos ao mesmo tempo) e ao mesmo tempo a disjun\u00e7\u00e3o entre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> \u00e9 um teorema, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> \u00e9 um teorema. Isso \u00e9 resumido formalmente da seguinte maneira:<\/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;\">A demonstra\u00e7\u00e3o desta t\u00e9cnica da l\u00f3gica cl\u00e1ssica \u00e9 a seguinte:<\/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>; Premissa<\/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>; Premissa<\/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>; Premissa<\/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>-Introdu\u00e7\u00e3o(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; Defini\u00e7\u00e3o<\/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 ambos os lados em (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>; Substituindo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A:=\\neg\\alpha<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B:=\\neg\\beta<\/span><\/span> em (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>Prova 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;\">Outra t\u00e9cnica da l\u00f3gica cl\u00e1ssica<\/span><\/a> \u00e9 a prova por casos. Se uma express\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> pode ser inferida a partir de outra express\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> e de sua nega\u00e7\u00e3o, ent\u00e3o a express\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> \u00e9 necessariamente um teorema. Isso \u00e9 representado formalmente como: <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>. Sua demonstra\u00e7\u00e3o \u00e9 a seguinte:<\/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;; Premissa\\\\\n\n(2) &amp; \\neg \\alpha \\vdash \\beta &amp;; Premissa \\\\\n\n(3) &amp; \\vdash \\alpha \\vee \\neg\\alpha &amp;; TAU \\\\\n\n(4) &amp; \\vdash \\beta &amp;; \\vee-Elimina\u00e7\u00e3o3(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>Redu\u00e7\u00e3o ao Absurdo (absurdo)<\/h3>\n<p style=\"text-align: justify; color: #000000;\">Uma das t\u00e9cnicas mais utilizadas da l\u00f3gica cl\u00e1ssica em demonstra\u00e7\u00f5es, especialmente na matem\u00e1tica, \u00e9 a redu\u00e7\u00e3o ao absurdo. Esta consiste em que se uma contradi\u00e7\u00e3o (uma afirma\u00e7\u00e3o e sua nega\u00e7\u00e3o) \u00e9 inferida a partir de uma express\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span>, ent\u00e3o a nega\u00e7\u00e3o de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> \u00e9 uma tautologia. Formalmente, \u00e9 expresso 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>. E pode ser demonstrado atrav\u00e9s do seguinte racioc\u00ednio:<\/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>; Premissa<\/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>; Premissa<\/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>Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica RESUMONesta aula, v\u00e1rias t\u00e9cnicas da l\u00f3gica cl\u00e1ssica para introduzir e eliminar conjun\u00e7\u00f5es e disjun\u00e7\u00f5es s\u00e3o apresentadas, junto com a lei do terceiro exclu\u00eddo e a lei da contradi\u00e7\u00e3o, tamb\u00e9m conhecida como o princ\u00edpio da explos\u00e3o. Al\u00e9m disso, a t\u00e9cnica de prova por casos e redu\u00e7\u00e3o ao absurdo s\u00e3o explicadas, [&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":8,"footnotes":""},"categories":[607,621,571],"tags":[],"class_list":["post-27464","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-logica-matematica-pt","category-logica-proposicional-pt","category-matematica-pt"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"L\u00f3gica Cl\u00e1ssica: introdu\u00e7\u00e3o e elimina\u00e7\u00e3o de conjun\u00e7\u00f5es e disjun\u00e7\u00f5es, terceiro exclu\u00eddo, explos\u00e3o, casos e redu\u00e7\u00e3o ao 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\/pt\/demonstracao-das-tecnicas-da-logica-classica\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica\" \/>\n<meta property=\"og:description\" content=\"L\u00f3gica Cl\u00e1ssica: introdu\u00e7\u00e3o e elimina\u00e7\u00e3o de conjun\u00e7\u00f5es e disjun\u00e7\u00f5es, terceiro exclu\u00eddo, explos\u00e3o, casos e redu\u00e7\u00e3o ao absurdo.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/\" \/>\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=\"2021-02-12T13:00:22+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-07-09T04:41:37+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/logicaclasica.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica\" \/>\n<meta name=\"twitter:description\" content=\"L\u00f3gica Cl\u00e1ssica: introdu\u00e7\u00e3o e elimina\u00e7\u00e3o de conjun\u00e7\u00f5es e disjun\u00e7\u00f5es, terceiro exclu\u00eddo, explos\u00e3o, casos e redu\u00e7\u00e3o ao 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=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/demonstracao-das-tecnicas-da-logica-classica\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/demonstracao-das-tecnicas-da-logica-classica\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica\",\"datePublished\":\"2021-02-12T13:00:22+00:00\",\"dateModified\":\"2024-07-09T04:41:37+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/demonstracao-das-tecnicas-da-logica-classica\\\/\"},\"wordCount\":1785,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/demonstracao-das-tecnicas-da-logica-classica\\\/#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\\\/pt\\\/demonstracao-das-tecnicas-da-logica-classica\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/demonstracao-das-tecnicas-da-logica-classica\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/demonstracao-das-tecnicas-da-logica-classica\\\/\",\"name\":\"Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica - toposuranos.com\\\/material\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/demonstracao-das-tecnicas-da-logica-classica\\\/#primaryimage\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/demonstracao-das-tecnicas-da-logica-classica\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/07\\\/logicaclasica.jpg\",\"datePublished\":\"2021-02-12T13:00:22+00:00\",\"dateModified\":\"2024-07-09T04:41:37+00:00\",\"description\":\"L\u00f3gica Cl\u00e1ssica: introdu\u00e7\u00e3o e elimina\u00e7\u00e3o de conjun\u00e7\u00f5es e disjun\u00e7\u00f5es, terceiro exclu\u00eddo, explos\u00e3o, casos e redu\u00e7\u00e3o ao absurdo.\",\"breadcrumb\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/demonstracao-das-tecnicas-da-logica-classica\\\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/demonstracao-das-tecnicas-da-logica-classica\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/demonstracao-das-tecnicas-da-logica-classica\\\/#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\\\/pt\\\/demonstracao-das-tecnicas-da-logica-classica\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/cursos-de-matematica-y-fisica\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica\"}]},{\"@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":"Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica - toposuranos.com\/material","description":"L\u00f3gica Cl\u00e1ssica: introdu\u00e7\u00e3o e elimina\u00e7\u00e3o de conjun\u00e7\u00f5es e disjun\u00e7\u00f5es, terceiro exclu\u00eddo, explos\u00e3o, casos e redu\u00e7\u00e3o ao 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\/pt\/demonstracao-das-tecnicas-da-logica-classica\/","og_locale":"es_ES","og_type":"article","og_title":"Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica","og_description":"L\u00f3gica Cl\u00e1ssica: introdu\u00e7\u00e3o e elimina\u00e7\u00e3o de conjun\u00e7\u00f5es e disjun\u00e7\u00f5es, terceiro exclu\u00eddo, explos\u00e3o, casos e redu\u00e7\u00e3o ao absurdo.","og_url":"http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2021-02-12T13:00:22+00:00","article_modified_time":"2024-07-09T04:41:37+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/logicaclasica.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica","twitter_description":"L\u00f3gica Cl\u00e1ssica: introdu\u00e7\u00e3o e elimina\u00e7\u00e3o de conjun\u00e7\u00f5es e disjun\u00e7\u00f5es, terceiro exclu\u00eddo, explos\u00e3o, casos e redu\u00e7\u00e3o ao 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":"1 minuto"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/#article","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/"},"author":{"name":"giorgio.reveco","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica","datePublished":"2021-02-12T13:00:22+00:00","dateModified":"2024-07-09T04:41:37+00:00","mainEntityOfPage":{"@id":"http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/"},"wordCount":1785,"commentCount":0,"publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/#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\/pt\/demonstracao-das-tecnicas-da-logica-classica\/#respond"]}]},{"@type":"WebPage","@id":"http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/","url":"http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/","name":"Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica - toposuranos.com\/material","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/#primaryimage"},"image":{"@id":"http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/07\/logicaclasica.jpg","datePublished":"2021-02-12T13:00:22+00:00","dateModified":"2024-07-09T04:41:37+00:00","description":"L\u00f3gica Cl\u00e1ssica: introdu\u00e7\u00e3o e elimina\u00e7\u00e3o de conjun\u00e7\u00f5es e disjun\u00e7\u00f5es, terceiro exclu\u00eddo, explos\u00e3o, casos e redu\u00e7\u00e3o ao absurdo.","breadcrumb":{"@id":"http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/pt\/demonstracao-das-tecnicas-da-logica-classica\/#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\/pt\/demonstracao-das-tecnicas-da-logica-classica\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"http:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"Demonstra\u00e7\u00e3o das T\u00e9cnicas da L\u00f3gica Cl\u00e1ssica"}]},{"@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\/27464","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=27464"}],"version-history":[{"count":0,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/27464\/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=27464"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=27464"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=27464"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}