{"id":28125,"date":"2021-04-08T13:00:29","date_gmt":"2021-04-08T13:00:29","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28125"},"modified":"2024-08-22T08:17:16","modified_gmt":"2024-08-22T08:17:16","slug":"completude-e-solvencia-na-logica-proposicional","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/completude-e-solvencia-na-logica-proposicional\/","title":{"rendered":"Completude e Solv\u00eancia na L\u00f3gica Proposicional"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Completude e Solv\u00eancia na L\u00f3gica Proposicional<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>RESUMO<\/strong><br \/><em>Esta aula aborda a rela\u00e7\u00e3o entre completude e solv\u00eancia na l\u00f3gica proposicional. Embora as t\u00e9cnicas de dedu\u00e7\u00e3o e sem\u00e2ntica na l\u00f3gica proposicional tenham sido amplamente discutidas, pouca aten\u00e7\u00e3o foi dada \u00e0 rela\u00e7\u00e3o entre essas duas facetas. A solv\u00eancia refere-se \u00e0 propriedade de um sistema l\u00f3gico onde, sempre que uma express\u00e3o G pode ser inferida a partir de um conjunto de express\u00f5es \u0393, G \u00e9 consequentemente uma consequ\u00eancia (sem\u00e2ntica) de \u0393. Por outro lado, a completude refere-se \u00e0 propriedade de um sistema l\u00f3gico onde, se G \u00e9 uma consequ\u00eancia sem\u00e2ntica de um conjunto de express\u00f5es \u0393, ent\u00e3o existe uma prova formal com premissas \u0393 a partir da qual G pode ser inferida. Mostra-se que a l\u00f3gica proposicional \u00e9 solvente e completa, e uma explica\u00e7\u00e3o detalhada de cada propriedade \u00e9 apresentada. Em particular, mostra-se como a solv\u00eancia decorre da constitui\u00e7\u00e3o do sistema dedutivo da l\u00f3gica proposicional, e como a completude \u00e9 inferida de forma simples. Esta an\u00e1lise \u00e9 de grande import\u00e2ncia para entender como a l\u00f3gica proposicional funciona e para aplic\u00e1-la efetivamente em v\u00e1rios campos do conhecimento.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>OBJETIVOS DE APRENDIZADO:<\/strong><br \/>\nAo final desta aula, o aluno ser\u00e1 capaz de:\n<\/p>\n<ol>\n<li><strong>Distinguir<\/strong> entre solv\u00eancia e completude em um sistema l\u00f3gico.<\/li>\n<li><strong>Aplicar<\/strong> a tabela verdade para os axiomas de \u0141ukasiewicz para demonstrar a solv\u00eancia da l\u00f3gica proposicional.<\/li>\n<li><strong>Explicar<\/strong> como o modus ponens pode ser reescrito usando a vers\u00e3o sem\u00e2ntica do teorema da dedu\u00e7\u00e3o.<\/li>\n<li><strong>Entender<\/strong> que solv\u00eancia e completude est\u00e3o relacionadas e podem ser inferidas uma da outra.<\/li>\n<li><strong>Analisar<\/strong> o conceito de tautologia e sua rela\u00e7\u00e3o com os teoremas na l\u00f3gica proposicional.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>\u00cdNDICE<\/strong><br \/>\n<a href=\"#1\">COMPLETUDE E SOLV\u00caNCIA NA L\u00d3GICA PROPOSICIONAL<\/a><br \/>\n<a href=\"#2\">A L\u00d3GICA PROPOSICIONAL \u00c9 SOLVENTE<\/a><br \/>\n<a href=\"#3\">A L\u00d3GICA PROPOSICIONAL \u00c9 COMPLETA<\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/dGrwsPMHa90\" title=\"YouTube video player\" 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><\/p>\n<h2>Completude e Solv\u00eancia na L\u00f3gica Proposicional<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=dGrwsPMHa90&amp;t=181s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Neste ponto, \u00e9 hora de falar sobre a completude e a solv\u00eancia da l\u00f3gica proposicional.<\/span><\/strong><\/a> Acontece que, at\u00e9 agora, muito foi dito sobre as t\u00e9cnicas de dedu\u00e7\u00e3o e a sem\u00e2ntica da l\u00f3gica proposicional, mas tudo foi feito de uma forma que faz parecer que s\u00e3o duas facetas completamente independentes, sem qualquer rela\u00e7\u00e3o entre si. A realidade \u00e9 completamente oposta.<\/p>\n<table>\n<tbody>\n<tr>\n<td>\n<p style=\"text-align: justify; color: #000000;\"><strong>SOLV\u00caNCIA:<\/strong> Por um lado, diz-se que um sistema l\u00f3gico \u00e9 solvente quando, sempre que uma express\u00e3o G pode ser inferida a partir de um conjunto de express\u00f5es <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/span>, segue-se que G \u00e9 uma consequ\u00eancia (sem\u00e2ntica) de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: justify; color: #000000;\"><strong>COMPLETUDE:<\/strong> Por outro lado, diz-se que \u00e9 completo quando, se G \u00e9 uma consequ\u00eancia sem\u00e2ntica de um conjunto de express\u00f5es <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/span>, ent\u00e3o existe uma prova formal com premissas <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/span> a partir da qual G pode ser inferida.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-VKgbesdSWbQ\/YG6s7C9kkdI\/AAAAAAAAE2g\/K5hUDwI6KkAs5FNmmkTZ1RRV_LqV_kxlQCLcBGAsYHQ\/s0\/completitud%2By%2Bsolvencia.PNG\" alt=\"Completude e Solv\u00eancia da L\u00f3gica Proposicional\" class=\"aligncenter lazyload\" width=\"285\" height=\"220\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-VKgbesdSWbQ\/YG6s7C9kkdI\/AAAAAAAAE2g\/K5hUDwI6KkAs5FNmmkTZ1RRV_LqV_kxlQCLcBGAsYHQ\/s0\/completitud%2By%2Bsolvencia.PNG\" alt=\"Completude e Solv\u00eancia da L\u00f3gica Proposicional\" class=\"aligncenter lazyload\" width=\"285\" height=\"220\" \/><\/noscript><\/p>\n<p style=\"text-align: justify; color: #000000;\">Revisando essas ideias, veremos que a completude e a solv\u00eancia s\u00e3o satisfeitas para a l\u00f3gica proposicional.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>A L\u00f3gica Proposicional \u00e9 Solvente<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=dGrwsPMHa90&amp;t=292s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u00c9 f\u00e1cil obter a solv\u00eancia da l\u00f3gica proposicional<\/span><\/strong><\/a> observando a constitui\u00e7\u00e3o do seu sistema dedutivo. Se fizermos a tabela verdade para os axiomas de \u0141ukasiewicz, veremos que eles t\u00eam uma estrutura tal que sempre resultam em verdadeiro como valor de verdade, ou seja:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\models (\\alpha \\rightarrow (\\beta \\rightarrow \\alpha))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\models ((\\alpha \\rightarrow (\\beta \\rightarrow \\gamma))\\rightarrow ((\\alpha \\rightarrow \\beta) \\rightarrow (\\alpha \\rightarrow \\gamma)))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\models ((\\neg\\beta \\rightarrow \\neg\\alpha)\\rightarrow(\\alpha \\rightarrow \\beta))<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Da mesma forma, o modus ponens pode ser reescrito como <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha,(\\alpha\\rightarrow \\beta)\\}\\models \\beta.<\/span><\/span>, o que pode ser obtido usando a vers\u00e3o sem\u00e2ntica do teorema da dedu\u00e7\u00e3o. De fato, por esse meio temos que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha\\rightarrow \\beta)\\}\\models (\\alpha\\rightarrow \\beta),<\/span><\/span> e ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\models ((\\alpha\\rightarrow \\beta)\\rightarrow (\\alpha\\rightarrow \\beta)),<\/span><\/span>, que, claro, \u00e9 uma tautologia mais do que \u00f3bvia.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>A L\u00f3gica Proposicional \u00e9 Completa<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=dGrwsPMHa90&amp;t=487s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">A completude da l\u00f3gica proposicional nos diz que,<\/span><\/strong><\/a> se B \u00e9 uma consequ\u00eancia sem\u00e2ntica de A, ent\u00e3o B \u00e9 inferida a partir de A. Em outras palavras: todas as express\u00f5es verdadeiras t\u00eam uma demonstra\u00e7\u00e3o. Isso \u00e9 o que chamamos de completude. Isso pode ser inferido de maneira simples.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Isso pode ser inferido de maneira simples. Suponha que B n\u00e3o possa ser inferido de A, ou melhor, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(A\\vdash B)<\/span><\/span>, pelo teorema da dedu\u00e7\u00e3o isso \u00e9 equivalente a dizer que: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg (\\vdash A\\rightarrow B)<\/span><\/span>; agora, se recorrermos \u00e0 solv\u00eancia, isso nos leva a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\models A \\rightarrow B)<\/span><\/span>, que, pela contrapartida do teorema da dedu\u00e7\u00e3o (vers\u00e3o sem\u00e2ntica), \u00e9 equivalente a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(A\\models B)<\/span><\/span>. Em resumo, o que temos \u00e9 que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(A\\vdash B) \\Rightarrow \\neg(A\\models B) <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Que \u00e9 equivalente a dizer<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(A\\models B) \\Rightarrow (A\\vdash B) <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Isso significa que, se A modela B, ent\u00e3o B \u00e9 inferido a partir de A. E se utilizarmos os respectivos teoremas da dedu\u00e7\u00e3o, podemos obter<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\models A\\rightarrow B) \\Rightarrow (\\vdash A \\rightarrow B) <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Ou seja: se uma express\u00e3o \u00e9 uma tautologia, ent\u00e3o \u00e9 um teorema; e como vimos, os teoremas s\u00e3o o resultado de uma demonstra\u00e7\u00e3o.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Completude e Solv\u00eancia na L\u00f3gica Proposicional RESUMOEsta aula aborda a rela\u00e7\u00e3o entre completude e solv\u00eancia na l\u00f3gica proposicional. Embora as t\u00e9cnicas de dedu\u00e7\u00e3o e sem\u00e2ntica na l\u00f3gica proposicional tenham sido amplamente discutidas, pouca aten\u00e7\u00e3o foi dada \u00e0 rela\u00e7\u00e3o entre essas duas facetas. A solv\u00eancia refere-se \u00e0 propriedade de um sistema l\u00f3gico onde, sempre que uma [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28119,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[607,621,571],"tags":[],"class_list":["post-28125","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 v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Completude e Solv\u00eancia na L\u00f3gica Proposicional - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Compreenda finalmente a rela\u00e7\u00e3o entre completude e solv\u00eancia na l\u00f3gica proposicional, e como essas propriedades fundamentais garantem a validade e coer\u00eancia nos 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