{"id":25257,"date":"2021-01-25T20:09:42","date_gmt":"2021-01-25T20:09:42","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25257"},"modified":"2025-07-31T01:08:26","modified_gmt":"2025-07-31T01:08:26","slug":"sistemas-dedutivos-formais-em-logica-proposicional","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/sistemas-dedutivos-formais-em-logica-proposicional\/","title":{"rendered":"Sistemas Dedutivos Formais em L\u00f3gica Proposicional"},"content":{"rendered":"<p><center><\/p>\n<h1 style=\"text-align:center;\">Sistemas Dedutivos Formais em L\u00f3gica Proposicional<\/h1>\n<p style=\"text-align:center;\"><em><strong>Resumo:<\/strong><\/br>Nesta aula, faz-se uma revis\u00e3o dos sistemas dedutivos formais. Explica-se como esses sistemas s\u00e3o usados para decifrar as rela\u00e7\u00f5es que podem existir entre diferentes express\u00f5es l\u00f3gicas, e os elementos b\u00e1sicos com os quais essas demonstra\u00e7\u00f5es s\u00e3o constru\u00eddas: a linguagem, os axiomas e as regras de infer\u00eancia. Mencionam-se os axiomas de \u0141ukasiewicz e explica-se o modus ponens como o motor dedutivo do c\u00e1lculo proposicional. Al\u00e9m disso, falamos sobre racioc\u00ednios, teoremas e premissas, e explicamos como as dedu\u00e7\u00f5es s\u00e3o executadas nos sistemas dedutivos.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Objetivos de Aprendizagem:<\/strong><\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Compreender<\/strong> o conceito de sistemas dedutivos formais na l\u00f3gica proposicional.<\/li>\n<li><strong>Identificar<\/strong> os componentes elementares dos sistemas dedutivos formais.<\/li>\n<li><strong>Conhecer<\/strong> os axiomas de \u0141ukasiewicz no c\u00e1lculo proposicional.<\/li>\n<li><strong>Entender<\/strong> o modus ponens como o motor dedutivo do c\u00e1lculo proposicional.<\/li>\n<li><strong>Compreender<\/strong> como as dedu\u00e7\u00f5es s\u00e3o executadas nos sistemas dedutivos e a diferen\u00e7a entre premissas, racioc\u00ednios e teoremas.<\/li>\n<li><strong>Compreender<\/strong> como as dedu\u00e7\u00f5es s\u00e3o geradas atrav\u00e9s de esquemas axiom\u00e1ticos e regras de infer\u00eancia.<\/li>\n<li><strong>Reconhecer<\/strong> a capacidade da l\u00f3gica de conectar express\u00f5es e substitu\u00ed-las por express\u00f5es da l\u00edngua usual.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>\u00cdNDICE DE CONTE\u00daDOS<\/u>:<\/strong><br \/>\n<a href=\"#1\">O QUE \u00c9 UM SISTEMA DEDUTIVO FORMAL?<\/a><br \/>\n<a href=\"#2\">OS AXIOMAS DE \u0141UKASIEWICZ PARA A L\u00d3GICA PROPOSICIONAL<\/a><br \/>\n<a href=\"#3\">O MODUS PONENS: O MOTOR DEDUTIVO DO C\u00c1LCULO PROPOSICIONAL<\/a><br \/>\n<a href=\"#4\">RACIOC\u00cdNIOS, TEOREMAS E PREMISSAS<\/a><br \/>\n<a href=\"#5\">COMO SE EXECUTA UMA DEMONSTRA\u00c7\u00c3O NA L\u00d3GICA PROPOSICIONAL?<\/a><br \/>\n<a href=\"#6\">O CONCEITO DE EQUIVAL\u00caNCIA PROVADA<\/a><br \/>\n<a href=\"#7\">O (META)TEOREMA DA DEDU\u00c7\u00c3O<\/a><br \/>\n<a href=\"#8\">O REC\u00cdPROCO DO TEOREMA DA DEDU\u00c7\u00c3O<\/a><br \/>\n<a href=\"#9\">DEDU\u00c7\u00d5ES SOBRE EXPRESS\u00d5ES E DEDU\u00c7\u00d5ES SOBRE DEDU\u00c7\u00d5ES<\/a><br \/>\n<a href=\"#10\">REGRA DE MONOTONIA<\/a><br \/>\n<a href=\"#11\">S\u00cdNTESE E REFLEX\u00d5ES SOBRE SISTEMAS DEDUTIVOS E A L\u00d3GICA PROPOSICIONAL<\/a>\n<\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/OvoEDefcSZg\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/center><\/p>\n<p style=\"text-align: justify;\">Chegamos, em nosso estudo da l\u00f3gica, a um ponto de inflex\u00e3o, porque aqui iniciamos a revis\u00e3o dos Sistemas Dedutivos da L\u00f3gica Proposicional. Aqui \u00e9 onde tudo o que vimos come\u00e7a a se tornar operativo e o verdadeiro esp\u00edrito da l\u00f3gica vem \u00e0 luz, pois estudaremos a ess\u00eancia das demonstra\u00e7\u00f5es. Neste ponto, assume-se que voc\u00ea j\u00e1 viu como escrever express\u00f5es e entende sobre o que se trata a l\u00f3gica proposicional; e, se n\u00e3o estiver totalmente claro, \u00e9 recomend\u00e1vel que reveja as aulas anteriores a esta.<\/p>\n<p style=\"text-align: justify;\">Feito isso, o que se segue agora \u00e9 revisar a forma como as express\u00f5es da l\u00f3gica proposicional se relacionam entre si para formar uma dedu\u00e7\u00e3o. O mecanismo atrav\u00e9s do qual essas rela\u00e7\u00f5es s\u00e3o constru\u00eddas \u00e9 o <strong>sistema dedutivo formal.<\/strong><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>O que \u00e9 um Sistema Dedutivo Formal?<\/h2>\n<p style=\"text-align: justify;\">Os sistemas dedutivos formais, ou sistemas de c\u00e1lculo dedutivo, t\u00eam tr\u00eas componentes elementares:<\/p>\n<ol style=\"color: #000000; text-align: justify;\">\n<li><strong>Uma Linguagem Formal.<\/strong><\/li>\n<li><strong>Um Esquema Axiom\u00e1tico.<\/strong><\/li>\n<li><strong>Regras de Infer\u00eancia Elementares.<\/strong><\/li>\n<\/ol>\n<p style=\"text-align: justify;\">J\u00e1 revisamos tudo relacionado \u00e0s linguagens formais. Agora, \u00e9 hora de introduzir os esquemas axiom\u00e1ticos e as regras de infer\u00eancia elementares.<\/p>\n<p style=\"text-align: justify;\">Para a constru\u00e7\u00e3o do sistema dedutivo do c\u00e1lculo proposicional, come\u00e7aremos montando o sistema dedutivo a partir dos <strong>Axiomas de <a href=\"https:\/\/es.wikipedia.org\/wiki\/Jan_%C5%81ukasiewicz\" rel=\"noopener\" target=\"_blank\">\u0141ukasiewicz<\/a><\/strong>, e como regra de infer\u00eancia elementar usaremos o <strong>Modus Ponens.<\/strong><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Os Axiomas de \u0141ukasiewicz para a L\u00f3gica Proposicional<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=206s\" target=\"_blank\" rel=\"noopener\"><strong>Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta<\/span><\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> s\u00e3o express\u00f5es do c\u00e1lculo proposicional,<\/strong><\/a> ent\u00e3o os seguintes s\u00e3o axiomas do c\u00e1lculo proposicional:<\/p>\n<table>\n<tbody>\n<tr>\n<td>[A1]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow (\\beta \\rightarrow \\alpha))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>[A2]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((\\alpha \\rightarrow (\\beta \\rightarrow \\gamma))\\rightarrow ((\\alpha\\rightarrow \\beta)\\rightarrow(\\alpha \\rightarrow \\gamma)))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>[A3]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((\\neg\\beta \\rightarrow \\neg\\alpha)\\rightarrow(\\alpha\\rightarrow \\beta))<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><\/p>\n<h2>O Modus Ponens: O motor dedutivo do c\u00e1lculo proposicional<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=392s\" target=\"_blank\" rel=\"noopener\"><strong>Se <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> s\u00e3o express\u00f5es v\u00e1lidas do c\u00e1lculo proposicional, <\/strong><\/a>ent\u00e3o o modus ponens estabelece que a partir de <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span> se deduz <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span>. Em forma de racioc\u00ednio, isso se escreve da seguinte maneira:<\/p>\n<table style=\"text-align: justify;\">\n<caption>Estrutura do Modus Ponens<\/caption>\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 dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/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<\/span><\/td>\n<td>; MP(1,2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Aqui, o Modus Ponens \u00e9 representado entre os passos (1) e (2) atrav\u00e9s da nota\u00e7\u00e3o \u00abMP(1,2)\u00bb, e a s\u00edntese de tudo isso se representa atrav\u00e9s da nota\u00e7\u00e3o:<\/p>\n<p style=\"text-align: center;\">Portanto<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha, (\\alpha \\rightarrow \\beta)\\}\\vdash \\beta <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Em breve veremos que, a partir dos axiomas de \u0141ukasiewicz e do Modus Ponens, podem ser constru\u00eddas todas as t\u00e9cnicas de dedu\u00e7\u00e3o do c\u00e1lculo proposicional, que sintetizam as regras b\u00e1sicas do racioc\u00ednio usual e servem de base para a <strong>l\u00f3gica cl\u00e1ssica.<\/strong><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Racioc\u00ednios, teoremas e premissas<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=506s\" target=\"_blank\" rel=\"noopener\"><strong>Nos sistemas dedutivos da l\u00f3gica proposicional s\u00e3o executados racioc\u00ednios<\/strong><\/a> (ou dedu\u00e7\u00f5es), e esses s\u00e3o qualquer sucess\u00e3o de express\u00f5es onde cada uma delas \u00e9 ou uma premissa ou uma express\u00e3o obtida a partir das premissas, utilizando apenas os axiomas de \u0141ukasiewicz e o modus ponens. Um teorema \u00e9 o resultado de uma dedu\u00e7\u00e3o sem premissas. Uma premissa pode ser qualquer express\u00e3o que n\u00e3o seja um axioma nem se deduza a partir deles. Em geral, quando temos um conjunto de premissas <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> e uma express\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> que \u00e9 obtida utilizando algum elemento de <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span>, os axiomas e o modus ponens, escreve-se \u00ab<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma \\vdash \\alpha<\/span><\/span>\u00bb e dizemos que<\/p>\n<p style=\"text-align: center;\"><em>de <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> deduz-se <\/em><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span>\n<p style=\"text-align: justify;\">Se <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> \u00e9 um conjunto vazio, ent\u00e3o em vez de escrever \u00ab<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset\\vdash \\alpha<\/span><\/span>\u00bb escreve-se \u00ab<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash \\alpha <\/span><\/span>. \u00bb Isso se l\u00ea \u00ab<span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> \u00e9 um teorema\u00bb. Essa forma de representar os teoremas pode ser estendida para a representa\u00e7\u00e3o dos axiomas de modo que, se <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> e <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, ent\u00e3o os axiomas de \u0141ukasiewicz podem ser escritos da seguinte forma<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>[A1]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow (\\beta \\rightarrow \\alpha))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>[A2]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash((\\alpha \\rightarrow (\\beta \\rightarrow \\gamma))\\rightarrow ((\\alpha\\rightarrow \\beta)\\rightarrow(\\alpha \\rightarrow \\gamma)))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>[A3]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash((\\neg\\beta \\rightarrow \\neg\\alpha)\\rightarrow(\\alpha\\rightarrow \\alpha))<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\u00c9 a partir disso que se diz que os axiomas s\u00e3o afirma\u00e7\u00f5es evidentes por si mesmas, ou que os teoremas s\u00e3o express\u00f5es que se inferem a partir do vazio, ou que axiomas e teoremas s\u00e3o propriedades do c\u00e1lculo proposicional.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Como se executa uma demonstra\u00e7\u00e3o na l\u00f3gica proposicional?<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=783s\" target=\"_blank\" rel=\"noopener\"><strong>Neste ponto, deixaremos de falar de teoria e passaremos \u00e0 pr\u00e1tica.<\/strong><\/a> E sobre a execu\u00e7\u00e3o de uma demonstra\u00e7\u00e3o pode-se dizer muitas coisas; mas, por mais que se digam coisas brilhantes sobre os sistemas dedutivos e a l\u00f3gica proposicional, e todas sejam entendidas, isso n\u00e3o implicar\u00e1 necessariamente que se estejam desenvolvendo as compet\u00eancias necess\u00e1rias para executar uma demonstra\u00e7\u00e3o. Por isso, para ensinar como fazer demonstra\u00e7\u00f5es, revisaremos a demonstra\u00e7\u00e3o de um teorema simples.<\/p>\n<p style=\"text-align: justify; color: #880000;\"><strong>Teorema<\/strong><\/p>\n<p style=\"text-align: justify;\">Se <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> \u00e9 uma express\u00e3o da l\u00f3gica proposicional, ent\u00e3o se cumpre que<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha\\rightarrow \\alpha)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000088;\"><strong>Demonstra\u00e7\u00e3o<\/strong><\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> (\\alpha\\rightarrow ( \\alpha \\rightarrow \\alpha)) <\/span><\/span><\/td>\n<td>; A1<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> (\\alpha\\rightarrow ((\\alpha\\rightarrow \\alpha)\\rightarrow\\alpha)) <\/span><\/span><\/td>\n<td>; A1<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ( (\\alpha\\rightarrow((\\alpha\\rightarrow\\alpha)\\rightarrow\\alpha)) \\rightarrow ((\\alpha\\rightarrow (\\alpha\\rightarrow\\alpha))\\rightarrow( \\alpha\\rightarrow \\alpha))) <\/span><\/span><\/td>\n<td>; A2<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ((\\alpha\\rightarrow (\\alpha\\rightarrow\\alpha))\\rightarrow( \\alpha\\rightarrow \\alpha)) <\/span><\/span><\/td>\n<td>; MP(2,3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ( \\alpha\\rightarrow \\alpha) <\/span><\/span><\/td>\n<td>; MP(1,5)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Portanto<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\alpha\\rightarrow\\alpha)<\/span><\/span><\/p>\n<p>Fim da demonstra\u00e7\u00e3o.<\/p>\n<p style=\"text-align: justify;\">Como se pode ver, nos sistemas dedutivos e na l\u00f3gica proposicional, as demonstra\u00e7\u00f5es n\u00e3o s\u00e3o triviais, mas uma vez constru\u00eddas s\u00e3o f\u00e1ceis de replicar.<\/p>\n<p style=\"text-align: justify;\">Agora, antes de nos lan\u00e7armos diretamente a fazer dedu\u00e7\u00f5es com essas t\u00e9cnicas, primeiro vamos desenvolver algumas propriedades e defini\u00e7\u00f5es que ser\u00e3o extremamente \u00fateis para esta tarefa, pois, se raciocinarmos apenas com isso, enfrentaremos terr\u00edveis problemas.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>O conceito de equival\u00eancia provada<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1191s\" target=\"_blank\" rel=\"noopener\"><strong>Se <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> s\u00e3o quaisquer express\u00f5es e se cumpre <\/strong><\/a>ao mesmo tempo que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash \\beta<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\beta\\} \\vdash \\alpha<\/span><\/span>, ent\u00e3o se diz que <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> s\u00e3o provadas equivalentes, e escreveremos <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha \\dashv \\vdash \\beta<\/span><\/span>. Isso \u00e9 resumido simbolicamente como:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\{\\alpha\\}\\vdash\\beta \\wedge \\{\\beta\\}\\vdash\\alpha \\right) \\Leftrightarrow \\left(\\alpha\\dashv\\vdash\\beta\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Isso se l\u00ea: de <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> se infere <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span>, e de <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> se infere <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> se, e somente se, <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> s\u00e3o provadas equivalentes.<\/p>\n<p style=\"text-align: justify;\">Isso \u00e9 uma meta-propriedade da l\u00f3gica proposicional<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h2>O (meta)Teorema da Dedu\u00e7\u00e3o<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1355s\" target=\"_blank\" rel=\"noopener\"><strong>Se <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> s\u00e3o express\u00f5es do c\u00e1lculo proposicional,<\/strong><\/a> e <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> \u00e9 um conjunto de premissas; ent\u00e3o temos que, se de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma \\cup \\{\\alpha\\}<\/span><\/span> se deduz <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span>, ent\u00e3o, a partir de <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span>, deduz-se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span>. Simbolicamente, isso seria expresso como:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\Gamma \\cup \\{\\alpha\\}\\vdash \\beta \\right) \\Rightarrow \\left( \\Gamma\\vdash(\\alpha\\rightarrow\\beta)\\right)\n\n<\/span>\n<p style=\"text-align: justify; color: #880000;\"><strong>Demonstra\u00e7\u00e3o:<\/strong><\/p>\n<p style=\"text-align: justify;\">Para que se cumpra <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma \\cup \\{\\alpha\\}\\vdash \\beta<\/span><\/span>, \u00e9 necess\u00e1rio ter uma dedu\u00e7\u00e3o da seguinte forma<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_1<\/span><\/span><\/td>\n<td>; Premissa 1 de <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(n)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_n<\/span><\/span><\/td>\n<td>; Premissa n de <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td>(n+1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\gamma}_1<\/span><\/span><\/td>\n<td>; Modus Ponens entre algum par de linhas anteriores<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(n+m)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\gamma}_m<\/span><\/span><\/td>\n<td>; Modus Ponens entre algum par de linhas anteriores<\/td>\n<\/tr>\n<tr>\n<td>(n+m+1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/td>\n<td>; Premissa<\/td>\n<\/tr>\n<tr>\n<td>(n+m+2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/td>\n<td>; Modus Ponens (n+m+1, algum dos passos anteriores, exceto o n+m+1)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Portanto<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\Gamma\\cup\\{\\alpha\\} \\vdash \\beta <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Para que isso seja poss\u00edvel, \u00e9 necess\u00e1rio que pelo menos uma das express\u00f5es <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_1, \\cdots \\gamma_n,\\overline{\\gamma_1},\\cdots,\\overline{\\gamma_m}<\/span><\/span> seja da forma <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\rightarrow \\beta)<\/span><\/span>, mas todas essas linhas envolvem apenas elementos de <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> e os axiomas de \u0141ukasiewicz em sua dedu\u00e7\u00e3o, portanto, deve-se cumprir que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma\\vdash (\\alpha \\rightarrow \\beta)<\/span><\/span>. Fica, portanto, demonstrado o teorema<\/p>\n<p>Fim da demonstra\u00e7\u00e3o.<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<h2>O Rec\u00edproco do Teorema da Dedu\u00e7\u00e3o<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1668s\" target=\"_blank\" rel=\"noopener\"><strong>Nas mesmas condi\u00e7\u00f5es que o teorema da dedu\u00e7\u00e3o, teremos que<\/strong><\/a><\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\left(\\Gamma\\vdash(\\alpha \\rightarrow \\beta)\\right) \\Rightarrow \\left( \\Gamma \\cup \\{\\alpha\\}\\vdash \\beta \\right)\n\n<\/span>\n<p style=\"text-align: justify; color: #880000;\"><strong>Demonstra\u00e7\u00e3o:<\/strong><\/p>\n<p style=\"text-align: justify;\">Se cumpre que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma\\vdash (\\alpha\\rightarrow \\beta)<\/span><\/span>, ent\u00e3o h\u00e1 uma dedu\u00e7\u00e3o da seguinte forma<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_1<\/span><\/span><\/td>\n<td>; Premissa 1 de <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(n)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_n<\/span><\/span><\/td>\n<td>; Premissa n de <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td>(n+1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span><\/td>\n<td>; Modus Ponens(entre algum par de linhas anteriores)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Agora, se adicionarmos <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> como premissa a esse racioc\u00ednio, ent\u00e3o teremos as seguintes linhas<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(n+2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/td>\n<td>; Premissa adicional<\/td>\n<\/tr>\n<tr>\n<td>(n+3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/td>\n<td>; MP(n+1,n+2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Portanto<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\Gamma \\cup \\{\\alpha\\} \\vdash \\beta<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Que \u00e9 o que se queria demonstrar.<\/p>\n<p>Fim da demonstra\u00e7\u00e3o.<\/p>\n<p><a name=\"9\"><\/a><\/p>\n<h2>Dedu\u00e7\u00f5es sobre Express\u00f5es e Dedu\u00e7\u00f5es sobre Dedu\u00e7\u00f5es<\/h2>\n<p style=\"text-align: justify;\">Demonstra\u00e7\u00f5es como a que fizemos antes para chegar ao resultado <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha\\rightarrow \\alpha)<\/span><\/span> s\u00e3o casos de dedu\u00e7\u00f5es baseadas em express\u00f5es, pois cada passo cont\u00e9m uma express\u00e3o espec\u00edfica. De forma an\u00e1loga, \u00e9 poss\u00edvel fazer dedu\u00e7\u00f5es baseadas em outras dedu\u00e7\u00f5es, onde cada passo \u00e9 uma dedu\u00e7\u00e3o por si s\u00f3. Na pr\u00e1tica, ambas s\u00e3o feitas de forma similar, mas a segunda permite o uso do teorema da dedu\u00e7\u00e3o e seu rec\u00edproco, dando grande flexibilidade \u00e0 t\u00e9cnica de racioc\u00ednio. Para ver isso, demonstremos novamente que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow \\alpha)<\/span><\/span>, mas agora usando dedu\u00e7\u00f5es em vez de express\u00f5es. Uma alternativa para isso \u00e9 a seguinte:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow (\\alpha \\rightarrow \\alpha))<\/span><\/span><\/td>\n<td>; A1<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash ( \\alpha \\rightarrow \\alpha)<\/span><\/span><\/td>\n<td>; RTD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\cup \\{\\alpha\\}\\vdash \\alpha<\/span><\/span><\/td>\n<td>; RTD(2)<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash \\alpha<\/span><\/span><\/td>\n<td>; Notemos que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\cup\\{\\alpha\\}=\\{\\alpha\\}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha\\rightarrow \\alpha)<\/span><\/span><\/td>\n<td>; TD(4)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Observe que este racioc\u00ednio n\u00e3o \u00e9 mais curto que o anterior, mas muito mais f\u00e1cil de realizar. Basta usar o teorema da dedu\u00e7\u00e3o, seu rec\u00edproco e o esquema axiom\u00e1tico A1 para construir a demonstra\u00e7\u00e3o.<\/p>\n<p style=\"text-align: justify;\">Na apar\u00eancia, no desenvolvimento que acabamos de fazer, utilizamos apenas um axioma de \u0141ukasiewicz e nos esquecemos de outros axiomas e do modus ponens. Isso significa que, ao raciocinarmos dessa forma, nos esquecemos dos outros axiomas e do modus ponens? A resposta \u00e9 sim e n\u00e3o. Podemos fingir que esquecemos de alguns axiomas e do modus ponens, apenas porque n\u00e3o os estamos usando explicitamente. No entanto, deve-se lembrar que tanto o teorema da dedu\u00e7\u00e3o quanto seu rec\u00edproco s\u00e3o consequ\u00eancias dos axiomas de \u0141ukasiewicz e do modus ponens, o que implica que, ao fazer uso deles, estamos fazendo um uso impl\u00edcito dos mesmos.<\/p>\n<p><a name=\"10\"><\/a><\/p>\n<h2>Regra de Monotonia<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1972s\" target=\"_blank\" rel=\"noopener\"><strong>Se <span class=\"katex-eq\" data-katex-display=\"false\">\\tau<\/span> \u00e9 um teorema,<\/strong><\/a> ent\u00e3o se ter\u00e1 que, dada qualquer express\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span>, se cumprir\u00e1 que<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\beta\\}\\vdash\\tau<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Essa \u00e9 uma regra muito f\u00e1cil de provar, pois sendo <span class=\"katex-eq\" data-katex-display=\"false\">\\tau<\/span> um teorema, se cumpre que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash \\tau<\/span><\/span>. Ou seja, existe um racioc\u00ednio que, sem a necessidade de adicionar premissas, leva \u00e0 express\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">\\tau<\/span>, ent\u00e3o adicionar uma express\u00e3o adicional \u00e0s premissas (vazia) n\u00e3o far\u00e1 diferen\u00e7a.<\/p>\n<p style=\"text-align: justify;\">De forma similar, pode-se apresentar o seguinte resultado: se de um conjunto de premissas <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> se infere <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span>, ent\u00e3o se cumpre que<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma\\cup\\{\\alpha\\}\\vdash\\gamma<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Onde <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> \u00e9 uma express\u00e3o qualquer.<\/p>\n<p><a name=\"11\"><\/a><\/p>\n<h2>S\u00edntese e Reflex\u00f5es sobre Sistemas Dedutivos e a L\u00f3gica Proposicional<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1933s\" target=\"_blank\" rel=\"noopener\"><strong>Quando fornecemos \u00e0 linguagem da l\u00f3gica proposicional uma regra de infer\u00eancia e express\u00f5es b\u00e1sicas:<\/strong><\/a> O Modus Ponens e os Axiomas de \u0141ukasiewicz, o que fazemos \u00e9 an\u00e1logo a montar uma \u00abm\u00e1quina dedutiva\u00bb e um \u00abmotor que a faz entrar em movimento\u00bb. A partir da\u00ed, as regras b\u00e1sicas de dedu\u00e7\u00e3o come\u00e7am a surgir naturalmente, e come\u00e7aremos a revis\u00e1-las nas entregas imediatamente ap\u00f3s esta.<\/p>\n<p style=\"text-align: justify;\">Outro detalhe importante. As express\u00f5es da l\u00f3gica proposicional s\u00e3o, na verdade, meta-express\u00f5es da linguagem de dois s\u00edmbolos que vimos antes. Lembremos que a utilidade dessas meta-express\u00f5es \u00e9 que nos permitem substituir suas meta-vari\u00e1veis por qualquer express\u00e3o da linguagem para obter uma nova que satisfa\u00e7a tal estrutura. Quando dotamos a linguagem da l\u00f3gica proposicional de esquemas axiom\u00e1ticos e regras de infer\u00eancia, constru\u00edmos os Sistemas Dedutivos da l\u00f3gica proposicional, que permitem gerar dedu\u00e7\u00f5es que conectam express\u00f5es. Como resultado, temos um esquema dedutivo capaz de englobar infinitas dedu\u00e7\u00f5es: todas as que podemos obter substituindo meta-vari\u00e1veis pelas express\u00f5es que quisermos. O poder da l\u00f3gica se revela, na verdade, quando nos damos conta de que, al\u00e9m dessas express\u00f5es da linguagem de dois s\u00edmbolos que utilizamos no in\u00edcio, podemos substituir por express\u00f5es da nossa l\u00edngua usual.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Sistemas Dedutivos Formais em L\u00f3gica Proposicional Resumo:Nesta aula, faz-se uma revis\u00e3o dos sistemas dedutivos formais. Explica-se como esses sistemas s\u00e3o usados para decifrar as rela\u00e7\u00f5es que podem existir entre diferentes express\u00f5es l\u00f3gicas, e os elementos b\u00e1sicos com os quais essas demonstra\u00e7\u00f5es s\u00e3o constru\u00eddas: a linguagem, os axiomas e as regras de infer\u00eancia. Mencionam-se os axiomas [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":33791,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":17,"footnotes":""},"categories":[621,571],"tags":[],"class_list":["post-25257","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","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>Sistemas Dedutivos Formais em L\u00f3gica Proposicional - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explora os fundamentos dos sistemas dedutivos formais em l\u00f3gica proposicional, incluindo axiomas, infer\u00eancias e teoremas.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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