{"id":443,"date":"2017-06-13T23:24:27","date_gmt":"2017-06-13T23:24:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/?post_type=chapter&#038;p=443"},"modified":"2018-08-14T20:27:04","modified_gmt":"2018-08-14T20:27:04","slug":"analyzing-arguments-with-logic","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-hccc-ma-124-1\/chapter\/analyzing-arguments-with-logic\/","title":{"raw":"Analyzing Arguments With Logic","rendered":"Analyzing Arguments With Logic"},"content":{"raw":"In the next section we will use what we have learned about constructing statements to build arguments with logical statements. We will also use more Venn diagrams to evaluate whether an argument is logical, and introduce how to use a truth table to evaluate a logical statement.\r\n<h2>Arguments<\/h2>\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 now<\/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=\"wp-image-256 aligncenter\" 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=\"Fig4_2_1\" width=\"200\" height=\"202\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it now<\/h3>\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=132642&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"450\"><\/iframe>\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 Answer[\/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 Answer[\/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 Answer[\/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 Now<\/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 Answer[\/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<h3>Logical Inference<\/h3>\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<h3>An Important Note<\/h3>\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.\r\n<h2>Logical Fallacies in Common Language<\/h2>\r\nIn the previous discussion, we saw that logical arguments can be invalid when the premises are not true, when the premises are not sufficient to guarantee the conclusion, or when there are invalid chains in logic. There are a number of other ways in which arguments can be invalid, a sampling of which are given here.\r\n<div class=\"textbox\">\r\n<h3>Ad hominem<\/h3>\r\nAn ad hominem argument attacks the person making the argument, ignoring the argument itself.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n\u201cJane says that whales aren\u2019t fish, but she\u2019s only in the second grade, so she can\u2019t be right.\u201d\r\n[reveal-answer q=\"683444\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"683444\"]\r\n\r\nHere the argument is attacking Jane, not the validity of her claim, so this is an ad hominem argument.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n\u201cJane says that whales aren\u2019t fish, but everyone knows that they\u2019re really mammals\u2014she\u2019s so stupid.\u201d\r\n[reveal-answer q=\"421872\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"421872\"]This certainly isn\u2019t very nice, but it is <em>not<\/em> ad hominem since a valid counterargument is made along with the personal insult.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it now<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25481&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>Appeal to ignorance<\/h3>\r\nThis type of argument assumes something it true because it hasn\u2019t been proven false.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n\u201cNobody has proven that photo isn\u2019t Bigfoot, so it must be Bigfoot.\u201d\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Appeal to authority<\/h3>\r\nThese arguments attempt to use the authority of a person to prove a claim. While often authority can provide strength to an argument, problems can occur when the person\u2019s opinion is not shared by other experts, or when the authority is irrelevant to the claim.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it now<\/h3>\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25600&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\n\u201cA diet high in bacon can be healthy \u2013 Doctor Atkins said so.\u201d\r\n[reveal-answer q=\"536253\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"536253\"]Here, an appeal to the authority of a doctor is used for the argument. This generally would provide strength to the argument, except that the opinion that eating a diet high in saturated fat runs counter to general medical opinion. More supporting evidence would be needed to justify this claim.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n\u201cJennifer Hudson lost weight with Weight Watchers, so their program must work.\u201d\r\n\r\n[reveal-answer q=\"764781\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"764781\"]\r\n\r\nHere, there is an appeal to the authority of a celebrity. While her experience does provide evidence, it provides no more than any other person\u2019s experience would.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Appeal to Consequence<\/h3>\r\nAn appeal to consequence concludes that a premise is true or false based on whether the consequences are desirable or not.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n\u201cHumans will travel faster than light: faster-than-light travel would be beneficial for space travel.\u201d\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it now<\/h3>\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25602&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>False dilemma<\/h3>\r\nA false dilemma argument falsely frames an argument as an \u201ceither or\u201d choice, without allowing for additional options.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n\u201cEither those lights in the sky were an airplane or aliens. There are no airplanes scheduled for tonight, so it must be aliens.\u201d\r\n\r\nThis argument ignores the possibility that the lights could be something other than an airplane or aliens.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it now<\/h3>\r\n<iframe id=\"mom15\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25484&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Circular reasoning<\/h3>\r\nCircular reasoning is an argument that relies on the conclusion being true for the premise to be true.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n\u201cI shouldn\u2019t have gotten a C in that class; I\u2019m an A student!\u201d\r\n\r\nIn this argument, the student is claiming that because they\u2019re an A student, though shouldn\u2019t have gotten a C. But because they got a C, they\u2019re not an A student.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it now<\/h3>\r\n<iframe id=\"mom20\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25603&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Straw man<\/h3>\r\nA straw man argument involves misrepresenting the argument in a less favorable way to make it easier to attack.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n\u201cSenator Jones has proposed reducing military funding by 10%. Apparently he wants to leave us defenseless against attacks by terrorists\u201d\r\n\r\nHere the arguer has represented a 10% funding cut as equivalent to leaving us defenseless, making it easier to attack.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Post hoc (post hoc ergo propter hoc)<\/h3>\r\nA post hoc argument claims that because two things happened sequentially, then the first must have caused the second.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n\u201cToday I wore a red shirt, and my football team won! I need to wear a red shirt everytime they play to make sure they keep winning.\u201d\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it now<\/h3>\r\n<iframe id=\"mom25\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25604&amp;theme=oea&amp;iframe_resize_id=mom25\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Correlation implies causation<\/h3>\r\nSimilar to post hoc, but without the requirement of sequence, this fallacy assumes that just because two things are related one must have caused the other. Often there is a third variable not considered.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n\u201cMonths with high ice cream sales also have a high rate of deaths by drowning. Therefore ice cream must be causing people to drown.\u201d\r\n\r\nThis argument is implying a causal relation, when really both are more likely dependent on the weather; that ice cream and drowning are both more likely during warm summer months.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it now<\/h3>\r\n<iframe id=\"mom35\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25482&amp;theme=oea&amp;iframe_resize_id=mom35\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It Now<\/h3>\r\nIdentify the logical fallacy in each of the arguments\r\n<ol>\r\n \t<li>Only an untrustworthy person would run for office. The fact that politicians are untrustworthy is proof of this.<\/li>\r\n \t<li>Since the 1950s, both the atmospheric carbon dioxide level and <a href=\"http:\/\/en.wikipedia.org\/wiki\/Obesity\">obesity<\/a> levels have increased sharply. Hence, atmospheric carbon dioxide causes obesity.<\/li>\r\n \t<li>The oven was working fine until you started using it, so you must have broken it.<\/li>\r\n \t<li>You can\u2019t give me a D in the class\u2014I can\u2019t afford to retake it.<\/li>\r\n \t<li>The senator wants to increase support for food stamps. He wants to take the taxpayers\u2019 hard-earned money and give it away to lazy people. This isn\u2019t fair so we shouldn\u2019t do it.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"912175\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"912175\"]\r\n\r\n1.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]A[\/latex]<\/td>\r\n<td>[latex]B[\/latex]<\/td>\r\n<td>[latex]{\\sim}A[\/latex]<\/td>\r\n<td>[latex]{\\sim}A{\\wedge}B[\/latex]<\/td>\r\n<td>[latex]{\\sim}B[\/latex]<\/td>\r\n<td>[latex]\\left({\\sim}A{\\wedge}B\\right){\\vee}{\\sim}B[\/latex]<\/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>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/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>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>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<td>T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n2. Since no cows are purple, we know there is no overlap between the set of cows and the set of purple things. We know Fido is not in the cow set, but that is not enough to conclude that Fido is in the purple things set.\r\n\r\n<img class=\"aligncenter wp-image-1939 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/20214932\/Screen-Shot-2017-03-20-at-2.48.21-PM.png\" alt=\"A blue circle labeled Cows. A yellow circle labeled Purple things, with an x followed by a question mark. Apart from both circles is a note asking Fido x?\" width=\"366\" height=\"215\" \/>\r\n\r\n3. Let S: have a shovel, D: dig a hole. The first premise is equivalent to [latex]S{\\rightarrow}D[\/latex]. The second premise is D. The conclusion is S.\r\n\r\nWe are testing [latex]\\left[\\left(S{\\rightarrow}D\\right){\\wedge}D\\right]{\\rightarrow}S[\/latex]\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]S[\/latex]<\/td>\r\n<td>[latex]D[\/latex]<\/td>\r\n<td>[latex]S{\\rightarrow}D[\/latex]<\/td>\r\n<td>[latex]\\left(S{\\rightarrow}D\\right){\\wedge}D[\/latex]<\/td>\r\n<td>[latex]\\left[\\left(S{\\rightarrow}D\\right){\\wedge}D\\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>T<\/td>\r\n<td>F<\/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\nThis is not a tautology, so this is an invalid argument.\r\n\r\n4.\u00a0Letting P = go to the party, T = being tired, and F = seeing friends, then we can represent this argument as P:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]P{\\rightarrow}T[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]P{\\rightarrow}F[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>[latex]{\\sim}F{\\rightarrow}{\\sim}T[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe could rewrite the second premise using the contrapositive to state [latex]{\\sim}F{\\rightarrow}{\\sim}P[\/latex], but that does not allow us to form a syllogism. If we don\u2019t see friends, then we didn\u2019t go the party, but that is not sufficient to claim I won\u2019t be tired tomorrow. Maybe I stayed up all night watching movies.\r\n\r\n5.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Circular<\/li>\r\n \t<li>Correlation does not imply causation<\/li>\r\n \t<li>Post hoc<\/li>\r\n \t<li>Appeal to consequence<\/li>\r\n \t<li>Straw man<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<p>In the next section we will use what we have learned about constructing statements to build arguments with logical statements. We will also use more Venn diagrams to evaluate whether an argument is logical, and introduce how to use a truth table to evaluate a logical statement.<\/p>\n<h2>Arguments<\/h2>\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 now<\/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=\"wp-image-256 aligncenter\" 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=\"Fig4_2_1\" width=\"200\" height=\"202\" \/><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it now<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=132642&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"450\"><\/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 Answer<\/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 Answer<\/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 Answer<\/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 Now<\/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 Answer<\/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<h3>Logical Inference<\/h3>\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<h3>An Important Note<\/h3>\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<h2>Logical Fallacies in Common Language<\/h2>\n<p>In the previous discussion, we saw that logical arguments can be invalid when the premises are not true, when the premises are not sufficient to guarantee the conclusion, or when there are invalid chains in logic. There are a number of other ways in which arguments can be invalid, a sampling of which are given here.<\/p>\n<div class=\"textbox\">\n<h3>Ad hominem<\/h3>\n<p>An ad hominem argument attacks the person making the argument, ignoring the argument itself.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>\u201cJane says that whales aren\u2019t fish, but she\u2019s only in the second grade, so she can\u2019t be right.\u201d<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q683444\">Show Answer<\/span><\/p>\n<div id=\"q683444\" class=\"hidden-answer\" style=\"display: none\">\n<p>Here the argument is attacking Jane, not the validity of her claim, so this is an ad hominem argument.\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>\u201cJane says that whales aren\u2019t fish, but everyone knows that they\u2019re really mammals\u2014she\u2019s so stupid.\u201d<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q421872\">Show Answer<\/span><\/p>\n<div id=\"q421872\" class=\"hidden-answer\" style=\"display: none\">This certainly isn\u2019t very nice, but it is <em>not<\/em> ad hominem since a valid counterargument is made along with the personal insult.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it now<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25481&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Appeal to ignorance<\/h3>\n<p>This type of argument assumes something it true because it hasn\u2019t been proven false.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>\u201cNobody has proven that photo isn\u2019t Bigfoot, so it must be Bigfoot.\u201d<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Appeal to authority<\/h3>\n<p>These arguments attempt to use the authority of a person to prove a claim. While often authority can provide strength to an argument, problems can occur when the person\u2019s opinion is not shared by other experts, or when the authority is irrelevant to the claim.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it now<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25600&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>\u201cA diet high in bacon can be healthy \u2013 Doctor Atkins said so.\u201d<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q536253\">Show Answer<\/span><\/p>\n<div id=\"q536253\" class=\"hidden-answer\" style=\"display: none\">Here, an appeal to the authority of a doctor is used for the argument. This generally would provide strength to the argument, except that the opinion that eating a diet high in saturated fat runs counter to general medical opinion. More supporting evidence would be needed to justify this claim.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>\u201cJennifer Hudson lost weight with Weight Watchers, so their program must work.\u201d<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q764781\">Show Answer<\/span><\/p>\n<div id=\"q764781\" class=\"hidden-answer\" style=\"display: none\">\n<p>Here, there is an appeal to the authority of a celebrity. While her experience does provide evidence, it provides no more than any other person\u2019s experience would.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Appeal to Consequence<\/h3>\n<p>An appeal to consequence concludes that a premise is true or false based on whether the consequences are desirable or not.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>\u201cHumans will travel faster than light: faster-than-light travel would be beneficial for space travel.\u201d<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it now<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25602&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>False dilemma<\/h3>\n<p>A false dilemma argument falsely frames an argument as an \u201ceither or\u201d choice, without allowing for additional options.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>\u201cEither those lights in the sky were an airplane or aliens. There are no airplanes scheduled for tonight, so it must be aliens.\u201d<\/p>\n<p>This argument ignores the possibility that the lights could be something other than an airplane or aliens.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it now<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom15\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25484&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Circular reasoning<\/h3>\n<p>Circular reasoning is an argument that relies on the conclusion being true for the premise to be true.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>\u201cI shouldn\u2019t have gotten a C in that class; I\u2019m an A student!\u201d<\/p>\n<p>In this argument, the student is claiming that because they\u2019re an A student, though shouldn\u2019t have gotten a C. But because they got a C, they\u2019re not an A student.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it now<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom20\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25603&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Straw man<\/h3>\n<p>A straw man argument involves misrepresenting the argument in a less favorable way to make it easier to attack.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>\u201cSenator Jones has proposed reducing military funding by 10%. Apparently he wants to leave us defenseless against attacks by terrorists\u201d<\/p>\n<p>Here the arguer has represented a 10% funding cut as equivalent to leaving us defenseless, making it easier to attack.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Post hoc (post hoc ergo propter hoc)<\/h3>\n<p>A post hoc argument claims that because two things happened sequentially, then the first must have caused the second.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>\u201cToday I wore a red shirt, and my football team won! I need to wear a red shirt everytime they play to make sure they keep winning.\u201d<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it now<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom25\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25604&amp;theme=oea&amp;iframe_resize_id=mom25\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Correlation implies causation<\/h3>\n<p>Similar to post hoc, but without the requirement of sequence, this fallacy assumes that just because two things are related one must have caused the other. Often there is a third variable not considered.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>\u201cMonths with high ice cream sales also have a high rate of deaths by drowning. Therefore ice cream must be causing people to drown.\u201d<\/p>\n<p>This argument is implying a causal relation, when really both are more likely dependent on the weather; that ice cream and drowning are both more likely during warm summer months.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it now<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom35\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25482&amp;theme=oea&amp;iframe_resize_id=mom35\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It Now<\/h3>\n<p>Identify the logical fallacy in each of the arguments<\/p>\n<ol>\n<li>Only an untrustworthy person would run for office. The fact that politicians are untrustworthy is proof of this.<\/li>\n<li>Since the 1950s, both the atmospheric carbon dioxide level and <a href=\"http:\/\/en.wikipedia.org\/wiki\/Obesity\">obesity<\/a> levels have increased sharply. Hence, atmospheric carbon dioxide causes obesity.<\/li>\n<li>The oven was working fine until you started using it, so you must have broken it.<\/li>\n<li>You can\u2019t give me a D in the class\u2014I can\u2019t afford to retake it.<\/li>\n<li>The senator wants to increase support for food stamps. He wants to take the taxpayers\u2019 hard-earned money and give it away to lazy people. This isn\u2019t fair so we shouldn\u2019t do it.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q912175\">Show Answer<\/span><\/p>\n<div id=\"q912175\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]A[\/latex]<\/td>\n<td>[latex]B[\/latex]<\/td>\n<td>[latex]{\\sim}A[\/latex]<\/td>\n<td>[latex]{\\sim}A{\\wedge}B[\/latex]<\/td>\n<td>[latex]{\\sim}B[\/latex]<\/td>\n<td>[latex]\\left({\\sim}A{\\wedge}B\\right){\\vee}{\\sim}B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/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>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/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<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>2. Since no cows are purple, we know there is no overlap between the set of cows and the set of purple things. We know Fido is not in the cow set, but that is not enough to conclude that Fido is in the purple things set.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1939 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/20214932\/Screen-Shot-2017-03-20-at-2.48.21-PM.png\" alt=\"A blue circle labeled Cows. A yellow circle labeled Purple things, with an x followed by a question mark. Apart from both circles is a note asking Fido x?\" width=\"366\" height=\"215\" \/><\/p>\n<p>3. Let S: have a shovel, D: dig a hole. The first premise is equivalent to [latex]S{\\rightarrow}D[\/latex]. The second premise is D. The conclusion is S.<\/p>\n<p>We are testing [latex]\\left[\\left(S{\\rightarrow}D\\right){\\wedge}D\\right]{\\rightarrow}S[\/latex]<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]S[\/latex]<\/td>\n<td>[latex]D[\/latex]<\/td>\n<td>[latex]S{\\rightarrow}D[\/latex]<\/td>\n<td>[latex]\\left(S{\\rightarrow}D\\right){\\wedge}D[\/latex]<\/td>\n<td>[latex]\\left[\\left(S{\\rightarrow}D\\right){\\wedge}D\\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>T<\/td>\n<td>F<\/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>This is not a tautology, so this is an invalid argument.<\/p>\n<p>4.\u00a0Letting P = go to the party, T = being tired, and F = seeing friends, then we can represent this argument as P:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]P{\\rightarrow}T[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]P{\\rightarrow}F[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>[latex]{\\sim}F{\\rightarrow}{\\sim}T[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We could rewrite the second premise using the contrapositive to state [latex]{\\sim}F{\\rightarrow}{\\sim}P[\/latex], but that does not allow us to form a syllogism. If we don\u2019t see friends, then we didn\u2019t go the party, but that is not sufficient to claim I won\u2019t be tired tomorrow. Maybe I stayed up all night watching movies.<\/p>\n<p>5.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Circular<\/li>\n<li>Correlation does not imply causation<\/li>\n<li>Post hoc<\/li>\n<li>Appeal to consequence<\/li>\n<li>Straw man<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\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-443\">\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>Introduction and Learning Objectives. <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><li>Math in Society. <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>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 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>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":21,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Introduction and Learning Objectives\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Math in Society\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 109528, 109527\",\"author\":\"Hartley, 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