{"id":27345,"date":"2021-01-27T13:00:27","date_gmt":"2021-01-27T13:00:27","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27345"},"modified":"2024-07-07T23:43:37","modified_gmt":"2024-07-07T23:43:37","slug":"4-tecnicas-de-deducao-essenciais","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/4-tecnicas-de-deducao-essenciais\/","title":{"rendered":"4 t\u00e9cnicas de dedu\u00e7\u00e3o essenciais"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Aprenda 4 T\u00e9cnicas Essenciais de Dedu\u00e7\u00e3o<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><em><strong>Resumo:<\/strong><\/br>Nesta aula, s\u00e3o descritas 4 t\u00e9cnicas de dedu\u00e7\u00e3o da l\u00f3gica proposicional para enriquecer o c\u00e1lculo proposicional rudimentar apresentado at\u00e9 agora. Apresenta-se a regra da presun\u00e7\u00e3o e sua combina\u00e7\u00e3o com a regra da monotonia, bem como o silogismo hipot\u00e9tico e duas formas de obter essa regra de dedu\u00e7\u00e3o. Tamb\u00e9m s\u00e3o explicadas as equival\u00eancias de dupla nega\u00e7\u00e3o e o contrarrec\u00edproco da implica\u00e7\u00e3o.<\/em><\/p>\n<p style=\"text-align:center;\"><strong><u>Objetivos de Aprendizagem<\/u>:<\/strong><br \/>Ao final desta aula, o aluno ser\u00e1 capaz de<\/p>\n<ol>\n<li><strong>Recordar<\/strong> a estrutura de um racioc\u00ednio e exemplos simples.<\/li>\n<li><strong>Compreender<\/strong> a regra da presun\u00e7\u00e3o e sua rela\u00e7\u00e3o com o teorema da dedu\u00e7\u00e3o.<\/li>\n<li><strong>Compreender<\/strong> a regra do silogismo hipot\u00e9tico e sua rela\u00e7\u00e3o com o modus ponens.<\/li>\n<li><strong>Aplicar<\/strong> o teorema da dedu\u00e7\u00e3o na l\u00f3gica proposicional.<\/li>\n<li><strong>Aplicar<\/strong> a regra da monotonia na dedu\u00e7\u00e3o de express\u00f5es.<\/li>\n<li><strong>Compreender<\/strong> as equival\u00eancias de dupla nega\u00e7\u00e3o e o contrarrec\u00edproco da implica\u00e7\u00e3o na l\u00f3gica proposicional.<\/li>\n<li><strong>Conhecer<\/strong> as demonstra\u00e7\u00f5es das t\u00e9cnicas de dedu\u00e7\u00e3o e ser capaz de aplic\u00e1-las na pr\u00e1tica.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>\u00cdNDICE DE CONTE\u00daDOS<\/strong><br \/>\n<a href=\"#1\">REGRA DA PRESUN\u00c7\u00c3O (PRE)<\/a><br \/>\n<a href=\"#2\">O SILOGISMO HIPOT\u00c9TICO (SH)<\/a><br \/>\n<a href=\"#3\">EQUIVAL\u00caNCIAS DE DUPLA NEGA\u00c7\u00c3O (DN)<\/a><br \/>\n<a href=\"#4\">EQUIVAL\u00caNCIA DO CONTRARREC\u00cdPROCO DA IMPLICA\u00c7\u00c3O (CPI)<\/a>\n<\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/6f_aavuC4E0\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\">J\u00e1 vimos como \u00e9 a estrutura de um racioc\u00ednio e exemplos simples. Agora colocaremos esse conhecimento \u00e0 prova <strong>raciocinando com 4 t\u00e9cnicas de dedu\u00e7\u00e3o da l\u00f3gica proposicional.<\/strong> Atrav\u00e9s disso, n\u00e3o apenas veremos que essas coisas funcionam, mas tamb\u00e9m come\u00e7aremos a dar uma certa riqueza de procedimentos que tirar\u00e1 o c\u00e1lculo proposicional do estado rudimentar em que foi apresentado at\u00e9 agora.<\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>Se <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> s\u00e3o express\u00f5es do c\u00e1lculo proposicional, ent\u00e3o \u00e9 poss\u00edvel inferir as seguintes t\u00e9cnicas de dedu\u00e7\u00e3o a partir dos fundamentos:<\/strong><\/p>\n<p><a name=\"1\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Regra da Presun\u00e7\u00e3o (Pre)<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=6f_aavuC4E0&amp;t=168s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">A regra de dedu\u00e7\u00e3o mais simples<\/span><\/strong><\/a> de todas \u00e9 a presun\u00e7\u00e3o. Esta \u00e9 obtida diretamente pela aplica\u00e7\u00e3o do <strong>rec\u00edproco do teorema da dedu\u00e7\u00e3o<\/strong> sobre o teorema <span class=\"katex-eq\" data-katex-display=\"false\">\\vdash(\\alpha\\rightarrow\\alpha)<\/span>. Se isso te pareceu linguagem arcaica, tudo o que voc\u00ea precisa saber est\u00e1 <a href=\"http:\/\/toposuranos.com\/material\/pt\/sistemas-dedutivos-formais-em-logica-proposicional\/\" rel=\"noopener\" target=\"_blank\"><strong>aqui<\/strong><\/a>.<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash \\alpha <\/span>\n<p style=\"text-align: justify; color: #000000;\">Combinada com a regra da monotonia, permitir\u00e1 adicionar express\u00f5es convenientes dentro de suas dedu\u00e7\u00f5es.<\/p>\n<p><a name=\"2\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>O Silogismo Hipot\u00e9tico (SH)<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=6f_aavuC4E0&amp;t=206s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">O silogismo hipot\u00e9tico<\/span><\/strong><\/a>, ou transitividade da implica\u00e7\u00e3o, \u00e9 uma esp\u00e9cie de evolu\u00e7\u00e3o do modus ponens. Sua formula\u00e7\u00e3o \u00e9 a seguinte:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha\\rightarrow\\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash (\\alpha\\rightarrow\\gamma)<\/span>\n<p style=\"text-align: justify; color: #000000;\">Existem v\u00e1rias formas de obter essa regra de dedu\u00e7\u00e3o, veremos algumas delas em breve.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Se raciocinarmos a partir de express\u00f5es, ser\u00e1 f\u00e1cil construir o seguinte racioc\u00ednio:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/td>\n<td>; Premissa<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/td>\n<td>; Premissa<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(\\beta\\rightarrow \\gamma)<\/span><\/td>\n<td>; Premissa<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/td>\n<td>; MP(1,2)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/td>\n<td>; MP(4,3)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Portanto<span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha,(\\alpha\\rightarrow\\beta),(\\beta\\rightarrow\\gamma)\\}\\vdash\\gamma<\/span>\n<p style=\"text-align: justify; color: #000000;\">Finalmente, aplicando o teorema da dedu\u00e7\u00e3o sobre esta \u00faltima express\u00e3o, temos que:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{(\\alpha\\rightarrow\\beta),(\\beta\\rightarrow\\gamma)\\}\\vdash(\\alpha\\rightarrow \\gamma)<\/span>\n<p style=\"text-align: justify; color: #000000;\">Outra forma de obter a demonstra\u00e7\u00e3o desta regra \u00e9 raciocinando a partir de dedu\u00e7\u00f5es, construindo atrav\u00e9s da presun\u00e7\u00e3o e da monotonia. Observe o seguinte racioc\u00ednio a partir de dedu\u00e7\u00f5es:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash \\alpha <\/span><\/td>\n<td>; Presun\u00e7\u00e3o e Monotonia<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash (\\alpha\\rightarrow \\beta) <\/span><\/td>\n<td>; Presun\u00e7\u00e3o e Monotonia<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash (\\beta\\rightarrow\\gamma) <\/span><\/td>\n<td>; Presun\u00e7\u00e3o e Monotonia<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash \\beta <\/span><\/td>\n<td>; MP(1,2)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha, (\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash \\gamma <\/span><\/td>\n<td>; MP(4,3)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha\\rightarrow \\beta), (\\beta\\rightarrow\\gamma)\\}\\vdash (\\alpha \\rightarrow \\gamma) <\/span><\/td>\n<td>; TD(5)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Voc\u00ea deve notar aqui que ambas as demonstra\u00e7\u00f5es s\u00e3o id\u00eanticas, apenas desenvolvidas em estilos diferentes. Na pr\u00e1tica, voc\u00ea pode alternar entre ambos os estilos dependendo do que achar mais confort\u00e1vel.<\/p>\n<p><a name=\"3\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Equival\u00eancias de Dupla Nega\u00e7\u00e3o (DN)<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=6f_aavuC4E0&amp;t=500s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">As equival\u00eancias de dupla nega\u00e7\u00e3o<\/span><\/strong><\/a> reproduzem a no\u00e7\u00e3o intuitiva de que a dupla nega\u00e7\u00e3o de uma afirma\u00e7\u00e3o \u00e9 equivalente \u00e0 pr\u00f3pria afirma\u00e7\u00e3o. Isso, escrito simbolicamente, ser\u00e1 na forma<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha\\dashv\\vdash\\neg\\neg\\alpha<\/span>\n<p style=\"text-align: justify; color: #000000;\">Vamos agora ver uma demonstra\u00e7\u00e3o:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg\\neg \\alpha \\rightarrow (\\neg\\neg\\neg\\neg \\alpha \\rightarrow\\neg\\neg\\alpha))<\/span><\/td>\n<td>; A1<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash ((\\neg\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\alpha)\\rightarrow(\\neg\\alpha \\rightarrow \\neg\\neg\\neg\\alpha))<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash ((\\neg\\alpha \\rightarrow \\neg\\neg\\neg\\alpha)\\rightarrow(\\neg\\neg\\alpha \\rightarrow \\alpha))<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash ((\\neg\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\alpha)\\rightarrow(\\neg\\neg\\alpha \\rightarrow \\alpha))<\/span><\/td>\n<td>; SH(2,3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash (\\neg\\neg\\neg\\neg \\alpha \\rightarrow\\neg\\neg\\alpha)<\/span><\/td>\n<td>; RTD(1)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash ((\\neg\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\alpha)\\rightarrow(\\neg\\neg\\alpha \\rightarrow \\alpha))<\/span><\/td>\n<td>; Monotonia(4)<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash (\\neg\\neg\\alpha \\rightarrow \\alpha)<\/span><\/td>\n<td>; MP(5,6)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash \\alpha<\/span><\/td>\n<td>; RTD(7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Portanto<span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\alpha \\} \\vdash \\alpha <\/span>\n<p style=\"text-align: justify; color: #000000;\">Para fazer a demonstra\u00e7\u00e3o na outra dire\u00e7\u00e3o, podemos usar esta que acabamos de fazer readaptando-a atrav\u00e9s de uma simples substitui\u00e7\u00e3o, obtendo o seguinte:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\neg \\alpha \\} \\vdash \\neg \\alpha <\/span>\n<p style=\"text-align: justify; color: #000000;\">E a partir disso, constru\u00edmos a demonstra\u00e7\u00e3o na outra dire\u00e7\u00e3o:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\neg\\neg \\neg \\alpha \\} \\vdash \\neg \\alpha <\/span><\/td>\n<td>; O que acabamos de provar<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash(\\neg\\neg \\neg \\alpha\\rightarrow \\neg \\alpha) <\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash((\\neg\\neg \\neg \\alpha\\rightarrow \\neg \\alpha) \\rightarrow(\\alpha \\rightarrow\\neg\\neg\\alpha)) <\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash(\\alpha \\rightarrow\\neg\\neg\\alpha) <\/span><\/td>\n<td>; MP(2,3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash\\neg\\neg\\alpha <\/span><\/td>\n<td>; RTD(4)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Portanto <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha \\} \\vdash \\neg\\neg \\alpha <\/span>\n<p style=\"text-align: justify; color: #000000;\">Finalmente, dessas duas demonstra\u00e7\u00f5es, temos que <span class=\"katex-eq\" data-katex-display=\"false\"> \\alpha \\dashv\\vdash \\neg\\neg \\alpha <\/span>.<\/p>\n<p><a name=\"4\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Equival\u00eancia do Contrarrec\u00edproco da Implica\u00e7\u00e3o (CpI)<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=6f_aavuC4E0&amp;t=948s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Isso corresponde<\/span><\/strong><\/a> \u00e0s seguintes equival\u00eancias<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta) \\dashv\\vdash (\\neg\\beta \\rightarrow \\neg\\alpha)<\/span>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n(\\neg\\alpha\\rightarrow\\beta)\\dashv\\vdash (\\neg\\beta\\rightarrow\\alpha)<\/span>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\rightarrow\\neg\\beta) \\dashv\\vdash (\\beta\\rightarrow\\neg\\alpha)<\/span>\n<p style=\"text-align: justify; color: #000000;\">A demonstra\u00e7\u00e3o dessa primeira rela\u00e7\u00e3o \u00e9 feita da seguinte maneira:<\/p>\n<p style=\"text-align: justify; color: #000000;\">De um lado, obt\u00e9m-se diretamente do terceiro axioma<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash ((\\neg\\beta\\rightarrow \\neg\\alpha) \\rightarrow (\\alpha \\rightarrow\\beta))<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow \\neg\\alpha)\\}\\vdash (\\alpha \\rightarrow \\beta)<\/span><\/td>\n<td>; RTD(1)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Portanto <span class=\"katex-eq\" data-katex-display=\"false\"> \\{(\\neg\\beta\\rightarrow \\neg\\alpha)\\}\\vdash (\\alpha \\rightarrow \\beta)<\/span>\n<p style=\"text-align: justify; color: #000000;\">E na outra dire\u00e7\u00e3o, a demonstra\u00e7\u00e3o pode ser obtida a partir do seguinte racioc\u00ednio:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg\\alpha \\dashv \\vdash \\alpha<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\neg \\alpha \\rightarrow \\alpha)<\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg\\beta \\dashv \\vdash \\beta<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\beta \\rightarrow \\neg\\neg \\beta)<\/span><\/td>\n<td>; TD(3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg \\alpha \\rightarrow \\alpha)<\/span><\/td>\n<td>; Mon(2)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\alpha \\rightarrow \\beta)<\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg \\alpha \\rightarrow\\beta)<\/span><\/td>\n<td>; SH(5,6)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\{(\\alpha \\rightarrow \\beta)\\} \\vdash (\\beta \\rightarrow \\neg\\neg \\beta)<\/span><\/td>\n<td>; Mon(4)<\/td>\n<\/tr>\n<tr>\n<td>(9)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg \\alpha \\rightarrow \\neg\\neg \\beta)<\/span><\/td>\n<td>; SH(7,8)<\/td>\n<\/tr>\n<tr>\n<td>(10)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\neg \\alpha \\rightarrow \\neg\\neg \\beta) \\rightarrow (\\neg \\beta \\rightarrow \\neg \\alpha )<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(11)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash ((\\neg\\neg \\alpha \\rightarrow \\neg\\neg \\beta) \\rightarrow (\\neg \\beta \\rightarrow \\neg \\alpha ))<\/span><\/td>\n<td>; Mon(10)<\/td>\n<\/tr>\n<tr>\n<td>(11)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg \\beta \\rightarrow \\neg \\alpha )<\/span><\/td>\n<td>; SH(10;11)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Portanto <span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg \\beta \\rightarrow \\neg \\alpha )<\/span>\n<p style=\"text-align: justify; color: #000000;\">Portanto, dos dois racioc\u00ednios anteriores, temos que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\"> (\\alpha \\rightarrow \\beta) \\dashv\\vdash (\\neg \\beta \\rightarrow \\neg \\alpha ) <\/span>\n<p style=\"text-align: justify; color: #000000;\">Para demonstrar a segunda, podemos fazer os seguintes dois racioc\u00ednios:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta \\dashv\\vdash \\neg\\neg\\beta<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg\\neg\\alpha \\dashv\\vdash \\neg\\alpha<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\beta \\rightarrow \\neg\\neg\\beta)<\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\alpha)<\/span><\/td>\n<td>; TD(2)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\alpha \\rightarrow \\beta)<\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\beta \\rightarrow \\neg\\neg\\beta)<\/span><\/td>\n<td>; Mon(3)<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\alpha)<\/span><\/td>\n<td>; Mon(4)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\alpha \\rightarrow \\neg\\neg\\beta)<\/span><\/td>\n<td>; SH(5,6)<\/td>\n<\/tr>\n<tr>\n<td>(9)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\beta)<\/span><\/td>\n<td>; SH(7,8)<\/td>\n<\/tr>\n<tr>\n<td>(10)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\beta) \\rightarrow (\\neg\\beta \\rightarrow \\neg\\neg\\alpha)<\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(11)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash ((\\neg\\neg\\neg\\alpha \\rightarrow \\neg\\neg\\beta) \\rightarrow (\\neg\\beta \\rightarrow \\neg\\neg\\alpha))<\/span><\/td>\n<td>; Mon(10)<\/td>\n<\/tr>\n<tr>\n<td>(12)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\}\\vdash (\\neg\\beta \\rightarrow \\neg\\neg\\alpha)<\/span><\/td>\n<td>; MP(9,11)<\/td>\n<\/tr>\n<tr>\n<td>(13)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg \\alpha \\dashv \\vdash \\alpha<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(14)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\neg\\neg \\alpha\\rightarrow \\alpha)<\/span><\/td>\n<td>; TD(13)<\/td>\n<\/tr>\n<tr>\n<td>(15)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\} \\vdash (\\neg\\neg \\alpha\\rightarrow \\alpha)<\/span><\/td>\n<td>; Mon(14)<\/td>\n<\/tr>\n<tr>\n<td>(16)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\} \\vdash(\\neg\\beta \\rightarrow \\alpha)<\/span><\/td>\n<td>; SH(12,15)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Portanto <span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\alpha \\rightarrow \\beta)\\} \\vdash(\\neg\\beta \\rightarrow \\alpha) <\/span>\n<p style=\"text-align: justify; color: #000000;\">Agora precisamos fazer a demonstra\u00e7\u00e3o na dire\u00e7\u00e3o inversa. Podemos faz\u00ea-lo atrav\u00e9s do seguinte racioc\u00ednio:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha \\dashv \\vdash \\neg\\neg\\alpha<\/span><\/td>\n<td>; DN<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow \\neg\\neg\\alpha)<\/span><\/td>\n<td>; TD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\neg\\beta\\rightarrow\\alpha)<\/span><\/td>\n<td>; Pre<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\alpha \\rightarrow \\neg\\neg\\alpha)<\/span><\/td>\n<td>; Mon(2)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\neg\\beta\\rightarrow\\neg\\neg\\alpha)<\/span><\/td>\n<td>; SH(3,4)<\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\neg\\beta\\rightarrow\\neg\\neg\\alpha)\\rightarrow (\\neg\\alpha \\rightarrow \\beta) <\/span><\/td>\n<td>; A3<\/td>\n<\/tr>\n<tr>\n<td>(7)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash ((\\neg\\beta\\rightarrow\\neg\\neg\\alpha)\\rightarrow (\\neg\\alpha \\rightarrow \\beta)) <\/span><\/td>\n<td>; Mon(6)<\/td>\n<\/tr>\n<tr>\n<td>(8)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\neg\\alpha \\rightarrow \\beta) <\/span><\/td>\n<td>; MP(5,7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">Portanto <span class=\"katex-eq\" data-katex-display=\"false\"> \\{(\\neg\\beta\\rightarrow\\alpha)\\}\\vdash (\\neg\\alpha \\rightarrow \\beta) <\/span>\n<p style=\"text-align: justify; color: #000000;\">Finalmente, desses dois racioc\u00ednios, conclui-se que <span class=\"katex-eq\" data-katex-display=\"false\"> (\\neg\\beta\\rightarrow\\alpha) \\dashv \\vdash (\\neg\\alpha \\rightarrow \\beta) <\/span>, que \u00e9 o que se queria demonstrar.<\/p>\n<p style=\"text-align: justify; color: #000000;\">A \u00faltima equival\u00eancia ficar\u00e1 como exerc\u00edcio. Para demonstr\u00e1-la, voc\u00ea pode se guiar com as duas demonstra\u00e7\u00f5es que j\u00e1 dei. Esta \u00e9 a melhor forma que existe para dominar as t\u00e9cnicas de dedu\u00e7\u00e3o.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Aprenda 4 T\u00e9cnicas Essenciais de Dedu\u00e7\u00e3o Resumo:Nesta aula, s\u00e3o descritas 4 t\u00e9cnicas de dedu\u00e7\u00e3o da l\u00f3gica proposicional para enriquecer o c\u00e1lculo proposicional rudimentar apresentado at\u00e9 agora. Apresenta-se a regra da presun\u00e7\u00e3o e sua combina\u00e7\u00e3o com a regra da monotonia, bem como o silogismo hipot\u00e9tico e duas formas de obter essa regra de dedu\u00e7\u00e3o. Tamb\u00e9m s\u00e3o [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27340,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":3,"footnotes":""},"categories":[607,621,571],"tags":[],"class_list":["post-27345","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>4 t\u00e9cnicas de dedu\u00e7\u00e3o essenciais - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Aprenda 4 t\u00e9cnicas de dedu\u00e7\u00e3o na l\u00f3gica proposicional. 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