{"id":185,"date":"2016-01-25T22:05:39","date_gmt":"2016-01-25T22:05:39","guid":{"rendered":"https:\/\/courses.candelalearning.com\/math4libarts\/?post_type=chapter&#038;p=185"},"modified":"2019-08-21T16:00:08","modified_gmt":"2019-08-21T16:00:08","slug":"truth-tables-and-analyzing-arguments-examples","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/truth-tables-and-analyzing-arguments-examples\/","title":{"raw":"Arguments","rendered":"Arguments"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Discern between an inductive argument and a deductive argument<\/li>\r\n \t<li>Evaluate deductive arguments<\/li>\r\n \t<li>Analyze arguments with Venn diagrams and truth tables<\/li>\r\n \t<li>Use logical inference to infer whether a statement is true<\/li>\r\n \t<li>Identify logical fallacies in common language including appeal to ignorance, appeal to authority, appeal to consequence, false dilemma, circular reasoning, post hoc, correlation implies causation, and straw man arguments<\/li>\r\n<\/ul>\r\n<\/div>\r\nA logical argument is a claim that a set of premises support a conclusion. There are two general types of arguments: inductive and deductive arguments.\r\n<div class=\"textbox\">\r\n<h3>Argument types<\/h3>\r\nAn <strong>inductive<\/strong> argument uses a collection of specific examples as its premises and uses them to propose a general conclusion.\r\n\r\nA <strong>deductive<\/strong> argument uses a collection of general statements as its premises and uses them to propose a specific situation as the conclusion.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=109526&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"150\"><\/iframe>\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=109527&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe argument \u201cwhen I went to the store last week I forgot my purse, and when I went today I forgot my purse. I always forget my purse when I go the store\u201d is an inductive argument.\r\n\r\nThe premises are:\r\n<p style=\"padding-left: 30px;\">I forgot my purse last week\r\nI forgot my purse today<\/p>\r\nThe conclusion is:\r\n<p style=\"padding-left: 30px;\">I always forget my purse<\/p>\r\nNotice that the premises are specific situations, while the conclusion is a general statement. In this case, this is a fairly weak argument, since it is based on only two instances.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe argument \u201cevery day for the past year, a plane flies over my house at 2pm. A plane will fly over my house every day at 2pm\u201d is a stronger inductive argument, since it is based on a larger set of evidence.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Evaluating inductive arguments<\/h3>\r\nAn inductive argument is never able to prove the conclusion true, but it can provide either weak or strong evidence to suggest it may be true.\r\n\r\n<\/div>\r\nMany scientific theories, such as the big bang theory, can never be proven. Instead, they are inductive arguments supported by a wide variety of evidence. Usually in science, an idea is considered a hypothesis until it has been well tested, at which point it graduates to being considered a theory. The commonly known scientific theories, like Newton\u2019s theory of gravity, have all stood up to years of testing and evidence, though sometimes they need to be adjusted based on new evidence. For gravity, this happened when Einstein proposed the theory of general relativity.\r\n\r\nA deductive argument is more clearly valid or not, which makes them easier to evaluate.\r\n<div class=\"textbox\">\r\n<h3>Evaluating deductive arguments<\/h3>\r\nA deductive argument is considered valid if all the premises are true, and the conclusion follows logically from those premises. In other words, the premises are true, and the conclusion follows necessarily from those premises.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe argument \u201cAll cats are mammals and a tiger is a cat, so a tiger is a mammal\u201d is a valid deductive argument.\r\n\r\nThe premises are:\r\n<p style=\"padding-left: 30px;\">All cats are mammals\r\nA tiger is a cat<\/p>\r\nThe conclusion is:\r\n<p style=\"padding-left: 30px;\">A tiger is a mammal<\/p>\r\nBoth the premises are true. To see that the premises must logically lead to the conclusion, one approach would be use a Venn diagram. From the first premise, we can conclude that the set of cats is a subset of the set of mammals. From the second premise, we are told that a tiger lies within the set of cats. From that, we can see in the Venn diagram that the tiger also lies inside the set of mammals, so the conclusion is valid.\r\n\r\n<img class=\"aligncenter wp-image-256\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155146\/Fig4_2_1.png\" alt=\"A Venn diagram with a large circle labeled Mammals and a smaller circle labeled Cats contained within the Mammals circle. A red X labeled Tiger is in the circle labeled Cats.\" width=\"200\" height=\"202\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]132642[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Analyzing Arguments with Venn\/Euler diagrams<\/h3>\r\nTo analyze an argument with a Venn\/ Euler diagram\r\n<ol>\r\n \t<li>Draw a Venn\/ Euler diagram based on the premises of the argument<\/li>\r\n \t<li>If the premises are insufficient to determine what determine the location of an element, indicate that.<\/li>\r\n \t<li>The argument is valid if it is clear that the conclusion must be true<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>All firefighters know CPR<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>Jill knows CPR<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>Jill is a firefighter<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFrom the first premise, we know that firefighters all lie inside the set of those who know CPR. From the second premise, we know that Jill is a member of that larger set, but we do not have enough information to know if she also is a member of the smaller subset that is firefighters.\r\n\r\n<img class=\"wp-image-257 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155147\/Fig4_2_2.png\" alt=\"Fig4_2_2\" width=\"200\" height=\"200\" \/>\r\n\r\nSince the conclusion does not necessarily follow from the premises, this is an invalid argument, regardless of whether Jill actually is a firefighter.\r\n\r\n<\/div>\r\nIt is important to note that whether or not Jill is actually a firefighter is not important in evaluating the validity of the argument; we are only concerned with whether the premises are enough to prove the conclusion.\r\n\r\nIn addition to these categorical style premises of the form \u201call ___,\u201d \u201csome ____,\u201d and \u201cno ____,\u201d it is also common to see premises that are implications.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If you live in Seattle, you live in Washington.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>Marcus does not live in Seattle.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>Marcus does not live in Washington.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFrom the first premise, we know that the set of people who live in Seattle is inside the set of those who live in Washington. From the second premise, we know that Marcus does not lie in the Seattle set, but we have insufficient information to know whether or not Marcus lives in Washington or not. This is an invalid argument.\r\n\r\n<img class=\"wp-image-258 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155147\/Fig4_2_3.png\" alt=\"Fig4_2_3\" width=\"200\" height=\"204\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider the argument \u201cYou are a married man, so you must have a wife.\u201d\r\n\r\n[reveal-answer q=\"383279\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"383279\"]\r\n\r\nThis is an invalid argument, since there are, at least in parts of the world, men who are married to other men, so the premise not insufficient to imply the conclusion.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSome arguments are better analyzed using truth tables.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider the argument:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If you bought bread, then you went to the store<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>You bought bread<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>You went to the store<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"23681\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"23681\"]\r\n\r\nWhile this example is hopefully fairly obviously a valid argument, we can analyze it using a truth table by representing each of the premises symbolically. We can then look at the implication that the premises together imply the conclusion. If the truth table is a tautology (always true), then the argument is valid.\r\n\r\nWe\u2019ll get B represent \u201cyou bought bread\u201d and S represent \u201cyou went to the store\u201d. Then the argument becomes:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]B{\\rightarrow}S[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]B[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>[latex]S[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo test the validity, we look at whether the combination of both premises implies the conclusion; is it true that [latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]?\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]B[\/latex]<\/td>\r\n<td>[latex]S[\/latex]<\/td>\r\n<td>[latex]B{\\rightarrow}S[\/latex]<\/td>\r\n<td>[latex]\\left(B{\\rightarrow}S\\right){\\wedge}B[\/latex]<\/td>\r\n<td>[latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince the truth table for [latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]\u00a0is always true, this is a valid argument.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Analyzing arguments using truth tables<\/h3>\r\nTo analyze an argument with a truth table:\r\n<ol>\r\n \t<li>Represent each of the premises symbolically<\/li>\r\n \t<li>Create a conditional statement, joining all the premises with and to form the antecedent, and using the conclusion as the consequent.<\/li>\r\n \t<li>Create a truth table for that statement. If it is always true, then the argument is valid.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If I go to the mall, then I\u2019ll buy new jeans.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If I buy new jeans, I\u2019ll buy a shirt to go with it.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>If I got to the mall, I\u2019ll buy a shirt.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet <em>M<\/em> = I go to the mall, <em>J <\/em>= I buy jeans, and <em>S<\/em> = I buy a shirt.\r\n\r\nThe premises and conclusion can be stated as:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]M{\\rightarrow}J[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]J{\\rightarrow}S[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>[latex]M{\\rightarrow}S[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can construct a truth table for [latex]\\left[\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)\\right]{\\rightarrow}\\left(M{\\rightarrow}S\\right)[\/latex]\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]M[\/latex]<\/td>\r\n<td>[latex]J[\/latex]<\/td>\r\n<td>[latex]S[\/latex]<\/td>\r\n<td>[latex]M{\\rightarrow}J[\/latex]<\/td>\r\n<td>[latex]J{\\rightarrow}S[\/latex]<\/td>\r\n<td>[latex]\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)[\/latex]<\/td>\r\n<td>[latex]M{\\rightarrow}S[\/latex]<\/td>\r\n<td>[latex]\\left[\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)\\right]{\\rightarrow}\\left(M{\\rightarrow}S\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFrom the truth table, we can see this is a valid argument.\r\n\r\n<\/div>\r\nThe previous problem is an example of a syllogism.\r\n<div class=\"textbox\">\r\n<h3>Syllogism<\/h3>\r\nA syllogism is an implication derived from two others, where the consequence of one is the antecedent to the other. The general form of a syllogism is:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]p{\\rightarrow}q[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]q{\\rightarrow}r[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>[latex]p{\\rightarrow}r[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis is sometime called the transitive property for implication.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If I work hard, I\u2019ll get a raise.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If I get a raise, I\u2019ll buy a boat.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>If I don\u2019t buy a boat, I must not have worked hard.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"880229\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"880229\"]\r\n\r\nIf we let <em>W<\/em> = working hard, <em>R<\/em> = getting a raise, and <em>B<\/em> = buying a boat, then we can represent our argument symbolically:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]H{\\rightarrow}R[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]R{\\rightarrow}B[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>[latex]\\sim{B}{\\rightarrow}{\\sim}H[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe could construct a truth table for this argument, but instead, we will use the notation of the contrapositive we learned earlier to note that the implication [latex]{\\sim}B{\\rightarrow}{\\sim}H[\/latex]\u00a0is equivalent to the implication [latex]H{\\rightarrow}B[\/latex]. Rewritten, we can see that this conclusion is indeed a logical syllogism derived from the premises.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nIs this argument valid?\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If I go to the party, I\u2019ll be really tired tomorrow.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If I go to the party, I\u2019ll get to see friends.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>If I don\u2019t see friends, I won\u2019t be tired tomorrow.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<iframe id=\"mom50\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25956&amp;theme=oea&amp;iframe_resize_id=mom50\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\nLewis Carroll, author of <em>Alice in Wonderland<\/em>, was a math and logic teacher, and wrote two books on logic. In them, he would propose premises as a puzzle, to be connected using syllogisms.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve the puzzle. In other words, find a logical conclusion from these premises.\r\n\r\nAll babies are illogical.\r\n\r\nNobody who can manage a crocodile is despised.\r\n\r\nIllogical persons are despised.\r\n[reveal-answer q=\"814448\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"814448\"]\r\n\r\nLet B = is a baby, D = is despised, I = is illogical, and M = can manage a crocodile.\r\n\r\nThen we can write the premises as:\r\n<p style=\"text-align: center;\">[latex]B{\\rightarrow}I\\\\M{\\rightarrow}{\\sim}D\\\\I{\\rightarrow}D[\/latex]<\/p>\r\nFrom the first and third premises, we can conclude that [latex]B{\\rightarrow}D[\/latex]; that babies are despised.\r\n\r\nUsing the contrapositive of the second premised, [latex]D{\\rightarrow}{\\sim}M[\/latex], we can conclude that [latex]B\\rightarrow\\sim{M}[\/latex]; that babies cannot manage crocodiles.\r\n\r\nWhile silly, this is a logical conclusion from the given premises.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Logical Inference<\/h2>\r\nSuppose we know that a statement of form [latex]P{\\rightarrow}Q[\/latex] is true. This tells us\u00a0that whenever <em>P<\/em> is true, <em>Q<\/em> will also be true. By itself, [latex]P{\\rightarrow}Q[\/latex]\u00a0being true\u00a0does not tell us that either <em>P<\/em> or <em>Q<\/em> is true (they could both be false, or <em>P<\/em>\u00a0could be false and <em>Q<\/em> true). However if in addition we happen to know\u00a0that <em>P<\/em> is true then it must be that <em>Q<\/em> is true. This is called a <strong>logical\u00a0inference<\/strong>: Given two true statements we can infer that a third statement\u00a0is true. In this instance true statements [latex]P{\\rightarrow}Q[\/latex] and <em>P<\/em> are \u201cadded together\u201d\u00a0to get <em>Q<\/em>. This is described below with [latex]P{\\rightarrow}Q[\/latex]\u00a0stacked one atop the\u00a0other with a line separating them from <em>Q<\/em>. The intended meaning is that [latex]P{\\rightarrow}Q[\/latex]\u00a0combined with <em>P<\/em> produces <em>Q<\/em>.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]P{\\rightarrow}Q\\\\\\underline{P\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\Q[\/latex]<\/td>\r\n<td>[latex]\\,\\,P{\\rightarrow}Q\\\\\\underline{{\\sim}Q\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\{\\sim}P[\/latex]<\/td>\r\n<td>[latex]\\,\\,P{\\vee}Q\\\\\\underline{{\\sim}P\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\Q[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTwo other logical inferences are listed above. In each case you should\u00a0convince yourself (based on your knowledge of the relevant truth tables)\u00a0that the truth of the statements above the line forces the statement below\u00a0the line to be true.\r\n\r\nFollowing are some additional useful logical inferences. The first\u00a0expresses the obvious fact that if <em>P<\/em> and <em>Q<\/em> are both true then the statement [latex]P{\\wedge}Q[\/latex] will be true. On the other hand, [latex]P{\\wedge}Q[\/latex]\u00a0being true forces <em>P<\/em> (also <em>Q<\/em>)\u00a0to be true. Finally, if <em>P<\/em> is true, then [latex]P{\\vee}Q[\/latex]\u00a0must be true, no matter what\u00a0statement <em>Q<\/em> is.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\,\\,P\\\\\\underline{\\,\\,Q\\,\\,\\,\\,\\,}\\\\P{\\wedge}Q[\/latex]<\/td>\r\n<td>[latex]\\underline{P{\\wedge}Q}\\\\P[\/latex]<\/td>\r\n<td>[latex]\\underline{\\,P\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\\\,P{\\vee}Q[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox shaded\">The first two statements in each case are called \u201cpremises\u201d and the final\u00a0statement is the \u201cconclusion.\u201d We combine premises with [latex]{\\wedge}[\/latex] (\u201cand\u201d). The\u00a0premises together imply the conclusion. Thus, the first argument would have [latex]\\left(\\left(P{\\rightarrow}Q\\right){\\wedge}P\\right){\\rightarrow}Q[\/latex]<\/div>\r\n<h2>An Important Note<\/h2>\r\nIt is important to be aware of the reasons that we study logic. There\u00a0are three very significant reasons. First, the truth tables we studied tell\u00a0us the exact meanings of the words such as \u201cand,\u201d \u201cor,\u201d \u201cnot,\u201d and so on.\u00a0For instance, whenever we use or read the \u201cIf..., then\u201d construction in\u00a0a mathematical context, logic tells us exactly what is meant. Second,\u00a0the rules of inference provide a system in which we can produce new\u00a0information (statements) from known information. Finally, logical rules\u00a0such as DeMorgan\u2019s laws help us correctly change certain statements into\u00a0(potentially more useful) statements with the same meaning. Thus logic\u00a0helps us understand the meanings of statements and it also produces new\u00a0meaningful statements.\r\n\r\nLogic is the glue that holds strings of statements together and pins down\u00a0the exact meaning of certain key phrases such as the \u201cIf..., then\u201d or \u201cFor\u00a0all\u201d constructions. Logic is the common language that all mathematicians\u00a0use, so we must have a firm grip on it in order to write and understand\u00a0mathematics.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Discern between an inductive argument and a deductive argument<\/li>\n<li>Evaluate deductive arguments<\/li>\n<li>Analyze arguments with Venn diagrams and truth tables<\/li>\n<li>Use logical inference to infer whether a statement is true<\/li>\n<li>Identify logical fallacies in common language including appeal to ignorance, appeal to authority, appeal to consequence, false dilemma, circular reasoning, post hoc, correlation implies causation, and straw man arguments<\/li>\n<\/ul>\n<\/div>\n<p>A logical argument is a claim that a set of premises support a conclusion. There are two general types of arguments: inductive and deductive arguments.<\/p>\n<div class=\"textbox\">\n<h3>Argument types<\/h3>\n<p>An <strong>inductive<\/strong> argument uses a collection of specific examples as its premises and uses them to propose a general conclusion.<\/p>\n<p>A <strong>deductive<\/strong> argument uses a collection of general statements as its premises and uses them to propose a specific situation as the conclusion.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=109526&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=109527&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The argument \u201cwhen I went to the store last week I forgot my purse, and when I went today I forgot my purse. I always forget my purse when I go the store\u201d is an inductive argument.<\/p>\n<p>The premises are:<\/p>\n<p style=\"padding-left: 30px;\">I forgot my purse last week<br \/>\nI forgot my purse today<\/p>\n<p>The conclusion is:<\/p>\n<p style=\"padding-left: 30px;\">I always forget my purse<\/p>\n<p>Notice that the premises are specific situations, while the conclusion is a general statement. In this case, this is a fairly weak argument, since it is based on only two instances.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The argument \u201cevery day for the past year, a plane flies over my house at 2pm. A plane will fly over my house every day at 2pm\u201d is a stronger inductive argument, since it is based on a larger set of evidence.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Evaluating inductive arguments<\/h3>\n<p>An inductive argument is never able to prove the conclusion true, but it can provide either weak or strong evidence to suggest it may be true.<\/p>\n<\/div>\n<p>Many scientific theories, such as the big bang theory, can never be proven. Instead, they are inductive arguments supported by a wide variety of evidence. Usually in science, an idea is considered a hypothesis until it has been well tested, at which point it graduates to being considered a theory. The commonly known scientific theories, like Newton\u2019s theory of gravity, have all stood up to years of testing and evidence, though sometimes they need to be adjusted based on new evidence. For gravity, this happened when Einstein proposed the theory of general relativity.<\/p>\n<p>A deductive argument is more clearly valid or not, which makes them easier to evaluate.<\/p>\n<div class=\"textbox\">\n<h3>Evaluating deductive arguments<\/h3>\n<p>A deductive argument is considered valid if all the premises are true, and the conclusion follows logically from those premises. In other words, the premises are true, and the conclusion follows necessarily from those premises.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The argument \u201cAll cats are mammals and a tiger is a cat, so a tiger is a mammal\u201d is a valid deductive argument.<\/p>\n<p>The premises are:<\/p>\n<p style=\"padding-left: 30px;\">All cats are mammals<br \/>\nA tiger is a cat<\/p>\n<p>The conclusion is:<\/p>\n<p style=\"padding-left: 30px;\">A tiger is a mammal<\/p>\n<p>Both the premises are true. To see that the premises must logically lead to the conclusion, one approach would be use a Venn diagram. From the first premise, we can conclude that the set of cats is a subset of the set of mammals. From the second premise, we are told that a tiger lies within the set of cats. From that, we can see in the Venn diagram that the tiger also lies inside the set of mammals, so the conclusion is valid.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-256\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155146\/Fig4_2_1.png\" alt=\"A Venn diagram with a large circle labeled Mammals and a smaller circle labeled Cats contained within the Mammals circle. A red X labeled Tiger is in the circle labeled Cats.\" width=\"200\" height=\"202\" \/><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm132642\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=132642&theme=oea&iframe_resize_id=ohm132642&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Analyzing Arguments with Venn\/Euler diagrams<\/h3>\n<p>To analyze an argument with a Venn\/ Euler diagram<\/p>\n<ol>\n<li>Draw a Venn\/ Euler diagram based on the premises of the argument<\/li>\n<li>If the premises are insufficient to determine what determine the location of an element, indicate that.<\/li>\n<li>The argument is valid if it is clear that the conclusion must be true<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>All firefighters know CPR<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>Jill knows CPR<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>Jill is a firefighter<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>From the first premise, we know that firefighters all lie inside the set of those who know CPR. From the second premise, we know that Jill is a member of that larger set, but we do not have enough information to know if she also is a member of the smaller subset that is firefighters.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-257 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155147\/Fig4_2_2.png\" alt=\"Fig4_2_2\" width=\"200\" height=\"200\" \/><\/p>\n<p>Since the conclusion does not necessarily follow from the premises, this is an invalid argument, regardless of whether Jill actually is a firefighter.<\/p>\n<\/div>\n<p>It is important to note that whether or not Jill is actually a firefighter is not important in evaluating the validity of the argument; we are only concerned with whether the premises are enough to prove the conclusion.<\/p>\n<p>In addition to these categorical style premises of the form \u201call ___,\u201d \u201csome ____,\u201d and \u201cno ____,\u201d it is also common to see premises that are implications.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>If you live in Seattle, you live in Washington.<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>Marcus does not live in Seattle.<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>Marcus does not live in Washington.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>From the first premise, we know that the set of people who live in Seattle is inside the set of those who live in Washington. From the second premise, we know that Marcus does not lie in the Seattle set, but we have insufficient information to know whether or not Marcus lives in Washington or not. This is an invalid argument.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-258 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155147\/Fig4_2_3.png\" alt=\"Fig4_2_3\" width=\"200\" height=\"204\" \/><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider the argument \u201cYou are a married man, so you must have a wife.\u201d<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q383279\">Show Solution<\/span><\/p>\n<div id=\"q383279\" class=\"hidden-answer\" style=\"display: none\">\n<p>This is an invalid argument, since there are, at least in parts of the world, men who are married to other men, so the premise not insufficient to imply the conclusion.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Some arguments are better analyzed using truth tables.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider the argument:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>If you bought bread, then you went to the store<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>You bought bread<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>You went to the store<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q23681\">Show Solution<\/span><\/p>\n<div id=\"q23681\" class=\"hidden-answer\" style=\"display: none\">\n<p>While this example is hopefully fairly obviously a valid argument, we can analyze it using a truth table by representing each of the premises symbolically. We can then look at the implication that the premises together imply the conclusion. If the truth table is a tautology (always true), then the argument is valid.<\/p>\n<p>We\u2019ll get B represent \u201cyou bought bread\u201d and S represent \u201cyou went to the store\u201d. Then the argument becomes:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]B{\\rightarrow}S[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>[latex]S[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To test the validity, we look at whether the combination of both premises implies the conclusion; is it true that [latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]?<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]B[\/latex]<\/td>\n<td>[latex]S[\/latex]<\/td>\n<td>[latex]B{\\rightarrow}S[\/latex]<\/td>\n<td>[latex]\\left(B{\\rightarrow}S\\right){\\wedge}B[\/latex]<\/td>\n<td>[latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since the truth table for [latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]\u00a0is always true, this is a valid argument.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Analyzing arguments using truth tables<\/h3>\n<p>To analyze an argument with a truth table:<\/p>\n<ol>\n<li>Represent each of the premises symbolically<\/li>\n<li>Create a conditional statement, joining all the premises with and to form the antecedent, and using the conclusion as the consequent.<\/li>\n<li>Create a truth table for that statement. If it is always true, then the argument is valid.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>If I go to the mall, then I\u2019ll buy new jeans.<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>If I buy new jeans, I\u2019ll buy a shirt to go with it.<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>If I got to the mall, I\u2019ll buy a shirt.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Let <em>M<\/em> = I go to the mall, <em>J <\/em>= I buy jeans, and <em>S<\/em> = I buy a shirt.<\/p>\n<p>The premises and conclusion can be stated as:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]M{\\rightarrow}J[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]J{\\rightarrow}S[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>[latex]M{\\rightarrow}S[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can construct a truth table for [latex]\\left[\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)\\right]{\\rightarrow}\\left(M{\\rightarrow}S\\right)[\/latex]<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]M[\/latex]<\/td>\n<td>[latex]J[\/latex]<\/td>\n<td>[latex]S[\/latex]<\/td>\n<td>[latex]M{\\rightarrow}J[\/latex]<\/td>\n<td>[latex]J{\\rightarrow}S[\/latex]<\/td>\n<td>[latex]\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)[\/latex]<\/td>\n<td>[latex]M{\\rightarrow}S[\/latex]<\/td>\n<td>[latex]\\left[\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)\\right]{\\rightarrow}\\left(M{\\rightarrow}S\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>From the truth table, we can see this is a valid argument.<\/p>\n<\/div>\n<p>The previous problem is an example of a syllogism.<\/p>\n<div class=\"textbox\">\n<h3>Syllogism<\/h3>\n<p>A syllogism is an implication derived from two others, where the consequence of one is the antecedent to the other. The general form of a syllogism is:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]p{\\rightarrow}q[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]q{\\rightarrow}r[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>[latex]p{\\rightarrow}r[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This is sometime called the transitive property for implication.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>If I work hard, I\u2019ll get a raise.<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>If I get a raise, I\u2019ll buy a boat.<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>If I don\u2019t buy a boat, I must not have worked hard.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q880229\">Show Solution<\/span><\/p>\n<div id=\"q880229\" class=\"hidden-answer\" style=\"display: none\">\n<p>If we let <em>W<\/em> = working hard, <em>R<\/em> = getting a raise, and <em>B<\/em> = buying a boat, then we can represent our argument symbolically:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]H{\\rightarrow}R[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]R{\\rightarrow}B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>[latex]\\sim{B}{\\rightarrow}{\\sim}H[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We could construct a truth table for this argument, but instead, we will use the notation of the contrapositive we learned earlier to note that the implication [latex]{\\sim}B{\\rightarrow}{\\sim}H[\/latex]\u00a0is equivalent to the implication [latex]H{\\rightarrow}B[\/latex]. Rewritten, we can see that this conclusion is indeed a logical syllogism derived from the premises.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Is this argument valid?<\/p>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>If I go to the party, I\u2019ll be really tired tomorrow.<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>If I go to the party, I\u2019ll get to see friends.<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>If I don\u2019t see friends, I won\u2019t be tired tomorrow.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><iframe loading=\"lazy\" id=\"mom50\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25956&amp;theme=oea&amp;iframe_resize_id=mom50\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Lewis Carroll, author of <em>Alice in Wonderland<\/em>, was a math and logic teacher, and wrote two books on logic. In them, he would propose premises as a puzzle, to be connected using syllogisms.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve the puzzle. In other words, find a logical conclusion from these premises.<\/p>\n<p>All babies are illogical.<\/p>\n<p>Nobody who can manage a crocodile is despised.<\/p>\n<p>Illogical persons are despised.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q814448\">Show Solution<\/span><\/p>\n<div id=\"q814448\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let B = is a baby, D = is despised, I = is illogical, and M = can manage a crocodile.<\/p>\n<p>Then we can write the premises as:<\/p>\n<p style=\"text-align: center;\">[latex]B{\\rightarrow}I\\\\M{\\rightarrow}{\\sim}D\\\\I{\\rightarrow}D[\/latex]<\/p>\n<p>From the first and third premises, we can conclude that [latex]B{\\rightarrow}D[\/latex]; that babies are despised.<\/p>\n<p>Using the contrapositive of the second premised, [latex]D{\\rightarrow}{\\sim}M[\/latex], we can conclude that [latex]B\\rightarrow\\sim{M}[\/latex]; that babies cannot manage crocodiles.<\/p>\n<p>While silly, this is a logical conclusion from the given premises.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Logical Inference<\/h2>\n<p>Suppose we know that a statement of form [latex]P{\\rightarrow}Q[\/latex] is true. This tells us\u00a0that whenever <em>P<\/em> is true, <em>Q<\/em> will also be true. By itself, [latex]P{\\rightarrow}Q[\/latex]\u00a0being true\u00a0does not tell us that either <em>P<\/em> or <em>Q<\/em> is true (they could both be false, or <em>P<\/em>\u00a0could be false and <em>Q<\/em> true). However if in addition we happen to know\u00a0that <em>P<\/em> is true then it must be that <em>Q<\/em> is true. This is called a <strong>logical\u00a0inference<\/strong>: Given two true statements we can infer that a third statement\u00a0is true. In this instance true statements [latex]P{\\rightarrow}Q[\/latex] and <em>P<\/em> are \u201cadded together\u201d\u00a0to get <em>Q<\/em>. This is described below with [latex]P{\\rightarrow}Q[\/latex]\u00a0stacked one atop the\u00a0other with a line separating them from <em>Q<\/em>. The intended meaning is that [latex]P{\\rightarrow}Q[\/latex]\u00a0combined with <em>P<\/em> produces <em>Q<\/em>.<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]P{\\rightarrow}Q\\\\\\underline{P\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\Q[\/latex]<\/td>\n<td>[latex]\\,\\,P{\\rightarrow}Q\\\\\\underline{{\\sim}Q\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\{\\sim}P[\/latex]<\/td>\n<td>[latex]\\,\\,P{\\vee}Q\\\\\\underline{{\\sim}P\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\Q[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Two other logical inferences are listed above. In each case you should\u00a0convince yourself (based on your knowledge of the relevant truth tables)\u00a0that the truth of the statements above the line forces the statement below\u00a0the line to be true.<\/p>\n<p>Following are some additional useful logical inferences. The first\u00a0expresses the obvious fact that if <em>P<\/em> and <em>Q<\/em> are both true then the statement [latex]P{\\wedge}Q[\/latex] will be true. On the other hand, [latex]P{\\wedge}Q[\/latex]\u00a0being true forces <em>P<\/em> (also <em>Q<\/em>)\u00a0to be true. Finally, if <em>P<\/em> is true, then [latex]P{\\vee}Q[\/latex]\u00a0must be true, no matter what\u00a0statement <em>Q<\/em> is.<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]\\,\\,P\\\\\\underline{\\,\\,Q\\,\\,\\,\\,\\,}\\\\P{\\wedge}Q[\/latex]<\/td>\n<td>[latex]\\underline{P{\\wedge}Q}\\\\P[\/latex]<\/td>\n<td>[latex]\\underline{\\,P\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\\\,P{\\vee}Q[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox shaded\">The first two statements in each case are called \u201cpremises\u201d and the final\u00a0statement is the \u201cconclusion.\u201d We combine premises with [latex]{\\wedge}[\/latex] (\u201cand\u201d). The\u00a0premises together imply the conclusion. Thus, the first argument would have [latex]\\left(\\left(P{\\rightarrow}Q\\right){\\wedge}P\\right){\\rightarrow}Q[\/latex]<\/div>\n<h2>An Important Note<\/h2>\n<p>It is important to be aware of the reasons that we study logic. There\u00a0are three very significant reasons. First, the truth tables we studied tell\u00a0us the exact meanings of the words such as \u201cand,\u201d \u201cor,\u201d \u201cnot,\u201d and so on.\u00a0For instance, whenever we use or read the \u201cIf&#8230;, then\u201d construction in\u00a0a mathematical context, logic tells us exactly what is meant. Second,\u00a0the rules of inference provide a system in which we can produce new\u00a0information (statements) from known information. Finally, logical rules\u00a0such as DeMorgan\u2019s laws help us correctly change certain statements into\u00a0(potentially more useful) statements with the same meaning. Thus logic\u00a0helps us understand the meanings of statements and it also produces new\u00a0meaningful statements.<\/p>\n<p>Logic is the glue that holds strings of statements together and pins down\u00a0the exact meaning of certain key phrases such as the \u201cIf&#8230;, then\u201d or \u201cFor\u00a0all\u201d constructions. Logic is the common language that all mathematicians\u00a0use, so we must have a firm grip on it in order to write and understand\u00a0mathematics.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-185\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 132642. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Logic. <strong>Authored by<\/strong>: David Lippman. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <strong>Project<\/strong>: Math In Society. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Question ID 25956. <strong>Authored by<\/strong>: Lippman,David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 109528, 109527. <strong>Authored by<\/strong>: Hartley,Josiah. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Logic\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"Math In Society\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 25956\",\"author\":\"Lippman,David\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 109528, 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