{"id":6800,"date":"2021-12-30T21:45:53","date_gmt":"2021-12-30T21:45:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/?post_type=chapter&#038;p=6800"},"modified":"2022-01-06T22:25:07","modified_gmt":"2022-01-06T22:25:07","slug":"1-13-inductive-and-deductive-reasoning","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/1-13-inductive-and-deductive-reasoning\/","title":{"raw":"1.13 Inductive and Deductive Reasoning","rendered":"1.13 Inductive and Deductive Reasoning"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Inductive Reasoning<\/li>\r\n \t<li>Deductive Reasoning<\/li>\r\n<\/ul>\r\n<\/div>\r\nThere are two general types of reasoning that we use when building and analyzing arguments: inductive and deductive.\r\n<div class=\"textbox\">\r\n<h3>Reasoning types<\/h3>\r\n<strong>Inductive Reasoning<\/strong>\u00a0uses a collection of specific examples as its premises and uses them to propose a general conclusion.\r\n\r\n<strong>Deductive<\/strong>\u00a0<strong>Reasoning<\/strong> uses a collection of general statements as its premises and uses them to propose a specific situation as the conclusion.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]109526[\/ohm_question]\r\n\r\n[ohm_question]109527[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n\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 example of inductive reasoning.\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 fairly weak, 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\n\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 example of inductive reasoning, since it is based on a larger set of evidence.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Evaluating inductive arguments<\/h3>\r\nAn inductive argument is never able to prove the conclusion true, but it can provide either weak or strong evidence to suggest it may be true.\r\n\r\n<\/div>\r\nMany scientific theories, such as the big bang theory, can never be proven. Instead, they are inductive arguments supported by a wide variety of evidence. Usually in science, an idea is considered a hypothesis until it has been well tested, at which point it graduates to being considered a theory. The commonly known scientific theories, like Newton\u2019s theory of gravity, have all stood up to years of testing and evidence, though sometimes they need to be adjusted based on new evidence. For gravity, this happened when Einstein proposed the theory of general relativity.\r\n\r\nA deductive argument is more clearly valid or not, which makes them easier to evaluate.\r\n<div class=\"textbox\">\r\n<h3>Evaluating deductive arguments<\/h3>\r\nA deductive argument is considered valid if all the premises are true, and the conclusion follows logically from those premises. In other words, the premises are true, and the conclusion follows necessarily from those premises.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe argument \u201cAll cats are mammals and a tiger is a cat, so a tiger is a mammal\u201d is a valid deductive argument.\r\n\r\nThe premises are:\r\n<p style=\"padding-left: 30px;\">All cats are mammals\r\nA tiger is a cat<\/p>\r\nThe conclusion is:\r\n<p style=\"padding-left: 30px;\">A tiger is a mammal<\/p>\r\nBoth the premises are true. To see that the premises must logically lead to the conclusion, one approach would be use a Venn diagram. From the first premise, we can conclude that the set of cats is a subset of the set of mammals. From the second premise, we are told that a tiger lies within the set of cats. From that, we can see in the Venn diagram that the tiger also lies inside the set of mammals, so the conclusion is valid.\r\n\r\n<img class=\"aligncenter wp-image-256\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155146\/Fig4_2_1.png\" alt=\"A Venn diagram with a large circle labeled Mammals and a smaller circle labeled Cats contained within the Mammals circle. A red X labeled Tiger is in the circle labeled Cats.\" width=\"200\" height=\"202\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]132642[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Analyzing Arguments with Venn diagrams<\/h3>\r\nTo analyze an argument with a Venn diagram\r\n<ol>\r\n \t<li>Draw a Venn diagram based on the premises of the argument<\/li>\r\n \t<li>If the premises are insufficient to determine what determine the location of an element, indicate that.<\/li>\r\n \t<li>The argument is valid if it is clear that the conclusion must be true<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>All firefighters know CPR<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>Jill knows CPR<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>Jill is a firefighter<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFrom the first premise, we know that firefighters all lie inside the set of those who know CPR. From the second premise, we know that Jill is a member of that larger set, but we do not have enough information to know if she also is a member of the smaller subset that is firefighters.\r\n\r\n<img class=\"wp-image-257 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155147\/Fig4_2_2.png\" alt=\"Fig4_2_2\" width=\"200\" height=\"200\" \/>\r\n\r\nSince the conclusion does not necessarily follow from the premises, this is an invalid argument, regardless of whether Jill actually is a firefighter.\r\n\r\n<\/div>\r\nIt is important to note that whether or not Jill is actually a firefighter is not important in evaluating the validity of the argument; we are only concerned with whether the premises are enough to prove the conclusion.\r\n\r\nIn addition to these categorical style premises of the form \u201call ___,\u201d \u201csome ____,\u201d and \u201cno ____,\u201d it is also common to see premises that are implications.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If you live in Seattle, you live in Washington.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>Marcus does not live in Seattle.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>Marcus does not live in Washington.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFrom the first premise, we know that the set of people who live in Seattle is inside the set of those who live in Washington. From the second premise, we know that Marcus does not lie in the Seattle set, but we have insufficient information to know whether or not Marcus lives in Washington or not. This is an invalid argument.\r\n\r\n<img class=\"wp-image-258 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155147\/Fig4_2_3.png\" alt=\"Fig4_2_3\" width=\"200\" height=\"204\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider the argument \u201cYou are a married man, so you must have a wife.\u201d\r\n\r\n[reveal-answer q=\"383279\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"383279\"]\r\n\r\nThis is an invalid argument, since there are, at least in parts of the world, men who are married to other men, so the premise not insufficient to imply the conclusion.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Inductive Reasoning<\/li>\n<li>Deductive Reasoning<\/li>\n<\/ul>\n<\/div>\n<p>There are two general types of reasoning that we use when building and analyzing arguments: inductive and deductive.<\/p>\n<div class=\"textbox\">\n<h3>Reasoning types<\/h3>\n<p><strong>Inductive Reasoning<\/strong>\u00a0uses a collection of specific examples as its premises and uses them to propose a general conclusion.<\/p>\n<p><strong>Deductive<\/strong>\u00a0<strong>Reasoning<\/strong> uses a collection of general statements as its premises and uses them to propose a specific situation as the conclusion.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm109526\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109526&theme=oea&iframe_resize_id=ohm109526&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm109527\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109527&theme=oea&iframe_resize_id=ohm109527&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>\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 example of inductive reasoning.<\/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 fairly weak, since it is based on only two instances.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>\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 example of inductive reasoning, since it is based on a larger set of evidence.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Evaluating inductive arguments<\/h3>\n<p>An inductive argument is never able to prove the conclusion true, but it can provide either weak or strong evidence to suggest it may be true.<\/p>\n<\/div>\n<p>Many scientific theories, such as the big bang theory, can never be proven. Instead, they are inductive arguments supported by a wide variety of evidence. Usually in science, an idea is considered a hypothesis until it has been well tested, at which point it graduates to being considered a theory. The commonly known scientific theories, like Newton\u2019s theory of gravity, have all stood up to years of testing and evidence, though sometimes they need to be adjusted based on new evidence. For gravity, this happened when Einstein proposed the theory of general relativity.<\/p>\n<p>A deductive argument is more clearly valid or not, which makes them easier to evaluate.<\/p>\n<div class=\"textbox\">\n<h3>Evaluating deductive arguments<\/h3>\n<p>A deductive argument is considered valid if all the premises are true, and the conclusion follows logically from those premises. In other words, the premises are true, and the conclusion follows necessarily from those premises.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The argument \u201cAll cats are mammals and a tiger is a cat, so a tiger is a mammal\u201d is a valid deductive argument.<\/p>\n<p>The premises are:<\/p>\n<p style=\"padding-left: 30px;\">All cats are mammals<br \/>\nA tiger is a cat<\/p>\n<p>The conclusion is:<\/p>\n<p style=\"padding-left: 30px;\">A tiger is a mammal<\/p>\n<p>Both the premises are true. To see that the premises must logically lead to the conclusion, one approach would be use a Venn diagram. From the first premise, we can conclude that the set of cats is a subset of the set of mammals. From the second premise, we are told that a tiger lies within the set of cats. From that, we can see in the Venn diagram that the tiger also lies inside the set of mammals, so the conclusion is valid.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-256\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155146\/Fig4_2_1.png\" alt=\"A Venn diagram with a large circle labeled Mammals and a smaller circle labeled Cats contained within the Mammals circle. A red X labeled Tiger is in the circle labeled Cats.\" width=\"200\" height=\"202\" \/><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm132642\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=132642&theme=oea&iframe_resize_id=ohm132642&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Analyzing Arguments with Venn diagrams<\/h3>\n<p>To analyze an argument with a Venn diagram<\/p>\n<ol>\n<li>Draw a Venn diagram based on the premises of the argument<\/li>\n<li>If the premises are insufficient to determine what determine the location of an element, indicate that.<\/li>\n<li>The argument is valid if it is clear that the conclusion must be true<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>All firefighters know CPR<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>Jill knows CPR<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>Jill is a firefighter<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>From the first premise, we know that firefighters all lie inside the set of those who know CPR. From the second premise, we know that Jill is a member of that larger set, but we do not have enough information to know if she also is a member of the smaller subset that is firefighters.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-257 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155147\/Fig4_2_2.png\" alt=\"Fig4_2_2\" width=\"200\" height=\"200\" \/><\/p>\n<p>Since the conclusion does not necessarily follow from the premises, this is an invalid argument, regardless of whether Jill actually is a firefighter.<\/p>\n<\/div>\n<p>It is important to note that whether or not Jill is actually a firefighter is not important in evaluating the validity of the argument; we are only concerned with whether the premises are enough to prove the conclusion.<\/p>\n<p>In addition to these categorical style premises of the form \u201call ___,\u201d \u201csome ____,\u201d and \u201cno ____,\u201d it is also common to see premises that are implications.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>If you live in Seattle, you live in Washington.<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>Marcus does not live in Seattle.<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>Marcus does not live in Washington.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>From the first premise, we know that the set of people who live in Seattle is inside the set of those who live in Washington. From the second premise, we know that Marcus does not lie in the Seattle set, but we have insufficient information to know whether or not Marcus lives in Washington or not. This is an invalid argument.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-258 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155147\/Fig4_2_3.png\" alt=\"Fig4_2_3\" width=\"200\" height=\"204\" \/><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider the argument \u201cYou are a married man, so you must have a wife.\u201d<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q383279\">Show Solution<\/span><\/p>\n<div id=\"q383279\" class=\"hidden-answer\" style=\"display: none\">\n<p>This is an invalid argument, since there are, at least in parts of the world, men who are married to other men, so the premise not insufficient to imply the conclusion.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>\n","protected":false},"author":359705,"menu_order":15,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-6800","chapter","type-chapter","status-publish","hentry"],"part":159,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6800","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/users\/359705"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6800\/revisions"}],"predecessor-version":[{"id":7003,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6800\/revisions\/7003"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/parts\/159"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6800\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/media?parent=6800"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapter-type?post=6800"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/contributor?post=6800"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/license?post=6800"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}