{"id":28003,"date":"2021-02-19T13:00:52","date_gmt":"2021-02-19T13:00:52","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28003"},"modified":"2024-08-16T04:16:39","modified_gmt":"2024-08-16T04:16:39","slug":"semantics-of-propositional-logic","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/semantics-of-propositional-logic\/","title":{"rendered":"Semantics of Propositional Logic"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Semantics of Propositional Logic<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>SUMMARY<\/strong><br \/><em>This class studies the semantics of propositional logic, specifically the assignment of truth values to an expression and how they propagate from one expression to another through logical connectors. The notion of truth tables is introduced, and the truth tables of derived connectors such as negation, disjunction, conjunction, implication, biconditional, and exclusive disjunction are shown. Additionally, the definition of an assignment over a set of atomic expressions is presented, and how it naturally extends over all expressions that can be constructed from that set is explained. Finally, the definitions of valid, satisfiable, and unsatisfiable expressions are established, and examples of tautologies and contradictions are provided. However, it is acknowledged that calculating truth tables for complex expressions can be inefficient, so alternative methods for solving validity or satisfiability problems are mentioned.<\/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>Explain<\/strong> the semantics of propositional logic<\/li>\n<li><strong>Use<\/strong> truth tables to represent the assignments of truth values of expressions in propositional logic<\/li>\n<li><strong>Model<\/strong> an expression with an assignment in propositional logic<\/li>\n<li><strong>Apply<\/strong> the semantic rules of propositional logic to determine whether an expression is a tautology, contradiction, or contingency<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\">ABOUT TRUTH VALUE ASSIGNMENTS<\/a><br \/>\n<a href=\"#2\">SEMANTICS OF PROPOSITIONAL LOGIC CONNECTORS<\/a><br \/>\n<a href=\"#3\">ASSIGNMENTS IN PROPOSITIONAL LOGIC<\/a><br \/>\n<a href=\"#4\">MODELS IN PROPOSITIONAL LOGIC<\/a><br \/>\n<a href=\"#5\">AN EFFICIENCY PROBLEM LURKS IN THE SEMANTICS OF PROPOSITIONAL LOGIC<\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/tX_JVhn-wl0\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/div>\n<p><a name=\"1\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>About Truth Value Assignments<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=tX_JVhn-wl0&amp;t=7s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">We previously reviewed<\/span><\/strong><\/a> the syntax and deductive systems of propositional logic. While this served us to review how one expression can be derived from another, we have not yet discussed the assignment of truth values. Since we have already said everything there is to say about deduction techniques in propositional logic, we will begin our study on the semantics of propositional logic, where we review how truth value assignments propagate from one expression to another.<\/p>\n<p><a name=\"2\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Semantics of Propositional Logic Connectors<\/h2>\n<p style=\"text-align: justify; color: #000000;\">The semantics of connectors is introduced through <strong>truth tables,<\/strong> as they provide a simple and organized way to represent all possible assignments on an expression.<\/p>\n<h3>Truth Table of Joint Denial<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=tX_JVhn-wl0&amp;t=282s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">We will start<\/span><\/strong><\/a> with the most fundamental connector of all, which is joint denial. Its truth table is as follows:<\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span><\/strong><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span><\/strong><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\downarrow\\beta)<\/span><\/span><\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">The values \u00ab1\u00bb and \u00ab0\u00bb correspond to \u00abTrue\u00bb and \u00abFalse,\u00bb respectively. Each row in the truth table is a possible assignment on the variables (or atomic expressions) that make up the expression (or expressions) to be studied. Similarly, each column where an expression formed by these variables is located has the possible results of those assignments. Thus, the interpretation of this table tells us that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha\\downarrow\\beta<\/span><\/span> is true only when <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> are both false at the same time, and false otherwise. For this reason, the connector <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span><\/span> is called <strong>joint denial.<\/strong><\/p>\n<h3>Truth Tables of Derived Connectors<\/h3>\n<p style=\"text-align: justify; color: #000000;\">From the semantics of joint denial, the semantics of other connectors can be obtained through their definitions. These are:<\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td><span><strong>Negation:<\/strong><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg \\alpha<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\downarrow\\alpha)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><strong>Inclusive Disjunction:<\/strong><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\vee \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\alpha\\downarrow\\beta)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><strong>Conjunction:<\/strong><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\wedge \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\neg\\alpha\\vee \\neg\\beta)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><strong>Implication:<\/strong><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\neg\\alpha\\vee \\beta)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><strong>Biconditional:<\/strong><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\leftrightarrow \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((\\alpha\\rightarrow \\beta)\\wedge(\\beta \\rightarrow \\alpha))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><strong>Exclusive Disjunction:<\/strong><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\underline{\\vee} \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\alpha\\leftrightarrow \\beta)<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">From these definitions, it is possible to calculate the truth tables of the other connectors:<\/p>\n<h4>Negation<\/h4>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span><\/strong><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg \\alpha<\/span><\/span><\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=tX_JVhn-wl0&amp;t=335s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Hence<\/span><\/strong><\/a> the fact that the negation connector has the property of inverting the truth value of the expression it applies to.<\/p>\n<h4>Inclusive Disjunction<\/h4>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span><\/strong><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span><\/strong><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\vee\\beta)<\/span><\/span><\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=tX_JVhn-wl0&amp;t=383s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">This is why<\/span><\/strong><\/a> the inclusive disjunction (or simply \u00abdisjunction\u00bb) between two expressions is true if at least one of the expressions is true, and false when both expressions are false simultaneously.<\/p>\n<h4>Conjunction<\/h4>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span><\/strong><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span><\/strong><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\wedge\\beta)<\/span><\/span><\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=tX_JVhn-wl0&amp;t=444s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">As a result,<\/span><\/strong><\/a> the conjunction between two expressions is true only if both expressions are true at the same time, and false otherwise. For this reason, an appropriate name for this connector could also be \u00abjoint affirmation.\u00bb<\/p>\n<h4>Implication<\/h4>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span><\/strong><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span><\/strong><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\rightarrow\\beta)<\/span><\/span><\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=tX_JVhn-wl0&amp;t=555s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Thus<\/span><\/strong><\/a> the truth table of implication summarizes the notion that a true expression can only imply a true expression, but a false one can imply anything.<\/p>\n<h4>Biconditional<\/h4>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span><\/strong><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span><\/strong><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\leftrightarrow\\beta)<\/span><\/span><\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=tX_JVhn-wl0&amp;t=641s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">The biconditional<\/span><\/strong><\/a> forms a true expression whenever the two expressions that form it have the same truth values, and false otherwise.<\/p>\n<h4>Exclusive Disjunction<\/h4>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span><\/strong><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span><\/strong><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\underline{\\vee}\\beta)<\/span><\/span><\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center;\">1<\/td>\n<td style=\"text-align: center; background-color: #ccffcc;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=tX_JVhn-wl0&amp;t=702s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">The exclusive disjunction<\/span><\/strong><\/a> between two expressions is true when one, and only one of the expressions is true, and false otherwise.<\/p>\n<p><a name=\"3\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Assignments in Propositional Logic<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=tX_JVhn-wl0&amp;t=857s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">With the above described<\/span><\/strong><\/a> we have a simple notion of what an assignment is; however, for the developments we will see later, we will need something more precise. If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S=\\{A_1, A_2, \\cdots, A_n\\}<\/span><\/span> is a set of atomic expressions and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{F}(S)<\/span><\/span> is the set of all expressions that can be constructed from the expressions of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span>, then the following definition holds:<\/p>\n<p><\/br><\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #880000;\"><strong>DEFINITION:<\/strong><\/span> An <strong>assignment<\/strong> over <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span> is a function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}: S \\longrightarrow \\{0,1\\}<\/span><\/span><\/p>\n<p><\/br><\/p>\n<p style=\"text-align: justify; color: #000000;\">In other words, an assignment over <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span> provides a truth value to each atomic expression in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span>. An assignment <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}<\/span><\/span> of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span> naturally extends over all the elements of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{F}(S)<\/span><\/span>. If we have an expression <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\in \\mathcal{F}(S)<\/span><\/span>, then an assignment <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}<\/span><\/span> over <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span> corresponds to a single row in the truth table of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> and is said to be the truth value of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> in that row.<\/p>\n<p style=\"text-align: justify; color: #000000;\">An assignment <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}<\/span><\/span> of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span> can also be extended to certain expressions that are not in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{F}(S)<\/span><\/span>. If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_0<\/span><\/span> is an expression that is not in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{F}(S)<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S_0<\/span><\/span> is the set of atomic subformulas of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_0<\/span><\/span>, then if all extensions of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}<\/span><\/span> to <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S\\cup S_0<\/span><\/span> have the same value for <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_0<\/span><\/span>, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}(F_0)<\/span><\/span> is defined as that value.<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000088;\">EXAMPLE:<\/span> If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span> are atomic expressions and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}<\/span><\/span> is an assignment over <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{A,B\\}<\/span><\/span> defined through <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}(A)=1<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}(B)=0<\/span><\/span>, then:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}(A\\wedge B)=0<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}(A\\vee B)=1<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">And even though an assignment has not been made over a variable <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C<\/span><\/span>, it can be said that:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}(A\\wedge (C\\vee \\neg C))=1<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}(B\\vee (C\\wedge \\neg C))=0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">This happens with the last two expressions because, regardless of the assignment of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C<\/span><\/span>, it will always be the case that<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}(C\\vee\\neg C)=1<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}(C\\wedge\\neg C)=0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Which we can quickly check by calculating their truth tables.<\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg C<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(C\\wedge \\neg C)<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(C \\vee \\neg C)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u25a0 End of Example<\/p>\n<p><a name=\"4\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Models in Propositional Logic<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=tX_JVhn-wl0&amp;t=1323s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Consider an assignment<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}<\/span><\/span> over a set of expressions <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span>. If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\in S<\/span><\/span> is such that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}(F)=1<\/span><\/span>, then the assignment <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}<\/span><\/span> models <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span>, or the expression <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> is sustained by the assignment <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}<\/span><\/span>, and we represent it through the notation<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}\\models F<\/span><\/span>.<\/p>\n<p style=\"text-align: justify; color: #000000;\">And from this, the following definitions are established:<\/p>\n<p><\/br><\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #880000;\"><strong>DEFINITION:<\/strong><\/span> An expression is said to be <strong>valid<\/strong> when it is sustained under any assignment. If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> is valid, then it is written as <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\models F <\/span><\/span>. Valid expressions are also called <strong>tautologies.<\/strong><\/p>\n<p><\/br><\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000088;\">EXAMPLE:<\/span> The expression <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(C\\vee \\neg C)<\/span><\/span>, which we have already seen, is a <strong>tautology.<\/strong><\/p>\n<p>\u25a0 End of Example<\/p>\n<p><\/br><\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #880000;\"><strong>DEFINITION:<\/strong><\/span> An expression is said to be <strong>satisfiable<\/strong> if it is sustained by some assignment. Satisfiable expressions that are not tautologies are called <strong>contingencies.<\/strong><\/p>\n<p><\/br><\/p>\n<p><\/br><\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #880000;\"><strong>DEFINITION:<\/strong><\/span> An expression is said to be <strong>unsatisfiable<\/strong> if it is not sustained by any assignment. Unsatisfiable expressions are called <strong>contradictions.<\/strong> If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> is a contradiction, then it is written as <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\not\\models F <\/span><\/span>.<\/p>\n<p><\/br><\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000088;\">EXAMPLE:<\/span> The expression <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(C\\wedge \\neg C)<\/span><\/span>, which we have already seen, is a <strong>contradiction.<\/strong>.<\/p>\n<p>\u25a0 End of Example<\/p>\n<h3>Examples of Tautologies and Contradictions<\/h3>\n<p style=\"text-align: justify; color: #000000;\">Suppose we want to see if an expression is valid or not. This question is a <strong>decision problem<\/strong> within the Semantics of Propositional Logic. A decision problem is any problem that, given certain input, results in a \u00abyes\u00bb or \u00abno.\u00bb If we are given an expression <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> and we ask whether it is valid or not, then we are facing a decision problem that we call a <strong>validity problem.<\/strong> Similarly, if we ask whether it is satisfiable or not, we are facing a <strong>satisfiability problem.<\/strong> In propositional logic, truth tables offer a systematic approach to solving these decision problems: if all possible truth values of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> are \u00ab1,\u00bb then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> is valid; if only some are \u00ab1,\u00bb then it is satisfiable; and finally, if all are \u00ab0,\u00bb then it is unsatisfiable.<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000088;\">EXAMPLE:<\/span> Consider the expression <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((A\\wedge (A \\rightarrow B)) \\rightarrow B)<\/span><\/span>. To determine whether this expression is valid, satisfiable, or contradictory, we create its truth table.<\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #eeeeee;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #eeeeee;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #eeeeee;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(A\\rightarrow B)<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #eeeeee;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(A\\wedge(A\\rightarrow B))<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((A\\wedge(A\\rightarrow B))\\rightarrow B)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #aaffaa;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #aaffaa;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #aaffaa;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #aaffaa;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">With this, we see that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((A\\wedge(A\\rightarrow B))\\rightarrow B)<\/span><\/span> has a truth value of \u00ab1\u00bb for all possible assignments, so the expression turns out to be a tautology.<\/p>\n<p>\u25a0 End of Example<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000088;\">EXAMPLE:<\/span> Now consider the expression <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(((A\\rightarrow B)\\rightarrow A)\\wedge \\neg A)<\/span><\/span>. The calculation of the truth table yields the following:<\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #eeeeee;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #eeeeee;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #eeeeee;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(A\\rightarrow B)<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #eeeeee;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((A\\rightarrow B)\\rightarrow A)<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #eeeeee;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg A<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(((A\\rightarrow B)\\rightarrow A)\\wedge \\neg A)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #aaffaa;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #aaffaa;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #aaffaa;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span><\/td>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<td style=\"text-align: center; background-color: #aaffaa;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">So the result is a contradiction.<\/p>\n<p>\u25a0 End of Example<\/p>\n<p><a name=\"5\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>An Efficiency Problem Lurks in the Semantics of Propositional Logic<\/h2>\n<p style=\"text-align: justify; color: #000000;\">In theory, we can determine whether an expression is valid, contingent, or unsatisfiable simply by calculating its truth table, which is not particularly difficult; unfortunately, the ease of execution comes at the cost of efficiency. If we have an expression <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> composed of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> atomic expressions, then we will have to calculate a truth table with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2^n<\/span><\/span> rows; so, for example, if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> is composed of 23 atomic expressions, then its truth table will have 8,388,608 rows that need to be calculated. Proceeding in this way, although mechanical and easy to perform, the calculation quickly becomes infeasible as the complexity of the expressions increases. For this reason, one of our future goals will be to find a way to solve validity or satisfiability problems without the need to calculate truth tables. The search for such methods is one of the central problems of any logic.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Semantics of Propositional Logic SUMMARYThis class studies the semantics of propositional logic, specifically the assignment of truth values to an expression and how they propagate from one expression to another through logical connectors. The notion of truth tables is introduced, and the truth tables of derived connectors such as negation, disjunction, conjunction, implication, biconditional, and [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28002,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":11,"footnotes":""},"categories":[605,567,619],"tags":[],"citadela-post-location":[],"class_list":["post-28003","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>Semantics of Propositional Logic - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Learn the semantics of propositional logic with this post. 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