{"id":35,"date":"2016-12-16T18:14:05","date_gmt":"2016-12-16T18:14:05","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/3-good-arguments\/"},"modified":"2025-10-13T17:12:03","modified_gmt":"2025-10-13T17:12:03","slug":"3-good-arguments","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/3-good-arguments\/","title":{"raw":"Good Arguments","rendered":"Good Arguments"},"content":{"raw":"<h2>3.1 \u00a0A historical example<\/h2>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>How can scientists use logic to support their arguments?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nAn important example of excellent reasoning can be found in the case of the medical advances of the Nineteenth Century physician, Ignaz Semmelweis. \u00a0Semmelweis was an obstetrician at the Vienna General Hospital. \u00a0Built on the foundation of a poor house, and opened in 1784, the General Hospital is still operating today. \u00a0Semmelweis, during his tenure as assistant to the head of one of two maternity clinics, noticed something very disturbing. \u00a0The hospital had two clinics, separated only by a shared anteroom, known as the First and the Second Clinics. \u00a0The mortality rate for mothers delivering babies in the First Clinic, however, was nearly three times as bad as the mortality for mothers in the Second Clinic (9.9 % average versus 3.4% average). \u00a0The same was true for the babies born in the clinics: \u00a0the mortality rate in the First Clinic was 6.1% versus 2.1% at the Second Clinic.[footnote]All the data cited here comes from Carter (1983) and additional biographical information comes from Carter and Carter (2008). These books are highly recommended to anyone interested in the history of science or medicine.[\/footnote]\u00a0 In nearly all these cases, the deaths were caused by what appeared to be the same illness, commonly called \u201cchildbed fever\u201d. \u00a0Worse, these numbers actually understated the mortality rate of the First Clinic, because sometimes very ill patients were transferred to the general treatment portion of the hospital, and when they died,\u00a0their death was counted as part of the mortality rate of the general hospital, not of the First Clinic.\r\n\r\nSemmelweis set about trying to determine why the First Clinic had the higher mortality rate. \u00a0He considered a number of hypotheses, many of which were suggested by or believed by other doctors.\r\n\r\nOne hypothesis was that cosmic-atmospheric-terrestrial influences caused childbed fever. \u00a0The idea here was that some kind of feature of the atmosphere would cause the disease. \u00a0But, Semmelweis observed, the First and Second Clinics were very close to each other, had similar ventilation, and shared a common anteroom. \u00a0So, they had similar atmospheric conditions. \u00a0He reasoned: \u00a0If childbed fever is caused by cosmic-atmospheric-terrestrial influences, then the mortality rate would be similar in the First and Second Clinics. \u00a0But the mortality rate was not similar in the First and Second Clinics. \u00a0So, the childbed fever was not caused by cosmic-atmospheric-terrestrial influences.\r\n\r\nAnother hypothesis was that overcrowding caused the childbed fever. \u00a0But, if overcrowding caused the childbed fever, then the more crowded of the two clinics should have the higher mortality rate. \u00a0But, the Second Clinic was more crowded (in part because, aware of its lower mortality rate, mothers fought desperately to be put there instead of in the First Clinic). \u00a0It did not have a higher mortality rate. \u00a0So, the childbed fever was not caused by overcrowding.\r\n\r\nAnother hypothesis was that fear caused the childbed fever. \u00a0In the Second Clinic, the priest delivering last rites could walk directly to a dying patient\u2019s room. \u00a0For reasons of the layout of the rooms, the priest delivering last rites in the First Clinic walked by all the rooms, ringing a bell announcing his approach. \u00a0This frightened patients; they could not tell if the priest was coming for them. \u00a0Semmelweis arranged a different route for the priest and asked him to silence his bell. \u00a0He reasoned: \u00a0if the higher rate of childbed fever was caused by fear of death resulting from the priest\u2019s approach, then the rate of childbed fever should decline if people could not tell when the priest was coming to the Clinic. \u00a0But it was not the case that the rate of childbed fever declined when people could not tell if the priest was coming to the First Clinic. \u00a0So, the higher rate of childbed fever in the First Clinic was not caused by fear of death resulting from the priest\u2019s approach.\r\n\r\nIn the First Clinic, male doctors were trained; this was not true in the Second Clinic. \u00a0These male doctors performed autopsies across the hall from the clinic, before delivering babies. \u00a0Semmelweis knew of a doctor who cut himself while performing an autopsy, and who then died a terrible death not unlike that of the mothers who died of childbed fever. \u00a0Semmelweis formed a hypothesis. \u00a0The childbed fever was caused by something on the hands of the doctors, something that they picked up from corpses during autopsies, but that infected the women and infants. \u00a0He reasoned that: \u00a0if the fever was caused by cadaveric matter on the hands of the doctors, then the mortality rate would drop when doctors washed their hands with chlorinated water before delivering babies. \u00a0He forced the doctors to do this. \u00a0The result was that the mortality rate dropped to a rate below that even of the Second Clinic.\r\n\r\nSemmelweis concluded that the best explanation of the higher mortality rate was this \u201ccadaveric matter\u201d on the hands of doctors. \u00a0He was the first person to see that washing of hands with sterilizing cleaners would save thousands of lives. \u00a0It is hard to overstate how important this contribution is to human well being. \u00a0Semmelweis\u2019s fine reasoning deserves our endless respect and gratitude.\r\n\r\nBut how can we be sure his reasoning was good? \u00a0Semmelweis was essentially considering a series of arguments. \u00a0Let us turn to the question: \u00a0how shall we evaluate arguments?\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Check for Understanding<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>Explain how the Nineteenth Century physician Ignaz Semmelweis used logic to assess the hypothesis presented in his investigation into mortality rates at the Vienna General Hospital.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>3.2 \u00a0Arguments<\/h2>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What makes a statement precise?<\/li>\r\n \t<li>How does logic get people to believe in things?<\/li>\r\n \t<li>When is an argument considered true?<\/li>\r\n \t<li>When is an argument considered valid?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Terms<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li><strong>Argument<\/strong> - an ordered list of sentences; we call one of these sentences the \u201cconclusion\u201d, and we call the other sentences \u201cpremises\u201d.<\/li>\r\n \t<li><strong>Valid Argument<\/strong> - an argument for which, necessarily, if the premises are true, then the conclusion is true.<\/li>\r\n \t<li><strong>Sound Argument<\/strong> - a valid argument with true premises.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nOur logical language now allows us to say conditional and negation statements. \u00a0That may not seem like much, but our language is now complex enough for us to develop the idea of using our logic not just to describe things, but also to reason about those things.\r\n\r\nWe will think of reasoning as providing an argument. \u00a0Here, we use the word \u201c<strong>argument<\/strong>\u201d not in the sense of two or more people criticizing each other, but rather in the sense we mean when we say, \u201cPythagoras\u2019s argument\u201d. \u00a0In such a case, someone is using language to try to convince us that something is true. \u00a0Our goal is to make this notion very precise, and then identify what makes an argument good.\r\n\r\nWe need to begin by making the notion of an argument precise. \u00a0Our logical language so far contains only sentences. \u00a0An argument will, therefore, consist of sentences. \u00a0In a natural language, we use the term \u201cargument\u201d in a strong way, which includes the suggestion that the argument should be good. \u00a0However, we want to separate the notion of a good argument from the notion of an argument, so we can identify what makes an argument good, and what makes an argument bad. \u00a0To do this, we will start with a minimal notion of what an argument is. Here is the simplest, most minimal notion:\r\n<p style=\"padding-left: 120px;\"><strong>Argument<\/strong>: \u00a0an ordered list of sentences; we call one of these sentences the \u201cconclusion\u201d, and we call the other sentences \u201cpremises\u201d.<\/p>\r\nThis is obviously very weak. \u00a0(There is\u00a0a famous Monty Python skit where one of the comedians ridicules the very idea that such a thing could be called an argument.) \u00a0But for our purposes, this is a useful notion because it is very clearly defined, and we can now ask, what makes an argument good?\r\n\r\nThe everyday notion of an argument is that it is used to convince us to believe something. \u00a0The thing that we are being encouraged to believe is the conclusion. \u00a0Following our definition of \u201cargument\u201d, the reasons that the person gives will be what we are calling \u201cpremises\u201d. \u00a0But <span class=\"em\">belief<\/span>\u00a0is a psychological notion. \u00a0We instead are interested only in truth. \u00a0So, we can reformulate this intuitive notion of what an argument should do, and think of an argument as being used to show that something is true. \u00a0The premises of the argument are meant to show us that the conclusion is true.\r\n\r\nWhat then should be this relation between the premises and the conclusion? \u00a0Intuitive notions include that the premises should support the conclusion, or corroborate the conclusion, or make the conclusion true. \u00a0But \u201csupport\u201d and \u201ccorroborate\u201d sound rather weak, and \u201cmake\u201d is not very clear. \u00a0What we can use in their place is a stronger standard: let us say as a first approximation that if the premises are true, the conclusion is true.\r\n\r\nBut even this seems weak, on reflection.\u00a0 The conclusion could be true by accident, for reasons unrelated to our premises. \u00a0Remember that we define the conditional as true if the antecedent and consequent are true. \u00a0But this could happen by accident. \u00a0For example, suppose I say, \u201cIf Tom wears blue then he will get an A on the exam\u201d. \u00a0Suppose also that Tom both wears blue and Tom gets an A on the exam. \u00a0This makes the conditional true, but (we hope) the color of his clothes really had nothing to do with his performance on the exam. \u00a0Just so, we want our definition of \u201cgood argument\u201d to be such that it cannot be an accident that the premises and conclusion are both true.\r\n\r\nA better and stronger standard would be that, necessarily, given true premises, the conclusion is true.\r\n\r\nThis points us to our definition of a good argument. \u00a0It is traditional to call a good argument \u201cvalid.\u201d\r\n<p style=\"padding-left: 120px;\"><strong>Valid argument<\/strong>: \u00a0an argument for which, necessarily, if the premises are true, then the conclusion is true.<\/p>\r\nThis is the single most important principle in this book. \u00a0Memorize it.\r\n\r\nA bad argument is an argument that is not valid. \u00a0Our name for this will be an \u201cinvalid argument\u201d.\r\n\r\nSometimes, a dictionary or other book will define or describe a \u201cvalid argument\u201d as an argument that follows the rules of logic. \u00a0This is a hopeless way to define \u201cvalid\u201d, because it is circular in a pernicious way: \u00a0we are going to create the rules of our logic in order to ensure that they construct valid arguments. \u00a0We cannot make rules of logical reasoning until we know what we want those rules to do, and what we want them to do is to create valid arguments. \u00a0So \u201cvalid\u201d must be defined before we can make our reasoning system.\r\n\r\nExperience shows that if a student is to err in understanding this definition of \u201cvalid argument\u201d, he or she will typically make the error of assuming that a valid argument has all true premises. \u00a0This is not required. \u00a0There are valid arguments with false premises and a false conclusion. \u00a0Here\u2019s one:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If Miami is the capital of Kansas, then Miami is in Canada. \u00a0Miami is the capital of Kansas. \u00a0Therefore, Miami is in Canada.<\/p>\r\nThis argument has at least one false premise: \u00a0Miami is not the capital of Kansas. \u00a0And the conclusion is false: \u00a0Miami is not in Canada. \u00a0But the argument is valid: \u00a0if the premises were both true, the conclusion would have to be true. \u00a0(If that bothers you, hold on a while and we will convince you that this argument is valid because of its form alone. \u00a0Also, keep in mind always that \u201cif\u2026then\u2026\u201d is interpreted as meaning the conditional.)\r\n\r\nSimilarly, there are invalid arguments with true premises, and with a true conclusion. \u00a0Here\u2019s one:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If Miami is the capital of Ontario, then Miami is in Canada. \u00a0Miami is not the capital of Ontario. \u00a0Therefore, Miami is not in Canada.<\/p>\r\n(If you find it confusing that this argument is invalid, look at it again after you finish reading this chapter.)\r\n\r\nValidity is about the relationship between the sentences in the argument. \u00a0It is not a claim that those sentences are true.\r\n\r\nAnother variation of this confusion seems to arise when we forgot to think carefully about the conditional. \u00a0The definition of valid is not \u201cAll the premises are true, so the conclusion is true.\u201d \u00a0If you don\u2019t see the difference, consider the following two sentences. \u00a0\u201cIf your house is on fire, then you should call the fire department.\u201d \u00a0In this sentence, there is no claim that your house is on fire. \u00a0It is rather advice about what you should do if your house is on fire. \u00a0In the same way, the definition of valid argument does not tell you that the premises are true. \u00a0It tells you what follows if they are true. \u00a0Contrast now, \u201cYour house is on fire, so you should call the fire department\u201d. \u00a0This sentence delivers very bad news. \u00a0It is not a conditional at all. \u00a0What it really means is, \u201cYour house is on fire and you should call the fire department\u201d. \u00a0Our definition of valid is not, \u201cAll the premises are true and the conclusion is true\u201d.\r\n\r\nFinally, another common mistake is to confuse <span class=\"em\">true<\/span>\u00a0and <span class=\"em\">valid<\/span>. \u00a0In the sense that we are using these terms in this book, only sentences can be true or false, and only arguments can be valid and invalid. \u00a0When discussing and using our logical language, it is nonsense to say, \u201ca true argument\u201d, and it is nonsense to say, \u201ca valid sentence\u201d.\r\n\r\nSomeone new to logic might wonder, why would we want a definition of \u201cgood argument\u201d that does not guarantee that our conclusion is true? \u00a0The answer is that logic is an enormously powerful tool for checking arguments, and we want to be able to identify what the good arguments are, independently of the particular premises that we use in the argument. \u00a0For example, there are infinitely many particular arguments that have the same form as the valid argument given above. \u00a0There are infinitely many particular arguments that have the same form as the invalid argument given above. \u00a0Logic lets us embrace all the former arguments at once, and reject all those bad ones at once.\r\n\r\nFurthermore, our propositional logic will not be able to tell us whether an atomic sentence is true. \u00a0If our argument is about rocks, we must ask the geologist if the premises are true. \u00a0If our argument is about history, we must ask the historian if the premises are true. \u00a0If our argument is about music, we must ask the music theorist if the premises are true. \u00a0But the logician can tell the geologist, the historian, and the musicologist whether her arguments are good or bad, independent of the particular premises.\r\n\r\nWe do have a common term for a good argument that has true premises. \u00a0This is called \u201csound\u201d. \u00a0It is a useful notion when we are applying our logic. \u00a0Here is our definition:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong>Sound argument<\/strong>: \u00a0a valid argument with true premises.<\/p>\r\nA sound argument must have a true conclusion, given the definition of \u201cvalid\u201d.\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Check for Understanding<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What makes an argument good?<\/li>\r\n \t<li>Explain the difference between belief and truth in argumentative statements.<\/li>\r\n \t<li>Can a statement be valid but not true? Please explain.<\/li>\r\n \t<li>Explain the difference between a valid argument and an invalid argument.<\/li>\r\n \t<li><strong>Self-Reflection:\u00a0<\/strong>What is a valid argument you have encountered recently in your day-to-day life? What is an invalid argument you commonly encounter?<\/li>\r\n \t<li><strong>Critical Thinking Task:\u00a0<\/strong>What is the single most important principle to remember in this book? Why?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>3.3 \u00a0Checking arguments semantically<\/h2>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What purpose does the word \u201cnecessarily\u201d serve in logic?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nEvery element of our definition of \u201cvalid\u201d is clear except for one. \u00a0We know what \u201cif\u2026then\u2026\u201d means. \u00a0We defined the semantics of the conditional in chapter 2. \u00a0We have defined \u201cargument\u201d, \u201cpremise\u201d, and \u201cconclusion\u201d. \u00a0We take <em>true<\/em> and <em>false<\/em> as primitives. \u00a0But what does \u201cnecessarily\u201d mean?\r\n\r\nWe define a valid argument as one where, <strong>necessarily<\/strong>, if the premises are true, then the conclusion is true. \u00a0It would seem the best way to understand this is to say, there is no situation in which the premises are true but the conclusion is false. \u00a0But then, what are these \u201csituations\u201d? \u00a0Fortunately, we already have a tool that looks like it could help us: \u00a0the truth table.\r\n\r\nRemember that in the truth table, we put on the bottom left side all the possible combinations of truth values of some set of atomic sentences. \u00a0Each row of the table then represents a kind of way the world could be. \u00a0Using this as a way to understand \u201cnecessarily\u201d, we could rephrase our definition of valid to something like this, \u201cIn any kind of situation in which all the premises are true, the conclusion is true.\u201d\r\n\r\nLet\u2019s try it out. \u00a0We will\u00a0need to use truth tables in a new way: \u00a0to check an argument. \u00a0That will require having not just one sentence, but several on the truth table. \u00a0Consider an argument that looks like it should be valid.\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If Jupiter is more massive than Earth, then Jupiter has a stronger gravitational field than Earth. \u00a0Jupiter is more massive than Earth. \u00a0In conclusion, Jupiter has a stronger gravitational field than Earth.<\/p>\r\nThis looks like it has the form of a valid argument, and it looks like an astrophysicist would tell us it is sound. \u00a0Let\u2019s translate it to our logical language using the following translation key. \u00a0(We\u2019ve used up our letters, so I\u2019m going to start over. \u00a0We\u2019ll do that often: \u00a0assume we are starting a new language each time we translate a new set of problems or each time we consider a new example.)\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong>: \u00a0Jupiter is more massive than Earth<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>: \u00a0Jupiter has a stronger gravitational field than Earth.<\/p>\r\nThis way of writing out sentences of logic and sentences of English we can call a \u201ctranslation key\u201d. \u00a0We can use this format whenever we want to explain what our sentences mean in English.\r\n\r\nUsing this key, our argument would be formulated\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong><\/span><span class=\"strong\">\u2192<strong>Q<\/strong>)<\/span><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">______<\/span><\/strong><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong><\/p>\r\nThat short line is not part of our language, but rather is a handy tradition. \u00a0When quickly writing down arguments, we write the premises, and then write the conclusion last, and draw a short line above the conclusion.\r\n\r\nThis is an argument: \u00a0it is an ordered list of sentences, the first two of which are premises and the last of which is the conclusion.\r\n\r\nTo make a truth table, we identify all the atomic sentences that constitute these sentences. \u00a0These are <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">Q<\/span><\/strong>. \u00a0There are four possible kinds of ways the world could be that matter to us then:\r\n<table class=\"grid\" style=\"height: 105px;\" width=\"75\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<th class=\"border\" style=\"width: 158.733px;\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">P \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\r\n<th class=\"border\" style=\"width: 66.7667px;\"><span class=\"strong\">Q<\/span><\/th>\r\n<th class=\"border\" style=\"width: 67.6167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/th>\r\n<th class=\"border\" style=\"width: 67.6333px;\" colspan=\"1\" rowspan=\"1\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/th>\r\n<th class=\"border\" style=\"width: 67.6167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 159.133px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border\" style=\"width: 67.5667px;\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" style=\"width: 68.4167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border\" style=\"width: 68.4333px;\" colspan=\"1\" rowspan=\"1\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border\" style=\"width: 68.0167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 159.133px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border\" style=\"width: 67.5667px;\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" style=\"width: 68.4167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border\" style=\"width: 68.4333px;\" colspan=\"1\" rowspan=\"1\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border\" style=\"width: 68.0167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 159.133px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border\" style=\"width: 67.5667px;\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" style=\"width: 68.4167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border\" style=\"width: 68.4333px;\" colspan=\"1\" rowspan=\"1\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border\" style=\"width: 68.0167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 159.133px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0\u00a0<\/span><\/td>\r\n<td class=\"border\" style=\"width: 67.5667px;\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" style=\"width: 68.4167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border\" style=\"width: 68.4333px;\" colspan=\"1\" rowspan=\"1\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border\" style=\"width: 68.0167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe\u2019ll write out the sentences, in the order of premises and then conclusion.\r\n<table class=\"grid\" style=\"height: 90px; width: 100px;\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 44.0833px;\"><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 47.75px;\"><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 209.1px;\"><strong>conclusion<\/strong><\/td>\r\n<\/tr>\r\n<tr class=\"border-bottom\" style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 44.0833px;\"><strong>P<\/strong><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 47.75px;\"><strong>Q<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong>(P\u2192Q)<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong>P<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 209.1px;\"><strong>Q<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 44.0833px;\"><strong><em>T<\/em><\/strong><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 47.75px;\"><strong><em>T<\/em><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong>\u00a0<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong>\u00a0<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 209.1px;\"><strong>\u00a0<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 44.0833px;\"><strong><em>T<\/em><\/strong><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 47.75px;\"><strong><em>F<\/em><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong><em>\u00a0<\/em><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong><em>\u00a0<\/em><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 209.1px;\"><strong><em>\u00a0<\/em><\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 44.0833px;\"><strong><em>F<\/em><\/strong><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 47.75px;\"><strong><em>T<\/em><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong><em>\u00a0<\/em><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong><em>\u00a0<\/em><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 209.1px;\"><strong><em>\u00a0<\/em><\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 44.0833px;\"><strong><em>F<\/em><\/strong><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 47.75px;\"><strong><em>F<\/em><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 209.1px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow we can fill in the columns for each sentence, identifying the truth value of the sentence for that kind of situation.\r\n<table class=\"grid\" style=\"height: 139px;\" width=\"100\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 33.7667px;\"><\/td>\r\n<td class=\"border-right\" style=\"width: 36.4833px;\"><\/td>\r\n<td class=\"border\" style=\"width: 110px;\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 110px;\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 141.317px;\"><strong>conclusion<\/strong><\/td>\r\n<\/tr>\r\n<tr class=\"border-bottom\">\r\n<td class=\"border\" style=\"width: 33.7667px;\"><strong>P<\/strong><\/td>\r\n<td class=\"border-right\" style=\"width: 36.4833px;\"><strong>Q<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 110px;\"><strong>(P\u2192Q)<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 110px;\"><strong>P<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 141.317px;\"><strong>Q<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 33.7667px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"width: 36.4833px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 110px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 110px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 141.317px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 33.7667px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"width: 36.4833px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 110px;\"><em><strong>\u00a0F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 110px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 141.317px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 33.7667px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"width: 36.4833px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 110px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 110px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 141.317px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 33.7667px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"width: 36.4833px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 110px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 110px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 141.317px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe know how to fill in the column for the conditional because we can refer back to the truth table used to define the conditional, to determine what its truth value is when the first part and second part are true; and so on. \u00a0<strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true in those kinds of situations where <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true, and <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is false in those kinds of situations where <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is false. \u00a0And the same is so for <strong><span class=\"strong\">Q<\/span><\/strong>.\r\n\r\nNow, consider all those kinds of ways the world could be such that all the premises are true. \u00a0Only the first row of the truth table is one where all the premises are true. \u00a0Note that the conclusion is true in that row. \u00a0That means, in any kind of situation in which all the premises are true, the conclusion will be true. \u00a0Or, equivalently: necessarily, if all the premises are true, then the conclusion is true.\r\n<table class=\"grid\" style=\"width: 100px;\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><\/td>\r\n<td class=\"border-right\"><\/td>\r\n<td class=\"border\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\"><strong>conclusion<\/strong><\/td>\r\n<\/tr>\r\n<tr class=\"border-bottom\">\r\n<td class=\"border\"><strong>P<\/strong><\/td>\r\n<td class=\"border-right\"><strong>Q<\/strong><\/td>\r\n<td class=\"border\"><strong>(P\u2192Q)<\/strong><\/td>\r\n<td class=\"border\"><strong>P<\/strong><\/td>\r\n<td class=\"border\"><strong>Q<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nConsider in contrast the second argument above, the invalid argument with all true premises and a true conclusion. \u00a0We\u2019ll use the following translation key.\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">R<\/span><\/strong>: \u00a0Miami is the capital of Ontario<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">S<\/span><\/strong>: \u00a0Miami is in Canada<\/p>\r\nAnd our argument is thus\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>R<\/strong>\u2192<strong>S<\/strong>)<\/span><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>R<\/strong><\/span><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong>_____<\/strong><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>S<\/strong><\/span><\/p>\r\nHere is the truth table.\r\n<table class=\"grid\" style=\"height: 90px; width: 125px;\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 45.7333px;\"><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 43.5667px;\"><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 161.65px;\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 161.633px;\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 210.083px;\"><strong>conclusion<\/strong><\/td>\r\n<\/tr>\r\n<tr class=\"border-bottom\" style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 45.7333px;\"><strong>R<\/strong><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 43.5667px;\"><strong>S<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 161.65px;\"><strong>(R\u2192S)<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 161.633px;\"><strong>\u00acR<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 210.083px;\"><strong>\u00acS<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 45.7333px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 43.5667px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 161.65px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 161.633px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 210.083px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 45.7333px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 43.5667px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 161.65px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 161.633px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 210.083px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 45.7333px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 43.5667px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 161.65px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 161.633px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 210.083px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 45.7333px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 43.5667px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 161.65px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 161.633px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 210.083px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNote that there are two kinds of ways that the world could be in which all of our premises are true. \u00a0These correspond to the third and fourth row of the truth table. \u00a0But for the third row of the truth table, the premises are true but the conclusion is false. \u00a0Yes, there is a kind of way the world could be in which all the premises are true and the conclusion is true; that is shown in the fourth row of the truth table. \u00a0But we are not interested in identifying arguments that will have true conclusions if we are lucky. \u00a0We are interested in valid arguments. \u00a0This argument is invalid. \u00a0There is a kind of way the world could be such that all the premises are true and the conclusion is false. \u00a0We can highlight this.\r\n<table class=\"grid\" style=\"width: 125px;\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><\/td>\r\n<td class=\"border-right\"><\/td>\r\n<td class=\"border\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\"><strong>conclusion<\/strong><\/td>\r\n<\/tr>\r\n<tr class=\"border-bottom\">\r\n<td class=\"border\"><strong>R<\/strong><\/td>\r\n<td class=\"border-right\"><strong>S<\/strong><\/td>\r\n<td class=\"border\"><strong>(R\u2192S)<\/strong><\/td>\r\n<td class=\"border\"><strong>\u00acR<\/strong><\/td>\r\n<td class=\"border\"><strong>\u00acS<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"shaded\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nHopefully it becomes clear why we care about validity. \u00a0Any argument of the form, <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>\u00a0and <strong><span class=\"strong\">P<\/span><\/strong>, therefore <strong><span class=\"strong\">Q<\/span><\/strong>, is valid. \u00a0We do not have to know what <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>mean to determine this. Similarly, any argument of the form, <span class=\"strong\">(<strong>R<\/strong>\u2192<strong>S<\/strong>)<\/span>\u00a0and <span class=\"strong\">\u00ac<strong>R<\/strong><\/span>, therefore <span class=\"strong\">\u00ac<strong>S<\/strong><\/span>, is invalid. \u00a0We do not have to know what <strong><span class=\"strong\">R<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">S<\/span>\u00a0<\/strong>mean to determine this. \u00a0So logic can be of equal use to the astronomer and the financier, the computer scientist or the sociologist.\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Check for Understanding<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>How can a truth table be used to examine the validity of an argument?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>3.4 Returning to our historical example<\/h2>\r\nWe described some (not all) of the hypotheses that Semmelweis tested when he tried to identify the cause of childbed fever, so that he could save thousands of women and infants. \u00a0Let us symbolize these and consider his reasoning.\r\n\r\nThe first case we considered was one where he reasoned: \u00a0If childbed fever is caused by cosmic-atmospheric-terrestrial influences, then the mortality rate would be similar in the First and Second Clinics. \u00a0But the mortality rate was not similar in the First and Second Clinics. \u00a0So, the childbed fever is not caused by cosmic-atmospheric-terrestrial influences.\r\n\r\nHere is a key to symbolize the argument.\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>T<\/strong>:<\/span>\u00a0 Childbed fever is caused by cosmic-atmospheric-terrestrial influences.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>U<\/strong>: \u00a0<\/span>The mortality rate is similar in the First and Second Clinics.<\/p>\r\nThis would mean the argument is:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>T<\/strong>\u2192<strong>U<\/strong>)<\/span><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>U<\/strong><\/span><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>T<\/strong><\/span><\/p>\r\nIs this argument valid? \u00a0We can check using a truth table.\r\n<table class=\"grid\" style=\"width: 125px;\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><\/td>\r\n<td class=\"border-right\"><\/td>\r\n<td class=\"border\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\"><strong>conclusion<\/strong><\/td>\r\n<\/tr>\r\n<tr class=\"border-bottom\">\r\n<td class=\"border\"><strong>T<\/strong><\/td>\r\n<td class=\"border-right\"><strong>U<\/strong><\/td>\r\n<td class=\"border\"><strong>(T\u2192U)<\/strong><\/td>\r\n<td class=\"border\"><strong>\u00acU<\/strong><\/td>\r\n<td class=\"border\"><strong>\u00acT<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe last row is the only row where all the premises are true. \u00a0For this row, the conclusion is true. \u00a0Thus, for all the kinds of ways the world could be in which the premises are true, the conclusion is also true. \u00a0This is a valid argument. \u00a0If we accept his premises, then we should accept that childbed fever was not caused by cosmic-atmospheric-terrestrial influences.\r\n\r\nThe second argument we considered was the concern that fear caused the higher mortality rates, particularly the fear of the priest coming to deliver last rites. \u00a0Semmelweis reasoned that if the higher rate of childbed fever is caused by fear of death resulting from the priest\u2019s approach, then the rate of childbed fever should decline if people cannot discern when the priest is coming to the Clinic. Here is a key:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>V<\/strong>:<\/span>\u00a0 the higher rate of childbed fever is caused by fear of death resulting from the priest\u2019s approach.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>W<\/strong>:<\/span>\u00a0 the rate of childbed fever will decline if people cannot discern when the priest is coming to the Clinic.<\/p>\r\nBut when Semmelweis had the priest silence his bell, and take a different route, so that patients could not discern that he was coming to the First Clinic, he found no difference in the mortality rate; the First Clinic remained far worse than the second clinic. \u00a0He concluded that the higher rate of childbed fever was not caused by fear of death resulting from the priest\u2019s approach.\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>V<\/strong>\u2192<strong>W<\/strong>)<\/span><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>W<\/strong><\/span><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>V<\/strong><\/span><\/p>\r\nIs this argument valid? \u00a0We can check using a truth table.\r\n<table class=\"grid\" style=\"height: 90px; width: 125px;\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 45.3167px;\"><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 55.6px;\"><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 205.55px;\"><strong>conclusion<\/strong><\/td>\r\n<\/tr>\r\n<tr class=\"border-bottom\" style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 45.3167px;\"><strong>V<\/strong><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 55.6px;\"><strong>W<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><strong>(V\u2192W)<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><strong>\u00acW<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 205.55px;\"><strong>\u00acV<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 45.3167px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 55.6px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 205.55px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 45.3167px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 55.6px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 205.55px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 45.3167px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 55.6px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 205.55px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"height: 15px; width: 45.3167px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\" style=\"height: 15px; width: 55.6px;\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"shaded\" style=\"height: 15px; width: 158.1px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"shaded\" style=\"height: 15px; width: 158.1px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"shaded\" style=\"height: 15px; width: 205.55px;\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAgain, we see that Semmelweis\u2019s reasoning was good. \u00a0He showed that it was not the case that the higher rate of childbed fever was caused by fear of death resulting from the Priest\u2019s approach.\r\n\r\nWhat about Semmelweis\u2019s positive conclusion, that the higher mortality rate was caused by some contaminant from the corpses that doctors had autopsied just before they assisted in a delivery? \u00a0To understand this step in his method, we need to reflect a moment on the scientific method and its relation to logic.\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Check for Understanding<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>Explain how argumentative truth tables were utilized to assess the validity of Semmelweis's argument?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>3.5 \u00a0Other kinds of arguments 1: \u00a0Scientific reasoning<\/h2>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What makes a statement deductive versus inductive?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Terms<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li><strong>Deductive Reasoning<\/strong> - reasoning in which necessarily our conclusions is true if our premises are true<\/li>\r\n \t<li><strong>Falsfiable<\/strong> - an argumentative statement that can use scientific evidence to determine its truth value<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nValid arguments, and the methods that we are developing, are sometimes called \u201c<strong>deductive reasoning<\/strong>.\u201d \u00a0This is the kind of reasoning in which necessarily our conclusions is true if our premises are true; these arguments can be shown to be good by way of our logical reasoning alone. \u00a0There are other kinds of reasoning, and understanding this may help clarify the relation of logic to other endeavors. \u00a0Two important, and closely related, alternatives to deductive reasoning are scientific reasoning and statistical generalizations. \u00a0We\u2019ll discuss statistical generalizations in the next section.\r\n\r\nScientific method relies upon logic, but science is not reducible to logic: \u00a0scientists do empirical research. \u00a0That is, they examine and test phenomena in the world. \u00a0This is a very important difference from pure logic. \u00a0To understand how this difference results in a distinct method, let us review Semmelweis\u2019s important discovery.\r\n\r\nThe details and nature of scientific reasoning are somewhat controversial. \u00a0I am going to provide here a basic\u2014many philosophers would say, oversimplified\u2014account of scientific reasoning. \u00a0My goal is to indicate the relation between logic and the kind of reasoning Semmelweis may have used.\r\n\r\nAs we noted, Semmelweis learned about the death of a colleague, Professor Jakob Kolletschka. \u00a0Kolletschka had been performing an autopsy, and he cut his finger. \u00a0Shortly thereafter, Kolletschka died with symptoms like those of childbed fever. \u00a0Semmelweis reasoned that something on the corpse caused the disease; he called this \u201ccadaveric matter\u201d. \u00a0In the First Clinic, where the mortality rate of women and babies was high, doctors were doing autopsies and then delivering babies immediately after. \u00a0If he could get this cadaveric matter off the hands of the doctors, the rate of childbed fever should fall.\r\n\r\nSo, he reasoned thus: \u00a0if the fever is caused by cadaveric matter on the hands of the doctors, then the mortality rate will drop when doctors wash their hands with chlorinated water before delivering babies. \u00a0He forced the doctors to do this. \u00a0The result was that the mortality rate dropped a very great deal, at times to below 1%.\r\n\r\nHere is a key:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>P<\/strong>:<\/span>\u00a0 The fever is caused by cadaveric matter on the hands of the doctors.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>Q<\/strong>: \u00a0<\/span>The mortality rate will drop when doctors wash their hands with chlorinated water before delivering babies.<\/p>\r\nAnd the argument appears to be something like this (as we will see, this isn\u2019t quite the right way to put it, but for now\u2026):\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong><\/p>\r\nIs this argument valid? \u00a0We can check using a truth table.\r\n<table class=\"grid\" style=\"width: 125px;\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><\/td>\r\n<td class=\"border-right\"><\/td>\r\n<td class=\"border\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\"><strong>premise<\/strong><\/td>\r\n<td class=\"border\"><strong>conclusion<\/strong><\/td>\r\n<\/tr>\r\n<tr class=\"border-bottom\">\r\n<td class=\"border\"><strong>P<\/strong><\/td>\r\n<td class=\"border-right\"><strong>Q<\/strong><\/td>\r\n<td class=\"border\"><strong>(P\u2192Q)<\/strong><\/td>\r\n<td class=\"border\"><strong>Q<\/strong><\/td>\r\n<td class=\"border\"><strong>P<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"shaded\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFrom this, it looks like Semmelweis has used an invalid argument!\r\n\r\nHowever, an important feature of scientific reasoning must be kept in mind. \u00a0There is some controversy over the details of the scientific method, but the most basic view goes something like this. \u00a0Scientists formulate hypotheses about the possible causes or features of a phenomenon. \u00a0They make predictions based on these hypotheses, and then they perform experiments to test those predictions. \u00a0The reasoning here uses the conditional: \u00a0if the hypotheses is true, then the particular prediction will be true. \u00a0If the experiment shows that the prediction is false, then the scientist rejects the hypothesis.[footnote]It would be more accurate to say, if the prediction proves false, the scientist must reject either the hypothesis or some other premise of her reasoning. For example, her argument may include the implicit premise that her scientific instruments were operating correctly. She might instead reject this premise that her instruments are working correctly, change one of her instruments, and try again to test the hypothesis. See Duhem (1991). Or, to return to the case of Semmelweis, he might wonder whether he sufficiently established that there were no differences in the atmosphere between the two clinics; or he might wonder whether he sufficiently muffled the Priest\u2019s approach; or whether he recorded his results accurately; and so on. As noted, my account of scientific reasoning here is simplified.[\/footnote]\u00a0 But if the prediction proved to be true, then the scientist has shown that the hypothesis may be true\u2014at least, given the information we glean from the conditional and the consequent alone.\r\n\r\nThis is very important. \u00a0Scientific conclusions are about the physical world, they are not about logic. \u00a0This means that scientific claims are not necessarily true, in the sense of \u201cnecessarily\u201d that we used in our definition of \u201cvalid\u201d. \u00a0Instead, science identifies claims that may be true, or (after some progress) are very likely to be true, or (after very much progress) are true.\r\n\r\nScientists keep testing their hypotheses, using different predictions and experiments. \u00a0Very often, they have several competing hypotheses that have, so far, survived testing. \u00a0To decide between these, they can use a range of criteria. \u00a0In order of their importance, these include: \u00a0choose the hypothesis with the most predictive power (the one that correctly predicts more kinds of phenomena); choose the hypothesis that will be most productive of other scientific theories; choose the hypothesis consistent with your other accepted hypotheses; choose the simplest hypothesis.\r\n\r\nWhat Semmelweis showed was that it could be true that cadaveric matter caused the childbed fever. \u00a0This hypothesis predicted more than any other hypothesis that the doctors had, and so for that reason alone this was the very best hypothesis. \u00a0\u201cBut,\u201d you might reason, \u201cdoesn\u2019t that mean his conclusion was true? \u00a0And don\u2019t we know now, given all that we\u2019ve learned, that his conclusion must be true?\u201d \u00a0No. \u00a0He was far ahead of other doctors, and his deep insights were of great service to all of humankind. \u00a0But the scientific method continued to refine Semmelweis\u2019s ideas. \u00a0For example, later doctors introduced the idea of microorganisms as the cause of childbed fever, and this refined and improved Semmelweis\u2019s insights: \u00a0it was not because the cadaveric matter came from corpses that it caused the disease; it was because the cadaveric matter contained particular micro-organisms that it caused the disease. \u00a0So, further scientific progress showed his hypothesis could be revised and improved.\r\n\r\nTo review and summarize, with the scientific method:\r\n<ol class=\"lst-kix_list_31-0 start\" start=\"1\">\r\n \t<li>We develop a hypothesis about the causes or nature of a phenomenon.<\/li>\r\n \t<li>We predict what (hopefully unexpected) effects are a consequence of this hypothesis.<\/li>\r\n \t<li>We check with experiments to see if these predictions come true:\r\n<ol class=\"lst-kix_list_31-0 start\" start=\"1\">\r\n \t<li>If the predictions prove false, we reject the hypothesis;[footnote]Or, as noted in note 6, we reject some other premise of the argument.[\/footnote]<\/li>\r\n \t<li>If the predictions prove true, we conclude that the hypothesis could be true. \u00a0We continue to test the hypothesis by making other predictions (that is, we return to step 2).<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nThis means that a hypothesis that does not make testable predictions (that is, a hypothesis that cannot possibly be proven false) is not a scientific hypothesis. \u00a0Such a hypothesis is called \u201cunfalsifiable\u201d and we reject it as unscientific.\r\n\r\nThis method can result in more than one hypothesis being shown to be possibly true. \u00a0Then, we chose between competing hypotheses by using criteria like the following (here ordered by their relative importance; \u201ctheory\u201d can be taken to mean a collection of one or more hypotheses):\r\n<ol class=\"lst-kix_list_30-0 start\" start=\"1\">\r\n \t<li>Predictive power: the more that a hypothesis can successfully predict, the better it is.<\/li>\r\n \t<li>Productivity: \u00a0a hypothesis that suggests more new directions for research is to be preferred.<\/li>\r\n \t<li>Coherence with Existing Theory: if two hypotheses predict the same amount and are equally productive, then the hypothesis that coheres with (does not contradict) other successful theories is preferable to one that does contradict them.<\/li>\r\n \t<li>Simplicity: if two hypotheses are equally predictive, productive, and coherent with existing theories, then the simpler hypothesis is preferable.<\/li>\r\n<\/ol>\r\nOut of respect to Ignaz Semmelweis we should tell the rest of his story, although it means we must end on a sad note. \u00a0Semmelweis\u2019s great accomplishment was not respected by his colleagues, who resented being told that their lack of hygiene was causing deaths. \u00a0He lost his position at the First Clinic, and his successors stopped the program of washing hands in chlorinated water. \u00a0The mortality rate leapt back to its catastrophically high levels. \u00a0Countless women and children died. \u00a0Semmelweis continued to promote his ideas, and this caused growing resentment. \u00a0Eventually, several doctors in Vienna\u2014not one of them a psychiatrist\u2014secretly signed papers declaring Semmelweis insane. \u00a0We do not know whether Semmelweis was mentally ill at this time. \u00a0These doctors took him to an asylum on the pretense of having him visit in his capacity as a doctor; when he arrived, the guards seized Semmelweis. \u00a0He struggled, and the guards at the asylum beat him severely, put him in a straightjacket, and left him alone in a locked room. \u00a0Neglected in isolation, the wounds from his beating became infected and he died a week later.\r\n\r\nIt was years before Semmelweis\u2019s views became widely accepted and his accomplishment properly recognized. \u00a0His life teaches many lessons, including unfortunately that even the most educated among us can be evil, petty, and willfully ignorant. \u00a0Let us repay Semmelweis, as those in his own time did not, by remembering and praising his scientific acumen and humanity.\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Check for Understanding<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What role does logic play in natural sciences?<\/li>\r\n \t<li><strong>Critical Thinking Task:<\/strong>\u00a0Can you explain the important difference between pure logic and empirical science? How does this difference apply to your life?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>3.6 Other kinds of arguments 2: \u00a0Statistical reasoning<\/h2>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What are common functions of statistics in logical reasoning?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Terms<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li><strong>Population<\/strong> - all the events or all the things we want to make a generalization about.<\/li>\r\n \t<li><strong>Sample<\/strong> - a portion of a population that is representative of the entire population?<\/li>\r\n \t<li><strong>Random Sample<\/strong> - every member of the population was equally likely to be in the sample.<\/li>\r\n \t<li><strong>Inductive Reasoning<\/strong> - method of testing claims about the world, it requires observations, and its conclusion is likely instead of being certain.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nHere we can say a few words about statistical generalizations\u2014our goal being only to provide a contrast with deductive reasoning.\r\n\r\nIn one kind of statistical generalization, we have a <strong>population<\/strong> of some kind that we want to make general claims about. \u00a0A population could be objects or events. \u00a0So, a population can be a group of organisms, or a group of weather events. \u00a0\u201cPopulation\u201d just means all the events or all the things we want to make a generalization about. \u00a0Often however it is impossible to examine every object or event in the population, so what we do is gather a <strong>sample<\/strong>. \u00a0A sample is some portion of the population. \u00a0Our hope is that the sample is representative of the population: \u00a0that whatever traits are shared by the members of the sample are also shared by the members of the population.\r\n\r\nFor a sample to representative, it must be <strong>random<\/strong> and large enough. \u00a0\u201cRandom\u201d in this context means that the sample was not chosen in any way that might distinguish members of the sample from the population, other than being members of the population. \u00a0In other words, every member of the population was equally likely to be in the sample. \u00a0\u201cLarge enough\u201d is harder to define. \u00a0Statisticians have formal models describing this, but suffice to say we should not generalize about a whole population using just a few members.\r\n\r\nHere\u2019s an example. \u00a0We wonder if all domestic dogs are descended from wolves. \u00a0Suppose we have some genetic test to identify if an organism was a descendent of wolves. \u00a0We cannot give the test to all domestic dogs\u2014this would be impractical and costly and unnecessary. \u00a0We pick a random sample of domestic dogs that is large enough, and we test them. \u00a0For the sample to be random, we need to select it without allowing any bias to influence our selection; all that should matter is that these are domestic dogs, and each member of the population must have an equal chance of being in the sample. \u00a0Consider the alternative: \u00a0if we just tested one family of dogs\u2014say, dogs that are large\u2014we might end up selecting dogs that differed from others in a way that matters to our test. \u00a0For example, maybe large dogs are descended from wolves, but small dogs are not. \u00a0Other kinds of bias can creep in less obviously. \u00a0We might just sample dogs in our local community, and it might just be that people in our community prefer large dogs, and again we would have a sample bias. \u00a0So, we randomly select dogs, and give them the genetic test.\r\n\r\nSuppose the results were positive. \u00a0We reason that if all the members of the randomly selected and large enough sample (the tested dogs) have the trait, then it is very likely that all the members of the population (all dogs) have the trait. \u00a0Thus: we could say that it appears very likely that all dogs have the trait. \u00a0(This likelihood can be estimated, so that we can also sometimes say how likely it is that all members of the population have the trait.)\r\n\r\nThis kind of reasoning obviously differs from a deductive argument very substantially. \u00a0It is a method of testing claims about the world, it requires observations, and its conclusion is likely instead of being certain.\r\n\r\nBut such reasoning is not unrelated to logic. \u00a0Deductive reasoning is the foundation of these and all other forms of reasoning. \u00a0If one must reason using statistics in this way, one relies upon deductive methods always at some point in one\u2019s arguments. \u00a0There was a conditional at the penultimate step of our reasoning, for example (we said \u201cif all the members of the randomly selected and large enough sample have the trait, then it is very likely that all the members of the population have the trait\u201d). \u00a0Furthermore, the foundations of these methods (the most fundamental descriptions of what these methods are) are given using logic and mathematics. \u00a0Logic, therefore, can be seen as the study of the most fundamental form of reasoning, which will be used in turn by all other forms of reasoning, including scientific and statistical reasoning.\\\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Check for Understanding<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li><strong>Self Reflection:<\/strong>What are instances in your day-to-day life you use statistical reasoning to make valid arguments about your decisions?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>3.7 \u00a0Problems<\/h2>\r\n<ol class=\"lst-kix_list_41-0 start\" start=\"1\">\r\n \t<li>Make truth tables to show that the following arguments are valid. \u00a0Circle or highlight the rows of the truth table that show the argument is valid (that is, all the rows where all the premises are true). \u00a0Note that you will need eight rows in the truth table for problems d-f, and sixteen rows in the truth table for problems g and h.\r\n<ol class=\"lst-kix_list_41-0 start\" start=\"1\">\r\n \t<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">\u00acQ<\/span>. Conclusion: \u00a0<span class=\"strong\">\u00acP<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">\u00acP<\/span>. Conclusion: <span class=\"strong\">(P\u2192Q)<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">Q<\/span>. Conclusion: <span class=\"strong\">(P\u2192Q)<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>. Conclusion: \u00a0<span class=\"strong\">(P\u2192R)<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">(P<\/span><span class=\"strong\">\u2192<\/span><span class=\"strong\">Q)<\/span>, <span class=\"strong\">(Q<\/span><span class=\"strong\">\u2192<\/span><span class=\"strong\">R)<\/span>, <span class=\"strong\">P<\/span>. Conclusion: \u00a0<span class=\"strong\">R<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">(P<\/span><span class=\"strong\">\u2192<\/span><span class=\"strong\">Q)<\/span>, <span class=\"strong\">(Q<\/span><span class=\"strong\">\u2192<\/span><span class=\"strong\">R)<\/span>, <span class=\"strong\">\u00ac<\/span><span class=\"strong\">R<\/span>. Conclusion: \u00a0<span class=\"strong\">\u00ac<\/span><span class=\"strong\">P<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>, <span class=\"strong\">(R\u2192S)<\/span>, <span class=\"strong\">P<\/span>. Conclusion: \u00a0<span class=\"strong\">S<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>, <span class=\"strong\">(R\u2192S)<\/span>. Conclusion: \u00a0<span class=\"strong\">(P\u2192S)<\/span>.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Make truth tables to show the following arguments are invalid. Circle or highlight the rows of the truth table that show the argument is invalid (that is, any row where all the premises are true but the conclusion is false).\r\n<ol class=\"lst-kix_list_41-0 start\" start=\"1\">\r\n \t<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>. Conclusion: \u00a0<span class=\"strong\">P<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>. Conclusion: \u00a0<span class=\"strong\">Q<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">P<\/span>. Conclusion: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">Q<\/span>. Conclusion: \u00a0<span class=\"strong\">P<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">\u00acQ<\/span>. Conclusion: <span class=\"strong\">(P\u2192Q)<\/span>.<\/li>\r\n \t<li>Premises:\u00a0<span class=\"strong\">(P\u2192Q)<\/span>. Conclusion: <span class=\"strong\">(Q\u2192P)<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>, <span class=\"strong\">\u00acP<\/span>. Conclusion: \u00a0<span class=\"strong\">\u00acR<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>, <span class=\"strong\">R<\/span>. Conclusion: \u00a0<span class=\"strong\">P<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>. Conclusion: \u00a0<span class=\"strong\">(R\u2192P)<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>, <span class=\"strong\">(R\u2192S)<\/span>. Conclusion: \u00a0<span class=\"strong\">(S\u2192P)<\/span>.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>In normal colloquial English, write your own valid argument with at least two premises. Your argument should just be a paragraph (not an ordered list of sentences or anything else that looks like logic). \u00a0Translate it into propositional logic and use a truth table to show it is valid.<\/li>\r\n \t<li>In normal colloquial English, write your own invalid argument with at least two premises. \u00a0Translate it into propositional logic and use a truth table to show it is invalid.<\/li>\r\n \t<li>For each of the following, state whether the argument described could be: valid, invalid, sound, unsound.\r\n<ol class=\"lst-kix_list_41-0 start\" start=\"1\">\r\n \t<li>An argument with false premises and a false conclusion.<\/li>\r\n \t<li>An argument with true premises and a false conclusion.<\/li>\r\n \t<li>An argument with false premises and a true conclusion.<\/li>\r\n \t<li>An argument with true premises and a true conclusion.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<div>\r\n\r\n<hr \/>\r\n\r\n<a id=\"ftnt5\" href=\"#ftnt_ref5\">[5]<\/a>\u00a0All the data cited here comes from Carter (1983) and additional biographical information comes from\u00a0Carter and Carter (2008). \u00a0These books are highly recommended to anyone interested in the history of science or medicine.\r\n\r\n<\/div>\r\n<div>\r\n\r\n<a id=\"ftnt6\" href=\"#ftnt_ref6\">[6]<\/a>\u00a0It would be more accurate to say, if the prediction proves false, the scientist must reject either the hypothesis or some other premise of her reasoning. \u00a0For example, her argument may include the implicit premise that her scientific instruments were operating correctly. \u00a0She might instead reject this premise that her instruments are working correctly, change one of her instruments, and try again to test the hypothesis. \u00a0See Duhem (1991). \u00a0Or, to return to the case of Semmelweis, he might wonder whether he sufficiently established that there were no differences in the atmosphere between the two clinics; or he might wonder whether he sufficiently muffled the Priest\u2019s approach; or whether he recorded his results accurately; and so on. \u00a0As noted, my account of scientific reasoning here is simplified.\r\n\r\n<\/div>\r\n<div>\r\n\r\n<a id=\"ftnt7\" href=\"#ftnt_ref7\">[7]<\/a>\u00a0Or, as noted in note 6, we reject some other premise of the argument.\r\n\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Check for Understanding<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>Explain the distinction between a population and a sample in propositional reasoning?<\/li>\r\n \t<li>What role does random sampling serve in statistical reasoning?<\/li>\r\n \t<li>What is the relationship between deductive and inductive reasoning in statistics?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>","rendered":"<h2>3.1 \u00a0A historical example<\/h2>\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>How can scientists use logic to support their arguments?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>An important example of excellent reasoning can be found in the case of the medical advances of the Nineteenth Century physician, Ignaz Semmelweis. \u00a0Semmelweis was an obstetrician at the Vienna General Hospital. \u00a0Built on the foundation of a poor house, and opened in 1784, the General Hospital is still operating today. \u00a0Semmelweis, during his tenure as assistant to the head of one of two maternity clinics, noticed something very disturbing. \u00a0The hospital had two clinics, separated only by a shared anteroom, known as the First and the Second Clinics. \u00a0The mortality rate for mothers delivering babies in the First Clinic, however, was nearly three times as bad as the mortality for mothers in the Second Clinic (9.9 % average versus 3.4% average). \u00a0The same was true for the babies born in the clinics: \u00a0the mortality rate in the First Clinic was 6.1% versus 2.1% at the Second Clinic.<a class=\"footnote\" title=\"All the data cited here comes from Carter (1983) and additional biographical information comes from Carter and Carter (2008). These books are highly recommended to anyone interested in the history of science or medicine.\" id=\"return-footnote-35-1\" href=\"#footnote-35-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0 In nearly all these cases, the deaths were caused by what appeared to be the same illness, commonly called \u201cchildbed fever\u201d. \u00a0Worse, these numbers actually understated the mortality rate of the First Clinic, because sometimes very ill patients were transferred to the general treatment portion of the hospital, and when they died,\u00a0their death was counted as part of the mortality rate of the general hospital, not of the First Clinic.<\/p>\n<p>Semmelweis set about trying to determine why the First Clinic had the higher mortality rate. \u00a0He considered a number of hypotheses, many of which were suggested by or believed by other doctors.<\/p>\n<p>One hypothesis was that cosmic-atmospheric-terrestrial influences caused childbed fever. \u00a0The idea here was that some kind of feature of the atmosphere would cause the disease. \u00a0But, Semmelweis observed, the First and Second Clinics were very close to each other, had similar ventilation, and shared a common anteroom. \u00a0So, they had similar atmospheric conditions. \u00a0He reasoned: \u00a0If childbed fever is caused by cosmic-atmospheric-terrestrial influences, then the mortality rate would be similar in the First and Second Clinics. \u00a0But the mortality rate was not similar in the First and Second Clinics. \u00a0So, the childbed fever was not caused by cosmic-atmospheric-terrestrial influences.<\/p>\n<p>Another hypothesis was that overcrowding caused the childbed fever. \u00a0But, if overcrowding caused the childbed fever, then the more crowded of the two clinics should have the higher mortality rate. \u00a0But, the Second Clinic was more crowded (in part because, aware of its lower mortality rate, mothers fought desperately to be put there instead of in the First Clinic). \u00a0It did not have a higher mortality rate. \u00a0So, the childbed fever was not caused by overcrowding.<\/p>\n<p>Another hypothesis was that fear caused the childbed fever. \u00a0In the Second Clinic, the priest delivering last rites could walk directly to a dying patient\u2019s room. \u00a0For reasons of the layout of the rooms, the priest delivering last rites in the First Clinic walked by all the rooms, ringing a bell announcing his approach. \u00a0This frightened patients; they could not tell if the priest was coming for them. \u00a0Semmelweis arranged a different route for the priest and asked him to silence his bell. \u00a0He reasoned: \u00a0if the higher rate of childbed fever was caused by fear of death resulting from the priest\u2019s approach, then the rate of childbed fever should decline if people could not tell when the priest was coming to the Clinic. \u00a0But it was not the case that the rate of childbed fever declined when people could not tell if the priest was coming to the First Clinic. \u00a0So, the higher rate of childbed fever in the First Clinic was not caused by fear of death resulting from the priest\u2019s approach.<\/p>\n<p>In the First Clinic, male doctors were trained; this was not true in the Second Clinic. \u00a0These male doctors performed autopsies across the hall from the clinic, before delivering babies. \u00a0Semmelweis knew of a doctor who cut himself while performing an autopsy, and who then died a terrible death not unlike that of the mothers who died of childbed fever. \u00a0Semmelweis formed a hypothesis. \u00a0The childbed fever was caused by something on the hands of the doctors, something that they picked up from corpses during autopsies, but that infected the women and infants. \u00a0He reasoned that: \u00a0if the fever was caused by cadaveric matter on the hands of the doctors, then the mortality rate would drop when doctors washed their hands with chlorinated water before delivering babies. \u00a0He forced the doctors to do this. \u00a0The result was that the mortality rate dropped to a rate below that even of the Second Clinic.<\/p>\n<p>Semmelweis concluded that the best explanation of the higher mortality rate was this \u201ccadaveric matter\u201d on the hands of doctors. \u00a0He was the first person to see that washing of hands with sterilizing cleaners would save thousands of lives. \u00a0It is hard to overstate how important this contribution is to human well being. \u00a0Semmelweis\u2019s fine reasoning deserves our endless respect and gratitude.<\/p>\n<p>But how can we be sure his reasoning was good? \u00a0Semmelweis was essentially considering a series of arguments. \u00a0Let us turn to the question: \u00a0how shall we evaluate arguments?<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Check for Understanding<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>Explain how the Nineteenth Century physician Ignaz Semmelweis used logic to assess the hypothesis presented in his investigation into mortality rates at the Vienna General Hospital.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>3.2 \u00a0Arguments<\/h2>\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What makes a statement precise?<\/li>\n<li>How does logic get people to believe in things?<\/li>\n<li>When is an argument considered true?<\/li>\n<li>When is an argument considered valid?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Terms<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li><strong>Argument<\/strong> &#8211; an ordered list of sentences; we call one of these sentences the \u201cconclusion\u201d, and we call the other sentences \u201cpremises\u201d.<\/li>\n<li><strong>Valid Argument<\/strong> &#8211; an argument for which, necessarily, if the premises are true, then the conclusion is true.<\/li>\n<li><strong>Sound Argument<\/strong> &#8211; a valid argument with true premises.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>Our logical language now allows us to say conditional and negation statements. \u00a0That may not seem like much, but our language is now complex enough for us to develop the idea of using our logic not just to describe things, but also to reason about those things.<\/p>\n<p>We will think of reasoning as providing an argument. \u00a0Here, we use the word \u201c<strong>argument<\/strong>\u201d not in the sense of two or more people criticizing each other, but rather in the sense we mean when we say, \u201cPythagoras\u2019s argument\u201d. \u00a0In such a case, someone is using language to try to convince us that something is true. \u00a0Our goal is to make this notion very precise, and then identify what makes an argument good.<\/p>\n<p>We need to begin by making the notion of an argument precise. \u00a0Our logical language so far contains only sentences. \u00a0An argument will, therefore, consist of sentences. \u00a0In a natural language, we use the term \u201cargument\u201d in a strong way, which includes the suggestion that the argument should be good. \u00a0However, we want to separate the notion of a good argument from the notion of an argument, so we can identify what makes an argument good, and what makes an argument bad. \u00a0To do this, we will start with a minimal notion of what an argument is. Here is the simplest, most minimal notion:<\/p>\n<p style=\"padding-left: 120px;\"><strong>Argument<\/strong>: \u00a0an ordered list of sentences; we call one of these sentences the \u201cconclusion\u201d, and we call the other sentences \u201cpremises\u201d.<\/p>\n<p>This is obviously very weak. \u00a0(There is\u00a0a famous Monty Python skit where one of the comedians ridicules the very idea that such a thing could be called an argument.) \u00a0But for our purposes, this is a useful notion because it is very clearly defined, and we can now ask, what makes an argument good?<\/p>\n<p>The everyday notion of an argument is that it is used to convince us to believe something. \u00a0The thing that we are being encouraged to believe is the conclusion. \u00a0Following our definition of \u201cargument\u201d, the reasons that the person gives will be what we are calling \u201cpremises\u201d. \u00a0But <span class=\"em\">belief<\/span>\u00a0is a psychological notion. \u00a0We instead are interested only in truth. \u00a0So, we can reformulate this intuitive notion of what an argument should do, and think of an argument as being used to show that something is true. \u00a0The premises of the argument are meant to show us that the conclusion is true.<\/p>\n<p>What then should be this relation between the premises and the conclusion? \u00a0Intuitive notions include that the premises should support the conclusion, or corroborate the conclusion, or make the conclusion true. \u00a0But \u201csupport\u201d and \u201ccorroborate\u201d sound rather weak, and \u201cmake\u201d is not very clear. \u00a0What we can use in their place is a stronger standard: let us say as a first approximation that if the premises are true, the conclusion is true.<\/p>\n<p>But even this seems weak, on reflection.\u00a0 The conclusion could be true by accident, for reasons unrelated to our premises. \u00a0Remember that we define the conditional as true if the antecedent and consequent are true. \u00a0But this could happen by accident. \u00a0For example, suppose I say, \u201cIf Tom wears blue then he will get an A on the exam\u201d. \u00a0Suppose also that Tom both wears blue and Tom gets an A on the exam. \u00a0This makes the conditional true, but (we hope) the color of his clothes really had nothing to do with his performance on the exam. \u00a0Just so, we want our definition of \u201cgood argument\u201d to be such that it cannot be an accident that the premises and conclusion are both true.<\/p>\n<p>A better and stronger standard would be that, necessarily, given true premises, the conclusion is true.<\/p>\n<p>This points us to our definition of a good argument. \u00a0It is traditional to call a good argument \u201cvalid.\u201d<\/p>\n<p style=\"padding-left: 120px;\"><strong>Valid argument<\/strong>: \u00a0an argument for which, necessarily, if the premises are true, then the conclusion is true.<\/p>\n<p>This is the single most important principle in this book. \u00a0Memorize it.<\/p>\n<p>A bad argument is an argument that is not valid. \u00a0Our name for this will be an \u201cinvalid argument\u201d.<\/p>\n<p>Sometimes, a dictionary or other book will define or describe a \u201cvalid argument\u201d as an argument that follows the rules of logic. \u00a0This is a hopeless way to define \u201cvalid\u201d, because it is circular in a pernicious way: \u00a0we are going to create the rules of our logic in order to ensure that they construct valid arguments. \u00a0We cannot make rules of logical reasoning until we know what we want those rules to do, and what we want them to do is to create valid arguments. \u00a0So \u201cvalid\u201d must be defined before we can make our reasoning system.<\/p>\n<p>Experience shows that if a student is to err in understanding this definition of \u201cvalid argument\u201d, he or she will typically make the error of assuming that a valid argument has all true premises. \u00a0This is not required. \u00a0There are valid arguments with false premises and a false conclusion. \u00a0Here\u2019s one:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If Miami is the capital of Kansas, then Miami is in Canada. \u00a0Miami is the capital of Kansas. \u00a0Therefore, Miami is in Canada.<\/p>\n<p>This argument has at least one false premise: \u00a0Miami is not the capital of Kansas. \u00a0And the conclusion is false: \u00a0Miami is not in Canada. \u00a0But the argument is valid: \u00a0if the premises were both true, the conclusion would have to be true. \u00a0(If that bothers you, hold on a while and we will convince you that this argument is valid because of its form alone. \u00a0Also, keep in mind always that \u201cif\u2026then\u2026\u201d is interpreted as meaning the conditional.)<\/p>\n<p>Similarly, there are invalid arguments with true premises, and with a true conclusion. \u00a0Here\u2019s one:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If Miami is the capital of Ontario, then Miami is in Canada. \u00a0Miami is not the capital of Ontario. \u00a0Therefore, Miami is not in Canada.<\/p>\n<p>(If you find it confusing that this argument is invalid, look at it again after you finish reading this chapter.)<\/p>\n<p>Validity is about the relationship between the sentences in the argument. \u00a0It is not a claim that those sentences are true.<\/p>\n<p>Another variation of this confusion seems to arise when we forgot to think carefully about the conditional. \u00a0The definition of valid is not \u201cAll the premises are true, so the conclusion is true.\u201d \u00a0If you don\u2019t see the difference, consider the following two sentences. \u00a0\u201cIf your house is on fire, then you should call the fire department.\u201d \u00a0In this sentence, there is no claim that your house is on fire. \u00a0It is rather advice about what you should do if your house is on fire. \u00a0In the same way, the definition of valid argument does not tell you that the premises are true. \u00a0It tells you what follows if they are true. \u00a0Contrast now, \u201cYour house is on fire, so you should call the fire department\u201d. \u00a0This sentence delivers very bad news. \u00a0It is not a conditional at all. \u00a0What it really means is, \u201cYour house is on fire and you should call the fire department\u201d. \u00a0Our definition of valid is not, \u201cAll the premises are true and the conclusion is true\u201d.<\/p>\n<p>Finally, another common mistake is to confuse <span class=\"em\">true<\/span>\u00a0and <span class=\"em\">valid<\/span>. \u00a0In the sense that we are using these terms in this book, only sentences can be true or false, and only arguments can be valid and invalid. \u00a0When discussing and using our logical language, it is nonsense to say, \u201ca true argument\u201d, and it is nonsense to say, \u201ca valid sentence\u201d.<\/p>\n<p>Someone new to logic might wonder, why would we want a definition of \u201cgood argument\u201d that does not guarantee that our conclusion is true? \u00a0The answer is that logic is an enormously powerful tool for checking arguments, and we want to be able to identify what the good arguments are, independently of the particular premises that we use in the argument. \u00a0For example, there are infinitely many particular arguments that have the same form as the valid argument given above. \u00a0There are infinitely many particular arguments that have the same form as the invalid argument given above. \u00a0Logic lets us embrace all the former arguments at once, and reject all those bad ones at once.<\/p>\n<p>Furthermore, our propositional logic will not be able to tell us whether an atomic sentence is true. \u00a0If our argument is about rocks, we must ask the geologist if the premises are true. \u00a0If our argument is about history, we must ask the historian if the premises are true. \u00a0If our argument is about music, we must ask the music theorist if the premises are true. \u00a0But the logician can tell the geologist, the historian, and the musicologist whether her arguments are good or bad, independent of the particular premises.<\/p>\n<p>We do have a common term for a good argument that has true premises. \u00a0This is called \u201csound\u201d. \u00a0It is a useful notion when we are applying our logic. \u00a0Here is our definition:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong>Sound argument<\/strong>: \u00a0a valid argument with true premises.<\/p>\n<p>A sound argument must have a true conclusion, given the definition of \u201cvalid\u201d.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Check for Understanding<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What makes an argument good?<\/li>\n<li>Explain the difference between belief and truth in argumentative statements.<\/li>\n<li>Can a statement be valid but not true? Please explain.<\/li>\n<li>Explain the difference between a valid argument and an invalid argument.<\/li>\n<li><strong>Self-Reflection:\u00a0<\/strong>What is a valid argument you have encountered recently in your day-to-day life? What is an invalid argument you commonly encounter?<\/li>\n<li><strong>Critical Thinking Task:\u00a0<\/strong>What is the single most important principle to remember in this book? Why?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>3.3 \u00a0Checking arguments semantically<\/h2>\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What purpose does the word \u201cnecessarily\u201d serve in logic?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>Every element of our definition of \u201cvalid\u201d is clear except for one. \u00a0We know what \u201cif\u2026then\u2026\u201d means. \u00a0We defined the semantics of the conditional in chapter 2. \u00a0We have defined \u201cargument\u201d, \u201cpremise\u201d, and \u201cconclusion\u201d. \u00a0We take <em>true<\/em> and <em>false<\/em> as primitives. \u00a0But what does \u201cnecessarily\u201d mean?<\/p>\n<p>We define a valid argument as one where, <strong>necessarily<\/strong>, if the premises are true, then the conclusion is true. \u00a0It would seem the best way to understand this is to say, there is no situation in which the premises are true but the conclusion is false. \u00a0But then, what are these \u201csituations\u201d? \u00a0Fortunately, we already have a tool that looks like it could help us: \u00a0the truth table.<\/p>\n<p>Remember that in the truth table, we put on the bottom left side all the possible combinations of truth values of some set of atomic sentences. \u00a0Each row of the table then represents a kind of way the world could be. \u00a0Using this as a way to understand \u201cnecessarily\u201d, we could rephrase our definition of valid to something like this, \u201cIn any kind of situation in which all the premises are true, the conclusion is true.\u201d<\/p>\n<p>Let\u2019s try it out. \u00a0We will\u00a0need to use truth tables in a new way: \u00a0to check an argument. \u00a0That will require having not just one sentence, but several on the truth table. \u00a0Consider an argument that looks like it should be valid.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If Jupiter is more massive than Earth, then Jupiter has a stronger gravitational field than Earth. \u00a0Jupiter is more massive than Earth. \u00a0In conclusion, Jupiter has a stronger gravitational field than Earth.<\/p>\n<p>This looks like it has the form of a valid argument, and it looks like an astrophysicist would tell us it is sound. \u00a0Let\u2019s translate it to our logical language using the following translation key. \u00a0(We\u2019ve used up our letters, so I\u2019m going to start over. \u00a0We\u2019ll do that often: \u00a0assume we are starting a new language each time we translate a new set of problems or each time we consider a new example.)<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong>: \u00a0Jupiter is more massive than Earth<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>: \u00a0Jupiter has a stronger gravitational field than Earth.<\/p>\n<p>This way of writing out sentences of logic and sentences of English we can call a \u201ctranslation key\u201d. \u00a0We can use this format whenever we want to explain what our sentences mean in English.<\/p>\n<p>Using this key, our argument would be formulated<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong><\/span><span class=\"strong\">\u2192<strong>Q<\/strong>)<\/span><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">______<\/span><\/strong><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong><\/p>\n<p>That short line is not part of our language, but rather is a handy tradition. \u00a0When quickly writing down arguments, we write the premises, and then write the conclusion last, and draw a short line above the conclusion.<\/p>\n<p>This is an argument: \u00a0it is an ordered list of sentences, the first two of which are premises and the last of which is the conclusion.<\/p>\n<p>To make a truth table, we identify all the atomic sentences that constitute these sentences. \u00a0These are <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">Q<\/span><\/strong>. \u00a0There are four possible kinds of ways the world could be that matter to us then:<\/p>\n<table class=\"grid\" style=\"height: 105px; width: 75px;\">\n<tbody>\n<tr class=\"border-bottom\">\n<th class=\"border\" style=\"width: 158.733px;\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">P \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\n<th class=\"border\" style=\"width: 66.7667px;\"><span class=\"strong\">Q<\/span><\/th>\n<th class=\"border\" style=\"width: 67.6167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/th>\n<th class=\"border\" style=\"width: 67.6333px;\" colspan=\"1\" rowspan=\"1\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/th>\n<th class=\"border\" style=\"width: 67.6167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/th>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 159.133px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border\" style=\"width: 67.5667px;\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" style=\"width: 68.4167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border\" style=\"width: 68.4333px;\" colspan=\"1\" rowspan=\"1\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border\" style=\"width: 68.0167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 159.133px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border\" style=\"width: 67.5667px;\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" style=\"width: 68.4167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border\" style=\"width: 68.4333px;\" colspan=\"1\" rowspan=\"1\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border\" style=\"width: 68.0167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 159.133px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border\" style=\"width: 67.5667px;\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" style=\"width: 68.4167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border\" style=\"width: 68.4333px;\" colspan=\"1\" rowspan=\"1\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border\" style=\"width: 68.0167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 159.133px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0\u00a0<\/span><\/td>\n<td class=\"border\" style=\"width: 67.5667px;\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" style=\"width: 68.4167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border\" style=\"width: 68.4333px;\" colspan=\"1\" rowspan=\"1\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border\" style=\"width: 68.0167px;\"><span class=\"space\">\u00a0\u00a0\u00a0<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We\u2019ll write out the sentences, in the order of premises and then conclusion.<\/p>\n<table class=\"grid\" style=\"height: 90px; width: 100px;\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 44.0833px;\"><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 47.75px;\"><\/td>\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong>premise<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong>premise<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 209.1px;\"><strong>conclusion<\/strong><\/td>\n<\/tr>\n<tr class=\"border-bottom\" style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 44.0833px;\"><strong>P<\/strong><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 47.75px;\"><strong>Q<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong>(P\u2192Q)<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong>P<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 209.1px;\"><strong>Q<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 44.0833px;\"><strong><em>T<\/em><\/strong><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 47.75px;\"><strong><em>T<\/em><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong>\u00a0<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong>\u00a0<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 209.1px;\"><strong>\u00a0<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 44.0833px;\"><strong><em>T<\/em><\/strong><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 47.75px;\"><strong><em>F<\/em><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong><em>\u00a0<\/em><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong><em>\u00a0<\/em><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 209.1px;\"><strong><em>\u00a0<\/em><\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 44.0833px;\"><strong><em>F<\/em><\/strong><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 47.75px;\"><strong><em>T<\/em><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong><em>\u00a0<\/em><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><strong><em>\u00a0<\/em><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 209.1px;\"><strong><em>\u00a0<\/em><\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 44.0833px;\"><strong><em>F<\/em><\/strong><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 47.75px;\"><strong><em>F<\/em><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><\/td>\n<td class=\"border\" style=\"height: 15px; width: 160.867px;\"><\/td>\n<td class=\"border\" style=\"height: 15px; width: 209.1px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now we can fill in the columns for each sentence, identifying the truth value of the sentence for that kind of situation.<\/p>\n<table class=\"grid\" style=\"height: 139px; width: 100px;\">\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 33.7667px;\"><\/td>\n<td class=\"border-right\" style=\"width: 36.4833px;\"><\/td>\n<td class=\"border\" style=\"width: 110px;\"><strong>premise<\/strong><\/td>\n<td class=\"border\" style=\"width: 110px;\"><strong>premise<\/strong><\/td>\n<td class=\"border\" style=\"width: 141.317px;\"><strong>conclusion<\/strong><\/td>\n<\/tr>\n<tr class=\"border-bottom\">\n<td class=\"border\" style=\"width: 33.7667px;\"><strong>P<\/strong><\/td>\n<td class=\"border-right\" style=\"width: 36.4833px;\"><strong>Q<\/strong><\/td>\n<td class=\"border\" style=\"width: 110px;\"><strong>(P\u2192Q)<\/strong><\/td>\n<td class=\"border\" style=\"width: 110px;\"><strong>P<\/strong><\/td>\n<td class=\"border\" style=\"width: 141.317px;\"><strong>Q<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 33.7667px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"width: 36.4833px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 110px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 110px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 141.317px;\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 33.7667px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"width: 36.4833px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 110px;\"><em><strong>\u00a0F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 110px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 141.317px;\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 33.7667px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"width: 36.4833px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 110px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 110px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 141.317px;\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 33.7667px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"width: 36.4833px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 110px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 110px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 141.317px;\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We know how to fill in the column for the conditional because we can refer back to the truth table used to define the conditional, to determine what its truth value is when the first part and second part are true; and so on. \u00a0<strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true in those kinds of situations where <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true, and <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is false in those kinds of situations where <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is false. \u00a0And the same is so for <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\n<p>Now, consider all those kinds of ways the world could be such that all the premises are true. \u00a0Only the first row of the truth table is one where all the premises are true. \u00a0Note that the conclusion is true in that row. \u00a0That means, in any kind of situation in which all the premises are true, the conclusion will be true. \u00a0Or, equivalently: necessarily, if all the premises are true, then the conclusion is true.<\/p>\n<table class=\"grid\" style=\"width: 100px;\">\n<tbody>\n<tr>\n<td class=\"border\"><\/td>\n<td class=\"border-right\"><\/td>\n<td class=\"border\"><strong>premise<\/strong><\/td>\n<td class=\"border\"><strong>premise<\/strong><\/td>\n<td class=\"border\"><strong>conclusion<\/strong><\/td>\n<\/tr>\n<tr class=\"border-bottom\">\n<td class=\"border\"><strong>P<\/strong><\/td>\n<td class=\"border-right\"><strong>Q<\/strong><\/td>\n<td class=\"border\"><strong>(P\u2192Q)<\/strong><\/td>\n<td class=\"border\"><strong>P<\/strong><\/td>\n<td class=\"border\"><strong>Q<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Consider in contrast the second argument above, the invalid argument with all true premises and a true conclusion. \u00a0We\u2019ll use the following translation key.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">R<\/span><\/strong>: \u00a0Miami is the capital of Ontario<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">S<\/span><\/strong>: \u00a0Miami is in Canada<\/p>\n<p>And our argument is thus<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>R<\/strong>\u2192<strong>S<\/strong>)<\/span><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>R<\/strong><\/span><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong>_____<\/strong><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>S<\/strong><\/span><\/p>\n<p>Here is the truth table.<\/p>\n<table class=\"grid\" style=\"height: 90px; width: 125px;\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 45.7333px;\"><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 43.5667px;\"><\/td>\n<td class=\"border\" style=\"height: 15px; width: 161.65px;\"><strong>premise<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 161.633px;\"><strong>premise<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 210.083px;\"><strong>conclusion<\/strong><\/td>\n<\/tr>\n<tr class=\"border-bottom\" style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 45.7333px;\"><strong>R<\/strong><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 43.5667px;\"><strong>S<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 161.65px;\"><strong>(R\u2192S)<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 161.633px;\"><strong>\u00acR<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 210.083px;\"><strong>\u00acS<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 45.7333px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 43.5667px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 161.65px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 161.633px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 210.083px;\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 45.7333px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 43.5667px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 161.65px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 161.633px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 210.083px;\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 45.7333px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 43.5667px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 161.65px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 161.633px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 210.083px;\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 45.7333px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 43.5667px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 161.65px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 161.633px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 210.083px;\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Note that there are two kinds of ways that the world could be in which all of our premises are true. \u00a0These correspond to the third and fourth row of the truth table. \u00a0But for the third row of the truth table, the premises are true but the conclusion is false. \u00a0Yes, there is a kind of way the world could be in which all the premises are true and the conclusion is true; that is shown in the fourth row of the truth table. \u00a0But we are not interested in identifying arguments that will have true conclusions if we are lucky. \u00a0We are interested in valid arguments. \u00a0This argument is invalid. \u00a0There is a kind of way the world could be such that all the premises are true and the conclusion is false. \u00a0We can highlight this.<\/p>\n<table class=\"grid\" style=\"width: 125px;\">\n<tbody>\n<tr>\n<td class=\"border\"><\/td>\n<td class=\"border-right\"><\/td>\n<td class=\"border\"><strong>premise<\/strong><\/td>\n<td class=\"border\"><strong>premise<\/strong><\/td>\n<td class=\"border\"><strong>conclusion<\/strong><\/td>\n<\/tr>\n<tr class=\"border-bottom\">\n<td class=\"border\"><strong>R<\/strong><\/td>\n<td class=\"border-right\"><strong>S<\/strong><\/td>\n<td class=\"border\"><strong>(R\u2192S)<\/strong><\/td>\n<td class=\"border\"><strong>\u00acR<\/strong><\/td>\n<td class=\"border\"><strong>\u00acS<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"shaded\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Hopefully it becomes clear why we care about validity. \u00a0Any argument of the form, <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>\u00a0and <strong><span class=\"strong\">P<\/span><\/strong>, therefore <strong><span class=\"strong\">Q<\/span><\/strong>, is valid. \u00a0We do not have to know what <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>mean to determine this. Similarly, any argument of the form, <span class=\"strong\">(<strong>R<\/strong>\u2192<strong>S<\/strong>)<\/span>\u00a0and <span class=\"strong\">\u00ac<strong>R<\/strong><\/span>, therefore <span class=\"strong\">\u00ac<strong>S<\/strong><\/span>, is invalid. \u00a0We do not have to know what <strong><span class=\"strong\">R<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">S<\/span>\u00a0<\/strong>mean to determine this. \u00a0So logic can be of equal use to the astronomer and the financier, the computer scientist or the sociologist.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Check for Understanding<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>How can a truth table be used to examine the validity of an argument?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>3.4 Returning to our historical example<\/h2>\n<p>We described some (not all) of the hypotheses that Semmelweis tested when he tried to identify the cause of childbed fever, so that he could save thousands of women and infants. \u00a0Let us symbolize these and consider his reasoning.<\/p>\n<p>The first case we considered was one where he reasoned: \u00a0If childbed fever is caused by cosmic-atmospheric-terrestrial influences, then the mortality rate would be similar in the First and Second Clinics. \u00a0But the mortality rate was not similar in the First and Second Clinics. \u00a0So, the childbed fever is not caused by cosmic-atmospheric-terrestrial influences.<\/p>\n<p>Here is a key to symbolize the argument.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>T<\/strong>:<\/span>\u00a0 Childbed fever is caused by cosmic-atmospheric-terrestrial influences.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>U<\/strong>: \u00a0<\/span>The mortality rate is similar in the First and Second Clinics.<\/p>\n<p>This would mean the argument is:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>T<\/strong>\u2192<strong>U<\/strong>)<\/span><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>U<\/strong><\/span><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>T<\/strong><\/span><\/p>\n<p>Is this argument valid? \u00a0We can check using a truth table.<\/p>\n<table class=\"grid\" style=\"width: 125px;\">\n<tbody>\n<tr>\n<td class=\"border\"><\/td>\n<td class=\"border-right\"><\/td>\n<td class=\"border\"><strong>premise<\/strong><\/td>\n<td class=\"border\"><strong>premise<\/strong><\/td>\n<td class=\"border\"><strong>conclusion<\/strong><\/td>\n<\/tr>\n<tr class=\"border-bottom\">\n<td class=\"border\"><strong>T<\/strong><\/td>\n<td class=\"border-right\"><strong>U<\/strong><\/td>\n<td class=\"border\"><strong>(T\u2192U)<\/strong><\/td>\n<td class=\"border\"><strong>\u00acU<\/strong><\/td>\n<td class=\"border\"><strong>\u00acT<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The last row is the only row where all the premises are true. \u00a0For this row, the conclusion is true. \u00a0Thus, for all the kinds of ways the world could be in which the premises are true, the conclusion is also true. \u00a0This is a valid argument. \u00a0If we accept his premises, then we should accept that childbed fever was not caused by cosmic-atmospheric-terrestrial influences.<\/p>\n<p>The second argument we considered was the concern that fear caused the higher mortality rates, particularly the fear of the priest coming to deliver last rites. \u00a0Semmelweis reasoned that if the higher rate of childbed fever is caused by fear of death resulting from the priest\u2019s approach, then the rate of childbed fever should decline if people cannot discern when the priest is coming to the Clinic. Here is a key:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>V<\/strong>:<\/span>\u00a0 the higher rate of childbed fever is caused by fear of death resulting from the priest\u2019s approach.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>W<\/strong>:<\/span>\u00a0 the rate of childbed fever will decline if people cannot discern when the priest is coming to the Clinic.<\/p>\n<p>But when Semmelweis had the priest silence his bell, and take a different route, so that patients could not discern that he was coming to the First Clinic, he found no difference in the mortality rate; the First Clinic remained far worse than the second clinic. \u00a0He concluded that the higher rate of childbed fever was not caused by fear of death resulting from the priest\u2019s approach.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>V<\/strong>\u2192<strong>W<\/strong>)<\/span><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>W<\/strong><\/span><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>V<\/strong><\/span><\/p>\n<p>Is this argument valid? \u00a0We can check using a truth table.<\/p>\n<table class=\"grid\" style=\"height: 90px; width: 125px;\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 45.3167px;\"><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 55.6px;\"><\/td>\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><strong>premise<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><strong>premise<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 205.55px;\"><strong>conclusion<\/strong><\/td>\n<\/tr>\n<tr class=\"border-bottom\" style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 45.3167px;\"><strong>V<\/strong><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 55.6px;\"><strong>W<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><strong>(V\u2192W)<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><strong>\u00acW<\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 205.55px;\"><strong>\u00acV<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 45.3167px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 55.6px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 205.55px;\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 45.3167px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 55.6px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 205.55px;\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 45.3167px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 55.6px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 158.1px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\" style=\"height: 15px; width: 205.55px;\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"height: 15px; width: 45.3167px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\" style=\"height: 15px; width: 55.6px;\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"shaded\" style=\"height: 15px; width: 158.1px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"shaded\" style=\"height: 15px; width: 158.1px;\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"shaded\" style=\"height: 15px; width: 205.55px;\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Again, we see that Semmelweis\u2019s reasoning was good. \u00a0He showed that it was not the case that the higher rate of childbed fever was caused by fear of death resulting from the Priest\u2019s approach.<\/p>\n<p>What about Semmelweis\u2019s positive conclusion, that the higher mortality rate was caused by some contaminant from the corpses that doctors had autopsied just before they assisted in a delivery? \u00a0To understand this step in his method, we need to reflect a moment on the scientific method and its relation to logic.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Check for Understanding<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>Explain how argumentative truth tables were utilized to assess the validity of Semmelweis&#8217;s argument?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>3.5 \u00a0Other kinds of arguments 1: \u00a0Scientific reasoning<\/h2>\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What makes a statement deductive versus inductive?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Terms<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li><strong>Deductive Reasoning<\/strong> &#8211; reasoning in which necessarily our conclusions is true if our premises are true<\/li>\n<li><strong>Falsfiable<\/strong> &#8211; an argumentative statement that can use scientific evidence to determine its truth value<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>Valid arguments, and the methods that we are developing, are sometimes called \u201c<strong>deductive reasoning<\/strong>.\u201d \u00a0This is the kind of reasoning in which necessarily our conclusions is true if our premises are true; these arguments can be shown to be good by way of our logical reasoning alone. \u00a0There are other kinds of reasoning, and understanding this may help clarify the relation of logic to other endeavors. \u00a0Two important, and closely related, alternatives to deductive reasoning are scientific reasoning and statistical generalizations. \u00a0We\u2019ll discuss statistical generalizations in the next section.<\/p>\n<p>Scientific method relies upon logic, but science is not reducible to logic: \u00a0scientists do empirical research. \u00a0That is, they examine and test phenomena in the world. \u00a0This is a very important difference from pure logic. \u00a0To understand how this difference results in a distinct method, let us review Semmelweis\u2019s important discovery.<\/p>\n<p>The details and nature of scientific reasoning are somewhat controversial. \u00a0I am going to provide here a basic\u2014many philosophers would say, oversimplified\u2014account of scientific reasoning. \u00a0My goal is to indicate the relation between logic and the kind of reasoning Semmelweis may have used.<\/p>\n<p>As we noted, Semmelweis learned about the death of a colleague, Professor Jakob Kolletschka. \u00a0Kolletschka had been performing an autopsy, and he cut his finger. \u00a0Shortly thereafter, Kolletschka died with symptoms like those of childbed fever. \u00a0Semmelweis reasoned that something on the corpse caused the disease; he called this \u201ccadaveric matter\u201d. \u00a0In the First Clinic, where the mortality rate of women and babies was high, doctors were doing autopsies and then delivering babies immediately after. \u00a0If he could get this cadaveric matter off the hands of the doctors, the rate of childbed fever should fall.<\/p>\n<p>So, he reasoned thus: \u00a0if the fever is caused by cadaveric matter on the hands of the doctors, then the mortality rate will drop when doctors wash their hands with chlorinated water before delivering babies. \u00a0He forced the doctors to do this. \u00a0The result was that the mortality rate dropped a very great deal, at times to below 1%.<\/p>\n<p>Here is a key:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>P<\/strong>:<\/span>\u00a0 The fever is caused by cadaveric matter on the hands of the doctors.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>Q<\/strong>: \u00a0<\/span>The mortality rate will drop when doctors wash their hands with chlorinated water before delivering babies.<\/p>\n<p>And the argument appears to be something like this (as we will see, this isn\u2019t quite the right way to put it, but for now\u2026):<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong><\/p>\n<p>Is this argument valid? \u00a0We can check using a truth table.<\/p>\n<table class=\"grid\" style=\"width: 125px;\">\n<tbody>\n<tr>\n<td class=\"border\"><\/td>\n<td class=\"border-right\"><\/td>\n<td class=\"border\"><strong>premise<\/strong><\/td>\n<td class=\"border\"><strong>premise<\/strong><\/td>\n<td class=\"border\"><strong>conclusion<\/strong><\/td>\n<\/tr>\n<tr class=\"border-bottom\">\n<td class=\"border\"><strong>P<\/strong><\/td>\n<td class=\"border-right\"><strong>Q<\/strong><\/td>\n<td class=\"border\"><strong>(P\u2192Q)<\/strong><\/td>\n<td class=\"border\"><strong>Q<\/strong><\/td>\n<td class=\"border\"><strong>P<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"shaded\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"shaded\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>From this, it looks like Semmelweis has used an invalid argument!<\/p>\n<p>However, an important feature of scientific reasoning must be kept in mind. \u00a0There is some controversy over the details of the scientific method, but the most basic view goes something like this. \u00a0Scientists formulate hypotheses about the possible causes or features of a phenomenon. \u00a0They make predictions based on these hypotheses, and then they perform experiments to test those predictions. \u00a0The reasoning here uses the conditional: \u00a0if the hypotheses is true, then the particular prediction will be true. \u00a0If the experiment shows that the prediction is false, then the scientist rejects the hypothesis.<a class=\"footnote\" title=\"It would be more accurate to say, if the prediction proves false, the scientist must reject either the hypothesis or some other premise of her reasoning. For example, her argument may include the implicit premise that her scientific instruments were operating correctly. She might instead reject this premise that her instruments are working correctly, change one of her instruments, and try again to test the hypothesis. See Duhem (1991). Or, to return to the case of Semmelweis, he might wonder whether he sufficiently established that there were no differences in the atmosphere between the two clinics; or he might wonder whether he sufficiently muffled the Priest\u2019s approach; or whether he recorded his results accurately; and so on. As noted, my account of scientific reasoning here is simplified.\" id=\"return-footnote-35-2\" href=\"#footnote-35-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>\u00a0 But if the prediction proved to be true, then the scientist has shown that the hypothesis may be true\u2014at least, given the information we glean from the conditional and the consequent alone.<\/p>\n<p>This is very important. \u00a0Scientific conclusions are about the physical world, they are not about logic. \u00a0This means that scientific claims are not necessarily true, in the sense of \u201cnecessarily\u201d that we used in our definition of \u201cvalid\u201d. \u00a0Instead, science identifies claims that may be true, or (after some progress) are very likely to be true, or (after very much progress) are true.<\/p>\n<p>Scientists keep testing their hypotheses, using different predictions and experiments. \u00a0Very often, they have several competing hypotheses that have, so far, survived testing. \u00a0To decide between these, they can use a range of criteria. \u00a0In order of their importance, these include: \u00a0choose the hypothesis with the most predictive power (the one that correctly predicts more kinds of phenomena); choose the hypothesis that will be most productive of other scientific theories; choose the hypothesis consistent with your other accepted hypotheses; choose the simplest hypothesis.<\/p>\n<p>What Semmelweis showed was that it could be true that cadaveric matter caused the childbed fever. \u00a0This hypothesis predicted more than any other hypothesis that the doctors had, and so for that reason alone this was the very best hypothesis. \u00a0\u201cBut,\u201d you might reason, \u201cdoesn\u2019t that mean his conclusion was true? \u00a0And don\u2019t we know now, given all that we\u2019ve learned, that his conclusion must be true?\u201d \u00a0No. \u00a0He was far ahead of other doctors, and his deep insights were of great service to all of humankind. \u00a0But the scientific method continued to refine Semmelweis\u2019s ideas. \u00a0For example, later doctors introduced the idea of microorganisms as the cause of childbed fever, and this refined and improved Semmelweis\u2019s insights: \u00a0it was not because the cadaveric matter came from corpses that it caused the disease; it was because the cadaveric matter contained particular micro-organisms that it caused the disease. \u00a0So, further scientific progress showed his hypothesis could be revised and improved.<\/p>\n<p>To review and summarize, with the scientific method:<\/p>\n<ol class=\"lst-kix_list_31-0 start\" start=\"1\">\n<li>We develop a hypothesis about the causes or nature of a phenomenon.<\/li>\n<li>We predict what (hopefully unexpected) effects are a consequence of this hypothesis.<\/li>\n<li>We check with experiments to see if these predictions come true:\n<ol class=\"lst-kix_list_31-0 start\" start=\"1\">\n<li>If the predictions prove false, we reject the hypothesis;<a class=\"footnote\" title=\"Or, as noted in note 6, we reject some other premise of the argument.\" id=\"return-footnote-35-3\" href=\"#footnote-35-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a><\/li>\n<li>If the predictions prove true, we conclude that the hypothesis could be true. \u00a0We continue to test the hypothesis by making other predictions (that is, we return to step 2).<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>This means that a hypothesis that does not make testable predictions (that is, a hypothesis that cannot possibly be proven false) is not a scientific hypothesis. \u00a0Such a hypothesis is called \u201cunfalsifiable\u201d and we reject it as unscientific.<\/p>\n<p>This method can result in more than one hypothesis being shown to be possibly true. \u00a0Then, we chose between competing hypotheses by using criteria like the following (here ordered by their relative importance; \u201ctheory\u201d can be taken to mean a collection of one or more hypotheses):<\/p>\n<ol class=\"lst-kix_list_30-0 start\" start=\"1\">\n<li>Predictive power: the more that a hypothesis can successfully predict, the better it is.<\/li>\n<li>Productivity: \u00a0a hypothesis that suggests more new directions for research is to be preferred.<\/li>\n<li>Coherence with Existing Theory: if two hypotheses predict the same amount and are equally productive, then the hypothesis that coheres with (does not contradict) other successful theories is preferable to one that does contradict them.<\/li>\n<li>Simplicity: if two hypotheses are equally predictive, productive, and coherent with existing theories, then the simpler hypothesis is preferable.<\/li>\n<\/ol>\n<p>Out of respect to Ignaz Semmelweis we should tell the rest of his story, although it means we must end on a sad note. \u00a0Semmelweis\u2019s great accomplishment was not respected by his colleagues, who resented being told that their lack of hygiene was causing deaths. \u00a0He lost his position at the First Clinic, and his successors stopped the program of washing hands in chlorinated water. \u00a0The mortality rate leapt back to its catastrophically high levels. \u00a0Countless women and children died. \u00a0Semmelweis continued to promote his ideas, and this caused growing resentment. \u00a0Eventually, several doctors in Vienna\u2014not one of them a psychiatrist\u2014secretly signed papers declaring Semmelweis insane. \u00a0We do not know whether Semmelweis was mentally ill at this time. \u00a0These doctors took him to an asylum on the pretense of having him visit in his capacity as a doctor; when he arrived, the guards seized Semmelweis. \u00a0He struggled, and the guards at the asylum beat him severely, put him in a straightjacket, and left him alone in a locked room. \u00a0Neglected in isolation, the wounds from his beating became infected and he died a week later.<\/p>\n<p>It was years before Semmelweis\u2019s views became widely accepted and his accomplishment properly recognized. \u00a0His life teaches many lessons, including unfortunately that even the most educated among us can be evil, petty, and willfully ignorant. \u00a0Let us repay Semmelweis, as those in his own time did not, by remembering and praising his scientific acumen and humanity.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Check for Understanding<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What role does logic play in natural sciences?<\/li>\n<li><strong>Critical Thinking Task:<\/strong>\u00a0Can you explain the important difference between pure logic and empirical science? How does this difference apply to your life?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>3.6 Other kinds of arguments 2: \u00a0Statistical reasoning<\/h2>\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What are common functions of statistics in logical reasoning?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Terms<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li><strong>Population<\/strong> &#8211; all the events or all the things we want to make a generalization about.<\/li>\n<li><strong>Sample<\/strong> &#8211; a portion of a population that is representative of the entire population?<\/li>\n<li><strong>Random Sample<\/strong> &#8211; every member of the population was equally likely to be in the sample.<\/li>\n<li><strong>Inductive Reasoning<\/strong> &#8211; method of testing claims about the world, it requires observations, and its conclusion is likely instead of being certain.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>Here we can say a few words about statistical generalizations\u2014our goal being only to provide a contrast with deductive reasoning.<\/p>\n<p>In one kind of statistical generalization, we have a <strong>population<\/strong> of some kind that we want to make general claims about. \u00a0A population could be objects or events. \u00a0So, a population can be a group of organisms, or a group of weather events. \u00a0\u201cPopulation\u201d just means all the events or all the things we want to make a generalization about. \u00a0Often however it is impossible to examine every object or event in the population, so what we do is gather a <strong>sample<\/strong>. \u00a0A sample is some portion of the population. \u00a0Our hope is that the sample is representative of the population: \u00a0that whatever traits are shared by the members of the sample are also shared by the members of the population.<\/p>\n<p>For a sample to representative, it must be <strong>random<\/strong> and large enough. \u00a0\u201cRandom\u201d in this context means that the sample was not chosen in any way that might distinguish members of the sample from the population, other than being members of the population. \u00a0In other words, every member of the population was equally likely to be in the sample. \u00a0\u201cLarge enough\u201d is harder to define. \u00a0Statisticians have formal models describing this, but suffice to say we should not generalize about a whole population using just a few members.<\/p>\n<p>Here\u2019s an example. \u00a0We wonder if all domestic dogs are descended from wolves. \u00a0Suppose we have some genetic test to identify if an organism was a descendent of wolves. \u00a0We cannot give the test to all domestic dogs\u2014this would be impractical and costly and unnecessary. \u00a0We pick a random sample of domestic dogs that is large enough, and we test them. \u00a0For the sample to be random, we need to select it without allowing any bias to influence our selection; all that should matter is that these are domestic dogs, and each member of the population must have an equal chance of being in the sample. \u00a0Consider the alternative: \u00a0if we just tested one family of dogs\u2014say, dogs that are large\u2014we might end up selecting dogs that differed from others in a way that matters to our test. \u00a0For example, maybe large dogs are descended from wolves, but small dogs are not. \u00a0Other kinds of bias can creep in less obviously. \u00a0We might just sample dogs in our local community, and it might just be that people in our community prefer large dogs, and again we would have a sample bias. \u00a0So, we randomly select dogs, and give them the genetic test.<\/p>\n<p>Suppose the results were positive. \u00a0We reason that if all the members of the randomly selected and large enough sample (the tested dogs) have the trait, then it is very likely that all the members of the population (all dogs) have the trait. \u00a0Thus: we could say that it appears very likely that all dogs have the trait. \u00a0(This likelihood can be estimated, so that we can also sometimes say how likely it is that all members of the population have the trait.)<\/p>\n<p>This kind of reasoning obviously differs from a deductive argument very substantially. \u00a0It is a method of testing claims about the world, it requires observations, and its conclusion is likely instead of being certain.<\/p>\n<p>But such reasoning is not unrelated to logic. \u00a0Deductive reasoning is the foundation of these and all other forms of reasoning. \u00a0If one must reason using statistics in this way, one relies upon deductive methods always at some point in one\u2019s arguments. \u00a0There was a conditional at the penultimate step of our reasoning, for example (we said \u201cif all the members of the randomly selected and large enough sample have the trait, then it is very likely that all the members of the population have the trait\u201d). \u00a0Furthermore, the foundations of these methods (the most fundamental descriptions of what these methods are) are given using logic and mathematics. \u00a0Logic, therefore, can be seen as the study of the most fundamental form of reasoning, which will be used in turn by all other forms of reasoning, including scientific and statistical reasoning.\\<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Check for Understanding<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li><strong>Self Reflection:<\/strong>What are instances in your day-to-day life you use statistical reasoning to make valid arguments about your decisions?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>3.7 \u00a0Problems<\/h2>\n<ol class=\"lst-kix_list_41-0 start\" start=\"1\">\n<li>Make truth tables to show that the following arguments are valid. \u00a0Circle or highlight the rows of the truth table that show the argument is valid (that is, all the rows where all the premises are true). \u00a0Note that you will need eight rows in the truth table for problems d-f, and sixteen rows in the truth table for problems g and h.\n<ol class=\"lst-kix_list_41-0 start\" start=\"1\">\n<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">\u00acQ<\/span>. Conclusion: \u00a0<span class=\"strong\">\u00acP<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">\u00acP<\/span>. Conclusion: <span class=\"strong\">(P\u2192Q)<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">Q<\/span>. Conclusion: <span class=\"strong\">(P\u2192Q)<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>. Conclusion: \u00a0<span class=\"strong\">(P\u2192R)<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">(P<\/span><span class=\"strong\">\u2192<\/span><span class=\"strong\">Q)<\/span>, <span class=\"strong\">(Q<\/span><span class=\"strong\">\u2192<\/span><span class=\"strong\">R)<\/span>, <span class=\"strong\">P<\/span>. Conclusion: \u00a0<span class=\"strong\">R<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">(P<\/span><span class=\"strong\">\u2192<\/span><span class=\"strong\">Q)<\/span>, <span class=\"strong\">(Q<\/span><span class=\"strong\">\u2192<\/span><span class=\"strong\">R)<\/span>, <span class=\"strong\">\u00ac<\/span><span class=\"strong\">R<\/span>. Conclusion: \u00a0<span class=\"strong\">\u00ac<\/span><span class=\"strong\">P<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>, <span class=\"strong\">(R\u2192S)<\/span>, <span class=\"strong\">P<\/span>. Conclusion: \u00a0<span class=\"strong\">S<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>, <span class=\"strong\">(R\u2192S)<\/span>. Conclusion: \u00a0<span class=\"strong\">(P\u2192S)<\/span>.<\/li>\n<\/ol>\n<\/li>\n<li>Make truth tables to show the following arguments are invalid. Circle or highlight the rows of the truth table that show the argument is invalid (that is, any row where all the premises are true but the conclusion is false).\n<ol class=\"lst-kix_list_41-0 start\" start=\"1\">\n<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>. Conclusion: \u00a0<span class=\"strong\">P<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>. Conclusion: \u00a0<span class=\"strong\">Q<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">P<\/span>. Conclusion: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">Q<\/span>. Conclusion: \u00a0<span class=\"strong\">P<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">\u00acQ<\/span>. Conclusion: <span class=\"strong\">(P\u2192Q)<\/span>.<\/li>\n<li>Premises:\u00a0<span class=\"strong\">(P\u2192Q)<\/span>. Conclusion: <span class=\"strong\">(Q\u2192P)<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>, <span class=\"strong\">\u00acP<\/span>. Conclusion: \u00a0<span class=\"strong\">\u00acR<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>, <span class=\"strong\">R<\/span>. Conclusion: \u00a0<span class=\"strong\">P<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>. Conclusion: \u00a0<span class=\"strong\">(R\u2192P)<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">(P\u2192Q)<\/span>, <span class=\"strong\">(Q\u2192R)<\/span>, <span class=\"strong\">(R\u2192S)<\/span>. Conclusion: \u00a0<span class=\"strong\">(S\u2192P)<\/span>.<\/li>\n<\/ol>\n<\/li>\n<li>In normal colloquial English, write your own valid argument with at least two premises. Your argument should just be a paragraph (not an ordered list of sentences or anything else that looks like logic). \u00a0Translate it into propositional logic and use a truth table to show it is valid.<\/li>\n<li>In normal colloquial English, write your own invalid argument with at least two premises. \u00a0Translate it into propositional logic and use a truth table to show it is invalid.<\/li>\n<li>For each of the following, state whether the argument described could be: valid, invalid, sound, unsound.\n<ol class=\"lst-kix_list_41-0 start\" start=\"1\">\n<li>An argument with false premises and a false conclusion.<\/li>\n<li>An argument with true premises and a false conclusion.<\/li>\n<li>An argument with false premises and a true conclusion.<\/li>\n<li>An argument with true premises and a true conclusion.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div>\n<hr \/>\n<p><a id=\"ftnt5\" href=\"#ftnt_ref5\">[5]<\/a>\u00a0All the data cited here comes from Carter (1983) and additional biographical information comes from\u00a0Carter and Carter (2008). \u00a0These books are highly recommended to anyone interested in the history of science or medicine.<\/p>\n<\/div>\n<div>\n<p><a id=\"ftnt6\" href=\"#ftnt_ref6\">[6]<\/a>\u00a0It would be more accurate to say, if the prediction proves false, the scientist must reject either the hypothesis or some other premise of her reasoning. \u00a0For example, her argument may include the implicit premise that her scientific instruments were operating correctly. \u00a0She might instead reject this premise that her instruments are working correctly, change one of her instruments, and try again to test the hypothesis. \u00a0See Duhem (1991). \u00a0Or, to return to the case of Semmelweis, he might wonder whether he sufficiently established that there were no differences in the atmosphere between the two clinics; or he might wonder whether he sufficiently muffled the Priest\u2019s approach; or whether he recorded his results accurately; and so on. \u00a0As noted, my account of scientific reasoning here is simplified.<\/p>\n<\/div>\n<div>\n<p><a id=\"ftnt7\" href=\"#ftnt_ref7\">[7]<\/a>\u00a0Or, as noted in note 6, we reject some other premise of the argument.<\/p>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Check for Understanding<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>Explain the distinction between a population and a sample in propositional reasoning?<\/li>\n<li>What role does random sampling serve in statistical reasoning?<\/li>\n<li>What is the relationship between deductive and inductive reasoning in statistics?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-35-1\">All the data cited here comes from Carter (1983) and additional biographical information comes from Carter and Carter (2008). These books are highly recommended to anyone interested in the history of science or medicine. <a href=\"#return-footnote-35-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-35-2\">It would be more accurate to say, if the prediction proves false, the scientist must reject either the hypothesis or some other premise of her reasoning. For example, her argument may include the implicit premise that her scientific instruments were operating correctly. She might instead reject this premise that her instruments are working correctly, change one of her instruments, and try again to test the hypothesis. See Duhem (1991). Or, to return to the case of Semmelweis, he might wonder whether he sufficiently established that there were no differences in the atmosphere between the two clinics; or he might wonder whether he sufficiently muffled the Priest\u2019s approach; or whether he recorded his results accurately; and so on. As noted, my account of scientific reasoning here is simplified. <a href=\"#return-footnote-35-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-35-3\">Or, as noted in note 6, we reject some other premise of the argument. <a href=\"#return-footnote-35-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":158,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-35","chapter","type-chapter","status-publish","hentry","license-cc-by-nc-sa"],"part":27,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/35","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":15,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/35\/revisions"}],"predecessor-version":[{"id":336,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/35\/revisions\/336"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/parts\/27"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/35\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/media?parent=35"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapter-type?post=35"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/contributor?post=35"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/license?post=35"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}