Responses to the following practice exercises from Chapter 1 here: pp. 24-26 Parts A, B.

For All X The Lorain County Remix P.D. Magnus University at Albany, State University of New York and Cathal Woods Virginia Wesleyan University and J. Robert Loftis Lorain County Community College Licensed under a Creative Commons license. (Attribution-NonComercial-ShareAlike 4.0 International ) https://creativecommons.org/licenses/by-nc-sa/4.0/ This is version 0.1 of An Open Introduction to Logic. It is current as of December 22, 2017. © 2005–2017 by Cathal Woods, P.D. Magnus, and J. Robert Loftis. Some rights reserved. Licensed under a Creative Commons license. (Attribution-NonComercial-ShareAlike 4.0 International ) https://creativecommons.org/licenses/by-nc-sa/4.0/ This book incorporates material from An Introduction to Reasoning by Cathal Woods, available at sites.google.com/site/anintroductiontoreasoning/ and For All X by P.D. Magnus (version 1.27 [090604]), available at www.fecundity.com/logic. Introduction to Reasoning © 2007–2014 by Cathal Woods. Some rights reserved. Licensed under a Creative Commons license: Attribution-NonCommercial-ShareAlike 3.0 Unported. http://creativecommons.org/licenses/by-nc-sa/3.0/ For All X © 2005–2010 by P.D. Magnus. Some rights reserved. Licensed under a Creative Commons license: Attribution ShareAlike http://creativecommons.org/licenses/by-sa/3.0/ J. Robert Loftis compiled this edition and wrote original material for it. He takes full responsibility for any mistakes remaining in this version of the text. Typesetting was carried out entirely in LATEX2ε. The style for typesetting proofs is based on fitch.sty (v0.4) by Peter Selinger, University of Ottawa. “When you come to any passage you don’t understand, read it again: if you still don’t understand it, read it again: if you fail, even after three readings, very likely your brain is getting a little tired. In that case, put the book away, and take to other occupations, and next day, when you come to it fresh, you will very likely find that it is quite easy.” – Charles Dodgson (Lewis Carroll) Symbolic Logic (1896) iii Contents About this Book ix Acknowledgments xi Part I Basic Concepts 1 What Is Logic? 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Statement, Argument, Premise, Conclusion . . . . . . . . . . . . . . . . . . . . . . . Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Arguments and Nonarguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Arguments and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 6 13 15 18 19 24 2 The Basics of Evaluating Argument 29 2.1 Two Ways an Argument Can Go Wrong . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Valid, Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3 Strong, Cogent, Deductive, Inductive . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3 What is Formal Logic? 49 3.1 Formal as in Concerned with the Form of Things . . . . . . . . . . . . . . . . . . . . 49 3.2 Formal as in Strictly Following Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3 More Logical Notions for Formal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Part II Categorical Logic 4 Categorical Statements 65 4.1 Quantified Categorical Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 Quantity, Quality, Distribution, and Venn Diagrams . . . . . . . . . . . . . . . . . . 68 iv 4.3 4.4 4.5 4.6 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Transforming English into Logically Structured English . . . . . . . . . . . . . . . . 75 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Conversion, Obversion, and Contraposition . . . . . . . . . . . . . . . . . . . . . . . 83 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 The Traditional Square of Opposition . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Existential Import and the Modern Square of Opposition . . . . . . . . . . . . . . . 105 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5 Categorical Syllogisms 113 5.1 Standard Form, Mood, and Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.2 Testing Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.3 Existential Import and Conditionally Valid Forms . . . . . . . . . . . . . . . . . . . 132 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.4 Rules and Fallacies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Part III Sentential Logic 6 Sentential Logic 151 6.1 Sentence Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.2 Sentential Connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.3 More Complicated Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.4 Recursive Syntax for SL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7 Truth Tables 177 7.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.2 Complete Truth Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.3 Using Truth Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.4 Partial Truth Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.5 Expressive Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8 Proofs in Sentential Logic 199 8.1 Substitution Instances and Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 8.2 Basic Rules for Sentential Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 8.3 Conditional Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 v 8.4 8.5 8.6 8.7 8.8 8.9 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Indirect Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Tautologies and Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Derived Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Rules of Replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Proof Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Soundness and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 Part IV Quantificational Logic 9 Quantified Logic 251 9.1 From Sentences to Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 9.2 Building Blocks of Quantified Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 9.3 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.4 Translating to Quantified Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 9.5 Recursive Syntax for QL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 9.6 Tricky Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 9.7 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 10 Semantics for Quantified Logic 285 10.1 Creating models in Quantified Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 10.2 Working with Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 11 Proofs in Quantified Logic 297 11.1 Rules for Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 11.2 Rules for Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 A Other Symbolic Notation 309 B Bibliography 313 vi C Glossary 317 D Quick Reference 327 vii viii About this Book This book was created by combining two previous books on logic and critical thinking, both made available under a Creative Commons license, and then adding some material so that the coverage matched that of commonly used logic textbooks. P.D. Magnus’ For All X (2008) formed the basis of Part B: Formal Logic. I began using For All X in my own logic classes in 2009, but I quickly realized I needed to make changes to make it appropriate for the community college students I was teaching. In 2010 I began developing For All X : The Lorain County Remix and using it in my classes. The main change I made was to separate the discussions of sentential and quantificational logic and to add exercises. It is this remixed version that became the basis for Part B: Formal Logic complete version of this text. Similarly, Part ??: Critical Thinking grew out of Cathal Woods’ Introduction to Reasoning. In the Spring of 2011, I began to use an early version of this text (Woods and Roche 2011) in my critical thinking courses. I kept up with the updates and changes to the text until the release of Woods 2014, all the while gradually merging the material with the work in For All X. After that point, my version forks from Woods’s. On May 20, 2016, I posted the combined textbook to Github and all subsequent changes have been tracked there: https://github.com/rob-helpy-chalk/openintroduction J. Robert Loftis Elyria, Ohio, USA ix x Acknowledgments Thanks first of all go to the authors of the textbooks here stitched together: P.D. Magnus for For All X and Cathal Woods for Introduction to Reasoning. My thanks go to them for writing the excellent textbooks that have been incorporated into this one, for making those publicly available under Creative Commons licenses, and for giving their blessing to this derivative work. In general, this book would not be possible without a culture of sharing knowledge. The book was typeset using LATEX2ε developed by Leslie Lamport. Lamport was building on TEXby Donald Knuth. Peter Selinger built on what Lamport made by developing the Fitch typesetting format that the proofs were laid out in. Diagrams were made in PikZ by Till Tantu. All of these are coding systems are not only freely available online, they have extensive user support communities. Add-on packages are designed, manuals are written, questions are answered in discussion forums, all by people who are donating their time and expertise. The culture of sharing isn’t just responsible for the typesetting of this book; it was essential to the content. Essential background information comes from the free online Stanford Encyclopedia of Philosophy. Primary sources from the history of logic came from Project Gutenberg. Logicians, too, can and should create free knowledge. Many early adopters of this text provided invaluable feedback, including Jeremy Dolan, Terry Winant, Benjamin Lennertz, Ben Sheredos, and Michael Hartsock. Lennertz, in particular, provided useful edits. Helpful comments were also made by Ben Cordry, John Emerson, Andrew Mills, Nathan Smith, Vera Tobin, Cathal Woods, and many more that I have forgot to mention, but whose emails are probably sitting on my computer somewhere. I would also like to thank Lorain County Community College for providing the sabbatical leave that allowed me to write the sections of this book on Aristotelian logic. Special thanks goes to all the students at LCCC who had to suffer through earlier versions of this work and provided much helpful feedback. Most importantly, I would like to thank Molly, Caroline and Joey for their incredible love and support. J. Robert Loftis Elyria, Ohio, USA xi Intellectual debts too great to articulate are owed to scholars too many to enumerate. At different points in the work, readers might detect the influence of various works of Aristotle, Toulmin (especially 1958), Fisher and Scriven (1997), Walton (especially 1996), Epstein (2002), Johnson-Laird (especially 2006), Scriven (1962), Giere (1997) and the works of the Amsterdam school of pragma-dialectics (Eemeren, Grootendorst, and Henkemans 2002). Thanks are due to Virginia Wesleyan College for providing me with Summer Faculty Development funding in 2008 and 2010 and a Batten professorship in 2011. These funds, along with some undergraduate research funds (also provided by VWC), allowed me to hire students Gaby Alexander (2008), Ksera Dyette (2009), Mark Jones (2008), Andrea Medrano (2011), Lauren Perry (2009), and Alan Robertson (2010). My thanks to all of them for their hard work and enthusiasm. For feedback on the text, thanks are due to James Robert (Rob) Loftis (Lorain County Community College) and Bill Roche (Texas Christian University). Answers (to exercises) marked with “(JRL)” are by James Robert Loftis. Particular thanks are due to my (once) Ohio State colleague Bill Roche. The book began as a collection of lecture notes, combining work by myself and Bill. Cathal Woods Norfolk, Virginia, USA (Taken from Introduction to Reasoning (2014)) The author would like to thank the people who made this project possible. Notable among these are Cristyn Magnus, who read many early drafts; Aaron Schiller, who was an early adopter and provided considerable, helpful feedback; and Bin Kang, Craig Erb, Nathan Carter, Wes McMichael, and the students of Introduction to Logic, who detected various errors in previous versions of the book. P.D. Magnus Albany, New York, USA (Taken from For All X (2008)) Part I Basic Concepts Chapter 1 What Is Logic? 1.1 Introduction Logic is a part of the study of human reason, the ability we have to think abstractly, solve problems, explain the things that we know, and infer new knowledge on the basis of evidence. Traditionally, logic has focused on the last of these items, the ability to make inferences on the basis of evidence. This is an activity you engage in every day. Consider, for instance, the game of Clue. (For those of you who have never played, Clue is a murder mystery game where players have to decide who committed the murder, what weapon they used, and where they were.) A player in the game might decide that the murder weapon was the candlestick by ruling out the other weapons in the game: the knife, the revolver, the rope, the lead pipe, and the wrench. This evidence lets the player know something they did not know previously, namely, the identity of the murderer. In logic, we use the word “argument” to refer to the attempt to show that certain evidence supports a conclusion. This is very different from the sort of argument you might have when you are mad at someone, which could involve screaming and throwing things. We are going to use the word “argument” a lot in this book, so you need to get used to thinking of it as a name for a rational process, and not a word that describes what happens when people disagree. A logical argument is structured to give someone a reason to believe some conclusion. Here is the argument about a game of Clue written out in a way that shows its structure. P1 : In a game of Clue, the possible murder weapons are the knife, the candlestick, the revolver, the rope, the lead pipe, and the wrench. P2 : The murder weapon was not the knife. P3 : The murder weapon was also not the revolver, the rope, the lead pipe, or the wrench. C: Therefore, the murder weapon was the candlestick. 3 4 CHAPTER 1. WHAT IS LOGIC? In the argument above, statements P1 –P3 are the evidence. We call these the premises. The word “therefore” indicates that the final statement, marked with a C, is the conclusion of the argument. If you believe the premises, then the argument provides you with a reason to believe the conclusion. You might use reasoning like this purely in your own head, without talking with anyone else. You might wonder what the murder weapon is, and then mentally rule out each item, leaving only the candlestick. On the other hand, you might use reasoning like this while talking to someone else, to convince them that the murder weapon is the candlestick. (Perhaps you are playing as a team.) Either way the structure of the reasoning is the same. We can define Logic then more precisely as the part of the study of reasoning that focuses on argument. In more casual situations, we will follow ordinary practice and use the word “logic” to either refer to the business of studying human reason or the thing being studied, that is, human reasoning itself. While logic focuses on argument, other disciplines, like decision theory and cognitive science, deal with other aspects of human reasoning, like abstract thinking and problem solving more generally. Logic, as the study of argument, has been pursued for thousands of years by people from civilizations all over the globe. The initial motivation for studying logic is generally practical. Given that we use arguments and make inferences all the time, it only makes sense that we would want to learn to do these things better. Once people begin to study logic, however, they quickly realize that it is a fascinating topic in its own right. Thus the study of logic quickly moves from being a practical business to a theoretical endeavor people pursue for its own sake. In order to study reasoning, we have to apply our ability to reason to our reason itself. This reasoning about reasoning is called metareasoning. It is part of a more general set of processes called metacognition, which is just any kind of thinking about thinking. When we are pursing logic as a practical discipline, one important part of metacognition will be awareness of your own thinking, especially its weakness and biases, as it is occurring. More theoretical metacognition will be about attempting to understand the structure of thought itself. Whether we are pursuing logical for practical or theoretical reasons, our focus is on argument. The key to studying argument is to set aside the subject being argued about and to focus on the way it is argued for. The section opened with an example that was about a game of Clue. However, the kind of reasoning used in that example was just the process of elimination. Process of elimination can be applied to any subject. Suppose a group of friends is deciding which restaurant to eat at, and there are six restaurants in town. If you could rule out five of the possibilities, you would use an argument just like the one above to decide where to eat. Because logic sets aside what an argument is about, and just looks at how it works rationally, logic is said to have content neutrality. If we say an argument is good, then the same kind of argument applied to a different topic will also be good. If we say an argument is good for solving murders, we will also say that the same kind of argument is good for deciding where to eat, what kind of disease is destroying your crops, or who to vote for. When logic is studied for theoretical reasons, it typically is pursued as formal logic. In formal logic we get content neutrality by replacing parts of the argument we are studying with abstract symbols. For instance, we could turn the argument above into a formal argument like this: 1.1. INTRODUCTION 5 P1 : There are six possibilities: A, B, C, D, E, and F. P2 : A is false. P3 : B, D, E, and F are also false. C: .˙. The correct answer is C. Here we have replaced the concrete possibilities in the first argument with abstract letters that could stand for anything. We have also replaced the English word “therefore” with the symbol “.˙.,” which means therefore. This lets us see the formal structure of the argument, which is why it works in any domain you can think of. In fact, we can think of formal logic as the method for studying argument that uses abstract notation to identify the formal structure of argument. Formal logic is closely allied with mathematics, and studying formal logic often has the sort of puzzle-solving character one associates with mathematics. You will see this when we get to Part B, which covers formal logic. When logic is studied for practical reasons, it is typically called critical thinking. We will define critical thinking narrowly as the use of metareasoning to improve our reasoning in practical situations. Sometimes we will use the term “critical thinking” more broadly to refer to the results of this effort at self-improvement. You are “thinking critically” when you reason in a way that has been sharpened by reflection and metareasoning. A critical thinker someone who has both sharpened their reasoning abilities using metareasoning and deploys those sharpened abilities in real world situations. Critical thinking is generally pursued as informal logic, rather than formal logic. This means that we will keep arguments in ordinary language and draw extensively on your knowledge of the world to evaluate them. In contrast to the clarity and rigor of formal logic, informal logic is suffused with ambiguity and vagueness. There are problems with multiple correct answers, or where reasonable people can disagree with what the correct answer is. This is because you will be dealing with reasoning in the real world, which is messy. You can think of the difference between formal logic and informal logic as the difference between a laboratory science and a field science. If you are studying, say, mice, you could discover things about them by running experiments in a lab, or you can go out into the field where mice live and observe them in their natural habitat. Informal logic is the field science for arguments: you go out and study arguments in their natural habitats, like newspapers, courtrooms, and scientific journal articles. Like studying mice scurrying around a meadow, the process takes patience, and often doesn’t yield clear answers but it lets you see how things work in the real world. Formal logic takes arguments out of their natural habitat and performs experiments on them to see what they are capable of. The arguments here are like lab mice. They are pumped full of chemicals and asked to perform strange tasks, as it were. They live lives very different than their wild cousins. Some of the arguments will wind up looking like the “ob/ob mouse”, a genetically engineered obese mouse scientists use to study type II diabetes (See Figure 1.1). These arguments will be huge, awkward, and completely unable to survive in the wild. But they will tell us a lot about the limits of logic as a process. Our main goal in studying arguments is to separate the good ones from the bad ones. The argument about Clue we saw earlier is a good one, based on the process of elimination. It is good 6 CHAPTER 1. WHAT IS LOGIC? Figure 1.1: The ob/ob mouse (left), a laboratory mouse which has been genetically engineered to be obese, and an ordinary mouse (right). Formal logic, which takes arguments out of their natural environment, often winds up studying arguments that look like the ob/ob mouse. They are huge, awkward, and unable to survive in the wild, but they tell us a lot about the limits of logic as a process. Photo from Wikimedia Commons 2006. because it leads to truth. If I’ve got all the premises right, the conclusion will also be right. The textbook Logic: Techniques of Formal Reasoning (Kalish, Montague, and Mar 1980) had a nice way of capturing the meaning of logic: “logic is the study of virtue in argument.” This textbook will accept this definition, with the caveat that an argument is virtuous if it helps us get to the truth. Logic is different from rhetoric, which is the study of effective persuasion. Rhetoric does not look at virtue in argument. It only looks at the power of arguments, regardless of whether they lead to truth. An advertisement might convince you to buy a new truck by having a gravelly voiced announcer tell you it is “ram tough” and showing you a picture of the truck on top of a mountain, where it no doubt actually had to be airlifted. This sort of persuasion is often more effective at getting people to believe things than logical argument, but it has nothing to do with whether the truck is really the right thing to buy. In this textbook we will only be interested in rhetoric to the extent that we need to learn to defend ourselves against the misleading rhetoric of others. This will not, however, be anything close to a full treatment of the study of rhetoric. 1.2 Statement, Argument, Premise, Conclusion So far we have defined logic as the study of argument and outlined its relationship to related fields. To go any further, we are going to need a more precise definition of what exactly an argument is. We have said that an argument is not simply two people disagreeing; it is an attempt to prove something using evidence. More specifically, an argument is composed of 1.2. STATEMENT, ARGUMENT, PREMISE, CONCLUSION 7 statements. In logic, we define a statement as a unit of language that can be true or false. To put it another way, it is some combination of words or symbols that have been put together in a way that lets someone agree or disagree with it. All of the items below are statements. (a) Tyrannosaurus rex went extinct 65 million years ago. (b) Tyrannosaurus rex went extinct last week. (c) On this exact spot, 100 million years ago, a T. rex laid a clutch of eggs. (d) George W. Bush is the king of Jupiter. (e) Murder is wrong. (f) Abortion is murder. (g) Abortion is a woman’s right. (h) Lady Gaga is pretty. (i) Murder is the unjustified killing of a person. (j) The slithy toves did gyre and gimble in the wabe. (k) The murder of logician Richard Montague was never solved. Because a statement is something that can be true or false, statements include truths like (a) and falsehoods like (b). A statement can also be something that that must either be true or false, but we don’t know which, like (c). A statement can be something that is completely silly, like (d). Statements in logic include statements about morality, like (e), and things that in other contexts might be called “opinions,” like (f) and (g). People disagree strongly about whether (f) or (g) are true, but it is definitely possible for one of them to be true. The same is true about (h), although it is a less important issue than (f) and (g). A statement in logic can also simply give a definition, like (i). This sort of statement announces that we plan to use words a certain way, which is different from statements that describe the world, like (a), or statements about morality, like (f). Statements can include nonsense words like (j), because we don’t really need to know what the statement is about to see that it is the sort of thing that can be true or false. All of this relates back to the content neutrality of logic. The statements we study can be about dinosaurs, abortion, Lady Gaga, and even the history of logic itself, as in statement (k), which is true. We are treating statements primarily as units of language or strings of symbols, and most of the time the statements you will be working with will just be words printed on a page. However, it is important to remember that statements are also what philosophers call “speech acts.” They are actions people take when they speak (or write). If someone makes a statement they are typically telling other people that they believe the statement to be true, and will back it up with evidence if asked to. When people make statements, they always do it in a context—they make statements at a place and a time with an audience. Often the context statements are made in will be important for us, so when we give examples, statements, or arguments we will sometimes include a description of the context. When we do that, we will give the context in italics. See Figure 1.2 8 CHAPTER 1. WHAT IS LOGIC? Figure 1.2: A statement in different contexts, or no context. for examples. For the most part, the context for a statement or argument will be important in the chapters on critical thinking, when we are pursing the study of logic for practical reasons. In the chapters on formal logic, context is less important, and we will be more likely to skip it. “Statements’ in this text does not include questions, commands, exclamations, or sentence fragments. Someone who asks a question like “Does the grass need to be mowed?” is typically not claiming that anything is true or false. Generally, questions will not count as statements, but answers will. “What is this course about?” is not a statement. “No one knows what this course is about,” is a statement. For the same reason commands do not count as statements for us. If someone bellows “Mow the grass, now!” they are not saying whether the grass has been mowed or not. You might infer that they believe the lawn has not been mowed, but then again maybe they think the lawn is fine and just want to see you exercise. An exclamation like “Ouch!” is also neither true nor false. On its own, it is not a statement. We will treat “Ouch, I hurt my toe!” as meaning the same thing as “I hurt my toe.” The “ouch” does not add anything that could be true or false. Finally, a lot of possible strings of words will fail to qualify as statements simply because they don’t form a complete sentence. In your composition classes, these were probably referred to as sentence fragments. This includes strings of words that are parts of sentences, such as noun phrases like “The tall man with the hat” and verb phrases, like “ran down the hall.” Phrases like these are missing something they need to make a claim about the world. The class of sentence fragments also includes completely random combinations of words, like “The up if blender route,” which don’t even have the form of a statement about the world. Other logic textbooks describe the components of argument as “propositions,” or “assertions,” and we will use these terms sometimes as well. There is actually a great deal of disagreement about what the differences between all of these things are and which term is best used to describe parts of arguments. However, none of that makes a difference for this textbook. We could have used any of the other terms in this text, and it wouldn’t change anything. Some textbooks will also use the term “sentence” here. We will not use the word “sentence” to mean the same thing as “statement.” Instead, we will use “sentence” the way it is used in ordinary grammar, to refer generally to statements, questions, and commands. 1.2. STATEMENT, ARGUMENT, PREMISE, CONCLUSION 9 Sometimes the outward form of a speech act does not match how it is actually being used. A rhetorical question, for instance, has the outward form of a question, but is really a statement or a command. If someone says “don’t you think the lawn needs to be mowed?” they may actually mean a statement like “the lawn needs to be mowed” or a command like “mow the lawn, now.” Similarly one might disguise a command as a statement. “You will respect my authority” is either true or false—either you will or you will not. But the speaker may intend this as an order—”Respect me!”—rather than a prediction of how you will behave. When we study argument, we need to express things as statements, because arguments are composed of statements. Thus if we encounter a rhetorical question while examining an argument, we need to convert it into a statement. “Don’t you think the lawn needs to be mowed” will become “the lawn needs to be mowed.” Similarly, commands will become should statements. “Mow the lawn, now!” will need to be transformed into “You should mow the lawn.” The latter kind of change will be important in critical thinking, because critical thinking often studies arguments whose goal is to an get audience to do something. These are called practical arguments. Most advertising and political speech consists of practical arguments, and these are crucial topics for critical thinking. Once we have a collection of statements, we can use them to build arguments. An argument is a connected series of statements designed to convince an audience of another statement. Here an audience might be a literal audience sitting in front of you at some public speaking engagement. Or it might be the readers of a book or article. The audience might even be yourself as you reason your way through a problem. Let’s start with an example of an argument given to an external audience. This passage is from an essay by Peter Singer called “Famine, Affluence, and Morality” in which he tries to convince people in rich nations that they need to do more to help people in poor nations who are experiencing famine. A contemporary philosopher writing in an academic journal If it is in our power to prevent something bad from happening, without thereby sacrificing anything of comparable moral importance, we ought, morally, to do so. Famine is something bad, and it can be prevented without sacrificing anything of comparable moral importance. So, we ought to prevent famine. (Singer 1972) Singer wants his readers to work to prevent famine. This is represented by the last statement of the passage, “we ought to prevent famine,” which is called the conclusion of the passage. The conclusion of an argument is the statement that the argument is trying to convince the audience of. The statements that do the convincing are called the premises. In this case, the argument has three premises: (1) “If it is in our power to prevent something bad from happening, without thereby sacrificing anything of comparable moral importance, we ought, morally, to do so”; (2) “Famine is something bad”; and (3) “it can be prevented without sacrificing anything of comparable moral importance.” Now let’s look at an example of internal reasoning. Jack arrives at the track, in bad weather. There is no one here. I guess the race is not 10 CHAPTER 1. WHAT IS LOGIC? Premise Indicators: because, as, for, since, given that, for the reason that Conclusion Indicators: therefore, thus, hence, so, consequently, it follows that, in conclusion, as a result, then, must, accordingly, this implies that, this entails that, we may infer that Table 1.2: Premise and Conclusion Indicators. happening. In the passage above, the words in italics explain the context for the reasoning, and the words in regular type represent what Jack is actually thinking to himself. This passage again has a premise and a conclusion. The premise is that no one is at the track, and the conclusion is that the race was canceled. The context gives another reason why Jack might believe the race has been canceled, the weather is bad. You could view this as another premise–it is very likely a reason Jack has come to believe that the race is canceled. In general, when you are looking at people’s internal reasoning, it is often hard to determine what is actually working as a premise and what is just working in the background of their unconscious. When people give arguments to each other, they typically use words like “therefore” and “because.” These are meant to signal to the audience that what is coming is either a premise or a conclusion in an argument. Words and phrases like “because” signal that a premise is coming, so we call these premise indicators. Similarly, words and phrases like “therefore” signal a conclusion and are called conclusion indicators. The argument from Peter Singer (on page 9) uses the conclusion indicator word, “so.” Table 1.2 is an incomplete list of indicator words and phrases in English. The two passages we have looked at in this section so far have been simply presented as quotations. But often it is extremely useful to rewrite arguments in a way that makes their logical structure clear. One way to do this is to use something called “canonical form.” An argument written in canonical form has each premise numbered and written on a separate line. Indicator words and other unnecessary material should be removed from the premises. Although you can shorten the premises and conclusion, you need to be sure to keep them all complete sentences with the same meaning, so that they can be true or false. The argument from Peter Singer, above, looks like this in canonical form: P1 : If we can stop something bad from happening, without sacrificing anything of comparable moral importance, we ought to do so. P2 : Famine is something bad. P3 : Famine can be prevented without sacrificing anything of comparable moral importance. C: We ought to prevent famine. Each statement has been written on its own line and given a number. The statements have been paraphrased slightly, for brevity, and the indicator word “so” has been removed. Also notice 1.2. STATEMENT, ARGUMENT, PREMISE, CONCLUSION 11 that the “it” in the third premise has been replaced by the word “famine,” so that statements reads naturally on its own. Similarly, we can rewrite the argument Jack gives at the racetrack, on page 10, like this: P: There is no one at the race track. C: The race is not happening. Notice that we did not include anything from the part of the passage in italics. The italics represent the context, not the argument itself. Also, notice that the “I guess” has been removed. When we write things out in canonical form, we write the content of the statements, ignore information about the speaker’s mental state, like “I believe” or “I guess.” One of the first things you have to learn to do in logic is to identify arguments and rewrite them in canonical form. This is a foundational skill for everything else we will be doing in this text, so we are going to run through a few examples now, and there will be more in the exercises. The passage below is paraphrased from the ancient Greek philosopher Aristotle. An ancient philosopher, writing for his students Again, our observations of the stars make it evident that the earth is round. For quite a small change of position to south or north causes a manifest alteration in the stars which are overhead. (Aristotle On the Heavens, 298a2-10) The first thing we need to do to put this argument in canonical form is to identify the conclusion. The indicator words are the best way to do this. The phrase “make it evident that” is a conclusion indicator phrase. He is saying that everything else is evidence for what follows. So we know that the conclusion is that the earth is round. “For” is a premise indicator word—it is sort of a weaker version of “because.” Thus the premise is that the stars in the sky change if you move north or south. In canonical form, Aristotle’s argument that the earth is round looks like this. P: There are different stars overhead in the northern and southern parts of the earth. C: The earth is spherical in shape. That one is fairly simple, because it just has one premise. Here’s another example of an argument, this time from the book of Ecclesiastes in the Bible. The speaker in this part of the bible is generally referred to as The Preacher, or in Hebrew, Koheleth. In this verse, Koheleth uses both a premise indicator and a conclusion indicator to let you know he is giving reasons for enjoying life. The words of the Preacher, son of David, King of Jerusalem There is something else meaningless that occurs on earth: the righteous who get what the wicked deserve, and 12 CHAPTER 1. WHAT IS LOGIC? the wicked who get what the righteous deserve. . . . So I commend the enjoyment of life, because there is nothing better for a person under the sun than to eat and drink and be glad. (Ecclesiastes 8:14-15, New International Version) Koheleth begins by pointing out that good things happen to bad people and bad things happen to good people. This is his first premise. (Most Bible teachers provide some context here by pointing that that the ways of God are mysterious and this is an important theme in Ecclesiastes.) Then Koheleth gives his conclusion, that we should enjoy life, which he marks with the word “so.” Finally he gives an extra premise, marked with a “because,” that there is nothing better for a person than to eat and drink and be glad. In canonical form, the argument would look like this. P1 : Good things happen to bad people and bad things happen to good people. P2 : There is nothing better for people than to eat, to drink and to enjoy life. C: You should enjoy life. Notice that in the original passages, Aristotle put the conclusion in the first sentence, while Koheleth put it in the middle of the passage, between two premises. In ordinary English, people can put the conclusion of their argument where ever they want. However, when we write the argument in canonical form, the conclusion goes last. Unfortunately, indicator words aren’t a perfect guide to when people are giving an argument. Look at this passage from a newspaper: From the general news section of a national newspaper The new budget underscores the consistent and paramount importance of tax cuts in the Bush philosophy. His first term cuts affected more money than any other initiative undertaken in his presidency, including the costs thus far of the war in Iraq. All told, including tax incentives for health care programs and the extension of other tax breaks that are likely to be taken up by Congress, the White House budget calls for nearly $300 billion in tax cuts over the next five years, and $1.5 trillion over the next 10 years. (Toner 2006) Although there are no indicator words, this is in fact an argument. The writer wants you to believe something about George Bush: tax cuts are his number one priority. The next two sentences in the paragraph give you reasons to believe this. You can write the argument in canonical form like this. P1 : Bush’s first term cuts affected more money than any other initiative undertaken in his presidency, including the costs thus far of the war in Iraq. P2 : The White House budget calls for nearly $300 billion in tax cuts over the next five years, and $1.5 trillion over the next 10 years. C: Tax cuts are of consistent and paramount importance of in the Bush philosophy. 1.2. STATEMENT, ARGUMENT, PREMISE, CONCLUSION 13 The ultimate test of whether something is an argument is simply whether some of the statements provide reason to believe another one of the statements. If some statements support others, you are looking at an argument. The speakers in these two cases use indicator phrases to let you know they are trying to give an argument. A final bit of terminology for this section. An inference is the act of coming to believe a conclusion on the basis of some set of premises. When Jack in the example above saw that no one was at the track, and came to believe that the race was not on, he was making an inference. We also use the term inference to refer to the connection between the premises and the conclusion of an argument. If your mind moves from premises to conclusion, you make an inference, and the premises and the conclusion are said to be linked by an inference. In that way inferences are like argument glue: they hold the premises and conclusion together. Practice Exercises Throughout the book, you will find a series of practice problems that review and explore the material covered in the chapter. There is no substitute for actually working through some problems, because logic is more about a way of thinking than it is about memorizing facts. Part A Decide whether the following passages are statements in the logical sense and give reasons for your answers. Example: Did you follow the instructions? Answer: Not a statement, a question. (1) England is smaller than China. (2) Greenland is south of Jerusalem. (3) Is New Jersey east of Wisconsin? (4) The atomic number of helium is 2. (5) The atomic number of helium is π. (6) I hate overcooked noodles. (7) Blech! Overcooked noodles! (8) Overcooked noodles are disgusting. (9) Take your time. (10) This is the last question. Part B Decide whether the following passages are statements in the logical sense and give reasons for your answers. (1) Is this a question? 14 CHAPTER 1. WHAT IS LOGIC? (2) Nineteen out of the 20 known species of Eurasian elephants are extinct. (3) The government of the United Kingdom has formally apologized for the way it treated the logician Alan Turing. (4) Texting while driving (5) Texting while driving is dangerous. (6) Insanity ran in the family of logician Bertrand Russell, and he had a life-long fear of going mad. (7) For the love of Pete, put that thing down before someone gets hurt! (8) Don’t try to make too much sense of this. (9) Never look a gift horse in the mouth. (10) The physical impossibility of death in the mind of someone living Part C Rewrite each of the following arguments in canonical form. Be sure to remove all indicator words and keep the premises and conclusion as complete sentences. Write the indicator words and phrases separately and state whether they are premise or conclusion indicators. Example: An ancient philosopher writes We should not be distressed or concerned by the thought of our our own death in any way. Why? Look back on the time before you were born: It is a time you did not exist, but it does not trouble you in any way. The time after you die is also a time when you will not exist, so it shouldn’t trouble you either. (Based on Lucretius De rerum natura 3.972–75) Answer: P1 : The time before you were born is a time you did not exist. P2 : You are not troubled by the time before you were born. P3 : The time after you die is also a time you will not exist. C: We should not be distressed or concerned by the thought of our our own death. Premise indicator: So (1) A detective is speaking: Henry’s finger-prints were found on the stolen computer. So, I infer that Henry stole the computer. (2) Monica is wondering about her co-workers political opinions You cannot both oppose abortion and support the death penalty, unless you think there is a difference between fetuses and felons. Steve opposes abortion and supports the death penalty. Therefore Steve thinks there is a difference between fetuses and felons. (3) The Grand Moff of Earth defense considers strategy We know that whenever people from one planet invade another, they always wind up being killed by the local diseases, because in 1938, when Martians invaded the Earth, they were defeated because they lacked immunity to Earth’s diseases. Also, in 1942, when Hitler’s forces landed on the Moon, they were killed by Moon diseases. (4) If you have slain the Jabberwock, my son, it will be a frabjous day. The Jabberwock lies there dead, its head cleft with your vorpal sword. This is truly a fabjous day. 1.3. ARGUMENTS AND NONARGUMENTS 15 (5) A detective trying to crack a case thinks to herself Miss Scarlett was jealous that Professor Plum would not leave his wife to be with her. Therefore she must be the killer, because she is the only one with a motive. Part D Rewrite each of the following arguments in canonical form. Be sure to remove all indicator words and keep the premises and conclusion as complete sentences. Write the indicator words and phrases separately and state whether they are premise or conclusion indicators. 1) A pundit is speaking on a Sunday political talk show Hillary Clinton should drop out of the race for Democratic Presidential nominee. For every day she stays in the race, McCain gets a day free from public scrutiny and the members of the Democratic party get to fight one another. 2) You have to be smart to understand the rules of Dungeons and Dragons. Most smart people are nerds. So, I bet most people who play D&D are nerds. 3) Any time the public receives a tax rebate, consumer spending increases. Since the public just received a tax rebate, consumer spending will increase. 4) Isabelle is taller than Jacob. Kate must also be taller than Jacob, because she is taller than Isabelle. 1.3 Arguments and Nonarguments We just saw that arguments are made of statements. However, there are lots of other things you can do with statements. Part of learning what an argument is involves learning what an argument is not, so in this section and the next we are going to look at some other things you can do with statements besides make arguments. The list below of kinds of nonarguments is not meant to be exhaustive: there are all sorts of things you can do with statements that are not discussed. Nor are the items on this list meant to be exclusive. One passage may function as both, for instance, a narrative and a statement of belief. Right now we are looking at real world reasoning, so you should expect a lot of ambiguity and imperfection. If your class is continuing on into the critical thinking portions of this textbook, you will quickly get used to this. Simple Statements of Belief An argument is an attempt to persuade an audience to believe something, using reasons. Often, though, when people try to persuade others to believe something, they skip the reasons, and give a simple statement of belief. This is a kind of nonargumentative passage where the speaker simply asserts what they believe without giving reasons. Sometimes simple statements of belief are prefaced with the words “I believe,” and sometimes they are not. A simple statements of belief can be a profoundly inspiring way to change people’s hearts and minds. Consider this 16 CHAPTER 1. WHAT IS LOGIC? passage from Dr. Martin Luther King’s Nobel acceptance speech. I believe that even amid today’s mortar bursts and whining bullets, there is still hope for a brighter tomorrow. I believe that wounded justice, lying prostrate on the blood-flowing streets of our nations, can be lifted from this dust of shame to reign supreme among the children of men. I have the audacity to believe that peoples everywhere can have three meals a day for their bodies, education and culture for their minds, and dignity, equality and freedom for their spirits. (King 2001) This actually is a part of a longer passage that consists almost entirely of statements that begin with some variation of “I believe.”It is incredibly powerful oration, because the audience, feeling the power of King’s beliefs, comes to share in those beliefs. The language King uses to describe how he believes is important, too. He says his belief in freedom and equality requires audacity, making the audience feel his courage and want to share in this courage by believing the same things. These statements are moving, but they do not form an argument. None of these statements provide evidence for any of the other statements. In fact, they all say roughly the same thing, that good will triumph over evil. So the study of this kind of speech belongs to the discipline of rhetoric, not of logic. Expository Passages Perhaps the most basic use of a statement is to convey information. Often if we have a lot of information to convey, we will sometimes organize our statements around a theme or a topic. Information organized in this fashion can often appear like an argument, because all of the statements in the passage relate back to some central statement. However, unless the other statements are given as reasons to believe the central statement, the passage you are looking at is not an argument. Consider this passage: From a college psychology textbook. Eysenck advocated three major behavior techniques that have been used successfully to treat a variety of phobias. These techniques are modeling, flooding, and systematic desensitization. In modeling phobic people watch nonphobics cope successfully with dreaded objects or situations.In flooding clients are exposed to dreaded objects or situations for prolonged periods of time in order to extinguish their fear. In contrast to flooding, systematic desensitization involves gradual, client-controlled exposure to the anxiety eliciting object or situation. (Adapted from Ryckman 2007) We call this kind of passage an expository passage. In an expository passage, statements are organized around a central theme or topic statement. The topic statement might look like a conclusion, but the other statements are not meant to be evidence for the topic statement. Instead, they elaborate on the topic statement by providing more details or giving examples. In 1.3. ARGUMENTS AND NONARGUMENTS 17 the passage above, the topic statement is “Eysenck advocated three major behavioral techniques . . . .” The statements describing these techniques elaborate on the topic statement, but they are not evidence for it. Although the audience may not have known this fact about Eysenk before reading the passage, they will typically accept the truth of this statement instantly, based on the textbook’s authority. Subsequent statements in the passage merely provide detail. Deciding whether a passage is an argument or an expository passage is complicated by the fact that sometimes people argue by example: Steve: Kenyans are better distance runners than everyone else. Monica: Oh come on, that sounds like an exaggeration of a stereotype that isn’t even true. Steve: What about Dennis Kimetto, the Kenyan who set the world record for running the marathon? And you know who the previous record holder was: Emmanuel Mutai, also Kenyan. Here Steve has made a general statement about all Kenyans. Monica clearly doubts this claim, so Steve backs it up with some examples that seem to match his generalization. This isn’t a very strong way to argue: moving from two examples to statement about all Kenyans is probably going to be a kind of bad argument known as a hasty generalization. (This mistake is covered in the complete version of this text in the chapter on induction) The point here however, is that Steve is just offering it as an argument. The key to telling the difference between expository passages and arguments by example is whether there is a conclusion that they audience needs to be convinced of. In the passage from the psychology textbook, “Eysenck advocated three major behavioral techniques” doesn’t really work as a conclusion for an argument. The audience, students in an introductory psychology course, aren’t likely to challenge this assertion, the way Monica challenges Steve’s overgeneralizing claim. Context is very important here, too. The Internet is a place where people argue in the ordinary sense of exchanging angry words and insults. In that context, people are likely to actually give some arguments in the logical sense of giving reasons to believe a conclusion. Narratives Statements can also be organized into descriptions of events and actions, as in this snippet from book V of Harry Potter. But she [Hermione] broke off; the morning post was arriving and, as usual, the Daily Prophet was soaring toward her in the beak of a screech owl, which landed perilously close to the sugar bowl and held out a leg. Hermione pushed a Knut into its leather pouch, took the newspaper, and scanned the front page critically as the owl took off again. (Rowling 2003) We will use the term narrative loosely to refer to any passage that gives a sequence of events 18 CHAPTER 1. WHAT IS LOGIC? or actions. A narrative can be fictional or nonfictional. It can be told in regular temporal sequence or it can jump around, forcing the audience to try to reconstruct a temporal sequence. A narrative can describe a short sequence of actions, like Hermione taking a newspaper from an owl, or a grand sweep of events, like this passage about the rise and fall of an empire in the ancient near east: The Guti were finally expelled from Mesopotamia by the Sumerians of Erech (c. 2100), but it was left to the kings of Ur’s famous third dynasty to re-establish the Sargonoid frontiers and write the final chapter of the Sumerian History. The dynasty lasted through the twenty first century at the close of which the armies of Ur were overthrown by the Elamites and Amorites (McEvedy and Woodcock 1967). This passage does not feature individual people performing specific actions, but it is still united by character and action. Instead of Hermione at breakfast, we have the Sumarians in Mesopotamia. Instead of retrieving a message from an owl, the conquer the Guti, but then are conquered by the Elamites and Amorites. The important thing is that the statements in a narrative are not related as premises and conclusion. Instead, they are all events which are united common characters acting in specific times and places. Practice Exercises Part A Identify each passage below as an argument or a nonargument, and give reasons for your answers. If it is a nonargument, say what kind of nonargument you think it is. If it is an argument, write it out in canonical form. Example: One student speaks to another student who has missed class: The instructor passed out the syllabus at 9:30. Then he went over some basic points about reasoning, arguments and explanations. Then he said we should call it a day. Answer: Not an argument, because none of the statements provide any support for any of the others. This is probably better classified as a narration because the events are in temporal sequence. (1) An anthropology teacher is speaking to her class Different gangs use different colors to distinguish themselves. Here are some illustrations: biologists tend to wear some blue, while the philosophy gang wears black. (2) The economy has been in trouble recently. And it’s certainly true that cell phone use has been rising during that same period. So, I suspect increasing cell phone use is bad for the economy. (3) At Widget-World Corporate Headquarters: We believe that our company must deliver a quality product to our customers. Our customers also expect first-class customer service. At the same time, we must make a profit. 1.4. ARGUMENTS AND EXPLANATIONS 19 (4) Jack is at the breakfast table and shows no sign of hurrying. Gill says: You should leave now. It’s almost nine a.m. and it takes three hours to get there. (5) In a text book on the brain: Axons are distinguished from dendrites by several features, including shape (dendrites often taper while axons usually maintain a constant radius), length (dendrites are restricted to a small region around the cell body while axons can be much longer), and function (dendrites usually receive signals while axons usually transmit them). Part B Identify each passage below as an argument or a nonargument, and give reasons for your answers. If it is a nonargument, say what kind of nonargument you think it is. If it is an argument, write it out in canonical form. (1) Suzi doesn’t believe she can quit smoking. Her friend Brenda says Some people have been able to give up cigarettes by using their will-power. Everyone can draw on their will-power. So, anyone who wants to give up cigarettes can do so. (2) The words of the Preacher, son of David, King of Jerusalem I have seen something else under the sun: The race is not to the swift or the battle to the strong, nor does food come to the wise or wealth to the brilliant or favor to the learned; but time and chance happen to them all. (Ecclesiastes 9:11, New International Version) (3) An economic development expert is speaking. The introduction of cooperative marketing into Europe greatly increased the prosperity of the farmers, so we may be confident that a similar system in Africa will greatly increase the prosperity of our farmers. (4) From the CBS News website, US section. Headline: “FBI nabs 5 in alleged plot to blow up Ohio bridge.” Five alleged anarchists have been arrested after a months-long sting operation, charged with plotting to blow up a bridge in the Cleveland area, the FBI announced Tuesday. CBS News senior correspondent John Miller reports the group had been involved in a series of escalating plots that ended with their arrest last night by FBI agents. The sting operation supplied the anarchists with what they thought were explosives and bomb-making materials. At no time during the case was the public in danger, the FBI said. (CBS News 2012) (5) At a school board meeting. Since creationism can be discussed effectively as a scientific model, and since evolutionism is fundamentally a religious philosophy rather than a science, it is unsound educational practice for evolution to be taught and promoted in the public schools to the exclusion or detriment of special creation. (Kitcher 1982, p. 177, citing Morris 1975.) 1.4 Arguments and Explanations Explanations are are not arguments, but they they share important characteristics with arguments, so we should devote a separate section to them. Both explanations and arguments are parts of reasoning, because both feature statements that act as reasons for other statements. The difference is that explanations are not used to convince an audience of a conclusion. 20 CHAPTER 1. WHAT IS LOGIC? Premises Prove Conclusion Explainers Reasons Clarify Explainee Target Proposition Figure 1.3: Arguments vs. Explanations. Let’s start with workplace example. Suppose you see your co-worker, Henry, removing a computer from his office. You think to yourself “Gosh, is he stealing from work?” But when you ask him about it later, Henry says, “I took the computer because I believed that it was scheduled for repair.” Henry’s statement looks like an argument. It has the indicator word “because” in it, which would mean that the statement “I believed it was scheduled for repairs” would be a premise. If it was, we could put the argument in canonical form, like this: P: I believed the computer was scheduled for repair C: I took the computer from the office. But this would be awfully weird as an argument. If it were an argument, it would be trying to convince us of the conclusion, that Henry took the computer from the office. But you don’t need to be convinced of this. You already know it—that’s why you were talking to him in the first place. Henry is giving reasons here, but they aren’t reasons that try to prove something. They are reasons that explain something. When you explain something with reasons, you increase your understanding of the world by placing something you already know in a new context. You already knew that Henry took the computer, but now you know why Henry took the computer, and can see that his action was completely innocent (if his story checks out). Both arguments and explanations both involve giving reasons, but the reasons function differently in each case. An explanation is defined as a kind of reasoning where reasons are used to provide a greater understanding of something that is already known. Because both arguments and explanations are parts of reasoning, we will use parallel language to describe them. In the case of an argument, we called the reasons “premises.” In the case of an explanation, we will call them explainers. Instead of a “conclusion,” we say that the explanation has an explainee. We can use the generic term reasons to refer to either premises or explainers and the generic term target proposition to refer to either conclusions or explainees. Figure 1.3 shows this relationship. We can put explanations in canonical form, just like arguments, but to distinguish the two, we will simply number the statements, rather than writing Ps and Cs, and we will put an E next to the line that separates explainers and exaplainee, like this: 1.4. ARGUMENTS AND EXPLANATIONS 21 1. Henry believed the computer was scheduled for repair E 2. Henry took the computer from the office. Cases where the target proposition is something that is completely common sense are clearcut cases of explanation. Consider the following passage. From Livescience, a science education website, under the headline “Why is grass green?” Like many plants, most species of grass produce a bright pigment called chlorophyll. Chlorophyll absorbs blue light (high energy, short wavelengths) and red light (low energy, longer wavelengths) well, but mostly reflects green light, which accounts for your lawn’s color. (Mauk 2013) The passage contains reasoning. The nature of chlorophyll “accounts for” the color of grass. But in this case the audience does not need to be convinced that grass is green. Everyone knows that. The audience went to the Livescience website because they wanted an explanation for why grass was green. Often the same piece of reasoning can work as either an argument or an explanation, depending on the situation where it is used. Consider this short dialogue Monica visits Steve’s cubical. Monica: All your plants are dead. Steve: It’s because I never water them. In the passage above, Steve uses the word “because,” which we’ve seen in the past is a premise indicator word. But if it were a premise, the conclusion would be “All Steve’s plants are dead.” But Steve can’t possibly be trying to convince Monica that all his plants are dead. It is something that Monica herself says, and that they both can see. The “because” here indicates a reason, but here Steve is giving an explanation, not an argument. He takes something that Steve and Monica already know—that the plants are dead—and puts it in a new light by explaining how it came to be. In this case, the plants died because they didn’t get water, rather than dying because they didn’t get enough light or were poisoned by a malicious co-worker. The reasoning is best represented like this: 1. Steve never waters his plants. E 2. All the plants are dead. But the same piece of reasoning can change form an explanation into an argument simply by putting it into a new situation: Monica and Steve are away from the office. Monica: Did you have someone water your plants while you were away? 22 CHAPTER 1. WHAT IS LOGIC? Steve: Monica: No. I bet they are all dead. Here Steve and Monica do not know that Steve’s plants are dead. Monica is inferring this idea based on the premise which she learns from Steve, that his plants are not being watered. This time “Steve’s plants are not being watered” is a premise and “The plants are dead” is a conclusion. We represent the argument like this: P. Steve never waters his plants. C. All the plants are dead. In the example of Steve’s plants, the same piece of reasoning can function either as an argument or an explanation, depending on the context where it is given. This is because the reasoning in the example of the plants is causal: the causes of the plants dying are given as reasons for the death, and we can appeal to causes either to explain something that we know happened or to predict something that we think might have happened. Not all kinds of reasoning are flexible like that, however. Reasoning from authority can be used in some kinds of argument, but often makes a lousy explanation. Consider another conversation between Steve and Monica: Monica: I saw on a documentary last night that the universe is expanding and probably will keep expanding for ever. Steve: Really? Monica: Yeah, Steven Hawking said so. There aren’t any indicator words here, but it looks like Monica is giving an argument. She states that the universe is expanding, and Steve gives a skeptical “really?” Monica then replies by saying that she got this information from the famous physicist Steven Hawking. It looks like Steve is supposed to believe that the universe will expand indefinitely because Hawking, an authority in the relevant field, said so. This makes for an ok argument: P: Steven Hawking said that the universe is expanding and will continue to do so indefinitely. C: The universe is expanding and will continue to do so indefinitely. Arguments from authority aren’t very reliable, but for very many things they are all we have to go on. We can’t all be experts on everything. But now try to imagine this argument as an explanation. What would it mean to say that the expansion of the universe can be explained by the fact that Steven Hawking said that it should expand. It would be as if Hawking were a god, and the universe obeyed his commands! Arguments from authority are acceptable, but not ideal. Explanations from authority, on the other hand, are completely illegitimate. 1.4. ARGUMENTS AND EXPLANATIONS 23 In general, arguments that appeal to how the world works are more satisfying than ones which appeals to the authority or expertise of others. Compare the following pair of arguments: (a) Jack says traffic will be bad this afternoon. So, traffic will be bad this afternoon. (b) Oh no! Highway repairs begin downtown today. And a bridge lift is scheduled for the middle of rush hour. Traffic is going to be terrible Even though the second passage is an argument, the reasons used to justify the conclusion could be used in an explanation. Someone who accepts this argument will also have an explanation ready to offer if someone should later ask “Traffic was terrible today! I wonder why?”. This is not true of the first passage: bad traffic is not explained by saying “Jack said it would be bad.” The argument that refers to the drawbridge going up is appealing to a more powerful sort of reason, one that works in both explanations and arguments. This simply makes for a more satisfying argument, one that makes for a deeper understanding of the world, than one that merely appeals to authority. Although arguments based on explanatory premises are preferred, we must often rely on other people for our beliefs, because of constraints on our time and access to evidence. But the other people we rely on should hopefully hold the belief on the basis of an empirical understanding. And if those people are just relying on authority, then we should hope that at some point the chain of testimony ends with someone who is relying on something more than mere authority. In [cross ref] we’ll look more closely at sources and how much you should trust them. We just have seen that they same set of statements can be used as an argument or an explanation depending on the context. This can cause confusion between speakers as to what is going on. Consider the following case: Bill and Henry have just finished playing basketball. Bill: Man, I was terrible today. Henry: I thought you played fine. Bill: Nah. It’s because I have a lot on my mind from work. Bill and Henry disagree about what is happening—arguing or explaining. Henry doubts Bill’s initial statement, which should provoke Bill to argue. But instead, he appears to plough ahead with his explanation. What Henry can do in this case, however, is take the reason that Bill offers as an explanation (that Bill is preoccupied by issues at work) and use it as a premise in an argument for the conclusion “Bill played terribly.” Perhaps Henry will argue (to himself) something like this: “It’s true that Bill has a lot on his mind from work. And whenever a person is preoccupied, his basketball performance is likely to be degraded. So, perhaps he did play poorly today (even though I didn’t notice).” In other situations, people can switch back and forth between arguing and explaining. Imagine that Jones says “The reservoir is at a low level because of several releases to protect the 24 CHAPTER 1. WHAT IS LOGIC? down-stream ecology.” Jones might intend this as an explanation, but since Smith does not share the belief that the reservoir’s water level is low, he will first have to be given reasons for believing that it is low. The conversation might go as follows: Jones: The reservoir is at a low level because of several releases to protect the down-stream ecology. Smith: Jones: Wait. The reservoir is low? Yeah. I just walked by there this morning. You haven’t been up there in a while? I guess not. Smith: Jones: Yeah, it’s because they’ve been releasing a lot of water to protect the ecology lately. When challenged, Smith offers evidence from his memory: he saw the reservoir that morning. Once Smith accepts that the water level is low, Jones can restate his explanation. Some forms of explanation overlap with other kinds of nonargumentative passages. We are dealing right now with thinking in the real world, and as we mentioned on page 5 the real world is full of messiness and ambiguity. One effect of this is that all the categories we are discussing will wind up overlapping. Narratives and expository passages, for instance, can also function as explanations. Consider this passage From the sports section Duke beat Butler 61-59 for the national championship Monday night. Gordon Hayward’s half-court, 3-point heave for the win barely missed to leave tiny Butler one cruel basket short of the Hollywood ending. (Based on Associated Press 2010) On the one hand, this is clearly a narrative—retelling a sequence of events united by time, place, and character. But it also can work as an explanation about how Duke won, if the audience immediately accepts the result. ’The last shot was a miss and then Duke won’ can be understood as ’the last shot was a miss and so Duke won’. Practice Exercises Part A Identify each of the passages below as an argument, an explanation, or neither, and justify your answer. If the passage is an argument write it in canonical form, with premises marked P1 etc., then a line, and then the conclusion marked with a C. If the argument is an explanation, write it in the canonical form for an explanation, with the explainers numbered and an “E” after the line that separates the explainers and the explainee. If the argument is neither an argument nor an explanation, state what kind of nonargument you think it is, such as a narrative or an expository passage. 1.4. ARGUMENTS AND EXPLANATIONS 25 Example: Henry arrives at work late: Bill is not here. He very rarely arrives late. So, he is not coming in today. Answer: Argument You can tell Henry is giving an argument to himself here because the conclusion is something that he did not already believe. P1 : Bill is not here. P2 : Bill very rarely arrives late. C: Bill is not coming in today (1) From a science education website run by NASA, also promoted by Google as the answer to the question “Why is the sky blue?” Blue light is scattered in all directions by the tiny molecules of air in Earth’s atmosphere. Blue is scattered more than other colors because it travels as shorter, smaller waves. This is why we see a blue sky most of the time. (NASA 2015) (2) Jack is reading a popular science magazine. Analyze Jack’s reasoning. The magazine says: Recent research has shown that people who rate themselves as “very happy” are less successful financially than those who rate themselves as “moderately happy.” Jack says, “Huh! It seems that a little unhappiness is good in life.” (3) An anthropologist is speaking. People get nicknames based on some distinctive feature they possess. And so, Mark, for example, who is 6’6” is (ironically) called “Smalls”, while Matt, who looks young, is called “Baby Face.” John looks just like his dad, and is called “Chip.” (4) Two teenaged friends are talking. Analyze Saida’s reasoning. Saida: I can’t go to the show tonight. Jordan: Saida: Bummer. I know! My mother wouldn’t let me go out when I asked. (5) A mother is speaking to her teenage son. You should always listen to your mother. I say “no.” So, you have to stay in tonight. (6) An economist is speaking. Any time the public receives a tax rebate, consumer spending increases. Since the public just received a tax rebate, consumer spending will increase. (7) In a letter to the editor. Today’s kids are all slackers. American society is doomed. (8) On Monday, Jack is told that his unit ships to Iraq in two days: I was hoping to go to Henry’s birthday party next weekend. But I’m shipping out on Wednesday. So, I will miss it. (9) A student is speaking to her instructor: I was late for class because the battery in my mobile phone, which I was using as an alarm clock, ran out. (10) There is a lot of positive talk concerning parenthood because people tend to think about the positive effects that have a child brings and they tend to exclude the numerous negatives that it brings. Part B Identify each of the passages below as an argument, an explanation, or neither, and 26 CHAPTER 1. WHAT IS LOGIC? justify your answer. If the passage is an argument write it in canonical form, with premises marked P1 etc., then a line, and then the conclusion marked with a C. If the argument is an explanation, write it in the canonical form for an explanation, with the explainers numbered and an “E” after the line that separates the explainers and the explainee. If the argument is neither an argument nor an explanation, state what kind of nonargument you think it is, such as a narrative or an expository passage. (1) You have to be smart to understand the rules of Dungeons and Dragons. Most smart people are nerds. So, I bet most people who play D&D are nerds. (2) A coach is emailing parents in a neighborhood youth soccer league. The game is canceled since it is raining heavily. (3) At the market. You know, granola bars generally aren’t healthy. The ingredients include lots of processed sugars. (4) At the pet store. Salesman: A small dog makes just as effective a guard dog for your home as a big dog. Henry: No way! Salesman: It might seem strange. But smaller “yappy” dogs bark readily and they also generate distinctive higher-pitched sounds. Most of a dog’s effectiveness as a guard is due to making a sound, not physical size. (5) A child is thinking out loud. I think my cat must be dead. It isn’t in any of its usual places. And when I asked my mother if she had seen it, she couldn’t look me in the eyes. (6) flurm Smith: Jones: Smith: I can solve any puzzle more quickly than you. Get out of here. It’s true! I’m a member of MENSA, and you’re not. (7) In the comments on a biology blog: According to Darwin’s theory, my ancestors were monkeys. But since that’s ridiculous, Darwin’s theory is false. (8) If you believe in [the Christian] God and turn out to be incorrect, you have lost nothing. But if you don’t believe in God and turn out to be incorrect, you will go to hell. Believing in God is better in both cases. One should therefore believe in God. (A formulation of “Pascal’s Wager” by Blaise Pascal.) (9) Bill and Henry are in Columbus. Bill: Good news—I just accepted a job offer in Omaha. Henry: That’s great. Congratulations! I suppose this means you’ll be leaving us, then? Bill: Yes, I’ll need to move sometime before September. (10) You already know that God kicked humanity out of Eden before they could eat of the tree of life but only after they had eaten of the tree of knowledge of good and evil. That was because Satan wanted to take over God’s throne and was responsible for their eating from the tree. If humans had eaten of both trees they could have been a threat to God. 1.4. ARGUMENTS AND EXPLANATIONS 27 Key Terms Argument Informal logic Canonical form Logic Conclusion Metacognition Conclusion indicator Metareasoning Content neutrality Narrative Critical thinker Practical argument Critical thinking Premise Explainee Premise indicator Explainer Reason Explanation Rhetoric Expository passage Simple statement of belief Formal logic Statement Inference Target proposition 28 CHAPTER 1. WHAT IS LOGIC? Chapter 2 The Basics of Evaluating Argument 2.1 Two Ways an Argument Can Go Wrong Arguments are supposed to lead us to the truth, but they don’t always succeed. There are two ways they can fail in their mission. First, they can simply start out wrong, using false premises. Consider the following argument. P1 : It is raining heavily. P2 : If you do not take an umbrella, you will get soaked. C: You should take an umbrella. If premise (1) is false—if it is sunny outside—then the argument gives you no reason to carry an umbrella.The argument has failed its job. Premise (2) could also be false: Even if it is raining outside, you might not need an umbrella. You might wear a rain poncho or keep to covered walkways and still avoid getting soaked. Again, the argument fails because a premise is false. Even if an argument has all true premises, there is still a second way it can fail. Suppose for a moment that both the premises in the argument above are true. You do not own a rain poncho. You need to go places where there are no covered walkways. Now does the argument show you that you should take an umbrella? Not necessarily. Perhaps you enjoy walking in the rain, and you would like to get soaked. In that case, even though the premises were true, the conclusion would be false. The premises, although true, do not support the conclusion. Back on page 13 we defined an inference, and said it was like argument glue: it holds the premises and conclusion together. When an argument goes wrong because the premises do not support the conclusion, we say there is something wrong with the inference. 29 30 CHAPTER 2. THE BASICS OF EVALUATING ARGUMENT Consider another example: P1 : You are reading this book. P2 : This is a logic book. C: You are a logic student. This is not a terrible argument. Most people who read this book are logic students. Yet, it is possible for someone besides a logic student to read this book. If your roommate picked up the book and thumbed through it, they would not immediately become a logic student. So the premises of this argument, even though they are true, do not guarantee the truth of the conclusion. Its inference is less than perfect. Again, for any argument, there are two ways that it could fail. First, one or more of the premises might be false. Second, the premises might fail to support the conclusion. Even if the premises were true, the form of the argument might be weak, meaning the inference is bad. 2.2 Valid, Sound In logic, we are mostly concerned with evaluating the quality of inferences, not the truth of the premises. The truth of various premises will be a matter of whatever specific topic we are arguing about, and, as we have said, logic is content neutral. The strongest inference possible would be one where the premises, if true, would somehow force the conclusion to be true. This kind of inference is called valid. There are a number of different ways to make this idea of the premises forcing the truth of the conclusion more precise. Here are a few: An argument is valid if and only if. . . (a) it is impossible to consistently both (i) accept the premises and (ii) reject the conclusion (b) it is impossible for the premises to be true and the conclusion false (c) the premises, if true, would necessarily make the conclusion true. (d) the conclusion is true in every imaginable scenario in which the premises are true (e) it is impossible to write a consistent story (even fictional) in which the premises are true and the conclusion is false In the glossary, we formally adopt item (b) as the definition for this textbook: an argument is valid if and only if it is impossible for the premises to be true and the conclusion false. However, 2.2. VALID, SOUND 31 P1 : Lady Gaga is from Mars. C: Lady Gaga is from the fourth planet from our sun. Figure 2.1: A valid argument. nothing will really ride on the differences between the definitions in the list above, and we can look at all of them in order to give us a sense of what logicians mean when they use the term “valid”. The important thing to see is that all the definitions in the list above try to get at what would happen if the premises were true. None of them assert that the premises actually are true. This is why definitions (d) and (e) talk about what would happen if you somehow pretend the premises are true, for instance by telling a story. The argument is valid if, when you pretend the premises are true, you also have to pretend the conclusion is true. Consider the argument in Figure 2.1 The American pop star Lady Gaga is not from Mars. (She’s from New York City.) Nevertheless, if you imagine she’s from Mars, you simply have to imagine that she is from the fourth planet from our sun, because mars simply is the fourth planet form our sun. Therefore this argument is valid. This way of understanding validity is based on what you can imagine, but not everyone is convinced that the imagination is a reliable tool in logic. That is why definitions like (c) and (b) talk about what is necessary or impossible. If the premises are true, the conclusion necessarily must be true. Alternately, it is impossible for the premises to be true and the conclusion false. The idea here is that instead of talking about the imagination, we will just talk about what can or cannot happen at the same time. The fundamental notion of validity remains the same, however: the truth of the premises would simply guarantee the truth of conclusion. So, assessing validity means wondering about whether the conclusion would be true if the premises were true. This means that valid arguments can have false conclusions. This is important to keep in mind because people naturally tend to think that any argument must be good if they agree with the conclusion. And the more passionately people believe in the conclusion, the more likely we are to think that any argument for it must be brilliant. Conversely, if the conclusion is something we don’t believe in, we naturally tend to think the argument is poor. And the more we don’t like the conclusion, the less likely we are to like the argument. But this is not the correct way to evaluate inferences at all. The quality of the inference is entirely independent of the truth of the conclusion. You can have great arguments for false conclusions and horrible arguments for true conclusions. We have trouble seeing this because of biases built deep in the way we think called “cognitive biases.” A cognitive bias is a habit of reasoning that can be dysfunctional in certain circumstances. Generally these biases developed for a reason, so they serve us well in many or most circumstances. But cognitive biases also systematically distort our reasoning in other circumstances, so we must be on guard against them. There is a particular cognitive bias that makes it hard for us to recognize when a poor argument is being given for a conclusion we agree with. It is called “confirmation bias” and it is in many ways the mother of all cognitive biases. Confirmation bias is the tendency to discount 32 CHAPTER 2. THE BASICS OF EVALUATING ARGUMENT P1 : Oranges are either fruits or musical instruments. P2 : Oranges are not fruits. C: Oranges are musical instruments. Figure 2.2: A valid argument or ignore evidence and arguments that contradict one’s current beliefs. It really pervades all of our thinking, right down to our perceptions. Because of confirmation bias, we need to train ourselves to recognize valid arguments for conclusions we think are false. Remember, an argument is valid if it is impossible for the premises to be true and the conclusion false. This means that you can have valid arguments with false conclusions, they just have to also have false premises. Consider the example in Figure 2.2 The conclusion of this argument is ridiculous. Nevertheless, it follows validly from the premises. This is a valid argument. If both premises were true, then the conclusion would necessarily be true. This shows that a valid argument does not need to have true premises or a true conclusion. Conversely, having true premises and a true conclusion is not enough to make an argument valid. Consider the example in Figure 2.3 The premises and conclusion of this argument are, as a matter of fact, all true. This is a terrible argument, however, because the premises have nothing to do with the conclusion. Imagine what would happen if Paris declared independence from the rest of France. Then the conclusion would be false, even though the premises would both still be true. Thus, it is logically possible for the premises of this argument to be true and the conclusion false. The argument is not valid. If an argument is not valid, it is called invalid. As we shall see, this term is a little misleading, because less than perfect arguments can be very useful. But before we do that, we need to look more at the concept of validity. In general, then, the actual truth or falsity of the premises, if known, do not tell you whether or not an inference is valid. There is one exception: when the premises are true and the conclusion is false, the inference cannot be valid, because valid reasoning can only yield a true conclusion when beginning from true premises. P1 : London is in England. P2 : Beijing is in China. C: Paris is in France. Figure 2.3: An invalid argument. 2.2. VALID, SOUND 33 P1 : All dogs are mammals P2 : All dogs are animals C: All animals are mammals. Figure 2.4: An invalid argument. Figure 2.4 has another invalid argument: In this case, we can see that the argument is invalid by looking at the truth of the premises and conclusion. We know the premises are true. We know that the conclusion is false. This is the one circumstance that a valid argument is supposed to make impossible. Some invalid arguments are hard to detect because they resemble valid arguments. Consider the one in Figure 2.5 This reasoning is not valid since the premises do not definitively support the conclusion. To see this, assume that the premises are true and then ask, ”Is it possible that the conclusion could be false in such a situation?”. There is no inconsistency in taking the premises to be true without taking the conclusion to be true. The first premise says that the stimulus package will allow the U.S. to avoid a depression, but it does not say that a stimulus package is the only way to avoid a depression. Thus, the mere fact that there is no stimulus package does not necessarily mean that a depression will occur. When an argument resembles a good argument but is actually a bad one, we say it is a .fallacy. Fallacies are similar to cognitive biases, in that they are ways our reasoning can go wrong. Fallacies, however, are always mistakes you can explicitly lay out as arguments in canonical form, as above. Here is another, trickier, example. I will give it first in ordinary language. A pundit is speaking on a cable news show If the U.S. economy were in recession and inflation were running at more than 4%, then the value of the U.S. dollar would be falling against other major currencies. But this is not happening — the dollar continues to be strong. So, the U.S. is not in recession. P1 : An economic stimulus package will allow the U.S. to avoid a depression. P2 : There is no economic stimulus package C: The U.S. will go into a depression. Figure 2.5: An invalid argument 34 CHAPTER 2. THE BASICS OF EVALUATING ARGUMENT P1 : If the U.S. were in a recession with more than 4% inflation, then the dollar would be falling P2 : The dollar is not falling C: The U.S. is not in a recession. Figure 2.6: An invalid argument The conclusion is ”The U.S. economy is not in recession.” If we put the argument in canonical form, it looks like figure 2.6 The conclusion does not follow necessarily from the premises. It does follow necessarily from the premises that (i) the U.S. economy is not in recession or (ii) inflation is running at more than 4%, but they do not guarantee (i) in particular, which is the conclusion. For all the premises say, it is possible that the U.S. economy is in recession but inflation is less than 4%. So, the inference does not necessarily establish that the U.S. is not in recession. A parallel inference would be ”Jack needs eggs and milk to make an omelet. He can’t make an omelet. So, he doesn’t have eggs.”. If an argument is not only valid, but also has true premises, we call it sound. “Sound” is the highest compliment you can pay an argument. If logic is the study of virtue in argument, sound arguments are the most virtuous. We said in Section 2.1 that there were two ways an argument could go wrong, either by having false premises or weak inferences. Sound arguments have true premises and undeniable inferences. If someone gives a sound argument in a conversation, you have to believe the conclusion, or else you are irrational. The argument on the left in Figure 2.7 is valid, but not sound. The argument on the right is both valid and sound. Both arguments have the exact same form. They say that a thing belongs to a general category and everything in that category has a certain property, so the thing has that property. Because the form is the same, it is the same valid inference each time. The difference in the arguments is not the validity of the inference, but the truth of the second premise. People are not carrots, therefore the argument on the left is not sound. People are mortal, so the argument on the right is sound. Often it is easy to tell the difference between validity and soundness if you are using completely silly examples. Things become more complicated with false premises that you might be tempted to believe, as in the argument in Figure 2.8. You might have a general sense that the argument in Figure 2.8 is bad—you shouldn’t assume that someone drinks Guinness just because they are Irish. But the argument is completely valid (at least when it is expressed this way.) The inference here is the same as it was in the previous two arguments. The problem is the first premise. Not all Irishmen drink Guinness, but if they did, and Smith was an Irishman, he would drink Guinness. P1 : Socrates is a person. P1 : Socrates is a person. P2 : All people are carrots. P2 : All people are mortal. C: Therefore, Socrates is a carrot. C: Therefore, Socrates is mortal. Valid, but not sound Valid and sound Figure 2.7: These two arguments are valid, but only the one on the right is sound 2.2. VALID, SOUND 35 P1 : Every Irishman drinks Guinness P2 : Smith is an Irishman C: Smith drinks Guinness. Figure 2.8: An argument that is valid but not sound The important thing to remember is that validity is not about the actual truth or falsity of the statements in the argument. Instead, it is about the way the premises and conclusion are put together. It is really about the form of the argument. A valid argument has perfect logical form. The premises and conclusion have been put together so that the truth of the premises is incompatible with the falsity of the conclusion. A general trick for determining whether an argument is valid is to try to come up with just one way in which the premises could be true but the conclusion false. If you can think of one (just one! anything at all! but no violating the laws of physics!), the reasoning is invalid. Practice Exercises Part A For each passage, (i) put the argument in canonical form and (ii) say whether it is valid or invalid. Example: Monica is looking for her coworker Jack is in his office. Jack’s office is on the second floor. So, Jack is on the second floor. Answer: (i) . P1 : Jack is in his office. P2 : Jack’s office is on the second floor. C: Jack is on the second floor. (ii) Valid (1) All dinosaurs are people, and all people are fruit. Therefore all dinosaurs are fruit. (2) All people are mortal. Socrates is mortal. Therefore all people are Socrates. (3) All dogs are mammals. Therefore, Fido is a mammal, because Fido is a dog. (4) Abe Lincoln must have been from France, because he was either from France or from Luxemborg, and we know was not from Luxemborg. (5) If the world were to end today, then I would not need to get up tomorrow morning. I will need to get up tomorrow morning. Therefore, the world will not end today. (6) If the triceratops were a dinosaur, it would be extinct. Therefore, the triceratops is extinct, because the triceratops was a dinosaur. (7) If George Washington was assassinated, he is dead. George Washington is dead. Therefore George Washington was assassinated. (8) Jack prefers Pepsi to Coke. After all, about 52% of people prefer Pepsi to Coke, and Jack is a person. (9) Steve thinks about the consequences of laziness. If I don’t mow the lawn, it will become a haven for all kinds of exotic insect species. If the lawn becomes a haven for all kinds of 36 CHAPTER 2….

Collepals.com Plagiarism Free Papers

Are you looking for custom essay writing service or even dissertation writing services? Just request for our write my paper service, and we'll match you with the best essay writer in your subject! With an exceptional team of professional academic experts in a wide range of subjects, we can guarantee you an unrivaled quality of custom-written papers.

Get ZERO PLAGIARISM, HUMAN WRITTEN ESSAYS

Why Hire Collepals.com writers to do your paper?

Quality- We are experienced and have access to ample research materials.

We write plagiarism Free Content

Confidential- We never share or sell your personal information to third parties.

Support-Chat with us today! We are always waiting to answer all your questions.