{"id":28123,"date":"2021-04-08T13:00:02","date_gmt":"2021-04-08T13:00:02","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28123"},"modified":"2024-08-22T08:15:08","modified_gmt":"2024-08-22T08:15:08","slug":"completeness-and-soundness-in-propositional-logic","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/completeness-and-soundness-in-propositional-logic\/","title":{"rendered":"Completeness and Soundness in Propositional Logic"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Completeness and Soundness in Propositional Logic<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>SUMMARY<\/strong><br \/><em>This class addresses the relationship between completeness and soundness in propositional logic. Although deduction techniques and semantics in propositional logic have been widely discussed, little attention has been paid to the relationship between these two facets. Soundness refers to the property of a logical system where, whenever an expression G can be inferred from a set of expressions \u0393, G is consequently a (semantic) consequence of \u0393. On the other hand, completeness refers to the property of a logical system where, if G is a semantic consequence of a set of expressions \u0393, then there exists a formal proof with premises \u0393 from which G can be inferred. It is shown that propositional logic is sound and complete, and a detailed explanation of each property is presented. In particular, it is shown how soundness derives from the constitution of the deductive system of propositional logic, and how completeness is inferred in a simple way. This analysis is of great importance for understanding how propositional logic works and for applying it effectively in various fields of knowledge.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>LEARNING OBJECTIVES:<\/strong><br \/>\nBy the end of this class, the student will be able to:\n<\/p>\n<ol>\n<li><strong>Distinguish<\/strong> between soundness and completeness in a logical system.<\/li>\n<li><strong>Apply<\/strong> the truth table for \u0141ukasiewicz&#8217;s axioms to demonstrate the soundness of propositional logic.<\/li>\n<li><strong>Explain<\/strong> how modus ponens can be rewritten using the semantic version of the deduction theorem.<\/li>\n<li><strong>Understand<\/strong> that soundness and completeness are related and can be inferred from one another.<\/li>\n<li><strong>Analyze<\/strong> the concept of tautology and its relationship with theorems in propositional logic.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\">COMPLETENESS AND SOUNDNESS IN PROPOSITIONAL LOGIC<\/a><br \/>\n<a href=\"#2\">PROPOSITIONAL LOGIC IS SOUND<\/a><br \/>\n<a href=\"#3\">PROPOSITIONAL LOGIC IS COMPLETE<\/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>Completeness and Soundness in Propositional Logic<\/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;\">At this point, it is time to talk about the completeness and soundness of propositional logic.<\/span><\/strong><\/a> It turns out that so far, a lot has been said about deduction techniques and the semantics of propositional logic, but everything has been done in a way that makes them seem like two completely independent facets without any relation to each other. The reality is completely opposite.<\/p>\n<table>\n<tbody>\n<tr>\n<td>\n<p style=\"text-align: justify; color: #000000;\"><strong>SOUNDNESS:<\/strong> On one hand, a logical system is said to be sound when, whenever an expression G can be inferred from a set of expressions <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/span>, it follows that G is a (semantic) consequence of <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>COMPLETENESS:<\/strong> On the other hand, it is said to be complete when, if G is a semantic consequence of a set of expressions <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/span>, then there exists a formal proof with premises <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/span> from which G can be inferred.<\/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=\"Completeness and Soundness of Propositional Logic\" 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=\"Completeness and Soundness of Propositional Logic\" class=\"aligncenter lazyload\" width=\"285\" height=\"220\" \/><\/noscript><\/p>\n<p style=\"text-align: justify; color: #000000;\">Reviewing these ideas, we will see that completeness and soundness are satisfied for propositional logic.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Propositional Logic is Sound<\/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;\">It is easy to obtain the soundness of propositional logic<\/span><\/strong><\/a> by observing the constitution of its deductive system. If we make the truth table for \u0141ukasiewicz&#8217;s axioms, we will see that they have a structure such that they always yield true as a truth value, that is:<\/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;\">Similarly, modus ponens can be rewritten as <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha,(\\alpha\\rightarrow \\beta)\\}\\models \\beta.<\/span><\/span> which can be obtained using the semantic version of the deduction theorem. In fact, by this means we have that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{(\\alpha\\rightarrow \\beta)\\}\\models (\\alpha\\rightarrow \\beta),<\/span><\/span> and then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\models ((\\alpha\\rightarrow \\beta)\\rightarrow (\\alpha\\rightarrow \\beta)),<\/span><\/span> which of course is a more than obvious tautology.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Propositional Logic is Complete<\/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;\">The completeness of propositional logic tells us that,<\/span><\/strong><\/a> if B is a semantic consequence of A, then B is inferred from A. In other words: all true expressions have a proof. This is what we call completeness. This can be inferred in a simple way.<\/p>\n<p style=\"text-align: justify; color: #000000;\">This can be inferred in a simple way. Suppose that B cannot be inferred from A, or rather <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(A\\vdash B)<\/span><\/span>, by the deduction theorem this is equivalent to saying that: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg (\\vdash A\\rightarrow B)<\/span><\/span>; now, if we resort to soundness, this leads us to <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\models A \\rightarrow B)<\/span><\/span>, which by the contrapositive of the deduction theorem (semantic version) is equivalent to <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(A\\models B)<\/span><\/span>. In summary, what we have is that<\/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;\">Which is equivalent to saying<\/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;\">This means that, if A models B, then B is inferred from A. And if we use the respective deduction theorems, we can obtain<\/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;\">That is: if an expression is a tautology, then it is a theorem; and as we have seen, theorems are the result of a proof.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Completeness and Soundness in Propositional Logic SUMMARYThis class addresses the relationship between completeness and soundness in propositional logic. Although deduction techniques and semantics in propositional logic have been widely discussed, little attention has been paid to the relationship between these two facets. Soundness refers to the property of a logical system where, whenever an expression [&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":38,"footnotes":""},"categories":[605,567,619],"tags":[],"citadela-post-location":[],"class_list":["post-28123","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematical-logic","category-mathematics","category-propositional-logic"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Completeness and Soundness in Propositional Logic - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Finally understand the relationship between completeness and soundness in propositional logic, and how these fundamental properties ensure the validity and coherence in logical 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