{"id":6787,"date":"2021-12-30T21:38:33","date_gmt":"2021-12-30T21:38:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/?post_type=chapter&#038;p=6787"},"modified":"2022-01-06T22:19:49","modified_gmt":"2022-01-06T22:19:49","slug":"1-6-the-language-of-logic","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/1-6-the-language-of-logic\/","title":{"raw":"1.6 The Language of Logic","rendered":"1.6 The Language of Logic"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3 style=\"padding-left: 30px;\">LEARNING OBJECTIVES<\/h3>\r\nLearning Objectives\r\n<ul>\r\n \t<li>Simple statements<\/li>\r\n \t<li>Compound statements<\/li>\r\n \t<li>Negation<\/li>\r\n \t<li>Translating English to Symbolic Logic<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Statements<\/h3>\r\nIn the English language there are many types of sentences.\u00a0 A few of the types are questions, exclamations and commands.\u00a0 In our study of logic we will only consider a declarative sentence.\r\n<div class=\"textbox\">\r\n<h3>Statement<\/h3>\r\nA <strong>statement<\/strong> is a declarative sentence that is either true or false.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nExamples of sentences that are statements:\r\n<ol>\r\n \t<li>George Washington is a man.<\/li>\r\n \t<li>A triangle has three sides.<\/li>\r\n \t<li>Denver is the capital of Colorado.<\/li>\r\n<\/ol>\r\nExamples of sentences that are not statements:\r\n<ol>\r\n \t<li>Who are you?<\/li>\r\n \t<li>Broccoli tastes good.<\/li>\r\n \t<li>Run for your life!<\/li>\r\n \t<li>Front Range Community College is the best.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3><strong><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\">Simple and Compound Statements<\/span><\/strong><\/h3>\r\nIn logic statements can be classified as simple or compound.\r\n\r\nA s<strong>imple statement<\/strong> contains only one idea while a <strong>compound statement<\/strong> is two or more simple statements joined together with a connective.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Connectives<\/h3>\r\nThere are 4 basic connectives used in logic\r\n<ol>\r\n \t<li>Conjunction (the word \"and\")<\/li>\r\n \t<li>Disjunction (the word \"or\")<\/li>\r\n \t<li>Conditional (if...then)<\/li>\r\n \t<li>Biconditional (if and only if)<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nHere are some examples of compound statements.\r\n<ul>\r\n \t<li>I went to Egypt and I rode a camel. (Conjunction)<\/li>\r\n \t<li>I will read my textbook or watch a reality show. (Disjunction)<\/li>\r\n \t<li>If the mountains are covered in clouds, then it will snow soon. (Conditional)<\/li>\r\n \t<li>I will graduate on time if and only if I take 15 credits each semester (Biconditional)<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>Negation<\/h3>\r\n<b>Negation<\/b>\u00a0tells us, \u201cIt is not the case that\u2026 \u201d\r\n\r\nFor example if we negate the statement \"It is snowing\", we would say \"It is not the case that it is snowing\".\u00a0 Or in more simple terms, \"It is not snowing\".\r\n\r\n<\/div>\r\n<h3>Symbolic Notation<\/h3>\r\nThe main goal in the study of logic is to be able to objectively evaluate logical arguments.\u00a0 In order to do this, we will need to translate English statements into symbolic form.\u00a0 We will use symbols to represent the negation and the connectives and, or, if...then and if and only if.\u00a0 Simple statements in logic are usually denoted by the lowercase letters p, q, and r.\r\n<div class=\"textbox\">\r\n<h3>Symbols<\/h3>\r\nThe symbol [latex]\\wedge[\/latex] is used for <em>and (also called a conjunction)<\/em>: <em>A<\/em> and <em>B<\/em> is notated [latex]A\\wedge{B}[\/latex].\r\n\r\nThe symbol [latex]\\vee[\/latex] is used for <em>or (also called a disjunction)<\/em>: <em>A<\/em> or <em>B<\/em> is notated [latex]A\\vee{B}[\/latex]\r\n\r\nThe symbol [latex]\\sim[\/latex] is used for <em>not (also called a negation)<\/em>: not <em>A<\/em> is notated [latex]\\sim{A}[\/latex]\r\n\r\nThe symbol\u00a0\u2192 is used for\u00a0<em>if ... then<\/em> (also called a conditional)\r\n\r\nThe symbol\u00a0\u2194 is used for\u00a0<em>if and only if\u00a0<\/em>(also called the biconditional)\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nTranslate each statement into symbolic notation. Let p represent \u201cI like Pepsi\u201d and let q\r\nrepresent \u201cI like Coke\u201d.\r\n\r\na. I like Pepsi or I like Coke.\r\nb. I like Pepsi and I like Coke.\r\nc. I do not like Pepsi.\r\nd. It is not the case that I like Pepsi or Coke.\r\ne. I like Pepsi and I do not like Coke.\r\n\r\nf.\u00a0 If I like Pepsi, then I don't like Coke.\r\n\r\ng.\u00a0 I will like Coke if and only if I don't like Pepsi\r\n\r\nSolution:\r\n\r\na. p \u22c1 q\r\nb. p \u22c0 q\r\nc. ~p\r\nd. ~(p \u22c1 q)\r\ne. p \u22c0 ~q\r\n\r\nf.\u00a0 p\u00a0\u2192 ~q\r\n\r\ng. q\u00a0\u2194 ~p\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n\r\n<strong>Logical Order of Operations<\/strong>\r\n\r\nWe often use parenthesis in logical statements when more than one connective is involved in order to specify the order.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nLet p represent \u201cThe resort is in Mexico\u201d and let q represent \u201cThe resort is all inclusive\u201d.\u00a0 Translate the following statements.\r\n\r\na)\u00a0 \u00a0(~p) \u22c0 q\r\n\r\nb)\u00a0 ~ (p V q)\r\n\r\nSolution:\r\n\r\na)\u00a0 The resort is not in Mexico and it is all inclusive.\r\n\r\nb)\u00a0 It is not the case that the resort is in Mexico or it is all inclusive\r\n\r\n<\/div>\r\n&nbsp;\r\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>","rendered":"<div class=\"textbox learning-objectives\">\n<h3 style=\"padding-left: 30px;\">LEARNING OBJECTIVES<\/h3>\n<p>Learning Objectives<\/p>\n<ul>\n<li>Simple statements<\/li>\n<li>Compound statements<\/li>\n<li>Negation<\/li>\n<li>Translating English to Symbolic Logic<\/li>\n<\/ul>\n<\/div>\n<h3>Statements<\/h3>\n<p>In the English language there are many types of sentences.\u00a0 A few of the types are questions, exclamations and commands.\u00a0 In our study of logic we will only consider a declarative sentence.<\/p>\n<div class=\"textbox\">\n<h3>Statement<\/h3>\n<p>A <strong>statement<\/strong> is a declarative sentence that is either true or false.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Examples of sentences that are statements:<\/p>\n<ol>\n<li>George Washington is a man.<\/li>\n<li>A triangle has three sides.<\/li>\n<li>Denver is the capital of Colorado.<\/li>\n<\/ol>\n<p>Examples of sentences that are not statements:<\/p>\n<ol>\n<li>Who are you?<\/li>\n<li>Broccoli tastes good.<\/li>\n<li>Run for your life!<\/li>\n<li>Front Range Community College is the best.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3><strong><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\">Simple and Compound Statements<\/span><\/strong><\/h3>\n<p>In logic statements can be classified as simple or compound.<\/p>\n<p>A s<strong>imple statement<\/strong> contains only one idea while a <strong>compound statement<\/strong> is two or more simple statements joined together with a connective.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Connectives<\/h3>\n<p>There are 4 basic connectives used in logic<\/p>\n<ol>\n<li>Conjunction (the word &#8220;and&#8221;)<\/li>\n<li>Disjunction (the word &#8220;or&#8221;)<\/li>\n<li>Conditional (if&#8230;then)<\/li>\n<li>Biconditional (if and only if)<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Here are some examples of compound statements.<\/p>\n<ul>\n<li>I went to Egypt and I rode a camel. (Conjunction)<\/li>\n<li>I will read my textbook or watch a reality show. (Disjunction)<\/li>\n<li>If the mountains are covered in clouds, then it will snow soon. (Conditional)<\/li>\n<li>I will graduate on time if and only if I take 15 credits each semester (Biconditional)<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Negation<\/h3>\n<p><b>Negation<\/b>\u00a0tells us, \u201cIt is not the case that\u2026 \u201d<\/p>\n<p>For example if we negate the statement &#8220;It is snowing&#8221;, we would say &#8220;It is not the case that it is snowing&#8221;.\u00a0 Or in more simple terms, &#8220;It is not snowing&#8221;.<\/p>\n<\/div>\n<h3>Symbolic Notation<\/h3>\n<p>The main goal in the study of logic is to be able to objectively evaluate logical arguments.\u00a0 In order to do this, we will need to translate English statements into symbolic form.\u00a0 We will use symbols to represent the negation and the connectives and, or, if&#8230;then and if and only if.\u00a0 Simple statements in logic are usually denoted by the lowercase letters p, q, and r.<\/p>\n<div class=\"textbox\">\n<h3>Symbols<\/h3>\n<p>The symbol [latex]\\wedge[\/latex] is used for <em>and (also called a conjunction)<\/em>: <em>A<\/em> and <em>B<\/em> is notated [latex]A\\wedge{B}[\/latex].<\/p>\n<p>The symbol [latex]\\vee[\/latex] is used for <em>or (also called a disjunction)<\/em>: <em>A<\/em> or <em>B<\/em> is notated [latex]A\\vee{B}[\/latex]<\/p>\n<p>The symbol [latex]\\sim[\/latex] is used for <em>not (also called a negation)<\/em>: not <em>A<\/em> is notated [latex]\\sim{A}[\/latex]<\/p>\n<p>The symbol\u00a0\u2192 is used for\u00a0<em>if &#8230; then<\/em> (also called a conditional)<\/p>\n<p>The symbol\u00a0\u2194 is used for\u00a0<em>if and only if\u00a0<\/em>(also called the biconditional)<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Translate each statement into symbolic notation. Let p represent \u201cI like Pepsi\u201d and let q<br \/>\nrepresent \u201cI like Coke\u201d.<\/p>\n<p>a. I like Pepsi or I like Coke.<br \/>\nb. I like Pepsi and I like Coke.<br \/>\nc. I do not like Pepsi.<br \/>\nd. It is not the case that I like Pepsi or Coke.<br \/>\ne. I like Pepsi and I do not like Coke.<\/p>\n<p>f.\u00a0 If I like Pepsi, then I don&#8217;t like Coke.<\/p>\n<p>g.\u00a0 I will like Coke if and only if I don&#8217;t like Pepsi<\/p>\n<p>Solution:<\/p>\n<p>a. p \u22c1 q<br \/>\nb. p \u22c0 q<br \/>\nc. ~p<br \/>\nd. ~(p \u22c1 q)<br \/>\ne. p \u22c0 ~q<\/p>\n<p>f.\u00a0 p\u00a0\u2192 ~q<\/p>\n<p>g. q\u00a0\u2194 ~p<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<p><strong>Logical Order of Operations<\/strong><\/p>\n<p>We often use parenthesis in logical statements when more than one connective is involved in order to specify the order.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Let p represent \u201cThe resort is in Mexico\u201d and let q represent \u201cThe resort is all inclusive\u201d.\u00a0 Translate the following statements.<\/p>\n<p>a)\u00a0 \u00a0(~p) \u22c0 q<\/p>\n<p>b)\u00a0 ~ (p V q)<\/p>\n<p>Solution:<\/p>\n<p>a)\u00a0 The resort is not in Mexico and it is all inclusive.<\/p>\n<p>b)\u00a0 It is not the case that the resort is in Mexico or it is all inclusive<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>\n","protected":false},"author":359705,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-6787","chapter","type-chapter","status-publish","hentry"],"part":159,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6787","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/users\/359705"}],"version-history":[{"count":18,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6787\/revisions"}],"predecessor-version":[{"id":6996,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6787\/revisions\/6996"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/parts\/159"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6787\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/media?parent=6787"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapter-type?post=6787"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/contributor?post=6787"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/license?post=6787"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}