{"id":51,"date":"2016-12-16T18:14:05","date_gmt":"2016-12-16T18:14:05","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/10-summary-of-propositional-logic\/"},"modified":"2025-10-13T18:16:36","modified_gmt":"2025-10-13T18:16:36","slug":"10-summary-of-propositional-logic","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/10-summary-of-propositional-logic\/","title":{"raw":"Summary of Propositional Logic","rendered":"Summary of Propositional Logic"},"content":{"raw":"<h2>10.1 Elements of the language<\/h2>\r\n<ul>\r\n \t<li>Principle of Bivalence: \u00a0each sentence is either true or false, never both, never neither.<\/li>\r\n \t<li>Each atomic sentence is a sentence.<\/li>\r\n \t<li>Syntax: \u00a0if <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>are sentences, then the following are also sentences\r\n<ul>\r\n \t<li><strong><span class=\"strong\">\u00ac<\/span><span class=\"strong\">\u03a6<\/span><\/strong><\/li>\r\n \t<li><strong><span class=\"strong\">(<\/span><span class=\"strong\">\u03a6<\/span><span class=\"strong\">\u2192<\/span><span class=\"strong\">\u03a8<\/span><span class=\"strong\">)<\/span><\/strong><\/li>\r\n \t<li><strong><span class=\"strong\">(<\/span><span class=\"strong\">\u03a6<\/span><span class=\"strong\">\u00a0^ <\/span><span class=\"strong\">\u03a8<\/span><span class=\"strong\">)<\/span><\/strong><\/li>\r\n \t<li><strong><span class=\"strong\">(<\/span><span class=\"strong\">\u03a6<\/span><span class=\"strong\">\u00a0v <\/span><span class=\"strong\">\u03a8<\/span><span class=\"strong\">)<\/span><\/strong><\/li>\r\n \t<li><strong><span class=\"strong\">(<\/span><span class=\"strong\">\u03a6<\/span><span class=\"strong\">\u2194<\/span><span class=\"strong\">\u03a8<\/span><span class=\"strong\">)<\/span><\/strong><\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Semantics: \u00a0if <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>are sentences, then the meanings of the connectives are fully given by their truth tables. \u00a0These truth tables are:<\/li>\r\n<\/ul>\r\n<table class=\"grid\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<td class=\"border-right\">\u03a6<\/td>\r\n<td class=\"border\"><strong>\u00ac\u03a6<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<table class=\"grid\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<td class=\"border\"><strong>\u03a6<\/strong><\/td>\r\n<td class=\"border-right\"><strong>\u03a8<\/strong><\/td>\r\n<td class=\"border\"><strong>(\u03a6\u2192\u03a8)<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<table class=\"grid\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<td class=\"border\"><strong>\u03a6<\/strong><\/td>\r\n<td class=\"border-right\"><strong>\u03a8<\/strong><\/td>\r\n<td class=\"border\"><strong>\u00a0(\u03a6 ^ \u03a8)<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<div class=\"keep\">\r\n<table class=\"grid\" style=\"height: 75px;\">\r\n<tbody>\r\n<tr class=\"border-bottom\" style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 145.4px;\"><strong>\u03a6<\/strong><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 138.383px;\"><strong>\u03a8<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 364.817px;\"><strong>(\u03a6 v \u03a8)<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 145.4px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 138.383px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 364.817px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 145.4px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 138.383px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 364.817px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 145.4px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 138.383px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 364.817px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 145.4px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 138.383px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 364.817px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<table class=\"grid\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<td class=\"border\"><strong>\u03a6<\/strong><\/td>\r\n<td class=\"border-right\"><strong>\u03a8<\/strong><\/td>\r\n<td class=\"border\"><strong>(\u03a6\u2194\u03a8)<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<ul>\r\n \t<li>A sentence of the propositional logic that must be true is a tautology.<\/li>\r\n \t<li>A sentence that must be false is a contradictory sentence.<\/li>\r\n \t<li>A sentence that is neither a tautology nor a contradictory sentence is a contingent sentence.<\/li>\r\n \t<li>Two sentences <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>are equivalent, or logically equivalent, when <strong><span class=\"strong\">(<\/span><span class=\"strong\">\u03a6<\/span><span class=\"strong\">\u2194<\/span><span class=\"strong\">\u03a8<\/span><span class=\"strong\">)<\/span><\/strong>\u00a0is a theorem.<\/li>\r\n<\/ul>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Check for Understanding<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li><strong>Self Reflection:<\/strong> What concepts of propositional logic are you the most comfortable with? What concepts are you the least comfortable with? Please provide some evidence to help support your thoughts.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>10.2 Reasoning with the language<\/h2>\r\n<ul>\r\n \t<li>An argument is an ordered list of sentences, one sentence of which we call the \u201cconclusion\u201d and the others of which we call the \u201cpremises\u201d.<\/li>\r\n \t<li>A valid argument is an argument in which: \u00a0necessarily, if the premises are true, then the conclusion is true.<\/li>\r\n \t<li>A sound argument is a valid argument with true premises.<\/li>\r\n \t<li>Inference rules allow us to write down a sentence that must be true, assuming that certain other sentences are true. \u00a0We say that the new sentence is \u201cderived from\u201d those other sentences using the inference rule.<\/li>\r\n \t<li>Schematically, we can write out the inference rules in the following way (think of these as saying, if you have written the sentence(s) above the line, then you can write the sentence below the line):<\/li>\r\n<\/ul>\r\n<div class=\"keep\">\r\n<table class=\"grid\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Modus ponens<\/strong><\/td>\r\n<td><strong>Modus tollens<\/strong><\/td>\r\n<td><strong>Double negation<\/strong><\/td>\r\n<td><strong>Double negation<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>(\u03a6\u2192\u03a8)<\/strong>\r\n\r\n<strong>\u03a6<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>\u03a8<\/strong><\/td>\r\n<td><strong>(\u03a6\u2192\u03a8)<\/strong>\r\n\r\n<strong>\u00ac\u03a8<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>\u00ac\u03a6<\/strong><\/td>\r\n<td><strong>\u03a6<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>\u00ac\u00ac\u03a6<\/strong><\/td>\r\n<td><strong>\u00ac\u00ac\u03a6<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>\u03a6<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Addition<\/strong><\/td>\r\n<td><strong>Addition<\/strong><\/td>\r\n<td><strong>Modus tollendo ponens<\/strong><\/td>\r\n<td><strong>Modus tollendo ponens<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>\u03a6<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>(\u03a6 v \u03a8)<\/strong><\/td>\r\n<td><strong>\u03a8<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>(\u03a6 v \u03a8)<\/strong><\/td>\r\n<td><strong>(\u03a6 v \u03a8)<\/strong>\r\n\r\n<strong>\u00ac\u03a6<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>\u03a8<\/strong><\/td>\r\n<td><strong>(\u03a6 v \u03a8)<\/strong>\r\n\r\n<strong>\u00ac\u03a8<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>\u03a6<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Adjunction<\/strong><\/td>\r\n<td><strong>Simplification<\/strong><\/td>\r\n<td><strong>Simplification<\/strong><\/td>\r\n<td><strong>Bicondition<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>\u03a6<\/strong>\r\n\r\n<strong>\u03a8<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>(\u03a6 ^ \u03a8)<\/strong><\/td>\r\n<td><strong>(\u03a6 ^ \u03a8)<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>\u03a6<\/strong><\/td>\r\n<td><strong>(\u03a6 ^ \u03a8)<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>\u03a8<\/strong><\/td>\r\n<td><strong>(\u03a6\u2192\u03a8)<\/strong>\r\n\r\n<strong>(\u03a8\u2192\u03a6)<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>(\u03a6\u2194\u03a8)<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Equivalence<\/strong><\/td>\r\n<td><strong>Equivalence<\/strong><\/td>\r\n<td><strong>Equivalence<\/strong><\/td>\r\n<td><strong>Equivalence<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>(\u03a6\u2194\u03a8)<\/strong>\r\n\r\n<strong>\u03a6<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>\u03a8<\/strong><\/td>\r\n<td><strong>(\u03a6\u2194\u03a8)<\/strong>\r\n\r\n<strong>\u03a8<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>\u03a6<\/strong><\/td>\r\n<td><strong>(\u03a6\u2194\u03a8)<\/strong>\r\n\r\n<strong>\u00ac\u03a6<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>\u00ac\u03a8<\/strong><\/td>\r\n<td><strong>(\u03a6\u2194\u03a8)<\/strong>\r\n\r\n<strong>\u00ac\u03a8<\/strong>\r\n\r\n<strong>_____<\/strong>\r\n\r\n<strong>\u00ac\u03a6<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<ul>\r\n \t<li>A proof (or derivation) is a syntactic method for showing an argument is valid. \u00a0Our system has three kinds of proof (or derivation): \u00a0direct, conditional, and indirect.<\/li>\r\n \t<li>A direct proof (or direct derivation) is an ordered list of sentences in which every sentence is either a premise or is derived from earlier lines using an inference rule. \u00a0 The last line of the proof is the conclusion.<\/li>\r\n \t<li>A conditional proof (or conditional derivation) is an ordered list of sentences in which every sentence is either a premise, is the special assumption for conditional derivation, or is derived from earlier lines using an inference rule. \u00a0If the assumption for conditional derivation is <strong><span class=\"strong\">\u03a6<\/span><\/strong>, and we derive as some step in the proof <strong><span class=\"strong\">\u03a8<\/span><\/strong>, then we can write after this <strong><span class=\"strong\">(<\/span><span class=\"strong\">\u03a6<\/span><span class=\"strong\">\u2192<\/span><span class=\"strong\">\u03a8<\/span><span class=\"strong\">)<\/span><\/strong>\u00a0as our conclusion.<\/li>\r\n \t<li>An indirect proof (or indirect derivation, and also known as a reductio ad absurdum) is: an ordered list of sentences in which every sentence is either 1) a premise, 2) the special assumption for indirect derivation (also sometimes called the \u201cassumption for reductio\u201d), or 3) derived from earlier lines using an inference rule. \u00a0If our assumption for indirect derivation is <strong><span class=\"strong\">\u00ac<\/span><span class=\"strong\">\u03a6<\/span><\/strong>, and we derive as some step in the proof <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>and also as some step of our proof <strong><span class=\"strong\">\u00ac<\/span><span class=\"strong\">\u03a8<\/span><\/strong>, then we conclude that <strong><span class=\"strong\">\u03a6<\/span><\/strong>.<\/li>\r\n \t<li>We can use Fitch bars to write out the three proof schemas in the following way:<\/li>\r\n<\/ul>\r\n<img class=\" wp-image-379 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-121530-164x300.png\" alt=\"\" width=\"199\" height=\"364\" \/>\r\n<ul>\r\n \t<li>A sentence that we can prove without premises is a theorem.<\/li>\r\n \t<li>Suppose <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is a theorem, and it contains the atomic sentences P<sub><span class=\"sub\">1<\/span><\/sub>\u2026P<sub><span class=\"sub\">n<\/span><\/sub>. \u00a0If we replace each and every occurrence of one of those atomic sentences P<sub><span class=\"sub\">i<\/span><\/sub>\u00a0in <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>with another sentence <strong><span class=\"strong\">\u03a8<\/span><\/strong>, the resulting sentence is also a theorem.\u00a0This can be repeated for any atomic sentences in the theorem.<\/li>\r\n<\/ul>","rendered":"<h2>10.1 Elements of the language<\/h2>\n<ul>\n<li>Principle of Bivalence: \u00a0each sentence is either true or false, never both, never neither.<\/li>\n<li>Each atomic sentence is a sentence.<\/li>\n<li>Syntax: \u00a0if <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>are sentences, then the following are also sentences\n<ul>\n<li><strong><span class=\"strong\">\u00ac<\/span><span class=\"strong\">\u03a6<\/span><\/strong><\/li>\n<li><strong><span class=\"strong\">(<\/span><span class=\"strong\">\u03a6<\/span><span class=\"strong\">\u2192<\/span><span class=\"strong\">\u03a8<\/span><span class=\"strong\">)<\/span><\/strong><\/li>\n<li><strong><span class=\"strong\">(<\/span><span class=\"strong\">\u03a6<\/span><span class=\"strong\">\u00a0^ <\/span><span class=\"strong\">\u03a8<\/span><span class=\"strong\">)<\/span><\/strong><\/li>\n<li><strong><span class=\"strong\">(<\/span><span class=\"strong\">\u03a6<\/span><span class=\"strong\">\u00a0v <\/span><span class=\"strong\">\u03a8<\/span><span class=\"strong\">)<\/span><\/strong><\/li>\n<li><strong><span class=\"strong\">(<\/span><span class=\"strong\">\u03a6<\/span><span class=\"strong\">\u2194<\/span><span class=\"strong\">\u03a8<\/span><span class=\"strong\">)<\/span><\/strong><\/li>\n<\/ul>\n<\/li>\n<li>Semantics: \u00a0if <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>are sentences, then the meanings of the connectives are fully given by their truth tables. \u00a0These truth tables are:<\/li>\n<\/ul>\n<table class=\"grid\">\n<tbody>\n<tr class=\"border-bottom\">\n<td class=\"border-right\">\u03a6<\/td>\n<td class=\"border\"><strong>\u00ac\u03a6<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table class=\"grid\">\n<tbody>\n<tr class=\"border-bottom\">\n<td class=\"border\"><strong>\u03a6<\/strong><\/td>\n<td class=\"border-right\"><strong>\u03a8<\/strong><\/td>\n<td class=\"border\"><strong>(\u03a6\u2192\u03a8)<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table class=\"grid\">\n<tbody>\n<tr class=\"border-bottom\">\n<td class=\"border\"><strong>\u03a6<\/strong><\/td>\n<td class=\"border-right\"><strong>\u03a8<\/strong><\/td>\n<td class=\"border\"><strong>\u00a0(\u03a6 ^ \u03a8)<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div class=\"keep\">\n<table class=\"grid\" style=\"height: 75px;\">\n<tbody>\n<tr class=\"border-bottom\" style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 145.4px;\"><strong>\u03a6<\/strong><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 138.383px;\"><strong>\u03a8<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 364.817px;\"><strong>(\u03a6 v \u03a8)<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 145.4px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 138.383px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 364.817px;\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 145.4px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 138.383px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 364.817px;\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 145.4px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 138.383px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 364.817px;\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 145.4px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 138.383px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 364.817px;\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<table class=\"grid\">\n<tbody>\n<tr class=\"border-bottom\">\n<td class=\"border\"><strong>\u03a6<\/strong><\/td>\n<td class=\"border-right\"><strong>\u03a8<\/strong><\/td>\n<td class=\"border\"><strong>(\u03a6\u2194\u03a8)<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<ul>\n<li>A sentence of the propositional logic that must be true is a tautology.<\/li>\n<li>A sentence that must be false is a contradictory sentence.<\/li>\n<li>A sentence that is neither a tautology nor a contradictory sentence is a contingent sentence.<\/li>\n<li>Two sentences <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>are equivalent, or logically equivalent, when <strong><span class=\"strong\">(<\/span><span class=\"strong\">\u03a6<\/span><span class=\"strong\">\u2194<\/span><span class=\"strong\">\u03a8<\/span><span class=\"strong\">)<\/span><\/strong>\u00a0is a theorem.<\/li>\n<\/ul>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Check for Understanding<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li><strong>Self Reflection:<\/strong> What concepts of propositional logic are you the most comfortable with? What concepts are you the least comfortable with? Please provide some evidence to help support your thoughts.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>10.2 Reasoning with the language<\/h2>\n<ul>\n<li>An argument is an ordered list of sentences, one sentence of which we call the \u201cconclusion\u201d and the others of which we call the \u201cpremises\u201d.<\/li>\n<li>A valid argument is an argument in which: \u00a0necessarily, if the premises are true, then the conclusion is true.<\/li>\n<li>A sound argument is a valid argument with true premises.<\/li>\n<li>Inference rules allow us to write down a sentence that must be true, assuming that certain other sentences are true. \u00a0We say that the new sentence is \u201cderived from\u201d those other sentences using the inference rule.<\/li>\n<li>Schematically, we can write out the inference rules in the following way (think of these as saying, if you have written the sentence(s) above the line, then you can write the sentence below the line):<\/li>\n<\/ul>\n<div class=\"keep\">\n<table class=\"grid\">\n<tbody>\n<tr>\n<td><strong>Modus ponens<\/strong><\/td>\n<td><strong>Modus tollens<\/strong><\/td>\n<td><strong>Double negation<\/strong><\/td>\n<td><strong>Double negation<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>(\u03a6\u2192\u03a8)<\/strong><\/p>\n<p><strong>\u03a6<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>\u03a8<\/strong><\/td>\n<td><strong>(\u03a6\u2192\u03a8)<\/strong><\/p>\n<p><strong>\u00ac\u03a8<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>\u00ac\u03a6<\/strong><\/td>\n<td><strong>\u03a6<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>\u00ac\u00ac\u03a6<\/strong><\/td>\n<td><strong>\u00ac\u00ac\u03a6<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>\u03a6<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Addition<\/strong><\/td>\n<td><strong>Addition<\/strong><\/td>\n<td><strong>Modus tollendo ponens<\/strong><\/td>\n<td><strong>Modus tollendo ponens<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>\u03a6<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>(\u03a6 v \u03a8)<\/strong><\/td>\n<td><strong>\u03a8<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>(\u03a6 v \u03a8)<\/strong><\/td>\n<td><strong>(\u03a6 v \u03a8)<\/strong><\/p>\n<p><strong>\u00ac\u03a6<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>\u03a8<\/strong><\/td>\n<td><strong>(\u03a6 v \u03a8)<\/strong><\/p>\n<p><strong>\u00ac\u03a8<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>\u03a6<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Adjunction<\/strong><\/td>\n<td><strong>Simplification<\/strong><\/td>\n<td><strong>Simplification<\/strong><\/td>\n<td><strong>Bicondition<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>\u03a6<\/strong><\/p>\n<p><strong>\u03a8<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>(\u03a6 ^ \u03a8)<\/strong><\/td>\n<td><strong>(\u03a6 ^ \u03a8)<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>\u03a6<\/strong><\/td>\n<td><strong>(\u03a6 ^ \u03a8)<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>\u03a8<\/strong><\/td>\n<td><strong>(\u03a6\u2192\u03a8)<\/strong><\/p>\n<p><strong>(\u03a8\u2192\u03a6)<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>(\u03a6\u2194\u03a8)<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Equivalence<\/strong><\/td>\n<td><strong>Equivalence<\/strong><\/td>\n<td><strong>Equivalence<\/strong><\/td>\n<td><strong>Equivalence<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>(\u03a6\u2194\u03a8)<\/strong><\/p>\n<p><strong>\u03a6<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>\u03a8<\/strong><\/td>\n<td><strong>(\u03a6\u2194\u03a8)<\/strong><\/p>\n<p><strong>\u03a8<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>\u03a6<\/strong><\/td>\n<td><strong>(\u03a6\u2194\u03a8)<\/strong><\/p>\n<p><strong>\u00ac\u03a6<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>\u00ac\u03a8<\/strong><\/td>\n<td><strong>(\u03a6\u2194\u03a8)<\/strong><\/p>\n<p><strong>\u00ac\u03a8<\/strong><\/p>\n<p><strong>_____<\/strong><\/p>\n<p><strong>\u00ac\u03a6<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<ul>\n<li>A proof (or derivation) is a syntactic method for showing an argument is valid. \u00a0Our system has three kinds of proof (or derivation): \u00a0direct, conditional, and indirect.<\/li>\n<li>A direct proof (or direct derivation) is an ordered list of sentences in which every sentence is either a premise or is derived from earlier lines using an inference rule. \u00a0 The last line of the proof is the conclusion.<\/li>\n<li>A conditional proof (or conditional derivation) is an ordered list of sentences in which every sentence is either a premise, is the special assumption for conditional derivation, or is derived from earlier lines using an inference rule. \u00a0If the assumption for conditional derivation is <strong><span class=\"strong\">\u03a6<\/span><\/strong>, and we derive as some step in the proof <strong><span class=\"strong\">\u03a8<\/span><\/strong>, then we can write after this <strong><span class=\"strong\">(<\/span><span class=\"strong\">\u03a6<\/span><span class=\"strong\">\u2192<\/span><span class=\"strong\">\u03a8<\/span><span class=\"strong\">)<\/span><\/strong>\u00a0as our conclusion.<\/li>\n<li>An indirect proof (or indirect derivation, and also known as a reductio ad absurdum) is: an ordered list of sentences in which every sentence is either 1) a premise, 2) the special assumption for indirect derivation (also sometimes called the \u201cassumption for reductio\u201d), or 3) derived from earlier lines using an inference rule. \u00a0If our assumption for indirect derivation is <strong><span class=\"strong\">\u00ac<\/span><span class=\"strong\">\u03a6<\/span><\/strong>, and we derive as some step in the proof <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>and also as some step of our proof <strong><span class=\"strong\">\u00ac<\/span><span class=\"strong\">\u03a8<\/span><\/strong>, then we conclude that <strong><span class=\"strong\">\u03a6<\/span><\/strong>.<\/li>\n<li>We can use Fitch bars to write out the three proof schemas in the following way:<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-379 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-121530-164x300.png\" alt=\"\" width=\"199\" height=\"364\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-121530-164x300.png 164w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-121530-65x119.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-121530-225x411.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-121530.png 339w\" sizes=\"auto, (max-width: 199px) 100vw, 199px\" \/><\/p>\n<ul>\n<li>A sentence that we can prove without premises is a theorem.<\/li>\n<li>Suppose <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is a theorem, and it contains the atomic sentences P<sub><span class=\"sub\">1<\/span><\/sub>\u2026P<sub><span class=\"sub\">n<\/span><\/sub>. \u00a0If we replace each and every occurrence of one of those atomic sentences P<sub><span class=\"sub\">i<\/span><\/sub>\u00a0in <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>with another sentence <strong><span class=\"strong\">\u03a8<\/span><\/strong>, the resulting sentence is also a theorem.\u00a0This can be repeated for any atomic sentences in the theorem.<\/li>\n<\/ul>\n","protected":false},"author":158,"menu_order":10,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-51","chapter","type-chapter","status-publish","hentry","license-cc-by-nc-sa"],"part":27,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/51","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":6,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/51\/revisions"}],"predecessor-version":[{"id":380,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/51\/revisions\/380"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/parts\/27"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/51\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/media?parent=51"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapter-type?post=51"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/contributor?post=51"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/license?post=51"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}