{"id":39,"date":"2023-01-25T16:33:55","date_gmt":"2023-01-25T16:33:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/analyzing-arguments-with-logic-dig-deeper-page-1\/"},"modified":"2024-10-18T20:50:17","modified_gmt":"2024-10-18T20:50:17","slug":"analyzing-arguments-with-logic-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/analyzing-arguments-with-logic-fresh-take\/","title":{"raw":"Analyzing Arguments With Logic: Fresh Take","rendered":"Analyzing Arguments With Logic: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand different types of arguments<\/li>\r\n\t<li>Use logic to see if a statement is true<\/li>\r\n\t<li>Identify logical fallacies<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Inductive and Deductive Reasoning<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>When we form arguments or draw conclusions, we often use reasoning, which can be classified into two major types: <strong>inductive<\/strong> and <strong>deductive<\/strong>.<\/p>\r\n<p><strong>Inductive reasoning<\/strong> is a form of logical thinking in which we make generalizations based on individual instances we have observed or experienced. It's about predicting future events or drawing broad conclusions from specific observations.<\/p>\r\n<p>For example, suppose you notice that every morning when you wake up, the sun has risen. This observation leads you to conclude, through inductive reasoning, that the sun will rise every morning.<\/p>\r\n<p>While inductive arguments can provide likely conclusions, they do not guarantee 100% truth in their conclusions, but it can provide either weak or strong evidence to suggest it may be true. This is due to the fact that they rely on observed patterns, and patterns sometimes change.<\/p>\r\n<p><strong>Deductive reasoning<\/strong>, on the other hand, starts with general statements or premises and leads to a specific, logical conclusion. It's about deriving a certain conclusion from a set of general principles or premises.<\/p>\r\n<p>For example, if all men are mortal (premise 1), and Socrates is a man (premise 2), then Socrates must be mortal (conclusion). Here, if the premises are true, then the conclusion must also be true. Deductive reasoning provides certainty, given that the initial premises are true.<\/p>\r\n<p>When assessing deductive arguments, two key concepts to consider are validity and soundness.<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li><strong>Validity<\/strong><br \/>\r\nA deductive argument is valid if the truth of its premises guarantees the truth of its conclusion. In other words, if the premises are true, the conclusion must also be true. The validity of an argument is purely about its logical structure or form, not about whether its premises or conclusions are actually true. For example, consider this argument:\r\n\r\n<ul>\r\n\t<li>If it rains, the ground will be wet.<\/li>\r\n\t<li>It's raining.<\/li>\r\n\t<li>Therefore, the ground is wet.<\/li>\r\n<\/ul>\r\n\r\nThis is a valid argument, as it follows a logical structure. If premises 1 and 2 are true, the conclusion (3) must also be true.<\/li>\r\n\t<li><strong>Soundness<\/strong><br \/>\r\nSoundness, on the other hand, is a characteristic of a deductive argument that is not only valid but also has true premises. A sound argument is thus a valid argument with true premises, which leads to a necessarily true conclusion. Using the previous example if it's indeed raining (which means premises 1 and 2 are true), and the argument structure is valid, then the argument is also sound, and the ground must indeed be wet.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205781&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=xu1792kg7wA&amp;video_target=tpm-plugin-2gl7c6rd-xu1792kg7wA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/3+7+arguments+and+Euler+Diagrams.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c3 7 arguments and Euler Diagrams\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Analyzing Arguments with Venn\/Euler Diagrams<\/h2>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205782&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=byOK9yv0dsc&amp;video_target=tpm-plugin-ezbhbgcp-byOK9yv0dsc\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/3.5+Analyzing+Arguments+with+Euler+Diagrams.txt\" target=\"_blank\" rel=\"noopener\">\u00a0transcript for \u201c3.5 Analyzing Arguments with Euler Diagrams\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox seeExample\">Analyze the following argument using a Venn\/Euler diagram.\r\n\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\n<p>[reveal-answer q=\"88485\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"88485\"]<br \/>\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.<\/p>\r\n<center><img class=\"aligncenter wp-image-6420 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10155416\/Arguments-logic-2-1-275x300.png\" alt=\"A Venn\/Euler diagram depicting a large circle labeled Washington and a smaller circle entirely contained within the larger one and labeled Seattle. In the larger circle, but outside of the smaller circle, the name Marcus is written, next to an x with a question mark. There is another x with a question mark outside of both circles.\" width=\"275\" height=\"300\" \/><\/center>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Analyzing Arguments with Truth Tables<\/h2>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205783&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=EfsbN5YbcPQ&amp;video_target=tpm-plugin-v4r9oz3d-EfsbN5YbcPQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Truth+Table+to+determine+if+an+argument+is+valid.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTruth Table to determine if an argument is valid\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205784&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=fkH3EcRDrnI&amp;video_target=tpm-plugin-bkx5ira3-fkH3EcRDrnI\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Truth+Tables+to+Analyze+Arguments.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTruth Tables to Analyze Arguments\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox seeExample\">Consider the argument:\r\n\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<p>[reveal-answer q=\"23681\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"23681\"]<\/p>\r\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>\r\n<p>We\u2019ll get [latex]B[\/latex] represent \u201cyou bought bread\u201d and [latex]S[\/latex] represent \u201cyou went to the store\u201d. Then the argument becomes:<\/p>\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\n<p>&nbsp;<\/p>\r\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>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]B[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]S[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]B{\\rightarrow}S[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\\left(B{\\rightarrow}S\\right){\\wedge}B[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\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>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox seeExample\">\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<p>[reveal-answer q=\"880229\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"880229\"]<\/p>\r\n<p>If we let [latex]W =[\/latex] working hard, [latex]R =[\/latex] getting a raise, and [latex]B =[\/latex] buying a boat, then we can represent our argument symbolically:<\/p>\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\n<p>&nbsp;<\/p>\r\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>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Logical Fallacies<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>A <strong>logical fallacy<\/strong> is an error in reasoning that results in an invalid argument or statement. These fallacies often sound persuasive and may seem to follow a logical line of thinking, which can make them tricky to spot. However, they don't hold up to rigorous scrutiny because they are based on flawed logic. Logical fallacies can occur either intentionally, as in the case of rhetorical strategies meant to sway opinion or manipulate, or unintentionally, as the result of faulty reasoning or misunderstanding. There are several different types of logical fallacies.<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li><strong>Ad hominem<\/strong>: This is a fallacy where someone attacks their opponent's character or personal traits rather than engaging with their argument or position. It is a way to divert attention from the issue at hand.<\/li>\r\n\t<li><strong>Appeal to ignorance<\/strong>: This fallacy assumes a claim is true because it has not been proven false, or false because it has not been proven true. It capitalizes on the lack of evidence to the contrary.<\/li>\r\n\t<li><strong>Appeal to authority<\/strong>: This fallacy asserts that because an authority figure or expert believes something, it must be true. It fails to account for the fact that even experts can be mistaken.<\/li>\r\n\t<li><strong>Appeal to consequence<\/strong>: This fallacy suggests that a premise is true or false based on whether its outcome is desirable or undesirable. The truth of a statement, however, isn't determined by its potential consequences.<\/li>\r\n\t<li><strong>False dilemma<\/strong>: Also known as false dichotomy, this fallacy presents only two options or outcomes when, in reality, there could be several. It artificially restricts possible choices to manipulate an outcome.<\/li>\r\n\t<li><strong>Circular reasoning<\/strong>: This fallacy occurs when the conclusion of an argument is used as one of its premises. It's essentially arguing in a circle, offering no new information or insight.<\/li>\r\n\t<li><strong>Straw man<\/strong>: This fallacy distorts an opponent's argument into an easily refutable position and then attacks this weaker proposition as if it were the original one. It's a way to sidestep the actual issue at hand.<\/li>\r\n\t<li><strong>Post hoc (post hoc ergo propter hoc)<\/strong>: The post hoc fallacy assumes that because event [latex]B[\/latex] followed event [latex]A[\/latex], [latex]A[\/latex] must have caused [latex]B[\/latex]. While often true in causality, it isn't always the case, and it's a fallacy to assume it without evidence.<\/li>\r\n\t<li><strong>Correlation implies causation<\/strong>: This fallacy, also known as \"cum hoc ergo propter hoc,\" suggests that because two events occur together, one must have caused the other. Correlation, however, does not necessarily mean causation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/UzT4L0Zt4ZY\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Understanding+Various+Types+of+Logical+Fallacies.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cUnderstanding Various Types of Logical Fallacies\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox seeExample\">\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 obesity 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<p>[reveal-answer q=\"912175\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"912175\"]<\/p>\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<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand different types of arguments<\/li>\n<li>Use logic to see if a statement is true<\/li>\n<li>Identify logical fallacies<\/li>\n<\/ul>\n<\/section>\n<h2>Inductive and Deductive Reasoning<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>When we form arguments or draw conclusions, we often use reasoning, which can be classified into two major types: <strong>inductive<\/strong> and <strong>deductive<\/strong>.<\/p>\n<p><strong>Inductive reasoning<\/strong> is a form of logical thinking in which we make generalizations based on individual instances we have observed or experienced. It&#8217;s about predicting future events or drawing broad conclusions from specific observations.<\/p>\n<p>For example, suppose you notice that every morning when you wake up, the sun has risen. This observation leads you to conclude, through inductive reasoning, that the sun will rise every morning.<\/p>\n<p>While inductive arguments can provide likely conclusions, they do not guarantee 100% truth in their conclusions, but it can provide either weak or strong evidence to suggest it may be true. This is due to the fact that they rely on observed patterns, and patterns sometimes change.<\/p>\n<p><strong>Deductive reasoning<\/strong>, on the other hand, starts with general statements or premises and leads to a specific, logical conclusion. It&#8217;s about deriving a certain conclusion from a set of general principles or premises.<\/p>\n<p>For example, if all men are mortal (premise 1), and Socrates is a man (premise 2), then Socrates must be mortal (conclusion). Here, if the premises are true, then the conclusion must also be true. Deductive reasoning provides certainty, given that the initial premises are true.<\/p>\n<p>When assessing deductive arguments, two key concepts to consider are validity and soundness.<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li><strong>Validity<\/strong><br \/>\nA deductive argument is valid if the truth of its premises guarantees the truth of its conclusion. In other words, if the premises are true, the conclusion must also be true. The validity of an argument is purely about its logical structure or form, not about whether its premises or conclusions are actually true. For example, consider this argument:<\/p>\n<ul>\n<li>If it rains, the ground will be wet.<\/li>\n<li>It&#8217;s raining.<\/li>\n<li>Therefore, the ground is wet.<\/li>\n<\/ul>\n<p>This is a valid argument, as it follows a logical structure. If premises 1 and 2 are true, the conclusion (3) must also be true.<\/li>\n<li><strong>Soundness<\/strong><br \/>\nSoundness, on the other hand, is a characteristic of a deductive argument that is not only valid but also has true premises. A sound argument is thus a valid argument with true premises, which leads to a necessarily true conclusion. Using the previous example if it&#8217;s indeed raining (which means premises 1 and 2 are true), and the argument structure is valid, then the argument is also sound, and the ground must indeed be wet.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205781&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=xu1792kg7wA&amp;video_target=tpm-plugin-2gl7c6rd-xu1792kg7wA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/3+7+arguments+and+Euler+Diagrams.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c3 7 arguments and Euler Diagrams\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Analyzing Arguments with Venn\/Euler Diagrams<\/h2>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205782&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=byOK9yv0dsc&amp;video_target=tpm-plugin-ezbhbgcp-byOK9yv0dsc\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/3.5+Analyzing+Arguments+with+Euler+Diagrams.txt\" target=\"_blank\" rel=\"noopener\">\u00a0transcript for \u201c3.5 Analyzing Arguments with Euler Diagrams\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox seeExample\">Analyze the following argument using a Venn\/Euler diagram.<\/p>\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><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q88485\">Show Solution<\/button><\/p>\n<div id=\"q88485\" class=\"hidden-answer\" style=\"display: none\">\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.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-6420 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10155416\/Arguments-logic-2-1-275x300.png\" alt=\"A Venn\/Euler diagram depicting a large circle labeled Washington and a smaller circle entirely contained within the larger one and labeled Seattle. In the larger circle, but outside of the smaller circle, the name Marcus is written, next to an x with a question mark. There is another x with a question mark outside of both circles.\" width=\"275\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10155416\/Arguments-logic-2-1-275x300.png 275w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10155416\/Arguments-logic-2-1-768x838.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10155416\/Arguments-logic-2-1-65x71.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10155416\/Arguments-logic-2-1-225x246.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10155416\/Arguments-logic-2-1-350x382.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10155416\/Arguments-logic-2-1.png 900w\" sizes=\"(max-width: 275px) 100vw, 275px\" \/><\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>Analyzing Arguments with Truth Tables<\/h2>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205783&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=EfsbN5YbcPQ&amp;video_target=tpm-plugin-v4r9oz3d-EfsbN5YbcPQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Truth+Table+to+determine+if+an+argument+is+valid.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTruth Table to determine if an argument is valid\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205784&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=fkH3EcRDrnI&amp;video_target=tpm-plugin-bkx5ira3-fkH3EcRDrnI\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Truth+Tables+to+Analyze+Arguments.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTruth Tables to Analyze Arguments\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox seeExample\">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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q23681\">Show Solution<\/button><\/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 [latex]B[\/latex] represent \u201cyou bought bread\u201d and [latex]S[\/latex] 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>&nbsp;<\/p>\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 style=\"text-align: center;\">[latex]B[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]S[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]B{\\rightarrow}S[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\left(B{\\rightarrow}S\\right){\\wedge}B[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\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<\/section>\n<section class=\"textbox seeExample\">\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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q880229\">Show Solution<\/button><\/p>\n<div id=\"q880229\" class=\"hidden-answer\" style=\"display: none\">\n<p>If we let [latex]W =[\/latex] working hard, [latex]R =[\/latex] getting a raise, and [latex]B =[\/latex] 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>&nbsp;<\/p>\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<\/section>\n<h2>Logical Fallacies<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>A <strong>logical fallacy<\/strong> is an error in reasoning that results in an invalid argument or statement. These fallacies often sound persuasive and may seem to follow a logical line of thinking, which can make them tricky to spot. However, they don&#8217;t hold up to rigorous scrutiny because they are based on flawed logic. Logical fallacies can occur either intentionally, as in the case of rhetorical strategies meant to sway opinion or manipulate, or unintentionally, as the result of faulty reasoning or misunderstanding. There are several different types of logical fallacies.<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li><strong>Ad hominem<\/strong>: This is a fallacy where someone attacks their opponent&#8217;s character or personal traits rather than engaging with their argument or position. It is a way to divert attention from the issue at hand.<\/li>\n<li><strong>Appeal to ignorance<\/strong>: This fallacy assumes a claim is true because it has not been proven false, or false because it has not been proven true. It capitalizes on the lack of evidence to the contrary.<\/li>\n<li><strong>Appeal to authority<\/strong>: This fallacy asserts that because an authority figure or expert believes something, it must be true. It fails to account for the fact that even experts can be mistaken.<\/li>\n<li><strong>Appeal to consequence<\/strong>: This fallacy suggests that a premise is true or false based on whether its outcome is desirable or undesirable. The truth of a statement, however, isn&#8217;t determined by its potential consequences.<\/li>\n<li><strong>False dilemma<\/strong>: Also known as false dichotomy, this fallacy presents only two options or outcomes when, in reality, there could be several. It artificially restricts possible choices to manipulate an outcome.<\/li>\n<li><strong>Circular reasoning<\/strong>: This fallacy occurs when the conclusion of an argument is used as one of its premises. It&#8217;s essentially arguing in a circle, offering no new information or insight.<\/li>\n<li><strong>Straw man<\/strong>: This fallacy distorts an opponent&#8217;s argument into an easily refutable position and then attacks this weaker proposition as if it were the original one. It&#8217;s a way to sidestep the actual issue at hand.<\/li>\n<li><strong>Post hoc (post hoc ergo propter hoc)<\/strong>: The post hoc fallacy assumes that because event [latex]B[\/latex] followed event [latex]A[\/latex], [latex]A[\/latex] must have caused [latex]B[\/latex]. While often true in causality, it isn&#8217;t always the case, and it&#8217;s a fallacy to assume it without evidence.<\/li>\n<li><strong>Correlation implies causation<\/strong>: This fallacy, also known as &#8220;cum hoc ergo propter hoc,&#8221; suggests that because two events occur together, one must have caused the other. Correlation, however, does not necessarily mean causation.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/UzT4L0Zt4ZY\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Understanding+Various+Types+of+Logical+Fallacies.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cUnderstanding Various Types of Logical Fallacies\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox seeExample\">\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 obesity 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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q912175\">Show Solution<\/button><\/p>\n<div id=\"q912175\" class=\"hidden-answer\" style=\"display: none\">\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<\/section>\n","protected":false},"author":15,"menu_order":24,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":24,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/39"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":40,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/39\/revisions"}],"predecessor-version":[{"id":15081,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/39\/revisions\/15081"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/24"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/39\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=39"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=39"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=39"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=39"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}