Philosophy of logic

Excerpts from

An Open Introduction to Logic

Selections for Critical Thinking (PHLY 174), Fall 2017

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 July 27, 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 LATEX2". 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 vii

Acknowledgments ix

Part I Basic Concepts

1 What Is Logic? 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Statement, Argument, Premise, Conclusion . . . . . . . . . . . . . . . . . . . . . . . 6

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Arguments and Nonarguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4 Arguments and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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 Sentential Logic

4 Sentential Logic 65 4.1 Sentence Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 Sentential Connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3 More Complicated Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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4.4 Recursive Syntax for SL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Truth Tables 91 5.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 Complete Truth Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.3 Using Truth Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.4 Partial Truth Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.5 Expressive Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Proofs in Sentential Logic 111 6.1 Substitution Instances and Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 Basic Rules for Sentential Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.3 Conditional Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.4 Indirect Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.5 Tautologies and Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.6 Derived Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.7 Rules of Replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.8 Proof Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.9 Soundness and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Part III Quantificational Logic

7 Quantified Logic 163 7.1 From Sentences to Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 7.2 Building Blocks of Quantified Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7.3 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.4 Translating to Quantified Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.5 Recursive Syntax for QL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.6 Tricky Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

v

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.7 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

8 Semantics for Quantified Logic 197 8.1 Creating models in Quantified Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 8.2 Working with Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

9 Proofs in Quantified Logic 209 9.1 Rules for Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 9.2 Rules for Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

A Other Symbolic Notation 221

B Bibliography 225

C Glossary 229

D Quick Reference 235

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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

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viii

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 LATEX2" 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 su↵er 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

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Intellectual debts too great to articulate are owed to scholars too many to enumerate. At di↵erent 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 di↵erent 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 di↵erent 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 e↵ort 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 su↵used 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 di↵erence between formal logic and informal logic as the di↵erence 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 di↵erent 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 di↵erent from rhetoric, which is the study of e↵ective 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 e↵ective 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 di↵erent 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 di↵erent 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 di↵erences between all of these things are and which term is best used to describe parts of arguments. However, none of that makes a di↵erence 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, A✏uence, 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

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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 di↵erent 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

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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 a↵ected 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 a↵ected 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 ⇡.

(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?

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(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 di↵erence between fetuses and felons. Steve opposes abortion and supports the death penalty. Therefore Steve thinks there is a di↵erence between fetuses and felons.

(3) The Grand Mo↵ 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 o↵ering it as an argument.

The key to telling the di↵erence 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 o↵; 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 o↵ again. (Rowling 2003)

We will use the term narrative loosely to refer to any passage that gives a sequence of events

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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 Di↵erent gangs use di↵erent 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 e↵ectively 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 di↵erence is that explanations are not used to convince an audience of a conclusion.

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Premises

Conclusion

Explainers

Explainee

Reasons

Target Proposition

Prove Clarify

Figure 1.3: Arguments vs. Explanations.

Let’s start with workplace example. Suppose you see your co-worker, Henry, removing a computer from his o�ce. 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 o�ce.

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 o�ce. 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 di↵erently 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 o�ce.

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 o�ce.

Monica: Did you have someone water your plants while you were away?

22 CHAPTER 1. WHAT IS LOGIC?

Steve: No.

Monica: 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 prob- ably 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 tra�c will be bad this afternoon. So, tra�c 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. Tra�c 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 o↵er if someone should later ask “Tra�c was terrible today! I wonder why?”. This is not true of the first passage: bad tra�c 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 o↵ers 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: Wait. The reservoir is low?

Jones: Yeah. I just walked by there this morning. You haven’t been up there in a while?

Smith: I guess not.

Jones: Yeah, it’s because they’ve been releasing a lot of water to protect the ecology lately.

When challenged, Smith o↵ers 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 e↵ect 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. It reads: Recent research has shown that people who rate themselves as “very happy” are less successful financially than those who rate themselves as “moderately happy.” He 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: Bummer.

Saida: 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 e↵ects 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 e↵ective 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 e↵ectiveness 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) flurmSmith: I can solve any puzzle more quickly than you.

Jones: Get out of here.

Smith: 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 o↵er 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

Canonical form

Conclusion

Conclusion indicator

Content neutrality

Critical thinker

Critical thinking

Explainee

Explainer

Explanation

Expository passage

Formal logic

Inference

Informal logic

Logic

Metacognition

Metareasoning

Narrative

Practical argument

Premise

Premise indicator

Reason

Rhetoric

Simple statement of belief

Statement

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 di↵erent 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 di↵erences 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 di↵erence 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 di↵erence 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.

P2: All people are carrots.

C: Therefore, Socrates is a carrot.

P1: Socrates is a person.

P2: All people are mortal.

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 o�ce. Jack’s o�ce is on the second floor. So, Jack is on the second floor.

Answer: (i) . P1: Jack is in his o�ce.

P2: Jack’s o�ce 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. THE BASICS OF EVALUATING ARGUMENT

exotic insect species, I will be protecting biodiversity. Therefore, if I don’t mow the lawn, I’ll be protecting biodiversity.

(10) A forest ranger is surveying the park I can tell that bears have been down by the river, because there are tracks in the mud. Tracks like these are made by bears in almost every case.

Part B For each passage, (i) put the argument in canonical form and (ii) say whether it is valid or invalid.

(1) Cindy Lou Who lives in Whoville. You can tell because Cindy Lou Who is a Who, and all Whos live in Whoville.

(2) If Frog and Toad like each other, they are friends. Frog and Toad like each other. Therefore, Frog and Toad are friends.

(3) If Cindy Lou Who is no more than two, then she is not five years old. Cindy Lou Who is not five. Therefore Cindy Lou Who is two or more.

(4) Jack’s suspicious house mate is in the kitchen Jack has moved my leftover slice of pizza. Jack must have moved it, because Jack is the only person who has been in the house, and the pizza is no longer in the fridge.

(5) Jack is Smith’s work colleague. So, Jack and Smith are friends.

(6) Abe Lincoln was either born in Illinois or he was once president. Therefore Abe Lincoln was born in Illinois, because he was never president.

(7) Politicians get a generous allowance for transportation costs. Enda Kenny is a politician. Therefore Kenny gets a generous transportation allowance.

(8) Jones is taller than Bill, because Smith is taller than Jones and Bill is shorter than Smith.

(9) If grass is green, then I am the pope. Grass is green. So, I am the pope.

(10) Smith is paid more than Jack. They are both paid weekly. So, Smith has more money than Jack.

Part C For each passage, (i) put the argument in canonical form and (ii) say whether it is valid or invalid.

(1) Jack is close to the pond. The pond is close to the playground. So, Jack is close to the playground.

(2) Jack is at work, and is unable to leave earlyI have up to half an hour to get to the bank, because work ends at 5:00 and the bank closes at 5:30.

(3) Jack and Gill ate at Guadalajara restaurant earlier and both of them feel nauseated now. So, something they ate there is making them sick.

(4) Zhaoquing must be west of Huizhou, because Zhaoquing is west of Guangzhou, which is west of Huizhou.

(5) Henry can’t find his glasses. I remember I had them when I came in from the car. So, they are in the house somewhere.

2.3. STRONG, COGENT, DEDUCTIVE, INDUCTIVE 37

(6) I was talking about tall John—the one who is over 6’4”—but Jack was talking about short John, who is at most 5’2”. So, we were talking about two di↵erent Johns.

(7) Tomorrow’s trip to Ensenada will take about 10 hours, because the last time I drove there from here it took 10 hours.

Part D For each passage, (i) put the argument in canonical form and (ii) say whether it is valid or invalid.

(1) Monica is surveying the crowd that showed up for her talk There must be at least 150 people here. That’s how many people the auditorium holds, and every seat is full and people are beginning to sit on the stairs at the side.

(2) The fire bell in the building is ringing. There is sometimes a fire in the building when the alarm goes o↵. So, there is a fire.

(3) I cannot drive on the motorways yet, because I just passed my driving test and anyone who passes can drive on the roads but not on the motorway for six months.

(4) Yesterday’s the temperature reached 91 degrees Fahrenheit. Today it is 94. So, today is warmer than yesterday.

(5) My car is functioning well at the moment. So, all of the parts in my car are functioning well.

(6) It has been sunny every day for the last five days. So, it will be sunny today.

(7) Jack is in front of Gill. So, Gill is behind Jack.

(8) Gill is returning home: The door to my house is still locked. So, my possessions are still inside.

2.3 Strong, Cogent, Deductive, Inductive

We have just seen that sound arguments are the very best arguments. Unfortunately, sound arguments are really hard to come by, and when you do find them, they often only prove things that were already quite obvious, like that Socrates (a dead man) is mortal. Fortunately, arguments can still be worthwhile, even if they are not sound. Consider this one:

P1: In January 1997, it rained in San Diego.

P2: In January 1998, it rained in San Diego.

P3: In January 1999, it rained in San Diego.

C: It rains every January in San Diego.

This argument is not valid, because the conclusion could be false even though the premises are true. It is possible, although unlikely, that it will fail to rain next January in San Diego.

38 CHAPTER 2. THE BASICS OF EVALUATING ARGUMENT

Moreover, we know that the weather can be fickle. No amount of evidence should convince us that it rains there every January. Who is to say that some year will not be a freakish year in which there is no rain in January in San Diego? Even a single counterexample is enough to make the conclusion of the argument false.

Still, this argument is pretty good. Certainly, the argument could be made stronger by adding additional premises: In January 2000, it rained in San Diego. In January 2001. . . and so on. Regardless of how many premises we add, however, the argument will still not be deductively valid. Instead of being valid, this argument is strong. An argument is strong if the premises would make the conclusion more likely, were they true. In a strong argument, the premises don’t guarantee the truth of the conclusion, but they do make it a good bet. If an argument is strong, and it has true premises, we say that it is cogent Cogency is the equivalent of soundness in strong arguments. If an inference is neither valid, nor strong, we say it is weak. In a weak argument, the premises would not even make the conclusion likely, even if they were true.

You may have noticed that the word “likely” is a little vague. How likely do the premises have to make the conclusion before we can count the argument as strong? The answer is a very unsatisfying “it depends.” It depends on what is at stake in the decision to believe the conclusion. What happens if you are wrong? What happens if you are right? The phrase “make the conclusion a good bet” is really quite apt. Whether something is a good bet depends a lot on how much money is at stake and how much you are willing to lose. Sometimes people feel comfortable taking a bet that has a 50% chance of doubling their money, sometimes they don’t.

The vagueness of the word “likely” brings out an interesting feature of strong arguments: some strong arguments are stronger than others. The argument about rain in San Diego, above, has three premises referring to three previous Januaries. The argument is pretty strong, but it can become stronger if we go back farther into the past, and find more years where it rains in January. The more evidence we have, the better a bet the conclusion is. Validity is not like this. Validity is a black-or-white matter. You either have it, and you’re perfect, or you don’t, and you’re nothing. There is no point in adding premises to an argument that is already valid.

Arguments that are valid, or at least try to be, are called deductive , and people who attempt to argue using valid arguments are said to be arguing deductively. The notion of validity we are using here is, in fact, sometimes called deductive validity. Deductive argument is di�cult, because, as we said, in the real world sound arguments are hard to come by, and people don’t always recognize them as sound when they find them. Arguments that purport to merely be strong rather than valid are called inductive. The most common kind of inductive argument includes arguments like the one above about rain in San Diego, which generalize from many cases to a conclusion about all cases.

Deduction is possible in only a few contexts. You need to have clear, fixed meanings for all of your terms and rules that are universal and have no exceptions. One can find situations like this if you are dealing with things like legal codes, mathematical systems or logical puzzles. One can also create, as it were, a context where deduction is possible by imagining a universal, exceptionless rule, even if you know that no such rule exists in reality. In the example above about rain in San Diego, we can change the argument from inductive to deductive by adding a universal, exceptionless premise like “It always rains in January in San Diego.” This premise is unlikely to

2.3. STRONG, COGENT, DEDUCTIVE, INDUCTIVE 39

be true, but it can make the inference valid. (For more about trade o↵s between the validity of the inference and the truth of the premise, see the chapter on incomplete arguments in the complete version of this text.

Here is an example in which the context is an artificial code — the tax code:

From a the legal code posted on a government website A tax credit for energy-e�cient home improvement is available at 30% of the cost, up to 1,500 total, in 2009 & 2010, ONLY for existing homes, NOT new construction, that are your ”principal residence” for Windows and Doors (including sliding glass doors, garage doors, storm doors and storm windows), Insulation, Roofs (Metal and Asphalt), HVAC: Central Air Conditioners, Air Source Heat Pumps, Furnaces and Boilers, Water Heaters: Gas, Oil, & Propane Water Heaters, Electric Heat Pump Water Heaters, Biomass Stoves.

This rule describes the conditions under which a person can or cannot take a certain tax credit. Such a rule can be used to reach a valid conclusion that the tax credit can or cannot be taken.

As another example of an inference in an artificial situation with limited and clearly defined options, consider a Sudoku puzzle. The rules of Sudoku are that each cell contains a single number from 1 to 9, and each row, each column and each 9-cell square contain one occurrence of each number from 1 to 9. Consider the following partially completed board:

The following inference shows that, in the first column, a 9 must be entered below the 7:

The 9 in the first column must go in one of the open cells in the column. It cannot go in the third cell in the column, because there is already a 9 in that 9-cell square. It cannot go in the eighth or ninth cell because each of these rows already contains a 9, and a row cannot contain two occurrences of the same number. Therefore, since there

40 CHAPTER 2. THE BASICS OF EVALUATING ARGUMENT

must be a 9 somewhere in this column, it must be entered in the seventh cell, below the 7.

The reasoning in this inference is valid: if the premises are true, then the conclusion must be true. Logic puzzles of all sorts operate by artificially restricting the available options in various ways. This then means that each conclusion arrived at (assuming the reasoning is correct) is necessarily true.

One can also create a context where deduction is possible by imagining a rule that holds without exception. This can be done with respect to any subject matter at all. Speakers often exaggerate the connecting premise in order to ensure that the justificatory or explanatory power of the inference is as strong as possible. Consider Smith’s words in the following passage:

Smith: I’m going to have some excellent pizza this evening.

Jones: I’m glad to hear it. How do you know?

Smith: I’m going to Adriatico’s. They always make a great pizza.

Here, Smith justifies his belief that the pizza will be excellent — it comes from Adriatico’s, where the pizza, he claims, is always great: in the past, present and future.

As stated by Smith, the inference that the pizza will be great this evening is valid. However, making the inference valid in this way often means making the general premise false: it’s not likely that the pizza is great every single time; Smith is overstating the case for emphasis. Note that Smith does not need to use a universal proposition in order to convince Jones that the pizza will very likely be good. The inference to the conclusion would be strong (though not valid) if he had said that the pizza is ”almost always” great, or that the pizza has been great on all of the many occasions he has been at that restaurant in the past. The strength of the inference would fall to some extent—it would not be guaranteed to be great this evening—but a slightly weaker inference seems appropriate, given that sometimes things go contrary to expectation.

Sometimes the laws of nature make constructing contexts for valid arguments more reasonable. Now consider the following passage, which involves a scientific law:

Jack is about to let go of Jim’s leash. The operation of gravity makes all unsupported objects near the Earth’s surface fall toward the center of the Earth. Nothing stands in the way. Therefore, Jim’s leash will fall.

(Or, as Spock said in a Star Trek episode, ”If I let go of a hammer on a planet that has a positive gravity, I need not see it fall to know that it has in fact fallen.”) The inference above is represented in canonical form as follows:

P1: Jack is about to let go of Jim’s leash. P2: The operation of gravity makes all unsupported objects near the Earth’s surface fall

toward the center of the Earth.

2.3. STRONG, COGENT, DEDUCTIVE, INDUCTIVE 41

P1: 92% of Republicans from Texas voted for Bush in 2000.

P2: Jack is a Republican from Texas.

C: Jack voted for Bush.

P1: Just over half of drivers are female.

P2: There’s a person driving the car that just cut me o↵.

C: The person driving the car that just cut me o↵ is female.

A strong argument A weak argument

Figure 2.9: Neither argument is valid, but one is strong and one is weak

P3: Nothing stands in the way of the leash falling.

C: Jim’s leash will fall toward the center of the Earth.

As stated, this argument is valid. That is, if you pretend that they are true or accept them ”for the sake of argument”, you would necessarily also accept the conclusion. Or, to put it another way, there is no way in which you could hold the premises to be true and the conclusion false.

Although this argument is valid, it involves idealizing assumptions similar to the ones we saw in the pizza example. P2 states a physical law which is about as well confirmed as any statement about the world around us you care to name. However, physical laws make assumptions about the situations they apply to—they typically neglect things like wind resistance. In this case, the idealizing assumption is just that nothing stands in the way of the leash falling. This can be checked just by looking, but this check can go wrong. Perhaps there is an invisible pillar underneath Jack’s hand? Perhaps a huge gust of wind will come? These events are much less likely than Adriatico’s making a lousy pizza, but they are still possible.

Thus we see that using scientific laws to create a context where deductive validity is possible is a much safer bet than simply asserting whatever exceptionless rule pops into your head. However, it still involves improving the quality of the inference by introducing premises that are less likely to be true.

So deduction is possible in artificial contexts like logical puzzles and legal codes. It is also possible in cases where we make idealizing assumptions or imagine exceptionless rules. The rest of the time we are dealing with induction. When we do induction, we try for strong inferences, where the premises, assuming they are true, would make the truth of the conclusion very likely, though not necessary. Consider the two arguments in Figure 2.9

Note that the premises in neither inference guarantee the truth of the conclusion. For all the premises in the first one say, Jack could be one of the 8% of Republicans from Texas who did not vote for Bush; perhaps, for example, Jack soured on Bush, but not on Republicans in general, when Bush served as governor. Likewise for the second; the driver could be one of the 49%.

42 CHAPTER 2. THE BASICS OF EVALUATING ARGUMENT

So, neither inference is valid. But there is a big di↵erence between how much support the premises, if true, would give to the conclusion in the first and how much they would in the second. The premises in the first, assuming they are true, would provide very strong reasons to accept the conclusion. This, however, is not the case with the second: if the premises in it were true then they would give only weak reasons for believing the conclusion. thus, the first is strong while the second is weak.

As we said earlier, there there are only two options with respect to validity—valid or not valid. On the other hand, strength comes in degrees, and sometimes arguments will have percentages that will enable you to exactly quantify their strength, as in the two examples in Figure 2.9.

However, even where the degree of support is made explicit by a percentage there is no firm way to say at what degree of support an inference can be classified as strong and below which it is weak. In other words, it is di�cult to say whether or not a conclusion is very likely to be true. For example, In the inference about whether Jack, a Texas Republican, voted for Bush. If 92% of Texas Republicans voted for Bush, the conclusion, if the premises are granted, would very probably be true. But what if the number were 85%? Or 75%? Or 65%? Would the conclusion very likely be true? Similarly, the second inference involves a percentage greater than 50%, but this does not seem su�cient. At what point, however, would it be su�cient?

In order to answer this question, go back to basics and ask yourself: ”If I accept the truth of the premises, would I then have su�cient reason to believe the conclusion?”. If you would not feel safe in adopting the conclusion as a belief as a result of the inference, then you think it is weak, that is, you do not think the premises give su�cient support to the conclusion.

Note that the same inference might be weak in one context but strong in another, because the degree of support needed changes. For example, if you merely have a deposit to make, you might accept that the bank is open on Saturday based on your memory of having gone to the bank on Saturday at some time in the past. If, on the other hand, you have a vital mortgage payment to make, you might not consider your memory su�cient justification. Instead, you will want to call up the bank and increase your level of confidence in the belief that it will be open on Saturday.

Most inferences (if successful) are strong rather than valid. This is because they deal with situations which are in some way open-ended or where our knowledge is not precise. In the example of Jack voting for Bush, we know only that 92% of Republicans voted for Bush, and so there is no definitive connection between being a Texas Republican and voting for Bush. Further, we have only statistical information to go on. This statistical information was based on polling or surveying a sample of Texas voters and so is itself subject to error (as is discussed in the chapter on induction in the complete version of this text. A more precise version of the premise might be ”92% ± 3% of Texas Republicans voted for Bush.”.

At the risk of redundancy, let’s consider a variety of examples of valid, strong and weak inferences, presented in standard form.

P1: David Duchovny weighs more than 200 pounds.

C: David Duchovny weighs more than 150 pounds.

2.3. STRONG, COGENT, DEDUCTIVE, INDUCTIVE 43

The inference here is valid. It is valid because of the number system (here applied to weight): 200 is more than 150. It might be false, as a matter of fact, that David Duchovny weighs more than 200 pounds, and false, as a matter of fact, that David Duchovny weighs more than 150 pounds. But if you suppose or grant or imagine that David Duchovny weighs more than 200 pounds, it would then have to be true that David Duchovny weighs more than 150 pounds. Next:

P1: Armistice Day is November 11th, each year.

P2: Halloween is October 31st, each year.

C: Armistice Day is later than Halloween, each year.

This inference is valid. It is valid because of order of the months in the Gregorian calendar and the placement of the New Year in this system. Next:

P1: All people are mortal.

P2: Professor Pappas is a person.

C: Professor Pappas is mortal.

As written, this inference is valid. If you accept for the sake of argument that all men are mortal (as the first premise says) and likewise that Professor Pappas is a man (as the second premise says), then you would have to accept also that Professor Pappas is mortal (as the conclusion says). You could not consistently both (i) a�rm that all men are mortal and that Professor Pappas is a man and (ii) deny that Professor Pappas is mortal. If a person accepted these premises but denied the conclusion, that person would be making a mistake in logic.

This inference’s validity is due to the fact that the first premise uses the word ”all”. You might, however, wonder whether or not this premise is true, given that we believe it to be true only on our experience of men in the past. This might be a case of over-stating a premise, which we mentioned earlier. Next:

P1: In 1933, it rained in Columbus, Ohio on 175 days.

P2: In 1934, it rained in Columbus, Ohio on 177 days.

P3: In 1935, it rained in Columbus, Ohio on 171 days.

C: In 1936, it rained in Columbus, Ohio on at least 150 days.

This inference is strong. The premises establish a record of days of rainfall that is well above 150. It is possible, however, that 1936 was exceptionally dry, and this possibility means that the inference does not achieve validity. Next:

P1: The Bible says that homosexuality is an abomination.

44 CHAPTER 2. THE BASICS OF EVALUATING ARGUMENT

C: Homosexuality is an abomination.

This inference is an appeal to a source. Appeals to sources are discussed in the sections on arguments from authority in the complete version of this text. of this book. In brief, you should think about whether the source is reliable, is biased, and whether the claim is consistent with what other authorities on the subject say. You should apply all these criteria to this argument for yourself. You should ask what issues, if any, the Bible is reliable on. If you believe humans had any role in writing the Bible, you can ask about what biases and agendas they might have had. And you can think about what other sources—religious texts or moral experts—say on this issue. You can certainly find many who disagree. Given the controversial nature of this issue, we will not give our evaluation. We will only encourage you to think it through systematically.

P1: Some professional philosophers published books in 2007.

P2: Some books published in 2007 sold more than 100,000 copies.

C: Some professional philosophers published books in 2007 that sold more than 100,000 copies.

This reasoning is weak. Both premises use the word ”some” which doesn’t tell you a lot about many professional philosophers published books and how many books sold more than 100,000 copies in 2007. This means that you cannot be confident that even one professional philosopher sold more than 100,000 copies. Next:

P1: Lots of Russians prefer vodka to bourbon.

C: George Bush was the President of the United States in 2006.

No one (in her right mind) would make an inference like this. It is presented here as an example only: it is clearly weak. It’s hard to see how the premise justifies the conclusion to any extent at all.

To sum up this section, we have seen that there are two styles of reasoning, deductive and inductive. The former tries to use valid arguments, while the latter contents itself to give arguments that are merely strong. The section of this book on formal logic will deal entirely with deductive reasoning. Historically, most of formal logic has been devoted to the study of deductive arguments, although many great systems have been developed for the formal treatment of inductive logic. On the other hand, the sections of this book on informal logic and critical thinking will focus mostly on inductive logic, because these arguments are more readily available in the real world.

2.3. STRONG, COGENT, DEDUCTIVE, INDUCTIVE 45

Practice Exercises

Part A For each inference, (i) say whether it is valid, strong, or weak and (ii) explain your answer.

Example: The patient has a red rash covering the extremities and head, but not the torso. The only cause of such a rash is a deficiency in vitamin K. So, the patient must have a vitamin K deficiency.

Answer: (i) Valid. (ii) The word ”only” means it must be vitamin K deficiency.

(1) On 2003-06-19 in Norfolk, VA, a violent storm blew through and the power went out over much of the city. So, the storm caused the power to go out.

(2) All human beings are things with purple hair, and all things with purple hair have nine legs. Therefore, all human beings have nine legs.

(3) Elvis Presley was known as The King. Elvis had 18 songs reach #1 in the Billboard charts. So, The King had 18 #1 hits.

(4) Most philosophers are right-handed. Terence Irwin is a philosopher. So, he is right-handed.

(5) Jack has purple hair, and purple toe nails. Hence, he has toe nails.

(6) The Ohio State football team beat the Miami football team on 2003-01-03 for the college national championship. So, the Ohio State football team was the best team in college football in the 2002-2003 season.

(7) Willie Mosconi made almost all of the pool shots he took from 1940-1945. He took a bunch of shots in 1941. So, he made almost every shot he took in 1941.

(8) Some philosophers are people who are right-handed. Therefore, some people who are right-handed are philosophers.

(9) U.S. President Obama firmly believed that Iran is planning a nuclear attack against Israel. We can conclude that Iran is planning a nuclear attack on Israel.

(10) Since the Spanish American War occurred before the American Civil War, and since the American Civil War occurred after the Korean War, it follows that the Spanish American War occurred before the Korean War.

(11) There are exactly 10 humans in Carnegie Hall right now. Every human in Carnegie Hall right now has exactly ten legs. And, of course, no human in Carnegie Hall shares any legs with another human. Thus, there are at least 100 legs in Carnegie Hall right now.

(12) Amy Bishop is an evolutionary biologist (who shot a number of her colleagues to death in 2010). Evolutionary biology is incompatible with [Christian] scriptural teaching. Scriptural teaching is the only grounding for morality. Thus, evolutionary biologists are immoral.

(13) Taxation means paying some of your earned income to the government. Some of this income is distributed to others. Paying so that someone else can benefit is slavery. Therefore, taxation is slavery.

46 CHAPTER 2. THE BASICS OF EVALUATING ARGUMENT

Part B For each inference, (i) say whether it is valid, strong, or weak and (ii) explain your answer.

(1) The sun has come up in the east every day in the past. So, the sun will come up in the east tomorrow.

(2) Jack’s dog Jim will die before the age of 73 (in human years). After all, you are familiar with lots of dogs, and lots of di↵erent kinds of dogs, and any dog that is now dead died before the age of 73 (in human years).

(3) Any time the public receives a tax rebate, consumer spending increases, and the economy is stimulated. Since the public just received a tax rebate, consumer spending will increase.

(4) 90% of the marbles in the box are blue. So, about 90% of the 20 I pick at random will be blue.

(5) According to the world-renowned physicist Stephen Hawking, quarks are one of the fundamental particles of matter. So, quarks are one of the fundamental particles of matter.

(6) Sean Penn, Susan Sarandon and Tim Robbins are actors, and Democrats. So, most actors are Democrats.

(7) The President’s approval rating has now fallen to 53%, employment is at a 10 year high, and he is in charge of two foreign wars. He would not win another term in two years’ time, if he were to run.

(8) If Bill Gates owns a lot of gold then Bill Gates is rich, and Bill Gates doesn’t own a lot of gold. So, Bill Gates isn’t rich.

(9) All birds have wings, and all vertebrates have wings. So, all birds are vertebrates.

(10) U.S. President Obama gave a speech in Berlin shortly after his inauguration. Berlin, of course, is where Hitler gave many speeches. Thus, Obama intends to establish a socialist system in the U.S.

(11) Einstein said that he believed in a god only in the sense of a pantheistic god equivalent with nature. Thus, there is no god in the Judeo-Christian sense.

(12) The United States Congress has more members than there are days in the year. Thus, at least two members of the United States Congress celebrate their birthdays on the same day of the year.

(13) The base at Guantanamo ought to be closed. The continued incarceration of prisoners without any move to try or release them provides terrorist organizations with an e↵ective recruiting tool, perhaps leading to attacks against Americans overseas.

(14) Smith and Jones surveyed teenagers (13-19 years old) at a local mall and found that 94% of this group owned a mobile phone. Therefore, they concluded, about 94% of all teenagers own mobile phone.

(15) Janice Brooks is an unfit mother. Her Facebook and Twitter records show that in the hour prior to the youngest son’s accident she had sent 50 messages — any parent who spends this much time on social media when they have kids is not giving them proper attention.

2.3. STRONG, COGENT, DEDUCTIVE, INDUCTIVE 47

Key Terms

Cogent

Cognitive Bias

Confirmation bias

Deductive

Fallacy

Inductive

Invalid

Sound

Strong

Valid

Weak

48 CHAPTER 2. THE BASICS OF EVALUATING ARGUMENT

Chapter 3

What is Formal Logic?

3.1 Formal as in Concerned with the Form of Things

The chapters in Part B deal with formal logic. Formal logic is distinguished from other branches of logic by the way it achieves content neutrality. Back on page 4, we said that a distinctive feature of logic is that it is neutral about the content of the argument it evaluates. If a kind of argument is strong—say, a kind of statistical argument—it will be strong whether it is applied to sports, politics, science or whatever. Formal logic takes radical measures to ensure content neutrality: it removes the parts of a statement that tie it to particular objects in the world and replaces them with abstract symbols. (See page 4)

Consider the two arguments from Figure 2.7 again:

P1: Socrates is a person.

P2: All persons are mortal.

C: Socrates is mortal.

P1: Socrates is a person.

P2: All people are carrots.

C: Socrates is a carrot.

These arguments are both valid. In each case, if the premises were true, the conclusion would have to be true. (In the case of the first argument, the premises are actually true, so the argument is sound, but that is not what we are concerned with right now.) What makes these arguments valid is that they are put together the right way. Another way of thinking about this is to say that they have the same logical form. Both arguments can be written like this:

P1: S is M .

P2: All M are P .

49

50 CHAPTER 3. WHAT IS FORMAL LOGIC?

C: S is P .

In both arguments S stands for Socrates and M stands for person. In the first argument, P stands for mortal; in the second, P stands for carrot. The letters ‘S’, ‘M’, and ‘P’ are variables. They are just like the variables you may have learned about in algebra class. In algebra, you had equations like y = 2x+ 3, where x and y were variables that could stand for any number. Just as x could stand for any number in algebra, ‘S’ can stand for any name in logic. In fact, this is one of the original uses of variables. Long before variables were used to stand for numbers in algebra, they were used to stand for classes of things, like people or carrots, by Aristotle in his book the Prior Analytics. At about the same time, over in India, the ancient grammarian and linguist Pān. ini was also using variables to represent possible sounds that could be used in di↵erent forms of a word (Pān. ini 2015). Both thinkers introduce their variables fairly causally, as if their readers would be familiar with the idea, so it may be that people prior to them actually invented the variable.

Whoever invented it, the variable was one of the most important conceptual innovations in human history, right up there with the invention of the zero, or alphabetic writing. The importance of the variable for the history of mathematics is obvious. But it was also incredibly important in one of its original fields of application, logic. For one thing, it allows logicians to be more content neutral. We can set aside any associations we have with people, or carrots, or whatever, when we are analyzing an argument. More importantly, once we set aside content in this way, we discover that something incredibly powerful is left over, the logical structure of the sentence itself. This is what we investigate when we study formal logic. In the case of the two arguments above, identifying the logical structure of statements reveals not only that the two arguments have the same logical form, but they have an impeccable logical form. Both arguments are valid, and any other arguments that have this form will be valid.

When Aristotle introduced the variable to the study of logic he used it the way we did in the argument above. His variables stood for names and categories in simple two-premise arguments called syllogisms. The system of logic Aristotle outlined became the dominant logic in the Western world for more than two millennia. It was studied and elaborated on by philosophers and logicians from Baghdad to Paris. The thinkers that carried on Aristotelian tradition were divided by language and religion. They were pagans, Muslims, Jews, and Christians writing typically in Greek, Latin or Arabic. But they were all united by the sense that the tools Aristotle had given them allowed them to see something profound about the nature of reality. They were looking at abstract structures which somehow seemed to be at the foundation of things. As the philosopher and historian of logic Catarina Dutilh Novaes points out, the logic that the thinkers of all these religious traditions were pursuing was formal in that it concerned the forms of things (Dutilh Novaes 2011). As formal logic evolved, however, the idea of being “formal” would take on an additional meaning.

3.2. FORMAL AS IN STRICTLY FOLLOWING RULES 51

3.2 Formal as in Strictly Following Rules

Despite its historical importance, Aristotelean logic has largely been superseded. Starting in the 19th century people learned to do more than simply replace categories with variables. They learned to replicate the whole structure of sentences with a formal system that brought out all sorts of features of the logical form of arguments. The result was the creation of entire artificial languages. An artificial language is a language that was consciously developed by identifiable individuals for some purpose. Esperanto, for instance, is an artificial language developed by Ludwig Lazarus Zamenhof in the 19th century with the hope of promoting world peace by creating a common language for all. J.R.R. Tolkien invented several languages to flesh out the fictional world of his fantasy novels, and even created timelines for their evolution. For Tolkien, the creation of languages was an art form in itself, “An art for which life is not long enough, indeed: the construction of imaginary languages in full or outline for amusement, for the pleasure of the constructor or even conceivably of any critic that might occur” (2016). And it is an art that is really beginning to catch on, especially with Hollywood commissioning languages to be constructed for blockbuster films.

Artificial languages contrast with natural languages, which develop spontaneously and are learned by infants as their first language. Natural languages include all the well-known languages spoken around the world, like English or Japanese or Arabic. It also includes more recently developed languages and evolved spontaneously amongst groups of people. For instance, whenever you put deaf children together, for instance in a boarding school, they will spontaneously develop their own sign language. This phenomenon was important for the development of American Sign Language (ASL) and is part of why ASL counts as a natural language. For similar reasons Nicaraguan Sign Language counts as a natural language, even though it emerged very recently—in the late 1970s and 80s, when the new Sandinista government set up schools for the deaf for the first time. Natural languages can also develop by creolization, when languages merge and children grow up speaking the merged language as their first language. Haitian Creole is the most famous example of this.

The languages developed by logicians are artificial, not natural. Their goal is not to promote global harmony, like Zamenhof’s Esperanto. Nor are they creating art for art’s sake, as Tolkein was, although logical languages can have a great deal of beauty. When the languages first started being developed in the late 19th and early 20th centuries, the goal was, in fact, to have a logically pure language, free of the irrationalities the plague natural languages. More specifically, they had two distinct goals: first, remove all ambiguity and vagueness, and second, to make the logical structure of the language immediately apparent, so that the language wore its logical structure on its face, as it were. If such a language could be developed, it would help us solve all kinds of problems. The logician and philosopher Rudolf Carnap, for instance, felt that the right artificial language could simply make philosophical problems disappear (Carnap 2003).

The languages developed by logicians in the late 19th and early 20th centuries got labeled formal languages, in part because the logicians in question were working in the tradition of formal logic that was already established. A shift began to happen here with the meaning of formal, however, a change which is well documented by Dutilh Novaes (2011). Logicians began to hope that the languages that were being developed were so logical that everything about them could be

52 CHAPTER 3. WHAT IS FORMAL LOGIC?

characterized by a machine. A machine could be used to create sentences in this language, and then again to identify all the valid arguments in this language. This brings out another sense of the word “formal.” As Dutilh Novaes puts it (2011) instead of being “formal” in the sense of concerning the forms of things, logic was formal in the sense that it followed rules perfectly precisely. You might compare this to the way a “formal hearing” in a court of law follows the rule of law to the letter.

For the purposes of this textbook, we will say that the core idea of a formal language is that it is an artificial language designed to bring out the logical structure of ideas and remove all the ambiguity and vagueness that plague natural languages like English. We will further add that sometimes, formal languages are languages that can be implemented by a machine. Creating formal languages always involves all kinds of trade o↵s. On the one hand, we are trying to create a language that makes a logical structure clear and obvious. This will require simplifying things, removing excess baggage from the language. On the other hand, we want to make the language perfectly precise, free of vagueness and ambiguity. This will mean adding complexity to the language. The other thing was that it was very important for the people developing these languages that you be able to prove the all the truths of mathematics in them. This meant that the languages had to have a certain scope.

This was a trade o↵ no logician was ever able to get perfectly correct, because, as it turns out, a logically pure language is impossible. No formal language can do everything that a natural language can do. Logicians became convinced of this, naturally enough, because of a pair of logical proofs. In 1931, the logician Kurt Gödel showed that you couldn’t do all of mathematics in a consistent logical system, which was enough to persuade most of the logicians engaged in the project to drop it. There is a more general problem with the idea of a purely logical language, though, which is that that many of the features logicians were trying to remove from language were actually necessary to make it function. Arika Okrent puts the point quite well. For Okrent, the failure of artificial languages is precisely what illuminates the virtues of natural language.

[By studying artificial languages we] gain a deeper appreciation of natural language and the messy qualities that give it so much flexibility and power and that a simple communication device. The ambiguity and lack of precision allow to serve as a instrument of thought formation, of experimentation and discovery. We don’t know exactly what we mean before we speak; we can figure it out as we go along,. We can talk just to talk, to be social, to feel connected, to participate. At the same time natural language still works as an instrument of thought transmission, one that can be made extremely precise and reliable when we need it to be, or left loose and sloppy when we can’t spare the time or e↵ort (Okrent 2009)

The languages developed in the late 19th and early 20th centuries had goals that were theoretical, rather than practical. They languages were meant to improve our understanding of the world for the sake of improving our understanding of the world. They failed at this theoretical goal, but they wound up having a practical spin-o↵ of world-historical proportions, which is why formal logic is a thriving discipline to this day. Remember that in this period people started thinking of formal languages as languages that could be implemented mechanically. At first, the idea of a a mechanistic language was a metaphor. The rules that were being followed to the letter

3.3. MORE LOGICAL NOTIONS FOR FORMAL LOGIC 53

were to be followed by a human being actually writing down symbols. This human being was generally referred to as a “computer,” because they were computing things. The world changed when a logician named Alan Turing started using literal machines to be computers.

In the 1930s, Turing developed the idea of a reasoning machine that could compute any function. At first, this was just an abstract idea: it involved an infinite stretch of tape. But during World War II, Turing went to work the British code breaking e↵ort at Bletchley Park. The Nazis encoded messages using a device called the Enigma Machine. The Allies had captured one, but since they settings on the machine were reshu✏ed for each message, it didn’t do them much good. Turing, together with people like the mathematicians Gordon Welchman and Joan Clarke, managed to build another machine that could test Enigma settings rapidly to identify the configuration being used. People had made computing machines before, but now the science of logic was so much more advanced that they real power of mechanical computing could be exploited. The human computers became the fully programmable machines we know today, and the formal languages logicians created for theoretical reasons came the computer languages the world of the 21st century depends on. (All of this information, plus lots of fascinating pictures and diagrams, is available at www.turing.org.uk.)

Part B of this book begins by exploring the world of Aristotelian logic, where logic is “formal” in the sense of being about the forms of things. Chapter ?? looks at the logical structure of the individual statements studied by the Aristotelian tradition. Chapter ?? then builds these into valid arguments. After we study Aristotelian logic, we will develop two formal languages, called SL and QL. Chapters 4 through 6 develop SL. In SL, the smallest units are individual statements. Simple statements are represented as letters and connected with logical connectives like and and not to make more complex statements. Chapters 7 through 9 develop QL. In QL, the basic units are objects, properties of objects, and relations between objects.

3.3 More Logical Notions for Formal Logic

Part I covered the basic concepts you need to study any kind of logic. When we study formal logic, we will be interested in some additional logical concepts, which we will explain here.

Truth values

A truth value is the status of a statement as true or false. Thus the truth value of the sentence “All dogs are mammals” is “True,” while the truth value of “All dogs are reptiles” is false. More precisely, a truth value is the status of a statement with relationship to truth. We have to say this, because there are systems of logic that allow for truth values besides “true” and “false,” like “maybe true,” or “approximately true,” or “kinda sorta true.” For instance, some philosophers have claimed that the future is not yet determined. If they are right, then statements about what will be the case are not yet true or false. Some systems of logic accommodate this by having an additional truth value. Other formal languages, so-called paraconsistent logics, allow for statements that are both true and false. We won’t be dealing with those in this textbook,

54 CHAPTER 3. WHAT IS FORMAL LOGIC?

however. For our purposes, there are two truth values, “true” and “false,” and every statement has exactly one of these. Logical systems like ours are called bivalent.

Tautology, contingent statement, contradiction

In considering arguments formally, we care about what would be true if the premises were true. Generally, we are not concerned with the actual truth value of any particular statements— whether they are actually true or false. Yet there are some statements that must be true, just as a matter of logic.

Consider these statements:

(a) It is raining.

(b) Either it is raining, or it is not.

(c) It is both raining and not raining.

In order to know if statement (a) is true, you would need to look outside or check the weather channel. Logically speaking, it might be either true or false. Statements like this are called contingent statements.

Statement (b) is di↵erent. You do not need to look outside to know that it is true. Regardless of what the weather is like, it is either raining or not. If it is drizzling, you might describe it as partly raining or in a way raining and a way not raining. However, our assumption of bivalence means that we have to draw a line, and say at some point that it is raining. And if we have not crossed this line, it is not raining. Thus the statement “either it is raining or it is not” is always going to be true, no matter what is going on outside. A statement that has to be true, as a matter of logic is called a tautology or logical truth.

You do not need to check the weather to know about statement (c), either. It must be false, simply as a matter of logic. It might be raining here and not raining across town, it might be raining now but stop raining even as you read this, but it is impossible for it to be both raining and not raining here at this moment. The third statement is logically false; it is false regardless of what the world is like. A logically false statement is called a contradiction.

We have already said that a contingent statement is one that could be true, or could be false, as far as logic is concerned. To be more precise, we should define a contingent statement as a statement that is neither a tautology nor a contradiction. This allows us to avoid worrying about what it means for something to be logically possible. We can just piggyback on the idea of being logically necessary or logically impossible.

A statement might always be true and still be contingent. For instance, it may be the case that in no time in the history of the universe was there ever an elephant with tiger stripes. Elephants only ever evolved on Earth, and there was never any reason for them to evolve tiger stripes. The

3.3. MORE LOGICAL NOTIONS FOR FORMAL LOGIC 55

statement “Some elephants have tiger stripes,” is therefore always false. It is, however, still a contingent statement. The fact that it is always false is not a matter of logic. There is no contradiction in considering a possible world in which elephants evolved tiger stripes, perhaps to hide in really tall grass. The important question is whether the statement must be true, just on account of logic.

When you combine the idea of tautologies and contradictions with the notion of deductive validity, as we have defined it, you get some curious results. For one thing, any argument with a tautology in the conclusion will be valid, even if the premises are not relevant to the conclusion. This argument, for instance, is valid.

P1: There is co↵ee in the co↵ee pot.

P2: There is a dragon playing bassoon on the armoire.

C: All bachelors are unmarried men.

The statement “All bachelors are unmarried men” is a tautology. No matter what happens in the world, all bachelors have to be unmarried men, because that is how the word “bachelor” is defined. But if the conclusion of the argument is a tautology, then there is no way that the premises could be true and the conclusion false. So the argument must be valid.

Even though it is valid, something seems really wrong with the argument above. The premises are not relevant to the conclusion. Each sentence is about something completely di↵erent. This notion of relevance, however, is something that we don’t have the ability to capture in the kind of simple logical systems we will be studying. The logical notion of validity we are using here will not capture everything we like about arguments.

Another curious result of our definition of validity is that any argument with a contradiction in the premises will also be valid. In our kind of logic, once you assert a contradiction, you can say anything you want. This is weird, because you wouldn’t ordinarily say someone who starts out with contradictory premises is arguing well. Nevertheless, an argument with contradictory premises is valid.

Logically Equivalent and Contradictory Pairs of Sentences

We can also ask about the logical relations between two statements. For example:

(a) John went to the store after he washed the dishes.

(b) John washed the dishes before he went to the store.

These two statements are both contingent, since John might not have gone to the store or washed dishes at all. Yet they must have the same truth value. If either of the statements is true,

56 CHAPTER 3. WHAT IS FORMAL LOGIC?

then they both are; if either of the statements is false, then they both are. When two statements necessarily have the same truth value, we say that they are logically equivalent.

On the other hand, if two sentences must have opposite truth values, we say that they are contradictories. Consider these two sentences

(a) Susan is taller than Monica.

(b) Susan is shorter or the same height as Monica.

One of these sentences must be true, and if one of the sentences is true, the other one is false. It is important to remember the di↵erence between a single sentence that is a contradiction and a pair of sentences that are contradictory. A single sentence that is a contradiction is in conflict with itself, so it is never true. When a pair of sentences is contradictory, one must always be true and the other false.

Consistency

Consider these two statements:

(a) My only brother is taller than I am.

(b) My only brother is shorter than I am.

Logic alone cannot tell us which, if either, of these statements is true. Yet we can say that if the first statement (a) is true, then the second statement (b) must be false. And if (b) is true, then (a) must be false. It cannot be the case that both of these statements are true. It is possible, however that both statements can be false. My only brother could be the same height as I am.

If a set of statements could not all be true at the same time, they are said to be inconsistent. Otherwise, they are consistent.

We can ask about the consistency of any number of statements. For example, consider the following list of statements:

(a) There are at least four gira↵es at the wild animal park.

(b) There are exactly seven gorillas at the wild animal park.

(c) There are not more than two Martians at the wild animal park.

(d) Every gira↵e at the wild animal park is a Martian.

3.3. MORE LOGICAL NOTIONS FOR FORMAL LOGIC 57

Statements (a) and (d) together imply that there are at least four Martian gira↵es at the park. This conflicts with (c), which implies that there are no more than two Martian gira↵es there. So the set of statements (a)–(d) is inconsistent. Notice that the inconsistency has nothing at all to do with (b). Statement (b) just happens to be part of an inconsistent set.

Sometimes, people will say that an inconsistent set of statements “contains a contradiction.” By this, they mean that it would be logically impossible for all of the statements to be true at once. A set can be inconsistent even when all of the statements in it are either contingent or tautologous. When a single statement is a contradiction, then that statement alone cannot be true.

Practice Exercises

Part A Label the following tautology, contradiction, or contingent statement.

Example: Caesar crossed the Rubicon.

Answer: Contingent statement.

The Rubicon is a river in Italy. When General Julius Caesar took his army across it, he was committing to a revolution against the Roman Republic. Since that time, “crossing the Rubicon” has been a expression referring to making an irreversible decision. This kind of decision certainly seems to be contingent. Caesar could have decided otherwise.

(1) Someone once crossed the Rubicon.

(2) No one has ever crossed the Rubicon.

(3) If Caesar crossed the Rubicon, then someone has.

(4) Even though Caesar crossed the Rubicon, no one has ever crossed the Rubicon.

(5) If anyone has ever crossed the Rubicon, it was Caesar.

Part B Label the following tautology, contradiction, or contingent statement.

(1) Elephants dissolve in water.

(2) Wood is a light, durable substance useful for building things.

(3) If wood were a good building material, it would be useful for building things.

(4) I live in a three story building that is two stories tall.

(5) If gerbils were mammals they would nurse their young.

Part C Label the following logically equivalent, contradictory, or neither.

Example: All students who study will pass the test.

If Jeremy studies, he will pass the test.

Answer: Neither.

58 CHAPTER 3. WHAT IS FORMAL LOGIC?

If the first statement is true, then the second statement has to be true, but the reverse is not the case. It might be that Jeremy will pass the test if he studies, but some other students are going to fail no matter what.

(1) Elephants dissolve in water. If you put an elephant in water, it will dissolve.

(2) All mammals dissolve in water. If you put an elephant in water, it will dissolve.

(3) Elephants are bigger than lions. Elephants are smaller or the same size as lions.

(4) The Eurasian elephant is an herbivore All the Eurasian elephant sometimes eats meat

(5) Elephants dissolve in water. All mammals dissolve in water.

Part D Label the following logically equivalent, contradictory, or neither.

(1) Thelonious Monk played piano. John Coltrane played tenor sax.

(2) Thelonious Monk played gigs with John Coltrane. John Coltrane played gigs with Thelonious Monk.

(3) All professional piano players have big hands. Piano player Bud Powell had big hands.

(4) Bud Powell su↵ered from severe mental illness. All piano players su↵er from severe mental illness.

(5) John Coltrane was deeply religious. John Coltrane was moderately or not at all religious

Part E Consider again the statements on p.56:

(a) There are at least four gira↵es at the wild animal park.

(b) There are exactly seven gorillas at the wild animal park.

(c) There are not more than two Martians at the wild animal park.

(d) Every gira↵e at the wild animal park is a Martian.

Now mark each of the following sets of statements consistent or inconsistent.

Example: Statements (a), (c), and (d)

Answer: Inconsistent. If there are at least four gira↵es, and every one of them is Martian, there can’t be no more than two Martians in the park.

3.3. MORE LOGICAL NOTIONS FOR FORMAL LOGIC 59

(1) Statements (b), (c), and (d)

(2) Statements (a), (b), (c), and (d)

(3) Statements (a), (b), and (d)

(4) Statements (a), (b), and (c)

Part F Consider the following set of statements.

(a) All people are mortal.

(b) Socrates is a person.

(c) Socrates will never die.

(d) Socrates is mortal.

Which combinations of statements form consistent sets? Mark each “consistent” or “inconsistent.”

(1) Statements (a), (b), and (c)

(2) Statements (b), (c), and (d)

(3) Statements (b) and (c)

(4) Statements (a) and (d)

(5) Statements (a), (b), (c), and (d)

Part G Which of the following is possible? If it is possible, give an example. If it is not possible, explain why.

Example: A valid argument that has one false premise and one true premise.

Answer: Possible: Example: If Taylor Swift were a kangaroo, she would be a marsupial (true). Taylor Swift is a kangaroo. (False.) Therefore Taylor Swift is a marsupial (false.)

Remember, if an argument is valid, the only thing that can’t happen is for it to have all true premises and a false conclusion. So if you don’t specify a false conclusion anything is possible.

(1) A false tautology.

(2) A valid argument that has a false conclusion

(3) A valid argument, the conclusion of which is a contradiction

(4) An invalid argument, the conclusion of which is a tautology

(5) A tautology that is contingent

(6) Two logically equivalent sentences, both of which are tautologies

(7) Two logically equivalent sentences, one of which is a tautology and one of which is contingent

60 CHAPTER 3. WHAT IS FORMAL LOGIC?

(8) Two logically equivalent sentences that together are an inconsistent set

(9) A consistent set of sentences that contains a contradiction

(10) An inconsistent set of sentences that contains a tautology

Part H Which of the following is possible? If it is possible, give an example. If it is not possible, explain why.

(1) A valid argument, whose premises are all tautologies, and whose conclusion is contingent

(2) A valid argument with true premises and a false conclusion

(3) A consistent set of sentences that contains two sentences that are not logically equivalent

(4) A consistent set of sentences, all of which are contingent

(5) A false tautology

(6) A valid argument with false premises

(7) A logically equivalent pair of sentences that are not consistent

(8) A tautological contradiction

(9) A consistent set of sentences that are all contradictions

(10) A valid argument, whose premises are all tautologies, and whose conclusion is contingent.

Key Terms

Artificial language

Bivalent

Consistent

Contingent statement

Contradiction

Contradictories

Formal language

Formal logic as concern for logical form

Formal logic as strictly following rules

Inconsistent

Logically equivalent

Natural language

3.3. MORE LOGICAL NOTIONS FOR FORMAL LOGIC 61

Tautology

Truth value

62 CHAPTER 3. WHAT IS FORMAL LOGIC?

Part II

Sentential Logic

Chapter 4

Sentential Logic

This chapter introduces a logical language called SL. It is a version of sentential logic, because the basic units of the language will represent statements, and a statement is usually given by a complete sentence in English.

4.1 Sentence Letters

The most basic unit in our formal language SL is an individual capital letter—A,B,C,D, etc. These letters, called sentence letters, are used to represent individual statements. Remember in section 1.2, we defined a statement as some bit of language that can be true or false, and listed all kinds of things that count as statements in English, from “Tyrannosaurus rex went extinct 65 million years ago” to “Lady Gaga is pretty.” In SL, all these statements are reduced to single capital letters.

Considered only as a symbol of SL, the letter A could mean any statement. In order to specify what we mean, we need to provide a key saying what the sentence letters represent. We will call a list that assigns English phrases or sentences to variable names a translation key. These are sometimes also called “symbolization keys” or simply just “dictionaries.”

Consider this argument (recall that the portion of the passage in italics establishes the context, and is not part of the passage):

A teacher is looking to see who has come to class There is an apple on the desk. If there is an apple on the desk, then Jenny made it to class. Therefore, Jenny made it to class.

In canonical form, the argument would look like this:

65

66 CHAPTER 4. SENTENTIAL LOGIC

1. There is an apple on the desk.

2. If there is an apple on the desk, then Jenny made it to class.

. .

.˙. Jenny made it to class.

A good symbolization key for this passage would look like this:

A: There is an apple on the desk. B: Jenny made it to class.

Why do the symbolization key this way? The argument we are looking at is obviously valid in English. In symbolizing it, we want to preserve the structure of the argument that makes it valid. We could have made each sentence in the original argument into its own letter. Then the symbolization key would look like this:

A: There is an apple on the desk. B: If there is an apple on the desk, then Jenny made it to class. C: Jenny made it to class.

But that would mean the argument would look like this:

1. A

2. B

. .

.˙. C

There is no necessary connection between some sentence A, which could be any statement, and some other sentences B and C, which could also be anything. The structure of the argument has been completely lost in this translation.

The important thing about the argument is that the second premise is not merely any statement, logically divorced from the other statement in the argument. The second premise contains the first premise and the conclusion as parts. Our original symbolization key allows us to write the argument like this.

1. A

2. If A, then B.

. .

.˙. B

4.2. SENTENTIAL CONNECTIVES 67

This preserves the structure of the argument that makes it valid, but it still makes use of the English expression “If. . . then. . ..” Although we ultimately want to replace all of the English expressions with logical notation, this is a good start.

The individual sentence letters in SL are called atomic sentences, because they are the basic building blocks out of which more complex sentences can be built. We can identify atomic sentences in English as well. An atomic sentence is one that cannot be broken into parts that are themselves sentences. “There is an apple on the desk” is an atomic sentence in English, because you can’t find any proper part of it that forms a complete sentence. For instance “an apple on the desk” is a noun phrase, not a complete sentence. Similarly “on the desk” is a prepositional phrase, and not a sentence, and “is an” is not any kind of phrase at all. This is what you will find no matter how you divide “There is an apple on the desk.” On the other hand you can find two proper parts of “If there is an apple on the desk, then Jenny made it to class” that are complete sentences: “There is an apple on the desk” and “Jenny made it to class.” As a general rule, we will want to use atomic sentences in SL (that is, the sentence letters) to represent atomic sentences in English. Otherwise, we will lose some of the logical structure of the English sentence, as we have just seen.

There are only 26 letters of the alphabet, but there is no logical limit to the number of atomic sentences. We can use the same letter to symbolize di↵erent atomic sentences by adding a subscript, a small number written after the letter. We could have a symbolization key that looks like this:

A1: The apple is under the armoire. A2: Arguments in SL always contain atomic sentences. A3: Adam Ant is taking an airplane from Anchorage to Albany.

... A294: Alliteration angers otherwise a↵able astronauts.

Keep in mind that each of these is a di↵erent sentence letter. When there are subscripts in the symbolization key, it is important to keep track of them.

4.2 Sentential Connectives

The previous section introduced the basic elements of SL, the sentence letters. But when we were looking at the argument involving Jenny and the apple, we saw that the best way to write a dictionary for the argument left the words “if” and “then” in English. In this section we will introduce ways to connect the sentence letters together that will allow us to form a complete artificial language.

The symbols used to connect sentence letters are called sentential connectives , naturally enough. SL uses five sentential connectives: & , _, ⇠, !, and $. To write the sentence about Jenny and the apple we use the symbol “!.” Using the dictionary above, “If there is an apple on

68 CHAPTER 4. SENTENTIAL LOGIC

Symbol What it is called What it means

⇠ negation “It is not the case that. . .” & conjunction “Both . . . and . . .”

_ disjunction “Either . . . or . . .” ! conditional “If . . . then . . .” $ biconditional “. . . if and only if . . .”

Table 4.1: The Sentential Connectives.

the desk, then Jenny made it to class” becomes A ! B. Table 4.1 summarizes the meaning of the five sentential connectives.

The sentential connectives are a kind of logical constant, because their meaning is fixed by the formal language that we have chosen. The other logical constants in SL are the parentheses. These are the things we cannot change in the symbolization key. The sentence letters, by contrast, are nonlogical symbols, because their meaning can change as we change the symbolization key. We can decide that A stands for “Arthur is an aardvark” in one translation key and “Apu is an anthropologist” in the next. But we can’t say that the the ⇠ symbol will mean “not” in one argument and “perhaps” in another.

The subsections below describe each connective in more detail.

Negation

Consider how we might symbolize these sentences:

1. Mary is in Barcelona. 2. Mary is not in Barcelona. 3. Mary is somewhere other than Barcelona.

In order to symbolize sentence 1, we will need one sentence letter. We can provide a symbolization key:

B: Mary is in Barcelona.

Note that here we are giving B a di↵erent interpretation than we did in the previous section. The symbolization key only specifies what B means in a specific context. It is vital that we continue to use this meaning of B so long as we are talking about Mary and Barcelona. Later, when we are symbolizing di↵erent sentences, we can write a new symbolization key and use B to mean something else.

4.2. SENTENTIAL CONNECTIVES 69

Now, sentence 1 is simply B. Sentence 2 is obviously related to sentence 1: it is basically 1 with a “not” added. We could put the sentence partly our symbolic language by writing “Not B.” This means we do not want to introduce a di↵erent sentence letter for 2. We just need a new symbol for the “not” part. Let’s use the symbol ‘⇠,’ which we will call negation. Now we can translate ‘Not B’ to ⇠B.

Sentence 3 is about whether or not Mary is in Barcelona, but it does not contain the word “not.” Nevertheless, it is obviously logically equivalent to sentence 2. They both say that if you are looking for Mary, you shouldn’t look in Barcelona. Remember that in section 3.3, we said that two sentences in English are logically equivalent if they always have the same truth value. For our purposes, this means that they basically say the same thing. It is clear then that 2 and 3 are logically equivalent, so we can translate them both as ⇠B.

Consider these further examples:

4. The widget can be replaced if it breaks. 5. The widget is irreplaceable. 6. The widget is not irreplaceable.

If we let R mean “The widget is replaceable”, then sentence 4 can be translated as R. Sentence 5 means the opposite of sentence 4, so we can translate it ⇠R. Sentence ?? adds another negation to sentence 5. We know, as competent English speakers, that the two negations cancel each other out, so that sentence ?? is equivalent to sentence 4. But the fact that two negations cancel each other out is a part of the logic of English that we actually want to capture with our formal language SL. So we will represent the two negations in sentence 6 as two negations in SL: ⇠⇠R. We will now have to be sure that in SL the sentences R and ⇠⇠R mean the same thing.

As the above examples begin to indicate, English has all kinds of ways to negate a sentence. Sometimes we use an explicit “not.” Sometimes we use a prefix like the “ir-” in “irreplaceable.” SL has just one way to form a negation: slap a ⇠in front of the sentence. There is an English expression, however, that always occurs in the same place in an English sentence as the ⇠occurs in the sentence SL. The English phrase is “It is not the case that.” Although this phrase sounds awkward, it always occurs in front of the sentence it is negating, just as the symbol ⇠does. This makes it useful in translating sentences from SL back into English. ⇠R can be translated “it is not the case that this widget is replaceable.” In the earlier example, ⇠B can be translated “It is not the case that Mary is in Barcelona.”

A sentence can be symbolized as ⇠A can always be paraphrased in English as “It is not the case that A .”

Sometimes negations in English do not function as neatly as the ⇠does in SL, because two things aren’t perfect opposites. Consider these sentences:

7. Elliott is happy.

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8. Elliott is unhappy.

If we let H mean “Elliot is happy”, then we can symbolize sentence 7 as H, but does 8 really mean the same thing as ⇠H? Saying “Elliott is unhappy” indicates that Elliott is actively sad. But ⇠H can be paraphrase as simply “It is not the case that Elliott is happy,” which might merely mean that Elliott is just feeling neutral. As we saw on page ??, the logics we discuss in this textbook are bivalent. Statements are only either true or false. Everything is in black and white, and issues like Elliott’s fine gradations in mood cannot be directly represented in our system. So in SL, sentences 7 and 8 would generally be represented by separate sentence letters.

One way of capturing the meaning of a sentential connective is to make a table which shows how the connective changes the meaning of the sentences it is applied to. The negation simply reverses the truth value of any sentence it is put in front of. For any sentence A : If A is true, then ⇠A is false. If ⇠A is true, then A is false. Using T for true and F for false, we can summarize this in a characteristic truth table for negation:

A ⇠A T F F T

We will discuss truth tables at greater length in the next chapter.

Conjunction

Consider these sentences:

9. Adam is athletic. 10. Barbara is athletic. 11. Adam is athletic, and Barbara is also athletic.

We will need separate sentence letters for 9 and 10, so we define this symbolization key:

A: Adam is athletic. B: Barbara is athletic.

Sentence 9 can be symbolized as A. Sentence 10 can be symbolized as B. Sentence 11 can be paraphrased as “A and B.” In order to fully symbolize this sentence, we need another symbol. We will use & . We translate “A and B” as A&B. The logical connective & is called the conjunction, and A and B are each called conjuncts.

Notice that we make no attempt to symbolize “also” in sentence 11. Words like “both” and “also” function to draw our attention to the fact that two things are being conjoined. They are not doing any further logical work, so we do not need to represent them in SL.

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Some more examples:

12. Barbara is athletic and energetic. 13. Barbara and Adam are both athletic. 14. Although Barbara is energetic, she is not athletic. 15. Barbara is athletic, but Adam is more athletic than she is.

Sentence 12 is obviously a conjunction. The sentence says two things about Barbara, so in English it is permissible to refer to Barbara only once. It might be tempting to try this when translating the argument: Since B means “Barbara is athletic”, one might paraphrase the sentences as “B and energetic.” This would be a mistake. Once we translate part of a sentence as B, any further structure is lost. B is an atomic sentence; it is nothing more than true or false. Conversely, “energetic” is not a sentence; on its own it is neither true nor false. We should instead paraphrase the sentence as “B and Barbara is energetic.” Now we need to add a sentence letter to the symbolization key. Let E mean “Barbara is energetic.” Now the sentence can be translated as B&E.

A sentence can be symbolized as A &B if it can be paraphrased in English as ‘Both A , and B .’ Each of the conjuncts must be a sentence.

Sentence 13 says one thing about two di↵erent subjects. It says of both Barbara and Adam that they are athletic, and in English we use the word “athletic” only once. In translating to SL, it is important to realize that the sentence can be paraphrased as, “Barbara is athletic, and Adam is athletic.” This translates as B&A.

Sentence 14 is a bit more complicated. The word “although” sets up a contrast between the first part of the sentence and the second part. Nevertheless, the sentence says both that Barbara is energetic and that she is not athletic. In order to make each of the conjuncts an atomic sentence, we need to replace “she” with “Barbara.”

So we can paraphrase sentence 14 as, “Both Barbara is energetic, and Barbara is not athletic.” The second conjunct contains a negation, so we paraphrase further: “Both Barbara is energetic and it is not the case that Barbara is athletic.” This translates as E&⇠B.

Sentence 15 contains a similar contrastive structure. It is irrelevant for the purpose of translating to SL, so we can paraphrase the sentence as “Both Barbara is athletic, and Adam is more athletic than Barbara.” (Notice that we once again replace the pronoun “she” with her name.) How should we translate the second conjunct? We already have the sentence letter A which is about Adam’s being athletic and B which is about Barbara’s being athletic, but neither is about one of them being more athletic than the other. We need a new sentence letter. Let R mean “Adam is more athletic than Barbara.” Now the sentence translates as B&R.

Sentences that can be paraphrased “A , but B” or “Although A , B” are best symbolized using conjunction A &B .

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It is important to keep in mind that the sentence letters A, B, and R are atomic sentences. Considered as symbols of SL, they have no meaning beyond being true or false. We have used them to symbolize di↵erent English language sentences that are all about people being athletic, but this similarity is completely lost when we translate to SL. No formal language can capture all the structure of the English language, but as long as this structure is not important to the argument there is nothing lost by leaving it out.

As with the negation, we can understand the meaning of the conjunction by making a table that shows how the conjunction a↵ects the truth value of the sentences it is bringing together. For any sentences A and B , A &B is true if and only if both A and B are true. We can summarize this in the characteristic truth table for conjunction:

A B A &B T T T T F F F T F F F F

Conjunction is symmetrical because we can swap the conjuncts without changing the truth value of the sentence. Regardless of what A and B are, A &B is logically equivalent to B &A .

Disjunction

Consider these sentences:

16. Either Denison will play golf with me, or he will watch movies. 17. Either Denison or Ellery will play golf with me.

For these sentences we can use this symbolization key:

D: Denison will play golf with me. E: Ellery will play golf with me. M: Denison will watch movies.

Sentence 16 is “Either D or M .” To fully symbolize this, we introduce a new symbol. The sentence becomes D _M . The _ connective is called disjunction, and D and M are called disjuncts.

Sentence 17 is only slightly more complicated. There are two subjects, but the English sentence only gives the verb once. In translating, we can paraphrase it as “Either Denison will play golf with me, or Ellery will play golf with me.” Now it obviously translates as D _ E.

4.2. SENTENTIAL CONNECTIVES 73

A sentence can be symbolized as A _ B if it can be paraphrased in English as “Either A or B .” Each of the disjuncts must be a sentence.

The English word “or” is somewhat ambiguous. Sometimes in English, when we say “this or that,” we mean that either option is possible, but not both. For instance, if a restaurant menu says, “Entrées come with either soup or salad” we naturally assume you can have soup, or you can have salad; but, if you want both soup and salad, then you will have to pay extra. This kind of disjunction is called an exclusive or , because it excludes the possibility that both disjuncts are true.

At other times, the word “or” allows for the possibility that both disjuncts might be true. This is probably the case with sentence 17, above. I might play with Denison, with Ellery, or with both Denison and Ellery. Sentence 17 merely says that I will play with at least one of them. The inclusive or is the kind of disjunction that allows for the possibility that both disjuncts are true. The inclusive or says “This or that, or both.”

To goal of a formal language is to remove ambiguity, so we need to pick one of these ors. SL follows tradition and uses the symbol _ to represent an inclusive or. This winds up being reflected in the characteristic truth table for the _. The sentence D _ E is true if D is true, if E is true, or if both D and E are true. It is false only if both D and E are false. The truth table looks like this:

A B A _B T T T T F T F T T F F F

Like conjunction, disjunction is symmetrical. A _B is logically equivalent to B _A .

Conditional

We already met the conditional at the start of this section, when we were discussing the sentence “If there is an apple on the table, Jenny made it to class,” which became A ! B. The symbol ! is called a conditional. The sentence on the left-hand side of the conditional (R in this example) is called the antecedent. . The sentence on the right-hand side (B) is called the consequent.

Like the English word “or,” the English phrase “if. . . then. . . ” has some ambiguity. Consider our original example, “If there is an apple on the table, Jenny made it to class.” The statements tells us what we should infer if there is an apple on the table, but what if there isn’t an apple on the table. Does that guarantee that Jenny did not make it to class? It could be that an apple on the table is a clear sign that Jenny made it to class, because no one else would put an apple on the table, but nevertheless Jenny sometimes comes to class without putting an apple on the table.

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We can get a good sense of the decision we face if we try to write up the characteristic truth table for the conditional. The first two lines are easy. The sentence“If A , then B” means that if A is true, then so is B . This would be confirmed by the situation where both A and B are true, but falsified by the situation where A is true and B is false. In terms of our example, if we came to class and found the apple there, but Jenny absent, we would know that the statement “If there is an apple on the table, Jenny made it to class” is false. But if we came to class and found both Jenny and the apple present, we could say that the statement “If there is an apple on the table, Jenny made it to class” is true. That gives us this much of a truth table.

A B A!B T T T T F F F T ? F F ?

How do we fill in the question marks in the last two lines? In real life, we would generally make judgments on a case by case basis, relying heavily on the context we are in. But for a formal language we just want to lay down a simple rule. The traditional solution for sentential logic is to say that the conditional is what logicians call a “material conditional.” If the antecedent of a material conditional is false, then the whole statement is automatically true, regardless of the truth value of B . In short, A !B is false if and only if A is true and B is false. We can summarize this with a characteristic truth table for the conditional.

A B A!B T T T T F F F T T F F T

The conditional is asymmetrical. You cannot swap the antecedent and consequent without changing the meaning of the sentence, because A !B and B !A are not logically equivalent.

Not all sentences of the form “If. . ., then. . .” are conditionals. Consider this sentence:

18. If anyone wants to see me, then I will be on the porch.

When I say this, it means that I will be on the porch, regardless of whether anyone wants to see me or not—but if someone did want to see me, then they should look for me there. If we let P mean “I will be on the porch,” then sentence 18 can be translated simply as P .

Biconditional

The conditional was an asymmetric connective. The sentence A ! B does not mean the same thing as the sentence B ! A. It is convenient to have a single symbol that combines the meaning

4.3. MORE COMPLICATED TRANSLATIONS 75

of these two sentences. The biconditional—written as double headed arrow, $—is a sentential connective used to represent a situation where A implies B and B implies A.

To draw up the characteristic truth table for the biconditional, we need to think about the situations where A ! B and B ! A are false. The sentence A ! B is only false when A is true and B is false. For B ! A the reverse is true. It is false when B is true and A is false. Our biconditional A $ B needs to avoid both of these situations to be true, because it is only true when A ! B and B ! A are true. This, then, is the characteristic truth table for the biconditional. It says that the biconditional is true when the truth values of the two sides match.

A B A $B T T T T F F F T F F F T

If the bioconditional holds between two sentences, we can that the two sentences are logically equivalent. Back on page 56, we said that two sentences were logically equivalent if they always have the same truth value. That is exactly what is happening here.

4.3 More Complicated Translations

The previous section introduced the five sentential connectives. Now we will look at some trickier translations involving those connectives

Combining connectives

A single sentence in SL can use multiple connectives.

Consider the English sentence “If it is not raining, we will have a picnic.” There are two aspects of this sentence we will want to represent with sentential connectives in SL, the “if. . . then. . . ” structure and the negation in the first part of the sentence. The rest of the sentence can be represented by these sentence letters

A: It is raining. M : We will have a picnic.

We can then translate the whole sentence into SL like this: ⇠A ! B. We can make sentences as complicated as we want this way, even to the point where the equivalent English sentence would be impossible to follow. The sentence ⇠(P &Q) ! [(R _ S) $ ⇠(T &U)] is perfectly acceptable in SL, even if any English sentence it translates into would be a monster. This is part

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of the power of a complete formal language like SL, but it is also why arguments in SL begin to resemble the ob/ob mouse more than they resemble any argument you might encounter in the wild. (See page 6)

Although sentences in SL can be as long as you like, you can’t just combine symbols any old way. There is a specific set of rules you have to follow. These are outlined in section 4.3, below.

The fact that we can write these more complicated sentences means we can actually do without some of the connectives we have given ourselves in SL. For instance, we don’t really need the biconditional. Any sentence of the form A $ B is going to be equivalent to the sentence (A ! B)& (B ! A). This just follows from the way we defined the biconditional earlier. Nevertheless, tradition and convenience mandate that we give the biconditional a separate symbol.

Unless

Because our connectives can be put together in di↵erent ways, some English sentences can be represented equally well by multiple sentences in SL. English sentences involving the word “unless” are a case in point.

19. Unless you wear a jacket, you will catch cold. 20. You will catch cold unless you wear a jacket.

These are basically two di↵erent version of the same English sentence. The only di↵erence is that in one case, the “unless” clause comes first, and in the other it comes second. Let J mean “You will wear a jacket” and let C mean “You will catch a cold.” We can paraphrase sentence 19 as “Unless J , C.” This means that if you do not wear a jacket, then you will catch cold. With this in mind, we might translate it as ⇠J ! C. It also means that if you do not catch a cold, then you must have worn a jacket; with this in mind, we might translate it as ⇠C ! J .

Which of these is the correct translation of sentence 19? Both translations are correct, because the two translations are logically equivalent in SL. Sentence 20, in English, is logically equivalent to sentence 19. So, it also can be translated as either ⇠J ! D or ⇠D ! J .

When symbolizing sentences like sentence 19 and sentence 20, it is easy to get turned around. We have two di↵erent versions of the English sentence and two di↵erent versions of the sentence in SL. The important thing to see here is that none of these sentences are equivalent to J ! ⇠D. The negated statement must be the antecedent to the conditional.

If this is too many options to keep track of, there is a simpler alternative. It turns out that any “unless” statement is actually equivalent to an “or” statement. Both statements 19 and 20 mean that you will wear a jacket or—if you do not wear a jacket—then you will catch a cold. So we can translate them as J _D. (You might worry that the “or” here should be an exclusive or. However, the sentences do not exclude the possibility that you might both wear a jacket and catch a cold; jackets do not protect you from all the possible ways that you might catch a cold.)

4.3. MORE COMPLICATED TRANSLATIONS 77

If a sentence can be paraphrased as “Unless A , B ,” then it can be symbolized as A _B .

Only

[The word “only” can reverse the meaning of a conditional sentence in SL.]For the following sentences, let R mean “You will cut the red wire” and B mean “The bomb will explode.”

21. If you cut the red wire, then the bomb will explode. 22. The bomb will explode only if you cut the red wire.

Sentence 21 can be translated partially as “If R, then B.” Sentence 22 is also a conditional. Since the word “if” appears in the second half of the sentence, it might be tempting to symbolize this in the same way as sentence 21. That would be a mistake.

The conditional R ! B says that if R were true, then B would also be true. It does not say that you cutting the red wire is the only way that the bomb could explode. Someone else might cut the wire, or the bomb might be on a timer. The sentence R ! B does not say anything about what to expect if R is false. Sentence 22 is di↵erent. It says that the only conditions under which the bomb will explode involve you having cut the red wire; i.e., if the bomb explodes, then you must have cut the wire. As such, sentence 22 should be symbolized as B ! R.

It is important to remember that the connective ! says only that, if the antecedent is true, then the consequent is true. It says nothing about the causal connection between the two events. Translating sentence 22 as B ! R does not mean that the bomb exploding would somehow have caused you cutting the wire. Both sentence 21 and 22 suggest that, if you cut the red wire, you cutting the red wire would be the cause of the bomb exploding. They di↵er on the logical connection. If sentence 22 were true, then an explosion would tell us—those of us safely away from the bomb—that you had cut the red wire. Without an explosion, sentence 22 tells us nothing.

The paraphrased sentence “A only if B” is logically equivalent to “If A , then B .”

Things can get a bit more complicated, because English also allows you to reverse the order of the clauses. Think about this sentence

23. The bomb will explode, if you cut the red wire

This is just sentence 21 with the order of the clauses reversed, so it still means R ! B. Changing the order of the English clauses does not change the sentence in SL, but adding the word “only” does.

If this gets confusing, just remember this rule:

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“If. . . ” introduces the antecedent. “Only if. . . ” introduces the consequent.

Because “if” and “only if” have opposite meanings, when we put them together, we get the biconditional. Consider these sentences:

24. The figure on the board is a triangle only if it has exactly three sides.

25. The figure on the board is a triangle if it has exactly three sides.

26. The figure on the board is a triangle if and only if it has exactly three sides.

Let T mean “The figure is a triangle” and S mean “The figure has three sides.” Sentence 24, for reasons discussed above, can be translated as T ! S. Sentence 25 is importantly di↵erent. It can be paraphrased as “If the figure has three sides, then it is a triangle.” So it can be translated as S ! T .

Sentence 26 says that T is true if and only if S is true; we can infer S from T , and we can infer T from S. In other words, 26 is equivalent to T ! S and S ! T , which is the same as T $ S

A final way to think about the way “only” e↵ects a conditional sentence is to think about the di↵erence between necessary and su�cient conditions. In a way, the terms are pretty much self explanatory. Nevertheless, it is really easy to get them confused, to the extent that even professional logicians and trained philosophers can get them mixed up.

A necessary condition is one that is needed for something else to be true, just like the name says. Having gas in the tank is a necessary condition for the car to move. It just doesn’t go anywhere without gas. However, having gas in the tank isn’t all you need to get the car moving. You also have to put the key in the ignition and turn it.

A sufficient condition, on the other hand, is all you need for something else to be true. If something is a dog, that is a su�cient condition for it to be a mammal. Once you know Cupcake (Fig. 4.1) is a dog, you have enough information to infer that she is a mammal. Being a dog is not a necessary condition for being a mammal however. You can also be a mammal being being a cat, or a human, or a wombat.

The conditional symbol in SL represents a su�cient condition, at least when read forward. That is, the antecedent is a su�cient condition for the consequent. If you have the antecedent, that is all you need to know to infer the consequent. So if D is “Cupcake is a dog” and M is “Cupcake is a mammal, then D ! C is true. Being a dog is su�cient for being a mammal. As it turns out, if the relationship is su�cient going one direction, it is necessary going the other. So being a mammal is a necessary condition for being a mammal. If cupcake weren’t a mammal, there would be no way for her to be a dog. Figure 4.2 shows this relationship.

4.3. MORE COMPLICATED TRANSLATIONS 79

Figure 4.1: This is Cupcake. The fact that she is a dog is a su�cient condition for her to be a mammal. She also likes socks.

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Figure 4.2: The antecedent of a material conditional is a su�cient condition for the consequent, while the consequent is a necessary condition for the antecedent.

Combining negation with conjunction and disjunction

Tricky things happen when you combine a negation with a conjunction or disjunction, so it is worth taking a closer look here. Consider these sentences

27. Either you will not have soup, or you will not have salad. 28. You will have neither soup nor salad.

We let S1 mean that you get soup and S2 mean that you get salad. Sentence 27 can be paraphrased in this way: “Either it is not the case that you get soup, or it is not the case that you get salad.” Translating this requires both disjunction and negation. It becomes ⇠S1 _ ⇠S2.

Sentence 28 also requires negation. It can be paraphrased as, “It is not the case that either you get soup or you get salad.” We need some way of indicating that the negation does not just negate the right or left disjunct, but rather negates the entire disjunction. In order to do this, we put parentheses around the disjunction: “It is not the case that (S1 _ S2).” This becomes simply ⇠(S1 _ S2). Notice that the parentheses are doing important work here. The sentence ⇠S1 _ S2 would mean “Either you will not have soup, or you will have salad.”

Something similar happens with negation and conjunction. Consider these sentences

29. You can’t have soup and you can’t have salad. 30. You can’t have both soup and salad.

In sentence 29, the two parts of the sentence are negated individually. We would translate it into SL like this: ⇠S1 &⇠S2. In sentence 30, the negation applies to soup and salad taken together. You are allowed to have soup only, or salad only. You just can’t have both together. We would translate sentence 30 like this: ⇠(S1 &S2).

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You can combine disjunction, conjunction, and negation to represent the exclusive or, as in this sentence.

31. You get either soup or salad, but not both.

Remember on page ??, we said that the _ in SL represented an inclusive or. It said “this or that or both.” If we want to represent an exclusive or, we need to combine disjunction, conjuction and negation. We can break the sentence into two parts. The first part says that you get one or the other. We translate this as (S1 _ S2). The second part says that you do not get both. We can paraphrase this as “It is not the case both that you get soup and that you get salad.” Using both negation and conjunction, we translate this as ⇠(S1 &S2). Now we just need to put the two parts together. As we saw above, “but” can usually be translated as a conjunction. Sentence 31 can thus be translated as (S1 _ S2)&⇠(S1 &S2).

4.4 Recursive Syntax for SL

The previous two sections gave you a rough, informal sense of how to create sentences in SL. If I give you an English sentence like “Grass is either green or brown,” you should be able to write a corresponding sentence in SL: “A_B.” In this section we want to give a more precise definition of a sentence in SL. When we defined sentences in English, we did so using the concept of truth: Sentences were units of language that can be true or false. (See page 7.) In SL, it is possible to define what counts as a sentence without talking about truth. Instead, we can just talk about the structure of the sentence. This is one respect in which a formal language like SL is more precise than a natural language like English.

The structure of a sentence in SL considered without reference to truth or falsity is called its syntax. More generally syntax refers to the study of the properties of language that are there even when you don’t consider meaning. Whether a sentence is true or false is considered part of its meaning. In this chapter, we will be giving a purely syntactical definition of a sentence in SL. The contrasting term is semantics the study of aspects of language that relate to meaning, including truth and falsity. (The word “semantics” comes from the Greek word for “mark”)

If we are going to define a sentence in SL just using syntax, we will need to carefully distinguish SL from the language that we use to talk about SL. When you create an artificial language like SL, the language that you are creating is called the object language. The language that we use to talk about the object language is called the metalanguage. Imagine building a house. The object language is like the house itself. It is the thing we are building. While you are building a house, you might put up sca↵olding around it. The sca↵olding isn’t part of the the house. You just use it to build the house. The metalanguage is like the sca↵olding.

The object language in this chapter is SL. For the most part, we can build this language just by talking about it in ordinary English. However we will also have to build some special sca↵olding that is not a part of SL, but will help us build SL. Our metalanguage will thus be ordinary English plus this sca↵olding.

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Element Symbols

sentence letters A,B,C, . . . , Z A1, B1, Z1, A2, A25, J375, . . .

connectives ⇠, & ,_,!,$ parentheses ( , )

Table 4.2: The basic elements of SL

An important part of the sca↵olding are the metavariables These are the fancy script letters we have been using in the characteristic truth tables for the connectives: A , B , C , etc. These are letters that can refer to any sentence in SL. They can represent sentences like P or Q, or they can represent longer sentences, like (((A _B)&G) ! (P $ Q)). Just as the sentence letters A, B, etc. are variables that range over any English sentence, the metavariables A , B , etc. are variables that range over any sentence in SL, including the sentence letters A, B, etc.

As we said, in this chapter we will give a syntactic definition for “sentence of SL.” The definition itself will be given in mathematical English, the metalanguage. Table 4.2 gives the basic elements of SL.

Most random combinations of these symbols will not count as sentences in SL. Any random connection of these symbols will just be called a “string” or “expression” Random strings only become meaningful sentences when the are structured according to the rules of syntax. We saw from the earlier two sections that individual sentence letters, like A and G13 counted as sentences. We also saw that we can put these sentences together using connectives so that ⇠A and ⇠G13 is a sentence. The problem is, we can’t simply list all the di↵erent sentences we can put together this way, because there are infinitely many of them. Instead, we will define a sentence in SL by specifying the process by which they are constructed.

Consider negation: Given any sentence A of SL, ⇠A is a sentence of SL. It is important here that A is not the sentence letter A. Rather, it is a metavariable: part of the metalanguage, not the object language. Since A is not a symbol of SL, ⇠A is not an expression of SL. Instead, it is an expression of the metalanguage that allows us to talk about infinitely many expressions of SL: all of the expressions that start with the negation symbol.

We can say similar things for each of the other connectives. For instance, if A and B are sentences of SL, then (A &B) is a sentence of SL. Providing clauses like this for all of the connectives, we arrive at the following formal definition for a sentence of SL:

1. Every atomic sentence is a sentence.

2. If A is a sentence, then ⇠A is a sentence of SL.

3. If A and B are sentences, then (A &B) is a sentence.

4. If A and B are sentences, then (A _ B) is a sentence.

4.4. RECURSIVE SYNTAX FOR SL 83

5. If A and B are sentences, then (A ! B) is a sentence.

6. If A and B are sentences, then (A $ B) is a sentence.

7. All and only sentences of SL can be generated by applications of these rules.

We can apply this definition to see whether an arbitrary string is a sentence. Suppose we want to know whether or not ⇠⇠⇠D is a sentence of SL. Looking at the second clause of the definition, we know that ⇠⇠⇠D is a sentence if ⇠⇠D is a sentence. So now we need to ask whether or not ⇠⇠D is a sentence. Again looking at the second clause of the definition, ⇠⇠D is a sentence if ⇠D is. Again, ⇠D is a sentence if D is a sentence. Now D is a sentence letter, an atomic sentence of SL, so we know that D is a sentence by the first clause of the definition. So for a compound formula like ⇠⇠⇠D, we must apply the definition repeatedly. Eventually we arrive at the atomic sentences from which the sentence is built up.

Definitions like this are called recursive. Recursive definitions begin with some specifiable base elements and define ways to indefinitely compound the base elements. Just as the recursive definition allows complex sentences to be built up from simple parts, you can use it to decompose sentences into their simpler parts. To determine whether or not something meets the definition, you may have to refer back to the definition many times. Recursive definitions are also sometimes called “inductive definitions.”

We are now in a position to define what it means for a system of logic to be a system of sentential logic. A sentential logic is a system of logic in which statements can be defined using a recursive definition with only sentences in the base class. This book defines on system of sentential logic, which we call SL. Other books use other systems.

When you use a connective to build a longer sentence from shorter ones, the shorter sentences are said to be in the scope of the connective. So in the sentence (A&B) ! C, the scope of the connective ! includes (A&B) and C. In the sentence ⇠(A&B) the scope of the ⇠ is (A&B). On the other hand, in the sentence ⇠A&B the scope of the ⇠ is just A.

The last connective that you add when you assemble a sentence using the recursive definition is the main connective of that sentence. For example: The main logical operator of ⇠(E _ (F ! G)) is negation, ⇠. The main logical operator of (⇠E _ (F ! G)) is disjunction, _. The main connective of any sentence will have all the rest of the sentence in its scope.

Because statement in our language is defined recursively, we can say it is “uniquely readable.” Unique readability is a property of formal languages which is present when each well formed formula can only be constructed in a single way. Every process of building up a sentence recursively yields a unique sentence, and every sentence is the product of a unique process of recursive definitions. This means that in an important sense our language SL is free of ambiguity, which is a key goal in the construction of any formal language. Every sentence in SL will have a unambiguous main connective and every connective in a sentence will have an unambiguous scope. This makes logicians happy.

84 CHAPTER 4. SENTENTIAL LOGIC

Notational conventions

A sentence like (Q&R) must be surrounded by parentheses, because we might apply the definition again to use this as part of a more complicated sentence. If we negate (Q&R), we get ⇠(Q&R). If we just had Q&R without the parentheses and put a negation in front of it, we would have ⇠Q&R. It is most natural to read this as meaning the same thing as (⇠Q&R), something very di↵erent than ⇠(Q&R). The sentence ⇠(Q&R) means that it is not the case that both Q and R are true; Q might be false or R might be false, but the sentence does not tell us which. The sentence (⇠Q&R) means specifically that Q is false and that R is true. As such, parentheses are crucial to the meaning of the sentence.

So, strictly speaking, Q&R without parentheses is not a sentence of SL. When using SL, however, we will often be able to relax the precise definition so as to make things easier for ourselves. We will do this in several ways.

First, we understand that Q&R means the same thing as (Q&R). As a matter of convention, we can leave o↵ parentheses that occur around the entire sentence.

Second, it can sometimes be confusing to look at long sentences with many nested pairs of parentheses. We adopt the convention of using square brackets [ and ] in place of parentheses. There is no logical di↵erence between (P _Q) and [P _Q], for example. The unwieldy sentence

(((H ! I) _ (I ! H))& (J _K))

could be written in this way:

⇥ (H ! I) _ (I ! H)

⇤ &(J _K)

Third, we will sometimes want to translate the conjunction of three or more sentences. For the sentence “Alice, Bob, and Candice all went to the party,” suppose we let A mean “Alice went,” B mean “Bob went,” and C mean “Candice went.” The definition only allows us to form a conjunction out of two sentences, so we can translate it as (A&B)&C or as A&(B&C). There is no reason to distinguish between these, since the two translations are logically equivalent. There is no logical di↵erence between the first, in which (A&B) is conjoined with C, and the second, in which A is conjoined with (B&C). So we might as well just write A&B&C. As a matter of convention, we can leave out parentheses when we conjoin three or more sentences.

Fourth, a similar situation arises with multiple disjunctions. “Either Alice, Bob, or Candice went to the party” can be translated as (A _B) _ C or as A _ (B _ C). Since these two translations are logically equivalent, we may write A _B _ C.

These latter two conventions only apply to multiple conjunctions or multiple disjunctions. If a series of connectives includes both disjunctions and conjunctions, then the parentheses are essential; as with (A&B) _ C and A&(B _ C). The parentheses are also required if there is a series of conditionals or biconditionals; as with (A ! B) ! C and A $ (B $ C).

We have adopted these four rules as notational conventions, not as changes to the definition of

4.4. RECURSIVE SYNTAX FOR SL 85

a sentence. Strictly speaking, A _B _ C is still not a sentence. Instead, it is a kind of shorthand. We write it for the sake of convenience, but we really mean the sentence (A _ (B _ C)).

If we had given a di↵erent definition for a sentence, then these could count as sentences. We might have written rule 3 in this way: “If A , B , . . . Z are sentences, then (A &B & . . . &Z), is a sentence .” This would make it easier to translate some English sentences, but would have the cost of making our formal language more complicated. We would have to keep the complex definition in mind when we develop truth tables and a proof system. We want a logical language that is expressively simple and allows us to translate easily from English, but we also want a formally simple language. Adopting notational conventions is a compromise between these two desires.

Practice Exercises

Part A Using the symbolization key given, translate each English-language sentence into SL.

M: Those creatures are men in suits. C: Those creatures are chimpanzees. G: Those creatures are gorillas.

Example: If those creatures are not men in suits, they are gorillas.

Answer: ⇠M ! G

(1) Those creatures are not men in suits.

(2) Those creatures are men in suits, or they are not.

(3) Those creatures are either gorillas or chimpanzees.

(4) Those creatures are not gorillas, but they are not chimpanzees either.

(5) Those creatures cannot be both gorillas and men in suits.

(6) If those creatures are not gorillas, then they are men in suits

(7) Those creatures are men in suits only if they are not gorillas.

(8) Those creatures are chimpanzees if and only if they are not gorillas.

(9) Those creatures are neither gorillas nor chimpanzees.

(10) Unless those creatures are men in suits, they are either chimpanzees or they are gorillas.

Part B Using the symbolization key given, translate each English-language sentence into SL.

A: Mister Ace was murdered. B: The butler did it. C: The cook did it. D: The Duchess is lying.

86 CHAPTER 4. SENTENTIAL LOGIC

E: Mister Edge was murdered. F: The murder weapon was a frying pan.

(1) Either Mister Ace or Mister Edge was murdered.

(2) If Mister Ace was murdered, then the cook did it.

(3) If Mister Edge was murdered, then the cook did not do it.

(4) Either the butler did it, or the Duchess is lying.

(5) The cook did it only if the Duchess is lying.

(6) If the murder weapon was a frying pan, then the culprit must have been the cook.

(7) If the murder weapon was not a frying pan, then the culprit was neither the cook nor the butler.

(8) Mister Ace was murdered if and only if Mister Edge was not murdered.

(9) The Duchess is lying, unless it was Mister Edge who was murdered.

(10) Mister Ace was murdered, but not with a frying pan.

(11) The butler and the cook did not both do it.

(12) Of course the Duchess is lying!

Part C Using the symbolization key given, translate each English-language sentence into SL.

E1: Ava is an electrician. E2: Harrison is an electrician. F1: Ava is a firefighter. F2: Harrison is a firefighter. S1: Ava is satisfied with her career. S2: Harrison is satisfied with his career.

(1) Ava and Harrison are both electricians.

(2) If Ava is a firefighter, then she is satisfied with her career.

(3) Ava is a firefighter, unless she is an electrician.

(4) Harrison is an unsatisfied electrician.

(5) Neither Ava nor Harrison is an electrician.

(6) Both Ava and Harrison are electricians, but neither of them find it satisfying.

(7) Harrison is satisfied only if he is a firefighter.

(8) If Ava is not an electrician, then neither is Harrison, but if she is, then he is too.

(9) Ava is satisfied with her career if and only if Harrison is not satisfied with his.

(10) If Harrison is both an electrician and a firefighter, then he must be satisfied with his work.

(11) It cannot be that Harrison is both an electrician and a firefighter.

4.4. RECURSIVE SYNTAX FOR SL 87

(12) Harrison and Ava are both firefighters if and only if neither of them is an electrician.

Part D Using the symbolization key given, translate each English-language sentence into SL.

J1: John Coltrane played tenor sax. J2: John Coltrane played soprano sax. J3: John Coltrane played tuba M1: Miles Davis played trumpet M2: Miles Davis played tuba

(1) John Coltrane played tenor and soprano sax.

(2) Neither Miles Davis nor John Coltrane played tuba.

(3) John Coltrane did not play both tenor sax and tuba.

(4) John Coltrane did not play tenor sax unless he also played soprano sax.

(5) John Coltrane did not play tuba, but Miles Davis did.

(6) Miles Davis played trumpet only if he also played tuba.

(7) If Miles Davis played trumpet, then John Coltrane played at least one of these three instruments: tenor sax, soprano sax, or tuba.

(8) If John Coltrane played tuba then Miles Davis played neither trumpet nor tuba.

(9) Miles Davis and John Coltrane both played tuba if and only if Coltrane did not play tenor sax and Miles Davis did not play trumpet.

Part E Give a symbolization key and symbolize the following sentences in SL.

(1) Alice and Bob are both spies.

(2) If either Alice or Bob is a spy, then the code has been broken.

(3) If neither Alice nor Bob is a spy, then the code remains unbroken.

(4) The German embassy will be in an uproar, unless someone has broken the code.

(5) Either the code has been broken or it has not, but the German embassy will be in an uproar regardless.

(6) Either Alice or Bob is a spy, but not both.

Part F Give a symbolization key and symbolize the following sentences in SL.

(1) If Gregor plays first base, then the team will lose.

(2) The team will lose unless there is a miracle.

(3) The team will either lose or it won’t, but Gregor will play first base regardless.

(4) Gregor’s mom will bake cookies if and only if Gregor plays first base.

(5) If there is a miracle, then Gregor’s mom will not bake cookies.

88 CHAPTER 4. SENTENTIAL LOGIC

Part G For each argument, write a symbolization key and translate the argument into SL, putting the argument in canonical form.

Example: If Dorothy plays the piano in the morning, then Roger wakes up cranky. Dorothy plays piano in the morning unless she is distracted. So if Roger does not wake up cranky, then Dorothy must be distracted.

Answer: . A: Dorothy plays the piano in the morning B: Roger wakes up cranky C: Dorothy is distracted

. P1: A ! B P2: A _ C

C: ⇠B ! C

(1) It will either rain or snow on Tuesday. If it rains on Tuesday, Neville will be sad. If it snows on Tuesday, Neville will be cold. Therefore, Neville will either be sad or cold on Tuesday.

(2) If Zoog remembered to do his chores, then things are clean but not neat. If he forgot, then things are neat but not clean. Therefore, things are either neat or clean—but not both.

Part H For each argument, write a symbolization key and translate the argument as well as possible into SL. The part of the passage in italics is there to provide context for the argument, and doesn’t need to be symbolized.

(1) It is going to rain soon. I know because my leg is hurting, and my leg hurts if it’s going to rain.

(2) Spider-man tries to figure out the bad guy’s plan. If Doctor Octopus gets the uranium, he will blackmail the city. I am certain of this because if Doctor Octopus gets the uranium, he can make a dirty bomb, and if he can make a dirty bomb, he will blackmail the city.

(3) A westerner tries to predict the policies of the Chinese government. If the Chinese government cannot solve the water shortages in Beijing, they will have to move the capital. They don’t want to move the capital. Therefore they must solve the water shortage. But the only way to solve the water shortage is to divert almost all the water from the Yangzi river northward. Therefore the Chinese government will go with the project to divert water from the south to the north.

Part I

(1) Are there any sentences of SL that contain no sentence letters? Why or why not?

(2) In the chapter, we symbolized an exclusive or using _, & , and ⇠. How could you translate an exclusive or using only two connectives? Is there any way to translate an exclusive or using only one kind of connective?

4.4. RECURSIVE SYNTAX FOR SL 89

Key Terms

Antecedent

Atomic sentence

Biconditional

Conditional

Conjunct

Conjunction

Consequent

Disjunct

Disjunction

Exclusive or

Inclusive or

Logical constant

Main connective

Metalanguage

Metavariables

Necessary condition

Negation

Nonlogical symbol

Object language

Recursive definition

Scope

Semantics

Sentence letter

Sentence of SL

Sentential connective

Su�cient condition

Symbolization key

Syntax

Translation key

90 CHAPTER 4. SENTENTIAL LOGIC

Chapter 5

Truth Tables

This chapter introduces a way of evaluating sentences and arguments of SL called the truth table method. As we shall see, the truth table method is semantic because it involves one aspect of the meaning of sentences, whether those sentences are true or false. As we saw on page 81, semantics is the study of aspects of language related to meaning, including truth and falsity. Although it can be laborious, the truth table method is a purely mechanical procedure that requires no intuition or special insight. When we get to Chapter 8,we will provide a parallel semantic method for QL; however, this method will not be purely mechanical.

5.1 Basic Concepts

In the previous chapter, we said that a formal language is built from two kinds of elements: logical constants and nonlogical symbols. The logical constants have their meaning fixed by the formal language, while the nonlogical symbols get their meaning in the symbolization key. The logical constants in SL are the sentential connectives and the parentheses, while the nonlogical symbols are the sentence letters.

When we assign meaning to the nonlogical symbols of a language using a dictionary, we say we are giving an “interpretation” of the language. More formally an interpretation of a language is a correspondence between elements of the object language and elements of some other language or logical structure. The symbolization keys we defined in Chapter 4 (p. 65) are one sort of interpretation. Fancier languages will have more complicated kinds of interpretations.

The truth table method will also involve giving an interpretation of sentences, but they will be much simpler than the translation keys we used in Chapter 4. We will not be concerned with what the individual sentence letters mean. We will only care whether they are true or false. In other words, our interpretations will assign truth values to the sentence letters. (See page 53.)

91

92 CHAPTER 5. TRUTH TABLES

We can get away with only worrying about the truth values of sentence letters because of the way that the meaning of larger sentences is generated by the meaning of their parts. Any larger sentence of SL is composed of atomic sentences with sentential connectives. The truth value of the compound sentence depends only on the truth value of the atomic sentences that it comprises. In order to know the truth value of D $ E, for instance, you only need to know the truth value of D and the truth value of E. Connectives that work in this way are called truth functional. More technically, we define a truth-functional connective as an operator that builds larger sentences out of smaller ones, and fixes the truth value of the resulting sentence based only on the truth value of the component sentences.

Because all of the logical symbols in SL are truth functional, we can study the the semantics of SL looking only at truth and falsity. If we want to know about the truth of the sentence A&B, the only thing we need to know is whether A and B are true. It doesn’t actually matter what else they mean. So if A is false, then A&B is false no matter what false sentence A is used to represent. It could be “I am the Pope” or “Pi is equal to 3.19.” The larger sentence A&B is still false. So to give an interpretation of sentences in SL, all we need to do is create a truth assignment. A truth assignment is a function that maps the sentence letters in SL onto our two truth values. In other words, we just need to assign Ts and Fs to all our sentence letters.

It is worth knowing that most languages are not built only out of truth functional connectives. In English, it is possible to form a new sentence from any simpler sentence X by saying “It is possible that X .” The truth value of this new sentence does not depend directly on the truth value of X . Even if X is false, perhaps in some sense X could have been true—then the new sentence would be true. Some formal languages, called modal logics, have an operator for possibility. In a modal logic, we could translate “It is possible that X ” as ⇧X . However, the ability to translate sentences like these comes at a cost: The ⇧ operator is not truth-functional, and so modal logics are not amenable to truth tables.

5.2 Complete Truth Tables

In the last chapter we introduced the characteristic truth tables for the di↵erent connectives. To put them all in one place, the truth tables for the connectives of SL are repeated in Table 5.2. On the left is the truth table for negation, and on the right is the truth table for the other four connectives. Notice that the truth table for the negation is shorter than the other table. This is because there is only one metavariable here, A , which can either be true or false. The other connectives involve two metavariables, which give us four possibilities of true and false. The columns to the left of the double line in these tables are called the reference columns. They just specify the truth values of the individual sentence letters. Each row of the table assigns truth values to all the variables. Each row is thus a truth assignment—a kind of interpretation—for that sentence. Because the full table gives all the possible truth assignments for the sentence, it gives all the possible interpretations of it.

The truth table of sentences that contain only one connective is given by the characteristic truth table for that connective. So the truth table for the sentence P &Q looks just like the

5.2. COMPLETE TRUTH TABLES 93

A ⇠A A B A &B A_B A!B A$B T F T T T T T T

F F T F F T F F

F T F T T F

F F F F T T

Table 5.2: The characteristic truth tables for the connectives of SL.

characteristic truth table for & , with the sentence letters P and Q substituted in. The truth tables for more complicated sentences can simply be built up out of the truth tables for these basic sentences. Consider the sentence (H & I) ! H. This sentence has two sentence letters, so we can represent all the possible truth assignments using a four line truth table. We can start by writing out all the possible combinations of true and false for H and I in the reference columns. We then copy the truth values for the sentence letters and write them underneath the letters in the sentence.

H I (H & I) ! H T T T T T T F T F T F T F T F F F F F F

Now consider just one part of the sentence above, the subsentence H & I. This is a conjunction A &B with H as A and with I as B . H and I are both true on the first row. Since a conjunction is true when both conjuncts are true, we write a T underneath the conjunction symbol. We continue for the other three rows and get this:

A &B z }| {

H I (H & I) ! H

T T T T T T T F T F F T F T F F T F F F F F F F

Next we need to fill in the final column under the conditional. The conditional is the main connective of the sentence, so the whole sentence is of the form A ! B with (H & I) as A and with H as B . So to fill the final column, we just need to look at the characteristic truth table for the conditional. For the first row, the sentence (H & I) is true and the sentence H is also true.

94 CHAPTER 5. TRUTH TABLES

The truth table for he conditional tells us this means that the whole sentence is true. Filling out the rest of the column gives us this:

A ! B z }| { z}|{ z}|{

H I (H & I) ! H

T T T T T T T T F T F F T T F T F F T T F F F F F F T F

The column of Ts underneath the conditional tells us that the sentence (H & I) ! H is true regardless of the truth values of H and I. They can be true or false in any combination, and the compound sentence still comes out true. It is crucial that we have considered all of the possible combinations. If we only had a two-line truth table, we could not be sure that the sentence was not false for some other combination of truth values.

In this example, the script letters over the table have just been there to indicate how the columns get filled in. We won’t need them in the final product. Also, the reference columns are redundant with the columns under the individual sentence letters, so we can eliminate those as well. Most of the time, when you see truth tables, we will just write them out this way:

(H & I) ! H T T T T T

T F F T T

F F T T F

F F F T F

The truth value of the sentence on each row is just the column underneath the main connective (see p. 83) of the sentence, in this case, the column underneath the conditional.

A complete truth table is a table that gives all the possible interpretations for a sentence or set of sentences in SL. It has a row for each possible assignment of T and F to all of the sentence letters. The size of the complete truth table depends on the number of di↵erent sentence letters in the table. A sentence that contains only one sentence letter requires only two rows, as in the characteristic truth table for negation. This is true even if the same letter is repeated many times, as in this sentence:

[(C $ C) ! C] &⇠(C ! C).

The complete truth table requires only two lines because there are only two possibilities: C can be true, or it can be false. A single sentence letter can never be marked both T and F on the same row. The truth table for this sentence looks like this:

5.2. COMPLETE TRUTH TABLES 95

[(C $ C) ! C] & ⇠ (C ! C) T T T T T F F T T T

F T F F F F F F T F

Looking at the column underneath the main connective, we see that the sentence is false on both rows of the table; i.e., it is false regardless of whether C is true or false.

A sentence that contains two sentence letters requires four lines for a complete truth table, as we saw above in the table for (H & I) ! I.

A sentence that contains three sentence letters requires eight lines, as in this example. Here the reference columns are included so you can see how to arrange the truth values for the individual sentence letters so that all the possibilities are covered.

M N P M & (N _ P ) T T T T T T T T

T T F T T T T F

T F T T T F T T

T F F T F F F F

F T T F F T T T

F T F F F T T F

F F T F F F T T

F F F F F F F F

From this table, we know that the sentence M &(N _ P ) might be true or false, depending on the truth values of M , N , and P .

A complete truth table for a sentence that contains four di↵erent sentence letters requires 16 lines. For five letters, 32 lines are required. For six letters, 64 lines, and so on. To be perfectly general: If a complete truth table has n di↵erent sentence letters, then it must have 2n rows.

By convention, the reference columns are filled in with the right most row alternating Ts and Fs. The next column over alternates sets of two Ts and two Fs. For the third column from the right, you have sets of four Ts and four Fs. This continues until you reach the leftmost column, which will always have the top have all Ts and the bottom half all Fs. This convention is completely arbitrary. There are other ways to be sure that all the possible combinations are covered, but everything is easier if we all stick to the same pattern.

Practice Exercises

Part A Identify the main connective in the each sentence.

96 CHAPTER 5. TRUTH TABLES

Example: (A ! C)&⇠D

Answer: (A ! C) & ⇠D

(1) ⇠(A _ ⇠B) (2) ⇠(A _ ⇠B) _ ⇠(A&D) (3) [⇠(A _ ⇠B) _ ⇠(A&D)] ! E (4) [(A ! B)&C] $ [A _ (B&C)] (5) ⇠⇠⇠[A _ (B&(C _D))]

Part B Identify the main connective in the each sentence.

(1) [(A $ B)&C] ! D (2) [(D&(E&F )) _G] $ ⇠[A ! (C _G)] (3) ⇠(⇠Z _ ⇠H) (4) (⇠(P &S) $ G)&Y (5) (A&(B ! C)) _ ⇠D

Part C Assume A, B, and C are true and X, Y, and Z are false and evaluate the truth of the each sentence by writing a one-line truth table.

Example: (A&⇠X) $ (B _ Y ) Answer: (A & ⇠ X) $ (B _ Y)

T T T F T T T F

(1) ⇠((A&B) ! X) (2) (Y _ Z) $ (⇠X $ B) (3) [(X ! A) _ (A ! X)] &Y (4) (X ! A) _ (A ! X) (5) [A&(Y &Z)] _A

Part D Assume A, B, and C are true and X, Y, and Z are false and evaluate the truth of the each sentence by writing a one-line truth table..

(1) ⇠⇠(⇠⇠⇠A _X) (2) (A ! B) ! X (3) ((A _B)& (C $ X)) _ Y (4) (A ! B) _ (X &(Y &Z)) (5) ((A _X) ! Y )&B

5.3. USING TRUTH TABLES 97

Part E Write complete truth tables for the following sentences and mark the column that represents the possible truth values for the whole sentence.

Example: D ! (D&(⇠F _ F )) Answer: D ! (D & (⇠ F _ F))

T T T T F T T T T T T T T F T F F T F F F T T T F T F F T F T F

(1) ⇠(S $ (P ! S)) (2) ⇠[(X &Y ) _ (X _ Y )] (3) (A ! B) $ (⇠B $ ⇠A) (4) [C $ (D _ E)] &⇠C (5) ⇠(G&(B&H)) $ (G _ (B _H))

Part F Write complete truth tables for the following sentences and mark the column that represents the possible truth values for the whole sentence.

(1) (D&⇠D) ! G (2) (⇠P _ ⇠M) $ M (3) ⇠⇠(⇠A&⇠B) (4) [(D&R) ! I] ! ⇠(D _R) (5) ⇠[(D $ O) $ A] ! (⇠D&O)

5.3 Using Truth Tables

Because truth table show all the possible interpretations of a sentence or set of sentences we can use them to explore the logical properties we first introduced in Chapter 3.

Tautologies, contradictions, and contingent sentences

We defined a tautology as a statement that must be true as a matter of logic, no matter how the world is (p. 54). A statement like “Either it is raining or it is not raining” is always true, no matter what the weather is like outside. Something similar goes on in truth tables. With a complete truth table, we consider all of the ways that the world might be. Each line of the truth table corresponds to a way the world might be. This means that if the sentence is true on every

98 CHAPTER 5. TRUTH TABLES

line of a complete truth table, then it is true as a matter of logic, regardless of what the world is like.

We can use this fact to create a test for whether a sentence is a tautology: if the column under the main connective of a sentence is a T on every row, the sentence is a tautology. We already have seen an example of this. On page 94 that the sentence (H & I) ! H had only T’s under its main connective, so it is a tautology.

Not every tautology in English will correspond to a tautology in SL. The sentence “All bachelors are unmarried” is a tautology in English, but we cannot represent it as a tautology in SL, because it just translates as a single sentence letter, like B. On the other hand, if something is a tautology in SL, it will also be a tautology in English. No matter how you translate A _ ⇠A, if you translate the As consistently, the statement will be a tautology.

Rather than thinking of complete truth tables as an imperfect test for the English notion of a tautology, we can define a separate notion of a tautology in SL based on truth tables. A statement is a semantic tautology in SL if and only if the column under the main connective in the complete truth table for the sentence contains only Ts. This is the semantic definition of a tautology in SL, because it uses truth tables. Later we will create a separate, syntactic definition and show that it is equivalent to the semantic definition. We will be doing the same thing for all the concepts defined in this section.

We defined a contradiction as a sentence that is false no matter how the world is (p. 54). This means we can define a semantic contradiction in SL as a sentence that has only Fs in the column under them main connective of its complete truth table. We saw on page 95 that the sentence [(C $ C) ! C] &⇠(C ! C) was a contradiction in this sense. As with the definition of a semantic tautology, this is a semantic definition because it uses truth tables.

Finally, a sentence is contingent if it is sometimes true and sometimes false (p. 54). Similarly, a sentence is semantically contingent in SL if and only if its complete truth table for has both Ts and Fs under the main connective. We saw on page 95 that the sentence M &(N _ P ) was contingent.

Logical equivalence

Two sentences are logically equivalent in English if they have the same truth value as a matter of logic (p. 56). Once again, we can use truth tables to define a similar property in SL: Two sentences are semantically logically equivalent in SL if they have the same truth value on every row of a complete truth table.

Consider the sentences ⇠(A _B) and ⇠A&⇠B. Are they logically equivalent? To find out, we construct a truth table.

5.3. USING TRUTH TABLES 99

⇠ (A _ B) ⇠ A & ⇠ B

F T T T F T F F T F T T F F T F T F F F T T T F F F T T F F F T F T T F

Look at the columns for the main connectives; negation for the first sentence, conjunction for the second. On the first three rows, both are F. On the final row, both are T. Since they match on every row, the two sentences are logically equivalent.

Consistency

A set of sentences in English is consistent if it is logically possible for them all to be true at once (p. 56). This means that a sentence is semantically consistent in SL if and only if there is at least one line of a complete truth table on which all of the sentences are true. It is semantically inconsistent otherwise.

Consider the three sentences A ! B, B ! C and C ! A. Since we are considering them as a set, we will put curly braces around them, as is done in set theory: {A ! B,B ! C,C ! A}. The conditionals in this set form a little loop, but it is possible for all the sentences to be true at the same time, as this truth table shows.

A ! B B ! C C ! A T T T T T T T T T

T T T T F F F T T

T F F F T T T T T

T F F F T F F T T

F T T T T T T F F

F T T T F F F T F

F T F F T T T F F

F T F F T F F T F

Validity

Logic is the study of argument, so the most important use of truth tables is to test the validity of arguments. An argument in English is valid if it is logically impossible for the premises to be true and for the conclusion to be false at the same time (p. 30). So we can define an argument as semantically valid in SL if there is no row of a complete truth table on which the premises are all marked “T” and the conclusion is marked “F.” An argument is invalid if there is such a row.

100 CHAPTER 5. TRUTH TABLES

Consider this argument:

1. ⇠L ! (J _ L) 2. ⇠L

. . .˙. J

Is it valid? To find out, we construct a truth table.

J L ⇠ L ! (J _ L) ⇠ L J T T F T T T T T F T T T F T F T T T F T F T F T F T T F T T F T F F F T F F F F F T F F

Yes, the argument is valid. The only row on which both the premises are T is the second row, and on that row the conclusion is also T.

In Chapters 1 and 2 we used the three dots .˙. to represent an inference in English. We used this symbol to represent any kind of inference. The truth table method gives us a more specific notion of a valid inference. We will call this semantic entailment and represent it using a new symbol, , called the “double turnstile.” The is like the .˙., except for arguments verified by truth tables. When you use the double turnstile, you write the premises as a set, using curly brackets, { and }, which mathematicians use in set theory. The argument above would be written {⇠L ! (J _ L),⇠L} J .

More formally, we can define the double turnstile this way: {A1 . . .An} B if and only if there is no truth value assignment for which A1. . .An are true and B is false. Put di↵erently, it means that B is true for any and all truth value assignments for which A1. . .An are true.

We can also use the double turnstile to represent other logical notions. Since a tautology is always true, it is like the conclusion of a valid argument with no premises. The string C means that C is true for all truth value assignments. This is equivalent to saying that the sentence is entailed by anything. We can represent logical equivalence by writing the double turnstile in both directions: A B For instance, if we want to point out that the sentence A&B is equivalent to B&A we would write this: A&B B&A.

Practice Exercises

If you want additional practice, you can construct truth tables for any of the sentences and arguments in the exercises for the previous chapter.

5.3. USING TRUTH TABLES 101

Part A Determine whether each sentence is a tautology, a contradiction, or a contingent sentence, using a complete truth table.

Example: (A ! B) _ (B ! A) Answer: (A ! B) _ (B ! A) Tautology

T T T T T T T T F F T F T T F T T T T F F F T F T F T F

(1) A ! A (2) C ! ⇠C (3) (A $ B) $ ⇠(A $ ⇠B) (4) (A&B) ! (B _A) (5) [(⇠A _A) _B] ! B (6) [(A _B)&⇠A] & (B ! A)

Part B Determine whether each sentence is a tautology, a contradiction, or a contingent sentence, using a complete truth table.

(1) ⇠B&B

(2) ⇠D _D

(3) (A&B) _ (B&A)

(4) ⇠[A ! (B ! A)]

(5) A $ [A ! (B&⇠B)]

(6) [(A&B) $ B] ! (A ! B)

Part C Determine whether each the following statements are equivalent using complete truth tables. If the two sentences really are logically equivalent, write ”Logically equivalent.” Otherwise write, ”Not logically equivalent.”

Example: A _B ⇠A ! B Answer: A _ B ⇠ A ! B Logically Equivalent

T T T F T T T T T F F T T F F T T T F T T F F F T F F F

102 CHAPTER 5. TRUTH TABLES

(1) A ⇠A

(2) A&⇠A ⇠B $ B

(3) [(A _B) _ C] [A _ (B _ C)]

(4) A _ (B&C) (A _B)& (A _ C)

(5) [A&(A _B)] ! B A ! B

Part D Determine whether each the following statements of equivalence are true or false using complete truth tables. If the two sentences really are logically equivalent, write ”Logically equivalent.” Otherwise write, ”Not logically equivalent.”

(1) A ! A A $ A

(2) ⇠(A ! B) ⇠A ! ⇠B

(3) A _B ⇠A ! B

(4) (A ! B) ! C A ! (B ! C)

(5) A $ (B $ C) A&(B&C)

Part E Determine whether each set of sentences is consistent or inconsistent using a complete truth table.

Example: {⇠(A _B), ⇠A _B, A _ ⇠B}

Answer: ⇠ (A _ B), ⇠ A _ B, A _ ⇠ B Consistent F T T T F T T T T T F T F T T F F T F F T T T F F F T T T F T T F F F T T F F F T F T F F T T F

(1) {A&⇠B, ⇠(A ! B), B ! A}

(2) {A _B, A ! ⇠A, B ! ⇠B}

(3) {⇠(⇠A _B), A ! ⇠C, A ! (B ! C)}

(4) {A ! B, A&⇠B}

(5) {A ! (B ! C), (A ! B) ! C, A ! C}

Part F Determine whether each set of sentences is consistent or inconsistent, using a complete truth table.

(1) {⇠B, A ! B, A}

5.3. USING TRUTH TABLES 103

(2) {⇠(A _B), A $ B, B ! A}

(3) { A _B, ⇠B, ⇠B ! ⇠A}

(4) {A $ B, ⇠B _ ⇠A, A ! B}

(5) {(A _B) _ C, ⇠A _ ⇠B, ⇠C _ ⇠B}

Part G Determine whether each argument is valid or invalid, using a complete truth table.

Example: A _B, C ! A, C ! B C

Answer: A _ B C ! A C ! B C Invalid T T T T T T T T T T T T T F T T F T T F T T F T T T T F F T T T F F T T F T F F F T T T F F T T T T F T T F T F F T T F F F F T F F T F F T F F F F T F F T F F

(1) A ! A A

(2) A ! B, B A

(3) A $ B, B $ C A $ C

(4) A ! B, A ! C B ! C

(5) A ! B, B ! A A $ B

Part H Determine whether each argument is valid or invalid, using a complete truth table.

(1) A _ ⇥ A ! (A $ A)

⇤ A

(2) A _B, B _ C, ⇠B A&C

(3) A ! B, ⇠A ⇠B

(4) A, B ⇠(A ! ⇠B)

(5) ⇠(A&B), A _B, A $ B C

104 CHAPTER 5. TRUTH TABLES

5.4 Partial Truth Tables

In order to show that a sentence is a tautology, we need to show that it is T on every row. So we need a complete truth table. To show that a sentence is not a tautology, however, we only need one line: a line on which the sentence is F. Therefore, in order to show that something is not a tautology, it is enough to provide a one-line partial truth table—regardless of how many sentence letters the sentence might have in it.

Consider, for example, the sentence (U &T ) ! (S&W ). We want to show that it is not a tautology by providing a partial truth table. We fill in F for the entire sentence. The main connective of the sentence is a conditional. In order for the conditional to be false, the antecedent must be true (T) and the consequent must be false (F). So we fill these in on the table:

S T U W (U & T ) ! (S & W ) T F F

In order for the (U &T ) to be true, both U and T must be true.

S T U W (U & T ) ! (S & W ) T T T T T F F

Now we just need to make (S&W ) false. To do this, we need to make at least one of S and W false. We can make both S and W false if we want. All that matters is that the whole sentence turns out false on this line. Making an arbitrary decision, we finish the table in this way:

S T U W (U & T ) ! (S & W ) F T T F T T T F F F F

Showing that something is a contradiction requires a complete truth table. Showing that something is not a contradiction requires only a one-line partial truth table, where the sentence is true on that one line.

A sentence is contingent if it is neither a tautology nor a contradiction. So showing that a sentence is contingent requires a two-line partial truth table: The sentence must be true on one line and false on the other. For example, we can show that the sentence above is contingent with this truth table:

S T U W (U & T ) ! (S & W ) F T T F T T T F F F F F T F F F F T T F F F

Note that there are many combinations of truth values that would have made the sentence true, so there are many ways we could have written the second line.

5.4. PARTIAL TRUTH TABLES 105

Property Truth table required to show presence

Truth table required to show absence

being a tautology complete one-line partial

being a contradiction complete one-line partial

contingency two-line partial complete truth

equivalence complete one-line partial

consistency one-line partial complete

validity complete one-line partial

Table 5.11: Complete or partial truth tables to test for di↵erent properties

Showing that a sentence is not contingent requires providing a complete truth table, because it requires showing that the sentence is a tautology or that it is a contradiction. If you do not know whether a particular sentence is contingent, then you do not know whether you will need a complete or partial truth table. You can always start working on a complete truth table. If you complete rows that show the sentence is contingent, then you can stop. If not, then complete the truth table. Even though two carefully selected rows will show that a contingent sentence is contingent, there is nothing wrong with filling in more rows.

Showing that two sentences are logically equivalent requires providing a complete truth table. Showing that two sentences are not logically equivalent requires only a one-line partial truth table: Make the table so that one sentence is true and the other false.

Showing that a set of sentences is consistent requires providing one row of a truth table on which all of the sentences are true. The rest of the table is irrelevant, so a one-line partial truth table will do. Showing that a set of sentences is inconsistent, on the other hand, requires a complete truth table: You must show that on every row of the table at least one of the sentences is false.

Showing that an argument is valid requires a complete truth table. Showing that an argument is invalid only requires providing a one-line truth table: If you can produce a line on which the premises are all true and the conclusion is false, then the argument is invalid.

Table 5.11 summarizes when a complete truth table is required and when a partial truth table will do.

Practice Exercises

Part A Determine whether each sentence is a tautology, a contradiction, or a contingent sentence. Justify your answer with a complete or partial truth table where appropriate.

106 CHAPTER 5. TRUTH TABLES

(1) A ! ⇠A

(2) A ! (A&(A _B))

(3) (A ! B) $ (B ! A)

(4) A ! ⇠(A&(A _B))

(5) ⇠B ! [(⇠A&A) _B]

(6) ⇠(A _B) $ (⇠A&⇠B)

(7) [(A&B)&C] ! B

(8) ⇠ ⇥ (C _A) _B

⇤

(9) ⇥ (A&B)&⇠(A&B)

⇤ &C

(10) (A&B)] ! [(A&C) _ (B&D)]

Part B Determine whether each sentence is a tautology, a contradiction, or a contingent sentence. Justify your answer with a complete or partial truth table where appropriate.

(1) ⇠(A _A)

(2) (A ! B) _ (B ! A)

(3) [(A ! B) ! A] ! A

(4) ⇠[(A ! B) _ (B ! A)]

(5) (A&B) _ (A _B)

(6) ⇠(A&B) $ A

(7) A ! (B _ C)

(8) (A&⇠A) ! (B _ C)

(9) (B&D) $ [A $ (A _ C)]

(10) ⇠[(A ! B) _ (C ! D)]

Part C Determine whether each the following statements of equivalence are true or false using complete truth tables. If the two sentences really are logically equivalent, write ”Logically equivalent.” Otherwise write, ”Not logically equivalent.”

(1) A ⇠A

(2) A ! A A $ A

(3) A&(B&C) A&⇠A

(4) A&⇠A ⇠B $ B

5.4. PARTIAL TRUTH TABLES 107

(5) ⇠(A ! B) ⇠A ! ⇠B

(6) A $ B ⇠[(A ! B) ! ⇠(B ! A)]

(7) (A&B) ! (⇠A _ ⇠B) ⇠(A&B)

(8) [(A _B) _ C] [A _ (B _ C)]

(9) (Z &(⇠R ! O)) ⇠(R ! ⇠O)

Part D Determine whether each the following statements of equivalence are true or false using complete truth tables. If the two sentences really are logically equivalent, write ”Logically equivalent.” Otherwise write, ”Not logically equivalent.”

(1) A A _A

(2) A A&A

(3) A _ ⇠B A ! B

(4) (A ! B) (⇠B ! ⇠A)

(5) ⇠(A&B) ⇠A _ ⇠B

(6) ((U ! (X _X)) _ U) ⇠(X &(X &U))

(7) ((C &(N $ C)) $ C) (⇠⇠⇠N ! C)

(8) [(A _B)&C] [A _ (B&C)]

(9) ((L&C)& I) L _ C

Part E Determine whether each set of sentences is consistent or inconsistent. Justify your answer with a complete or partial truth table where appropriate.

(1) {A ! A, ⇠A ! ⇠A, A&A, A _A}

(2) {A ! ⇠A, ⇠A ! A}

(3) {A _B, A ! C, B ! C}

(4) {A _B, A ! C, B ! C, ⇠C}

(5) {B&(C _A), A ! B, ⇠(B _ C)}

(6) {(A $ B) ! B, B ! ⇠(A $ B), A _B}

(7) {A $ (B _ C), C ! ⇠A, A ! ⇠B}

(8) {A $ B, ⇠B _ ⇠A, A ! B}

(9) {A $ B, A ! C, B ! D, ⇠(C _D)}

108 CHAPTER 5. TRUTH TABLES

(10) {⇠(A&⇠B), B ! ⇠A, ⇠B }

Part F Determine whether each set of sentences is consistent or inconsistent. Justify your answer with a complete or partial truth table where appropriate.

(1) {A&B, C ! ⇠B, C}

(2) {A ! B, B ! C, A, ⇠C}

(3) {A _B, B _ C, C ! ⇠A}

(4) {A, B, C, ⇠D, ⇠E, F}

(5) {A&(B _ C), ⇠(A&C), ⇠(B&C)}

(6) {A ! B, B ! C, ⇠(A ! C)}

Part G Determine whether each argument is valid or invalid. Justify your answer with a complete or partial truth table where appropriate.

(1) A ! (A&⇠A) ⇠A

(2) A _B, A ! B, B ! A A $ B

(3) A _ (B ! A) ⇠A ! ⇠B

(4) A _B, A ! B, B ! A A&B

(5) (B&A) ! C, (C &A) ! B (C &B) ! A

(6) ⇠(⇠A _ ⇠B), A ! ⇠C A ! (B ! C)

(7) A&(B ! C), ⇠C &(⇠B ! ⇠A) C &⇠C

(8) A&B, ⇠A ! ⇠C, B ! ⇠D A _B

(9) A ! B (A&B) _ (⇠A&⇠B)

(10) ⇠A ! B,⇠B ! C,⇠C ! A ⇠A ! (⇠B _ ⇠C)

Part H Determine whether each argument is valid or invalid. Justify your answer with a complete or partial truth table where appropriate.

(1) A $ ⇠(B $ A) A

(2) A _B, B _ C, ⇠A B&C

(3) A ! C, E ! (D _B), B ! ⇠D (A _ C) _ (B ! (E&D))

(4) A _B, C ! A, C ! B A ! (B ! C)

(5) A ! B, ⇠B _A A $ B

Part I Answer each of the questions below and justify your answer.

5.5. EXPRESSIVE COMPLETENESS 109

(1) Suppose that A and B are logically equivalent. What can you say about A $ B? (2) Suppose that (A &B) ! C is contingent. What can you say about the argument “A , B , .˙.

C”? (3) Suppose that {A ,B ,C} is inconsistent. What can you say about (A &B &C )? (4) Suppose that A is a contradiction. What can you say about the argument {A , B} C? (5) Suppose that C is a tautology. What can you say about the argument {A , B} C”? (6) Suppose that A and B are not logically equivalent. What can you say about (A _ B)?

5.5 Expressive Completeness

We could leave the biconditional ($) out of the language. If we did that, we could still write “A $ B” so as to make sentences easier to read, but that would be shorthand for (A ! B)& (B ! A). The resulting language would be formally equivalent to SL, since A $ B and (A ! B)& (B ! A) are logically equivalent in SL. If we valued formal simplicity over expressive richness, we could replace more of the connectives with notational conventions and still have a language equivalent to SL.

There are a number of equivalent languages with only two connectives. You could do logic with only the negation and the material conditional. Alternately you could just have the negation and the disjunction. You will be asked to prove that these things are true in the last problem set. You could even have a language with only one connective, if you designed the connective right. The She↵er stroke is a logical connective with the following characteristic truth table:

A B A |B T T F T F T F T T F F T

The She↵er stroke has the unique property that it is the only connective you need to have a complete system of logic. You will be asked to prove that this is true in the last problem set also.

Practice Exercises

Part A

(1) In section 5.5, we said that you could have a language that only used the negation and the material conditional. Prove that this is true by writing sentences that are logically

110 CHAPTER 5. TRUTH TABLES

equivalent to each of the following using only parentheses, sentence letters, negation (⇠), and the material conditional (!). (a) A _B (b) A&B

(c) A $ B (2) We also said in section 3.5 that you could have a language which used only the negation and

the disjunction. Show this: Using only parentheses, sentence letters, negation (⇠), and disjunction (_), write sentences that are logically equivalent to each of the following. (a) A&B

(b) A ! B (c) A $ B

(3) Write a sentence using the connectives of SL that is logically equivalent to (A|B). (4) Every sentence written using a connective of SL can be rewritten as a logically equivalent

sentence using one or more She↵er strokes. Using only the She↵er stroke, write sentences that are equivalent to each of the following.

(a) ⇠A (b) (A&B)

(c) (A _B) (d) (A ! B) (e) (A $ B)

Key Terms

Complete truth table

Interpretation

Logical constant

Nonlogical symbol

Semantically consistent in SL

Semantically contingent in SL

Semantically logically equivalent in SL

Semantically valid in SL

Semantic contradiction in SL

Semantic tautology in SL

Truth-functional connective

Truth assignment

Chapter 6

Proofs in Sentential Logic

6.1 Substitution Instances and Proofs

In the last chapter, we introduced the truth table method, which allowed us to check to see if various logical properties were present, such as whether a statement is a tautology or whether an argument is valid. The method in that chapter was semantic, because it relied on the meaning of symbols, specifically, whether they were interpreted as true or false. The nice thing about that method was that it was completely mechanical. If you just followed the rules like a robot, you would eventually get the right answer. You didn’t need any special insight and there were no tough decisions to make. The downside to this method was that the tables quickly became way too long. It just isn’t practical to make a 32 line table every time you have to deal with five di↵erent sentence letters.

In this chapter, we are going to introduce a new method for checking for validity and other logical properties. This time our method is going to be purely syntactic. We won’t be at all concerned with what our symbols mean. We are just going to look at the way they are arranged. Our method here will be called a system of natural deduction. When you use a system of natural deduction, you won’t do it mechanically. You will need to understand the logical structure of the argument and employ your insight. This is actually one of the reasons people like systems of natural deduction. They let us represent the logical structure of arguments in a way we can understand. Learning to represent and manipulate arguments this way is a core mental skill, used in fields like mathematics and computer programming.

Consider two arguments in SL:

111

112 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

Argument A Argument B

P1: P _Q P2: ⇠P

C: Q

P1: P ! Q P2: P

C: Q

These are both valid arguments. Go ahead and prove that for yourself by constructing the four-line truth tables. These particular valid arguments are examples of important kinds of arguments that are given special names. Argument A is an example of a kind of argument traditionally called disjunctive syllogism. In the system of proof we will develop later in the chapter, it will be given a newer name, disjunction elimination (_-E). Given a disjunction and the negation of one of the disjuncts, the other disjunct follows as a valid consequence. Argument B makes use of a di↵erent valid form: Given a conditional and its antecedent, the consequent follows as a valid consequence. This is traditionally called modus ponens. In our system it will be called conditional elimination (!-E).

Both of the arguments above remain valid even if we substitute di↵erent sentence letters. You don’t even need to run the truth tables again to see that these arguments are valid:

Argument A* Argument B*

P1: A _B P2: ⇠A

C: B

P1: A ! B P2: A

C: B

Replacing P with A and Q with B changes nothing (so long as we are sure to replace every P with an A and every Q with a B). What’s more interesting is that we can replace the individual sentence letters in Argument A and Argument B with longer sentences in SL and the arguments will still be valid, as long as we do the substitutions consistently. Here are two more perfectly valid instances of disjunction and conditional elimination.

Argument A** Argument B**

P1: (C &D) _ (E _ F ) P2: ⇠(C &D)

C: E _ F

P1: (G ! H) ! (I _ J) P2: (G ! H)

C: I _ J

Again, you can check these using truth tables, although the 16 line truth tables begin to get tiresome. All of these arguments are what we call substitution instances of the same two logical

6.1. SUBSTITUTION INSTANCES AND PROOFS 113

forms. We call them that because you get them by replacing the sentence letters with other sentences, either sentence letters or longer sentences in SL. A substitution instance cannot change the sentential connectives of a sentence, however. The sentential connectives are what make the logical form of the sentence. We can write these logical forms using fancy script letters.

Disjunction Elimination (Disjunctive Syllogism)

Conditional Elimination (Modus Ponens)

P1: A _ B P2: ⇠A

C: B

P1: A ! B P2: A

C: B

As we explained in Chapter 4, the fancy script letters are metavariables. They are a part of our metalanguage and can refer to single sentence letters like P or longer sentences like A $ (B&(C _D)).

Formally, we can define a sentence form as a sentence in SL that contains one or more metavariables in place of sentence letters. A substitution instance of that sentence form is then a sentence created by consistently substituting sentences for one or more of the metavariables in the sentence form. Here “consistently substituting” means replacing all instances of the metavariable with the same sentence. You cannot replace instances of the same metavariable with di↵erent sentences, or leave a metavariable as it is, if you have replaced other metavariables of that same type. An argument form is an argument that includes one or more sentence forms, and a substitution instance of the argument form is the argument obtained by consistently replacing the sentence forms in the argument form with their substitution instances.

Once we start identifying valid argument forms like this, we have a new way of showing that longer arguments are valid. Truth tables are fun, but doing the 1028 line truth table for an argument with 10 sentence letters would be tedious. Worse, we would never be sure we hadn’t made a little mistake in all those Ts and Fs. Part of the problem is that we have no way of knowing why the argument is valid. The table gives you very little insight into how the premises work together.

The aim of a proof system is to show that particular arguments are valid in a way that allows us to understand the reasoning involved in the argument. Instead of representing all the premises and the conclusion in one table, we break the argument up into steps. Each step is a basic argument form of the sort we saw above, like disjunctive syllogism or modus ponens. Suppose we are given the premises ⇠L ! (J _ L) and ⇠L and wanted to show J . We can break this up into two smaller arguments, each of which is a substitution inference of a form we know is correct.

114 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

Argument 1 Argument 2

P1: ⇠L ! (J _ L) P2: ⇠L

C: J _ L

P1: J _ L P2: ⇠L

C: J

The first argument is a substitution instance of modus ponens and the second is a substitution instance of disjunctive syllogism, so we know they are both valid. Notice also that the conclusion of the first argument is the first premise of the second, and the second premise is the same in both arguments. Together, these arguments are enough to get us from ⇠L ! (J _ L) and ⇠L to J .

These two arguments take up a lot of space, though. To complete our proof system, we need a system for showing clearly how simple steps can combine to get us from premises to conclusions. The system we will use in this book was devised by the American logician Frederic Brenton Fitch (1908–1987). We begin by writing our premises on numbered lines with a bar on the left and a little bar underneath to represent the end of the premises. Then we write “Want” on the side followed by the conclusion we are trying to reach. If we wanted to write out arguments 1 and 2 above, we would begin like this.

1 ⇠L ! (J _ L)

2 ⇠L Want: J

We then add the steps leading to the conclusion below the horizontal line, each time explaining o↵ to the right why we are allowed to write the new line. This explanation consists of citing a rule and the prior lines the rule is applied to. In the example we have been working with we would begin like this

1 ⇠L ! (J _ L)

2 ⇠L Want: J

3 J _ L !E 1, 2

and then go like this

1 ⇠L ! (J _ L)

2 ⇠L Want: J

3 J _ L !E 1, 2

4 J _E 2, 3

The little chart above is a proof that J follows from ⇠L ! (J _ L) and ⇠L. We will also call proofs like this derivations. Formally, a proof is a sequence of sentences. The first sentences of the sequence are assumptions; these are the premises of the argument. Every sentence later in the

6.1. SUBSTITUTION INSTANCES AND PROOFS 115

sequence follows from earlier sentences by one of the rules of proof. The final sentence of the sequence is the conclusion of the argument.

In the remainder of this chapter, we will develop a system for proving sentences in SL. Later, in Chapter 9, this will be expanded to cover Quantified Logic (QL). First, though, you should practice identifying substitution instances of sentences and longer rules.

Practice Exercises

Part A For each problem, a sentence form is given in metavariables. Identify which of the sentences after it are legitimate substitution instances of that form.

(1) A &B : a. P _Q b. (A ! B)&C c. [(A&B) ! (B&A)]

& (⇠A&⇠B) d. [((A&B)&C)&D] &F

e. (A&B) ! C

(2) ⇠(P &Q ) a. ⇠(A&B) b. ⇠(A&A) c. ⇠A&B d. ⇠((⇠A&B)& (B&⇠A)) e. ⇠(A ! B)

(3) ⇠A a. ⇠A ! B b. ⇠(A ! B) c. ⇠[(G ! (H _ I)) ! G] d. ⇠G&(⇠B&⇠H) e. ⇠(G&(B&H))

(4) ⇠A ! B a. ⇠A&B b. ⇠B ! A c. ⇠(X &Y ) ! (Z _B) d. ⇠(A ! B) e. A ! ⇠B

(5) ⇠A $ ⇠Z a. ⇠(P $ Q) b. ⇠(P $ Q) $ ⇠(Q $ P ) c. ⇠H ! ⇠G d. ⇠(A&B) $ C e. ⇠[⇠(P $ Q) $ R] $ ⇠S

(6) (A &B) _ C a. (P _Q)&R b. (⇠M &⇠D) _ C c. (D&R)& (I _D) d. [(D ! O) _A] &D e. [(A&B)&C] _ (D _A)

116 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

(7) (A &B) _ A

a. ((C ! D)&E) _A b. (A&A) _A c. ((C ! D)&E) _ (C ! D) d. ((G&B)& (Q _R)) _ (G&B) e. (P _Q)&P

(8) P ! (P ! Q )

a. A ! (B ! C) b. (A&B) ! [(A&B) ! C] c. (G ! B) ! [(G ! B) ! (G ! B)] d. M ! [M ! (D&(C &M))] e. (S _O) ! [(O _ S) ! A]

(9) ⇠A _ (B &⇠B)

a. ⇠P _ (Q&⇠P ) b. ⇠A _ (A&⇠A) c. (P ! Q) _ [(P ! Q)&⇠R] d. ⇠E&(F &⇠F ) e. ⇠G _ [(H ! G)&⇠(H ! G)]

(10) (P _ Q ) ! ⇠(P &Q )

a. A ! ⇠B b. (A _B) ! ⇠(A&B) c. (A _A) ! ⇠(A&A) d. [(A&B) _ (D ! E)] !

⇠[(A&B)& (D ! E)] e. (A&B) ! ⇠(A _B)

Part B For each problem, a sentence form is given in sentence variables. Identify which of the sentences after it are legitimate substitution instances of that form.

(1) P &P

a. A&B

b. D _D c. Z &Z

d. (Z _B)& (Z &B) e. (Z _B)& (Z _B)

(2) O &(N &N )

a. A&(B&C)

b. A&(A&B)

c. (A&B)&B

d. A&(B&B)

e. (C ! D)& (Q&Q)

(3) H ! Z

a. E ! E b. G ! H c. G ! (I ! K) d. [(I ! K) ! G] ! A e. G&(I ! K)

(4) ⇠H &C

a. H &C

b. ⇠(H &C) c. ⇠Q&R d. R&⇠Q e. ⇠(X $ Y )& (Y ! Z)

6.1. SUBSTITUTION INSTANCES AND PROOFS 117

(5) ⇠(G $ M )

a. ⇠(K $ K) b. ⇠K $ K c. ⇠((I $ K) $ (S&S)) d. ⇠(H ! (I _ J) e. ⇠((H _ F ) $ (Z ! D))

(6) (I ! W ) _ W

a. (D _ E) ! E b. (D ! E) _ E c. D ! (E _ E) d. ((W &L) ! L) _W e. ((W &L) ! J) _ J

(7) M _ (A _ A)

a. A _ (A _A) b. (A _A) _A c. C _ (C _D) d. (R ! K) _ ((D&G) _ (D&G)) e. (P &P ) _ ((⇠H &C) _ (⇠H &C))

(8) A ! ⇠(G &G)

a. B $ ⇠(G&G) b. O ! ⇠(R&D) c. (H ! Z) ! (⇠D&D) d. (O&(N &N)) ! ⇠(F &F ) e. ⇠D&⇠((J ! J)& (O $ O)

(9) ⇠((K ! K ) _ K )&G

a. ⇠(D ! D)(_D&L) b. ⇠(D ! (D _ (D&L)) c. ⇠((D ! D) _D)&L d. ((⇠K ! ⇠K) _K)&L e. ⇠((D ! D) _D)& ((D ! D) _D)

(10) (B $ (N $ N )) _ N

a. (B $ (N $ (N &N))) _N b. ((E&T ) $ (V $ V )) _ V c. (B $ (N &N)) _B d. A $ (N $ (N _N))) e. ((X $ N) $ N) _N

Part C Use the following symbolization key in the gray bubble to create substitution instances of the sentences below.

A : B B : ⇠C C : A ! B D : ⇠(B&C) E : D $ E

(1) ⇠(A $ B) (2) (B ! C )&D

(3) D ! (B &⇠B) (4) ⇠⇠(C _ E) (5) ⇠C $ (⇠⇠D &E)

Part D Use the following symbolization key in the gray bubble to create substitution instances of the sentences below.

118 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

A : I _ (I $ V ) B : C $ V C : L ! X D : V E : U

(1) ⇠A ! ⇠B (2) ⇠(B &D)

(3) (A ! A) _ (C ! A) (4) [(A ! B) ! A ] ! A

(5) A &(B &(C &(D &E)))

Part E Decide whether the following are examples of !E (modus ponens).

(1) P1: A ! B P2: B ! C

C: A ! C

(2) P1: P &Q P2: P

C: Q

(3) P1: P ! Q

C: Q

(4) P1: D ! E P2: E

C: D

(5) P1: (P &Q) ! (Q&V ) P2: P &Q

C: Q&V

Part F Decide whether the following are examples of !E (modus ponens).

(1) P1: C ! D

C: C

(2) P1: (C &L) ! (E _ C) P2: C &L

C: E _ C

(3) P1: ⇠A ! B P2: ⇠B

C: B

(4) P1: X ! ⇠Y P2: ⇠Y

C: .˙. X

(5) P1: G ! H P2: ⇠H

C: ⇠G

Part G Decide whether the following are examples of disjunctive syllogism.

6.2. BASIC RULES FOR SENTENTIAL LOGIC 119

(1) P1: (A ! B) _ (X ! Y ) P2: ⇠A

C: X ! Y

(2) P1: [(S _ T ) _ U ] _ V P2: ⇠[(S _ T ) _ U ]

C: V

(3) P1: P _Q P2: P

C: ⇠Q

(4) P1: ⇠(A _B) P2: ⇠A

C: B

(5) P1: (P _Q) _R

C: R

Part H Decide whether the following are examples of disjunctive syllogism.

(1) P1: (C &D) _ E P2: (C &D)

C: E

(2) P1: (P _Q) ! R P2: ⇠(P _Q)

C: R

(3) P1: X _ (Y ! Z) P2: ⇠X

C: Y ! Z

(4) P1: (P _Q) _R P2: ⇠P

C: Q

(5) P1: A _ (B _ C) P2: ⇠A

C: B _ C

6.2 Basic Rules for Sentential Logic

In designing a proof system, we could just start with disjunctive syllogism and modus ponens. Whenever we discovered a valid argument that could not be proved with rules we already had, we could introduce new rules. Proceeding in this way, we would have an unsystematic grab bag of rules. We might accidentally add some strange rules, and we would surely end up with more rules than we need.

120 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

Instead, we will develop what is called a system of natural deduction. In a natural deduction system, there will be two rules for each logical operator: an introduction, and an elimination rule. The introduction rule will allow us to prove a sentence that has the operator you are “introducing” as its main connective. The elimination rule will allow us to prove something given a sentence that has the operator we are “eliminating” as the main logical operator.

In addition to the rules for each logical operator, we will also have a reiteration rule. If you already have shown something in the course of a proof, the reiteration rule allows you to repeat it on a new line. We can define the rule of reiteration like this

Reiteration (R)

m A

n A R m

This diagram shows how you can add lines to a proof using the rule of reiteration. As before, the script letters represent sentences of any length. The upper line shows the sentence that comes earlier in the proof, and the bottom line shows the new sentence you are allowed to write and how you justify it. The reiteration rule above is justified by one line, the line that you are reiterating. So the “R m” on line 2 of the proof means that the line is justified by the reiteration rule (R) applied to line m. The letters m and n are variables, not real line numbers. In a real proof, they might be lines 5 and 7, or lines 1 and 2, or whatever. When we define the rule, however, we use variables to underscore the point that the rule may be applied to any line that is already in the proof.

Obviously, the reiteration rule will not allow us to show anything new. For that, we will need more rules. The remainder of this section will give six basic introduction and elimination rules. This will be enough to do some basic proofs in SL. Sections 4.3 through 4.5 will explain introduction rules involved in fancier kinds of derivation called conditional proof and indirect proof. The remaining sections of this chapter will develop our system of natural deduction further and give you tips for playing in it.

All of the rules introduced in this chapter are summarized starting on p.237.

Conjunction

Think for a moment: What would you need to show in order to prove E&F?

Of course, you could show E&F by proving E and separately proving F . This holds even if the two conjuncts are not atomic sentences. If you can prove [(A _ J) ! V ] and [(V ! L) $ (F _N)], then you have e↵ectively proved [(A _ J) ! V ] & [(V ! L) $ (F _N)]. So this will be our conjunction introduction rule, which we abbreviate & I:

6.2. BASIC RULES FOR SENTENTIAL LOGIC 121

m A

n B

A &B &I m, n

m A

n B

B &A &I m, n

A line of proof must be justified by some rule, and here we have “& I m, n.” This means: Conjunction introduction applied to line m and line n. Again, these are variables, not real line numbers; m is some line and n is some other line. If you have K on line 8 and L on line 15, you can prove (K &L) at some later point in the proof with the justification “& I 8, 15.”

We have written two versions of the rule to indicate that you can write the conjuncts in any order. Even though K occurs before L in the proof, you can derive (L&K) from them using the right-hand version & I. You do not need to mark this in any special way in the proof.

Now, consider the elimination rule for conjunction. What are you entitled to conclude from a sentence like E&F? Surely, you are entitled to conclude E; if E&F were true, then E would be true. Similarly, you are entitled to conclude F . This will be our conjunction elimination rule, which we abbreviate &E:

m A &B

A &E m

m A &B

B &E m

When you have a conjunction on some line of a proof, you can use &E to derive either of the conjuncts. Again, we have written two versions of the rule to indicate that it can be applied to either side of the conjunction. The &E rule requires only one sentence, so we write one line number as the justification for applying it. For example, both of these moves are acceptable in derivations.

4 A&(B _ C)

5 A &E 4

10 A&(B _ C)

. . . . . .

15 (B _ C) &E 10

Some textbooks will only let you use &E on one side of a conjunction. They then make you prove that it works for the other side. We won’t do this, because it is a pain in the neck.

Even with just these two rules, we can provide some proofs. Consider this argument.

[(A _B) ! (C _D)] & [(E _ F ) ! (G _H)] .˙. [(E _ F ) ! (G _H)] & [(A _B) ! (C _D)]

122 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

The main logical operator in both the premise and conclusion is a conjunction. Since the conjunction is symmetric, the argument is obviously valid. In order to provide a proof, we begin by writing down the premise. After the premises, we draw a horizontal line—everything below this line must be justified by a rule of proof. So the beginning of the proof looks like this:

1 [(A _B) ! (C _D)] & [(E _ F ) ! (G _H)]

From the premise, we can get each of the conjuncts by &E. The proof now looks like this:

1 [(A _B) ! (C _D)] & [(E _ F ) ! (G _H)]

2 [(A _B) ! (C _D)] &E 1

3 [(E _ F ) ! (G _H)] &E 1

The rule & I requires that we have each of the conjuncts available somewhere in the proof. They can be separated from one another, and they can appear in any order. So by applying the & I rule to lines 3 and 2, we arrive at the desired conclusion. The finished proof looks like this:

1 [(A _B) ! (C _D)] & [(E _ F ) ! (G _H)]

2 [(A _B) ! (C _D)] &E 1

3 [(E _ F ) ! (G _H)] &E 1

4 [(E _ F ) ! (G _H)] & [(A _B) ! (C _D)] & I 3, 2

This proof is trivial, but it shows how we can use rules of proof together to demonstrate the validity of an argument form. Also: Using a truth table to show that this argument is valid would have required a staggering 256 lines, since there are eight sentence letters in the argument.

Disjunction

If M were true, then M _N would also be true. So the disjunction introduction rule (_I) allows us to derive a disjunction if we have one of the two disjuncts:

m A

A _ B _I m

m A

B _ A _I m

Like the rule of conjunction elimination, this rule can be applied two ways. Also notice that B can be any sentence whatsoever. So the following is a legitimate proof:

1 M

2 M _ ([(A $ B) ! (C &D)] $ [E&F ]) _I 1

6.2. BASIC RULES FOR SENTENTIAL LOGIC 123

This might seem odd. How can we prove a sentence that includes A, B, and the rest, from the simple sentence M—which has nothing to do with the other letters? The secret here is to remember that all the new letters are on just one side of a disjunction, and nothing on that side of the disjunction has to be true. As long as M is true, we can add whatever we want after a disjunction and the whole thing will continue to be true.

Now consider the disjunction elimination rule. What can you conclude from M _N? You cannot conclude M . It might be M ’s truth that makes M _N true, as in the example above, but it might not. From M _N alone, you cannot conclude anything about either M or N specifically. If you also knew that N was false, however, then you would be able to conclude M .

m A _ B

n ⇠B

A _E m, n

m A _ B

n ⇠A

B _E m, n

We’ve seen this rule before: it is just disjunctive syllogism. Now that we are using a system of natural deduction, we are going to make it our rule for disjunction elimination (_E). Once again, the rule works on both sides of the sentential connective.

Conditionals and biconditionals

The rule for conditional introduction is complicated because it requires a whole new kind of proof, called conditional proof. We will deal with this in the next section. For now, we will only use the rule of conditional elimination.

Nothing follows from M ! N alone, but if we have both M ! N and M , then we can conclude N . This is another rule we’ve seen before: modus ponens. It now enters our system of natural deduction as the conditional elimination rule (!E).

m A ! B

n A

B !E m, n

Biconditional elimination ($E) will be a double-barreled version of conditional elimination. If you have the left-hand subsentence of the biconditional, you can derive the right-hand subsentence. If you have the right-hand subsentence, you can derive the left-hand subsentence. This is the rule:

m A $ B

n A

B $E m, n

124 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

m A $ B

n B

A $E m, n

Invalid argument forms

In section 4.1, in the last two problem parts (p. 118), we saw that sometimes an argument looks like a legitimate substitution instance of a valid argument form, but really isn’t. For instance, the problem set C asked you to identify instances of modus ponens. Below I’m giving you two of the answers.

(5) Modus ponens

1. (C &L) ! (E _ C) 2. C &L

. . .˙. E _ C

(7) Not modus ponens.

1. D ! E 2. E

. . .˙. D

The argument on the left is an example of a valid argument, because it is an instance of modus ponens, while the argument on the right is an example of an invalid argument, because it is not an example of modus ponens. (We originally defined the terms valid and invalid on p. 30). Arguments like the one on the right, which try to trick you into thinking that they are instances of valid arguments, are called deductive fallacies. The argument on the right is specifically called the fallacy of affirming the consequent. In the system of natural deduction we are using in this textbook, modus ponens has been renamed “conditional elimination,” but it still works the same way. So you will need to be on the lookout for deductive fallacies like a�rming the consequent as you construct proofs.

Notation

The rules we have learned in this chapter give us enough to start doing some basic derivations in SL. This will allow us to prove things syntactically which would have been too cumbersome to prove using the semantic method of truth tables. We now need to introduce a few more symbols to be clear about what methods of proof we are using.

In Chapter 1, we used the three dots .˙. to indicate generally that one thing followed from another. In chapter 3 we introduced the double turnstile, , to indicate that one statement could

be proven some others using truth tables. Now we are going to use a single turnstile, , to indicate that we can derive a statement from a bunch of premises, using the system of natural deduction we have begun to introduce in this section. Thus we will write {A ,B ,C} D, to

6.2. BASIC RULES FOR SENTENTIAL LOGIC 125

indicate that there is a derivation going from the premises A , B , and C to the conclusion D. Note that these are metavariables, so I could be talking about any sentences in SL.

The single turnstile will work the same way the double turnstile did. So, in addition to the uses of the single turnstile above we can write A to indicate that A can be proven a tautology using syntactic methods. We can write A B to say that A and B can be proven logically equivalent using these derivations. You will learn how to do these later things at the end of the chapter. In the meantime, we need to practice our basic rules of derivation.

Practice Exercises

Part A Some of the following arguments are legitimate instances of our six basic inference rules. The others are either invalid arguments or valid arguments that are still illegitimate because they would take multiple steps using our basic inference rules. For those that are legitimate, mark the rule that they are instances of. Mark those that are not “Not a single inference.”

(1) P1: R _ S

C: S

(2) P1: (A ! B) _ (B ! A) P2: A ! B

C: B ! A

(3) P1: P &(Q _R)

C: R

(4) P1: P &(Q&R)

C: P

(5) P1: A

C: P &(Q ! A)

(6) P1: A P2: B&C

C: (A&B)&C

(7) P1: (X &Y ) $ (Z &W ) P2: Z &W

C: X &Y

(8) P1: ((L ! M) ! N) ! O P2: L

C: M

Part B Some of the following arguments are legitimate instances of our six basic inference rules. The others are either invalid arguments or valid arguments that are still illegitimate because they would take multiple steps using our basic inference rules. For those that are legitimate, mark the rule that they are instances of. Mark those that are not “Not a single inference.”

126 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

(1) P1: A&B

C: A

(2) P1: A ! (B&(C _D)) P2: A

C: B&(C _D)

(3) P1: P &(Q&R)

C: R

(4) P1: P

C: P _ [A&(B $ C)]

(5) P1: M P2: D&C

C: M &(D&C)

(6) P1: (X &Y ) ! (Z &W ) P2: Z &W

C: X &Y

(7) P1: (X &Y ) ! (Z &W ) P2: ⇠(X &Y )

C: ⇠(Z &W )

(8) P1: ((L ! M) ! N) ! O P2: (L ! M) ! N

C: O

Part C Fill in the missing pieces in the following proofs. Some are missing the justification column on the right. Some are missing the left column that contains the actual steps, and some are missing lines from both columns.

(1) . 1 W ! ⇠B

2 A&W

3 B _ (J &K) Want: K

4 W

5 ⇠B

6 J &K

7 K

(2) . 1 W &B

2 E&Z Want: W &Z

3 &E 1

4 &E 2

5 W &Z

6.2. BASIC RULES FOR SENTENTIAL LOGIC 127

(3) . 1 (A&B)&C Want: A&(B&C)

2 A&B

3 C

4 A

5 B

6 B&C

7 A&(B&C)

(4) . 1 (⇠A&B) ! C

2 ⇠A

3 A _B Want: C

4 _E 2, 3

5 & I 2, 4

6 !E 1, 5

(5) . 1 ⇠A&(⇠B&C)

2 C ! (D&(B _ E))

3 (E&⇠A) ! F Want: D&F

4 &E 1

5 ⇠B&C

6 &E 5

7 C

8 !E 2, 7

9 D

10 &E 8

11 E

12 & I 4, 11

13 F

14 & I 9, 13

128 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

Part D Fill in the missing pieces in the following proofs. Some are missing the justification column on the right. Some are missing the left column that contains the actual steps, and some are missing lines from both columns.

(1) . 1 A&⇠B

2 A ! ⇠C

3 B _ (C _D) Want: D

4 &E 1

5 &E 1

6 !E 2, 4

7 _E 3, 5

8 _E 6, 7

(2) . 1 W _ V

2 I &(⇠Z ! ⇠W )

3 I ! ⇠Z Want: I &V

4 &E 2

5 &E 2

6 ⇠Z

7 !E 5, 6

8 V

9 & I 4, 8

(3) . 1 ⇠P &S) $ S

2 S&(P _Q) Want: Q

3 S

4 P _Q

5 ⇠P &S

6 ⇠P

7 Q

(4) . 1 C ! (A ! B)

2 D _ C

3 ⇠D Want: A !B

4 _E 2, 3

5 !E 1, 4

6.2. BASIC RULES FOR SENTENTIAL LOGIC 129

(5) . 1 X &(Y &Z) Want: (X _A)& [(Y _B)& (Z &C)]

2 &E 1

3 &E 1

4 &E 3

5 &E 3

6 _I 2

7 _I 4

8 _I 5

9 & I 7, 8

10 & I 6, 9

Part E Derive the following.

(1) {A ! B,A} A&B

(2) {A $ D,C, [(A $ D)&C] ! (C $ B)} B

(3) {A $ B,B $ C,C ! D,A} D

(4) {(A ! ⇠B)&A,B _ C} C

(5) {(A ! B) _ (C ! (D&E)),⇠(A ! B), C} D

(6) {C _ (B&A),⇠C} A _A

(7) {A _B,⇠A,⇠B} C

Part F Derive the following.

(1) {A&B,B ! C} A&(B&C)

(2) {(P _R)& (S _R),⇠R&Q} P &(Q _R)

130 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

(3) {(X &Y ) ! Z,X &W,W ! Y } Z

(4) {A _ (B _G), A _ (B _H),⇠A&⇠B} G&H

(5) {P &(Q&⇠R), R _ T} T _ S

(6) {((A ! D) _B) _ C,⇠C,⇠B,A} D

(7) {A _ ⇠⇠B,⇠B _ ⇠C,C _A,⇠A} D

Part G Derive the following.

(1) H &A A&H

(2) {P _Q,D ! E,⇠P &D} E&Q

(3) {⇠A ! (A _ ⇠C),⇠A,⇠C $ D} D

(4) {⇠A&C,A _B, (B&C) ! (D&E)} D

(5) {A ! (B ! (C ! D)), A&(B&C)} D

(6) {E _ F, F _G,⇠F} E&G

(7) {X &(Z _ Y ),⇠Z, Y ! ⇠X} A

Part H Derive the following.

(1) {P $ (Q $ R),P ,P ! R} Q

(2) {A ! (B ! C), A,B} C

(3) {(X _A) ! ⇠Y, Y _ (Z &Q), X} Z

(4) {A&(B&C), A&D,B&E} D&(E&C)

(5) {A&(B _ ⇠C),⇠B&(C _ E), E ! D} D

(6) {A ! B,B ! C,C ! A,B,⇠A} D

(7) {⇠A&B,A _ P,A _Q,B ! R} P &(Q&R)

Part I Translate the following arguments into SL and then show that they are valid. Be sure to write out your dictionary.

(1) If Professor Plum did it, he did it with the rope in the kitchen. Either Professor Plum or Miss Scarlett did it, and it wasn’t Miss Scarlett. Therefore the murder was in the kitchen.

6.3. CONDITIONAL PROOF 131

(2) If you are going to replace the bathtub, you might as well redo the whole bathroom. If you redo the whole bathroom, you will have to replace all the plumbing on the north side of the house. You will spend a lot of money on this project if and only if you replace the plumbing on the north side of the house. You are definitely going to replace the bathtub. Therefore you will spend a lot of money on this project.

Part J Translate the following arguments into SL and then show that they are valid. Be sure to write out your dictionary.

(1) Either Caroline is happy, or Joey is happy, but not both. If Joey teases Caroline, she is not happy. Joey is teasing Caroline. Therefore, Joey is happy.

(2) Either grass is green or one of two other things: the sky is blue or snow is white. If my lawn is brown, the sky is gray, and if the sky is gray, it is not blue. If my lawn is brown, then grass is not green, and on top of that my lawn is brown. Therefore snow is white.

6.3 Conditional Proof

So far we have introduced introduction and elimination rules for the conjunction and disjunction, and elimination rules for the conditional and biconditional, but we have no introduction rules for conditionals and biconditionals, and no rules at all for negations. That’s because these other rules require fancy kinds of derivations that involve putting proofs inside proofs. In this section, we will look at one of these kinds of proof, called conditional proof.

Conditional introduction

Consider this argument:

1. R _ F . .

.˙. ⇠R ! F

The argument is valid. You can use the truth table to check it. Unfortunately, we don’t have a way to prove it in our syntactic system of derivation. To help us see what our rule for conditional introduction should be, we can try to figure out what new rule would let us prove this obviously true argument.

Let’s start the proof in the usual way, like this:

1 R _ F Want: ⇠R !F

132 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

If we had ⇠R as a further premise, we could derive F by the _E rule. But sadly, we do not have ⇠R as a premise, and we can’t derive it directly from the premise we do have—so we cannot simply prove F . What we will do instead is start a subproof, a proof within the main proof. When we start a subproof, we draw another vertical line to indicate that we are no longer in the main proof. Then we write in an assumption for the subproof. This can be anything we want. Here, it will be helpful to assume ⇠R. Our proof now looks like this:

1 R _ F Want: ⇠R !F

2 ⇠R Assumption for CD, Want: F

It is important to notice that we are not claiming to have proved ⇠R. We do not need to write in any justification for the assumption line of a subproof. You can think of the subproof as posing the question: What could we show if ⇠R were true? For one thing, we can derive F . To make this completely clear, I have annotated line 2 “Assumption for CD,” to indicate that this is an additional assumption we are making because we are using conditional derivation (CD). I have also added “Want: F” because that is what we will want to show during the subderivation. In the future I won’t always include all this information in the annotation. But for now we will use it to be completely clear on what we will be doing.

So now let’s go ahead and show F in the subderivation.

1 R _ F Want: ⇠R !F

2 ⇠R Assumption for CD, Want: F

3 F _E 1, 2

This has shown that if we had ⇠R as a premise, then we could prove F . In e↵ect, we have proven ⇠R ! F . So the conditional introduction rule (!I) will allow us to close the subproof and derive ⇠R ! F in the main proof. Our final proof looks like this:

1 R _ F Want: ⇠R !F

2 ⇠R Assumption for CD, Want: F

3 F _E 1, 2

4 ⇠R ! F !I 2–3

Notice that the justification for applying the !I rule is the entire subproof. Usually that will be more than just two lines.

Now that we have that example, let’s lay out more precisely the rules for subproofs and then give the formal schemes for the rule of conditional and biconditional introduction.

6.3. CONDITIONAL PROOF 133

rule 1 You can start a subproof on any line, except the last one, and introduce any assumptions with that subproof.

rule 2 All subproofs must be closed by the time the proof is over.

rule 3 Subproofs may closed at any time. Once closed, they can be used to justify !I, $I, ⇠E, and ⇠I.

rule 4 Nested subproofs must be closed before the outer subproof is closed.

rule 5 Once the subproof is closed, lines in the subproof cannot be used in later justifications.

Rule 1 gives you great power. You can assume anything you want, at any time. But with great power, comes great responsibility, and rules 2–5 explain what your responsibilities are. Making an assumption creates the burden of starting a subproof, and subproofs must end before the proof is done. (That’s why we can’t start a subproof on the last line.) Closing a subproof is called discharging the assumptions of that subproof. So we can summarize your responsibilities this way: You cannot complete a proof until you have discharged all of the assumptions introduced in subproofs. Once the assumptions are discharged, you can use the whole subproof as a justification, but not the individual lines. So you need to know going into the subproof what you are going to use it for once you get out. As in so many parts of life, you need an exit strategy.

With those rules for subproofs in mind, the !I rule looks like this:

m A want B

n B

A ! B !I m–n

You still might think this gives us too much power. In logic, the ultimate sign you have too much power is that given any premise A you can prove any conclusion B . Fortunately, our rules for subproofs don’t let us do this. Imagine a proof that looks like this:

1 A

2 B

It may seem as if a proof like this will let you reach any conclusion B from any premise A . But this is not the case. By rule 2, in order to complete a proof, you must close all of the subproofs, and we haven’t done that. A subproof is only closed when the vertical line for that subproof ends. To put it another way, you can’t end a proof and still have two vertical lines going.

You still might think this system gives you too much power. Maybe we can try closing the subproof and writing B in the main proof, like this

134 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

1 A

2 B

3 B R 2

4 B R 3

But this is wrong, too. By rule 5, once you close a subproof, you cannot refer back to individual lines inside it.

Of course, it is legitimate to do this:

1 A

2 B

3 B R 2

4 B ! B !I 2–3

This should not seem so strange, though. Since B!B is a tautology, no particular premises should be required to validly derive it. (Indeed, as we will see, a tautology follows from any premises.)

When we introduce a subproof, we typically write what we want to derive in the right column, just like we did in the first example in this section. This is just so that we do not forget why we started the subproof if it goes on for five or ten lines. There is no “want” rule. It is a note to ourselves and not formally part of the proof.

Having an exit strategy when you launch a subproof is crucial. Even if you discharge an assumption properly, you might wind up with a final line that doesn’t do you any good. In order to derive a conditional by !I, for instance, you must assume the antecedent of the conditional in a subproof. The last line of the subproof must be the consequent of the conditional, and the whole conditional is the first line after the end of the subproof. Pick your assumptions so that you wind up with a conditional that you actually need. It is always permissible to close a subproof and discharge its assumptions, but it will not be helpful to do so until you get what you want.

Now that we have the rule for conditional introduction, consider this argument:

1. P ! Q 2. Q ! R

. . .˙. P ! R

We begin the proof by writing the two premises as assumptions. Since the main logical operator in the conclusion is a conditional, we can expect to use the !I rule. For that, we need a subproof—so we write in the antecedent of the conditional as an assumption of a subproof:

6.3. CONDITIONAL PROOF 135

1 P ! Q

2 Q ! R

3 P

We made P available by assuming it in a subproof, allowing us to use !E on the first premise. This gives us Q, which allows us to use !E on the second premise. Having derived R, we close the subproof. By assuming P we were able to prove R, so we apply the !I rule and finish the proof.

1 P ! Q

2 Q ! R

3 P want R

4 Q !E 1, 3

5 R !E 2, 4

6 P ! R !I 3–5

Biconditional introduction

Just as the rule for biconditional elimination was a double-headed version of conditional elimination, our rule for biconditional introduction is a double-headed version of conditional introduction. In order to derive W $ X, for instance, you must be able to prove X by assuming W and prove W by assuming X. The biconditional introduction rule ($I) requires two subproofs. The subproofs can come in any order, and the second subproof does not need to come immediately after the first—but schematically, the rule works like this:

m A want B

n B

p B want A

q A

A $ B $I m–n, p–q

We will call any proof that uses subproofs and either !I or $I conditional proof. By contrast, the first kind of proof you learned, where you only use the six basic rules, will be called direct proof. In section 4.3 we will learn the third and final kind of proof indirect proof. But for now you should practice conditional proof.

136 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

Practice Exercises

Part A Fill in the blanks in the following proofs. Be sure to include the “Want” line for each subproof.

(1) . 1 ⇠P ! (Q _R)

2 P _ ⇠Q Want: ⇠P !R

3 ⇠P Want:

4 Q _R

5 _E 2, 3

6 R

7 ⇠P ! R

(2) . 1 A _B

2 B ! (B ! ⇠A) Want: ⇠A $B

3 ⇠A Want:

4 _E 1, 3

5 B Want:

6 B ! ⇠A

7 !E 5, 6

8 ⇠A $ B

Part B Fill in the blanks in the following proofs. Be sure to include the “Want” line for each subproof.

(1) .

6.3. CONDITIONAL PROOF 137

1 B ! ⇠D

2 A ! (D _ C) Want: A ! (B ! C)

3 A

4 Want: C

5 !E 2, 3

6 !E 1, 4

7 _E 5, 6

8 !I 4–7

9 !I 3–8

(2) . 1 (G _H) ! (S&T )

2 (T _ U) ! (C &D) Want: G ! C

3 Want: C

4 G _H

5 !E 1, 4

6 T

7 _I 6

8 C &D

9 &E 8

10 !I 3–9

Part C Derive the following

(1) {K &L} K $ L

(2) {A ! (B ! C)} (A&B) ! C

(3) {A $ B,B $ C} A $ C

(4) {X $ (A&B), B $ Y,B ! A} X $ Y

(5) {B ! ⇠E,A ! ⇠D,D _ (E _R), (R&A) ! C} A ! (B ! C)

(6) {⇠W &⇠E,Q $ D} (W _Q) $ (E _D)

(7) {⇠A, (B&C) ! D} (A _B) ! (C ! D)

138 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

(8) {(A&B) $ D,D $ (X &Y ), C $ Z} A&(B&C) $ X &(Y &Z)

6.4 Indirect Proof

The last two rules we need to discuss are negation introduction (⇠I) and negation elimination (⇠E). As with the rules of conditional and biconditional introduction, we have put o↵ explaining the rules, because they require launching subproofs. In the case of negation introduction and elimination, these subproofs are designed to let us perform a special kind of derivation classically known as reductio ad absurdum, or simply reductio.

A reductio in logic is a variation on a tactic we use in ordinary arguments all the time. In arguments we often stop to imagine, for a second, that what our opponent is saying is true, and then realize that it has unacceptable consequences. In so-called “slippery slope” arguments or “arguments from consequences,” we claim that doing one thing will will lead us to doing another thing which would be horrible. For instance, you might argue that legalizing physician assisted suicide for some patients might lead to the involuntary termination of lots of other sick people. These arguments are typically not very good, but they have a basic pattern whcih we can make rigorous in our logical system. These arguments say “if my opponent wins, all hell will break loose.” In logic the equivalent of all hell breaking loose is asserting a contradiction. The worst thing you can do in logic is contradict yourself. The equivalent of our opponent being right in logic would be that a sentence we are trying to prove true turns out to be false (or alternately, that a sentence we are trying to prove false turns out to be true.) So, in developing the rules for reductio ad absurdum, we need to find a way to say “if this sentence were false (or true), we would have to assert a contradiction.”

In our system of natural deduction, this kind of proof will be known as indirect proof.The basic rules for negation will allow for arguments like this. If we assume something and show that it leads to contradictory sentences, then we have proven the negation of the assumption. This is the negation introduction (⇠I) rule:

m A for reductio

n B

n+ 1 ⇠B

n+ 2 ⇠A ⇠I m–n+ 1

m A for reductio

n ⇠B

n+ 1 B

n+ 2 ⇠A ⇠I m–n+ 1

For the rule to apply, the last two lines of the subproof must be an explicit contradiction: either the second sentence is the direct negation of the first, or vice versa. We write “for reductio” as a note to ourselves, a reminder of why we started the subproof. It is not formally part of the proof, and you can leave it out if you find it distracting.

To see how the rule works, suppose we want to prove a law of double negation A .˙. ⇠⇠A

6.4. INDIRECT PROOF 139

1 A Want: ⇠⇠A

2 ⇠A for reductio

3 A R 1

4 ⇠A R 2

5 ⇠⇠A ⇠I 2–4

The ⇠E rule will work in much the same way. If we assume ⇠A and show that it leads to a contradiction, we have e↵ectively proven A . So the rule looks like this:

m ⇠A for reductio

n B

n+ 1 ⇠B

n+ 2 A ⇠E m–n+ 1

m ⇠A for reductio

n ⇠B

n+ 1 B

n+ 2 A ⇠E m–n+ 1

Here is a simple example of negation elimination at work. We can show L $ ⇠O,L _ ⇠O.˙. L by assuming ⇠L, deriving a contradiction, and then using ⇠E.

1 L $ ⇠O

2 L _ ⇠O Want: L

3 ⇠L for reductio

4 ⇠O _E 2, 3

5 L $E 1, 4

6 ⇠L R 3

7 L ⇠E 3–6

With the addition of ⇠E and ⇠I, our system of natural deduction is complete. We can now prove that any valid argument is actually valid. This is really where the fun begins.

One important bit of strategy. Sometimes, you will launch a subproof right away by assuming the negation of the conclusion to the whole argument. Other times, you will use a subproof to get a piece of the conclusion you want, or some stepping stone to the conclusion you want. Here’s a simple example. Suppose you were asked to show that this argument is valid: ⇠(A _B).˙.⇠A&⇠B. (The argument, by the way, is part of DeMorgan’s Laws, some very useful equivalences which we will see more of later on.)

You need to set up the proof like this.

140 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

1 ⇠(A _B) Want ⇠A&⇠B

Since you are trying to show ⇠A&⇠B, you could open a subproof with ⇠(⇠A&⇠B) and try to derive a contradiction, but there is an easier way to do things. Since you are tying to prove a conjunction, you can set out to prove each conjunct separately. Each conjunct, then, would get its own reductio. Let’s start by assuming A in order to show ⇠A

1 ⇠(A _B) Want ⇠A&⇠B

2 A for reductio

3 A _B _I 2

4 ⇠(A _B) R 1

5 ⇠A ⇠I 2–4

We can then finish the proof by showing ⇠B and putting it together with ⇠A and conjunction introduction.

1 ⇠(A _B) Want ⇠A&⇠B

2 A for reductio

3 A _B _I 2

4 ⇠(A _B) R 1

5 ⇠A ⇠I 2–4

6 B for reductio

7 A _B _I 6

8 ⇠(A _B) R 1

9 ⇠B ⇠I 7–9

10 ⇠A&⇠B &I 6, 9

Practice Exercises

Part A Fill in the blanks in the following proofs.

(1) . 1 ⇠A&⇠B Want:

2 ⇠A

3 ⇠B

4 A _B for reductio

5 B

6 ⇠B

7 ⇠(A _B)

6.4. INDIRECT PROOF 141

(2) . 1 ⇠A _ ⇠B Want: ⇠(A&B)

2 A&B for reductio

3 &E 2

4 &E 2

5 for reductio

6 R 3

7 R 5

8 ⇠⇠A

9 B

10 ⇠B

11 ⇠(A&B)

Part B Fill in the blanks in the following proofs.

(1) .

1 P ! Q Want: ⇠P _Q

2 for reductio

3 for reductio

4 !E 1, 3

5 _I 4

6 R 2

7 ⇠P ⇠I 3–6

8 _I 7

9 R 2

10 ⇠P _Q ⇠E 2–9

(2) .

142 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

1 (X &Y ) _ (X &Z)

2 ⇠(X &D)

3 D _M Want: M

4 for reductio

5 D _E

6 for reductio

7 ⇠X _ ⇠Y

8 for reductio

9 X

10 ⇠X

11 ⇠(X &Y ) ⇠I 8–10

12 _E 1, 11

13 &E 12

14 R 6

15 X

16 X &D

17 ⇠(X &D)

18 M ⇠E 4–17

Part C Derive the following using indirect derivation. You may also have to use conditional derivation.

(1) ⇠⇠A A

(2) {A ! B,⇠B} ⇠A

(3) A ! (⇠B _ ⇠C) A ! ⇠(B&C)

(4) ⇠(A&B) ⇠A _ ⇠B

(5) {⇠F ! G,F ! H} G _H

(6) {(T &K) _ (C &E), E ! ⇠C} T &K

(7) {(A ! B)} (A ! ⇠B) ! ⇠A

6.5. TAUTOLOGIES AND EQUIVALENCES 143

Part D Derive the following using indirect derivation. You may also have to use conditional derivation.

(1) {P ! Q,P ! ⇠Q} ⇠P

(2) (C &D) _ E E _D

(3) M _ (N ! M) ⇠M ! ⇠N

(4) {A _B,A ! C,B ! C} C

(5) A ! (B _ (C _D)) ⇠[A&(⇠B&(⇠C &⇠D))]

6.5 Tautologies and Equivalences

So far all we’ve looked at is whether conclusions follow validly from sets of premises. However, as we saw in the chapter on truth tables, there are other logical properties we want to investigate: whether a statement is a tautology, a contradiction or a contingent statement, whether two statements are equivalent, and whether sets of sentences are consistent. In this section, we will look at using derivations to test for two properties which will be important in later sections, logical equivalence and being a tautology.

We can say that two statements are syntactically logically equivalent in SL if you can derive each of them from the other. We can symbolize this the same way we symbolized semantic equivalence. When we introduced the double turnstile (p. 100), we said we would write the symbol facing both directions to indicate that two sentences were semantically equivalent, like this: A&B B&A. We can do the same thing with the single turnstile for syntactic

equivalence, like this: A&B B&A.

For an example of how we can show two sentences to be syntactically equivalent, consider the sentences P ! (Q ! R) and (P ! Q) ! (P ! R). To prove these logically equivalent using derivations, we simply use derivations to prove the equivalence one way, from P !(Q !R) to (P !Q) !(P !R). And then we prove it going the other way, from (P !Q) !(P !R) to P !(Q !R). We set up the proof going left to right like this:

1 P ! (Q ! R) Want: (P !Q) !(P !R)

Since our want line is a conditional, we can set this up as a conditional proof. Once we set up the conditional proof, we also have a conditional in next want line, which means that we can put a conditional proof inside a conditional proof, like this.

144 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

1 P ! (Q ! R) Want: (P !Q) !(P !R)

2 P ! Q Want: P !R

3 P Want: R

The completed proof for the equivalence going in one direction will look like this.

1 P ! (Q ! R) Want: (P !Q) !(P !R)

2 P ! Q Want: P !R

3 P Want: R

4 Q ! R !E 1, 3

5 Q !E 2, 3

6 R !E 4, 5

7 P ! R !I 3–6

8 (P ! Q) ! (P ! R) !I 2–7

This shows that P ! (Q ! R) (P ! Q) ! (P ! R). In order to show P ! (Q ! R) (P ! Q) ! (P ! R), we need to prove the equivalence going the other direction. That proof will look like this:

1 (P ! Q) ! (P ! R) Want: P !(Q !R)

2 P Want: Q !R

3 Q Want: R

4 P Want: Q

5 Q R 3

6 P ! Q !I 4–5

7 P ! R !E 1, 6

8 R !E 2, 7

9 Q ! R !I 3–8

10 P ! (Q ! R) !I 2–9

You might think it is strange that we assume P twice in this proof, but that is the way we have to do it. When we assume P on line 2, our goal is to prove P ! Q ! R. Before we can finish that proof, we also need to know that P ! Q. This requires a di↵erent subproof.

6.5. TAUTOLOGIES AND EQUIVALENCES 145

These two proofs show that P ! (Q ! R) and (P ! Q) ! (P ! R) are equivalent, so we can write P ! (Q ! R) (P ! Q) ! (P ! R).

We can also prove that a sentence is a tautology using a derivation. A tautology is something that must be true as a matter of logic. If we want to put this in syntactic terms, we would say that syntactic tautology in SL is a statement that can be derived without any premises, because its truth doesn’t depend on anything else. Now that we have all of our rules for starting and ending subproofs, we can actually do this. Rather than listing any premises, we simply start a subproof at the beginning of the derivation. The rest of the proof can work only using premises assumed for the purposes of subproofs. By the end of the proof, you have discharged all these assumptions, and are left knowing a tautological statement without relying on any leftover premises. Consider this proof of the law of noncontradiction: ⇠(G&⇠G).

1 G&⇠G for reductio

2 G &E 1

3 ⇠G &E 1

4 ⇠(G&⇠G) ⇠I 1–3

This statement simply says that any sentence G cannot be both true and not true at the same time. We prove it by imagining what would happen if G were actually both true and not true, and then pointing out that we already have our contradiction.

In the previous chapter, we expressed the fact that something could be proven a tautology using truth tables by writing the double turnstile in front of it. The law of noncontradiction above could have been proven using truth tables, so we could write: ⇠(G&⇠G) In this chapter, we will use the single turnstile the same way, to indicate that a sentence can be proven to be a tautology using a derivation. Thus the above proof entitles us to write ⇠(G&⇠G).

Practice Exercises

Part A Prove each of the following equivalences

(1) J J _ (L&⇠L)

(2) P ! (Q ! R) Q ! (P ! R)

(3) P ! ⇠P ⇠P

(4) ⇠(P $ Q) (P $ ⇠Q)

Part B Prove each of the following equivalences

146 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

(1) (P ! R)& (Q ! R) (P _Q) ! R

(2) (P ! (Q _R)) (P ! Q) _ (P ! R)

(3) (P $ Q) ⇠P $ ⇠Q

Part C Prove each of the following tautologies

(1) O ! O

(2) N _ ⇠N

(3) ⇠(A ! ⇠C) ! (A ! C)

(4) P $ (P _ (Q&P ))

Part D Prove each of the following tautologies

(1) (B ! ⇠B) $ ⇠B

(2) (P ! [P ! Q]) ! (P ! Q)

(3) (P _ ⇠P )& (Q $ Q)

(4) (P &⇠P ) _ (Q $ Q)

6.6 Derived Rules

Now that we have our five rules for introduction and our five rules for elimination, plus the rule of reiteration, our system is complete. If an argument is valid, and you can symbolize that argument in SL, you can prove that the argument is valid using a derivation. (We will say a bit more about this in section 6.7.) Now that our system is complete, we can really begin to play around with it and explore the exciting logical world it creates.

There’s an exciting logical world created by these eleven rules? Yes, yes there is. You can begin to see this by noticing that there are a lot of other interesting rules that we could have used for our introduction and elimination rules, but didn’t. In many textbooks, the system of natural deduction has a disjunction elimination rule that works like this:

m A _ B

n A ! C

o B ! C

C _⇤ m, n, o

6.6. DERIVED RULES 147

You might think our system is incomplete because it lacks this alternative rule of disjunction elimination. Yet this is not the case. If you can do a proof with this rule, you can do a proof with the basic rules of the natural deduction system. You actually proved this rule in problem (4) of part D in the exercises for section 6.4. Furthermore, once you have a proof of this rule, you can use it inside other proofs whenever you think you would need a rule like _⇤. Simply use the proof you gave in the last homework as a sort of recipe for generating a new series of steps to get you to a line saying C _ D

But adding lines to a proof using this recipe all the time would be a pain in the neck. What’s worse, there are dozens of interesting possible rules out there, which we could have used for our introduction and elimination rules, and which we now find ourselves replacing with recipes like the one above.

Fortunately our basic set of introduction and elimination rules, plus reiteration, was meant to be expanded on. That’s part of the game we are playing here. The first system of deduction created in the Western tradition was the system of geometry created by Euclid (c 300 BCE). Euclid’s Elements began with 10 basic laws, along with definitions of terms like “point,” “line,” and “plane.” He then went on to prove hundreds of di↵erent theorems about geometry, and each time he proved a theorem he could use that theorem to help him prove later theorems.

We can do the same thing in our system of natural deduction. What we need is a rule that will allow us to make up new rules. The new rules we add to the system will be called derived rules. Our ten rules for adding and eliminating connectives are then the axioms of SL. Now here is our rule for adding rules.

Rule of Derived Theorem Introduction: Given a derivation in SL of some argument A1 . . .An B, create the rule A1 . . .An B and assign a name to it of the form “Tn”, to be read “theorem n.” Now given a derivation of some theorem Tm, where n < m, if A1 . . .An occur as earlier lines x1 . . .xn in a proof, one may infer B , and justify it “Tn, x1 . . .xn”, so long as none of lines x1 . . .xn are in a closed subproof.

Let’s make our rule _⇤ above our first theorem. The proof of T1 is derived simply from the recipe above.

T1 (Constructive Dilemma, CD): {A _ B ,A ! C ,B ! C} C

Proof:

148 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

1 A _B

2 A ! C

3 B ! C want: C

4 ⇠C for reductio

5 ⇠A for reductio

6 B _E 2, 5

7 C !I 3, 6

8 ⇠C R 4

9 A ⇠I 5–8

10 C !E 2, 10

11 ⇠C R 4

12 C ⇠I 4–12

Informally, we will refer to T1 as “Constructive Dilemma” or by the abbreviation “CD.” Most theorems will have names and easy abbreviations like this. We will generally use the abbreviations to refer to the proofs when we use them in derivations, because they are easier to remember.

Several other important theorems have already appeared as examples or in homework problems. We’ll talk about most of them in the next section, when we discuss rules of replacement. In the meantime, there is one important one we need to introduce now

T2 (Modus Tollens, MT): {A ! B ,⇠B} ⇠A

Proof: See page 142

Now that we have some theorems, let’s close by looking at how they can be used in a proof.

T3 (Destructive Dilemma, DD): {A ! B ,A ! C ,⇠B _ ⇠C} ⇠A

6.6. DERIVED RULES 149

1 A ! B

2 A ! C

3 ⇠B _ ⇠C Want: ⇠A

4 A for reductio

5 B !E 1, 4

6 ⇠B For reductio

7 B R 5

8 ⇠B R 6

9 ⇠⇠B ⇠I 6–8

10 ⇠C _E 3, 9

11 A R 4

12 ⇠A MT 2, 10

13 ⇠A ⇠I 4–12

Practice Exercises

Part A Prove the following theorems

(1) T4 (Hypothetical Syllogism, HS): {A ! B ,B ! C} A ! C

(2) T5 (Idempotence of _, Idem_): A _ A A

(3) T6 (Idempotence of & , Idem& ): A A &A

(4) T7 (Weakening, WK): A B ! A

Part B Provide proofs using both axioms and derived rules to show each of the following.

(1) {M &(⇠N ! ⇠M)} (N &M) _ ⇠M

(2) {C ! (E&G), ⇠C ! G} G

(3) {(Z &K) $ (Y &M), D&(D ! M)} Y ! Z

(4) {(W _X) _ (Y _ Z), X ! Y , ⇠Z} W _ Y

150 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

(5) {(B ! C)& (C ! D), (B ! D) ! A } A

Part C

(1) If you know that A B , what can you say about (A &C ) B? Explain your answer.

(2) If you know that A B , what can you say about (A _ C ) B? Explain your answer.

6.7 Rules of Replacement

Very often in a derivation, you have probably been tempted to apply a rule to a part of a line. For instance, if you knew F ! (G&H) and wanted F ! G, you would be tempted to apply &E to just the G&H part of F ! (G&H). But, of course you aren’t allowed to do that. We will now introduce some new derived rules where you can do that. These are called rules of replacement, because they can be used to replace part of a sentence with a logically equivalent expression. What makes the rules of replacement di↵erent from other derived rules is that they draw on only one previous line and are symmetrical, so that you can reverse premise and conclusion and still have a valid argument. Some of the most simple examples are Theorems 8� 10, the rules of commutativity for & , _, and $.

T8 (Commutativity of & , Comm& ): (A &B) (B &A) T9 (Commutativity of _, Comm_): (A _ B) (B _ A) T10 (Commutativity of $, Comm$): (A $ B) (B $ A)

You will be asked to prove these in the homework. In the meantime, let’s see an example of how they work in a proof. Suppose you wanted to prove (M _ P ) ! (P &M), .˙. (P _M) ! (M &P ) You could do it using only the basic rules, but it will be long and inconvenient. With the Comm rules, we can provide a proof easily:

1 (M _ P ) ! (P &M)

2 (P _M) ! (P &M) Comm& 1

3 (P _M) ! (M &P ) Comm_ 2

Formally, we can put our rule for deploying rules of replacement like this

Inserting rules of replacement: Given a theorem T of the form A B and a line in a derivation C which contains in it a sentence D, where D is a substitution instance of either A or B , replace D with the equivalent substitution instance of the other side of theorem T.

Here are some other important theorems that can act as rules of replacement. Some are theorems we have already proved, while you will be asked to prove others in the homework.

6.7. RULES OF REPLACEMENT 151

T11 (Double Negation, DN): A ⇠⇠A

Proof: See pages 138 and 142.

T12: ⇠(A _ B) ⇠A &⇠B

Proof: See page 140

T13: ⇠(A &B) ⇠A _ ⇠B

Proof: See pages 141 and 143.

T12 and T13 are collectively known as demorgan’s laws, and we will use the abbreviation DeM to refer to either of them in proofs.

T14: (A ! B) (⇠A _ B)

T15: (A _ B) (⇠A ! B)

T14 and T15 are collectively known as the rule of Material Conditional (MC). You will prove them in the homework.

T16 (Biconditional Exportation, ex): A $ B (A ! B)& (B ! A)

Proof: See the homework.

T17 (Transposition, trans): A ! B ⇠B ! ⇠A

Proof: See the homework.

To see how much these theorems can help us, consider this argument:

⇠(P ! Q) P &⇠Q

As always, we could prove this argument using only the basic rules. With rules of replacement, though, the proof is much simpler:

1 ⇠(P ! Q)

2 ⇠(⇠P _Q) MC 1

3 ⇠⇠P &⇠Q DeM 2

4 P &⇠Q DN 3

152 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

Practice Exercises

Part A Prove T8 through T10. You may use T1 through T7 in your proofs.

Part B Prove T11 through T17. You may use T1 through T12 in your proofs.

6.8 Proof Strategy

There is no simple recipe for proofs, and there is no substitute for practice. Here, though, are some rules of thumb and strategies to keep in mind.

Work backwards from what you want. The ultimate goal is to derive the conclusion. Look at the conclusion and ask what the introduction rule is for its main logical operator. This gives you an idea of what should happen just before the last line of the proof. Then you can treat this line as if it were your goal. Ask what you could do to derive this new goal. For example: If your conclusion is a conditional A ! B , plan to use the !I rule. This requires starting a subproof in which you assume A . In the subproof, you want to derive B . Similarly, if your conclusion is a biconditional, A $ B , plan on using $I and be prepared to launch two subproofs. If you are trying to prove a single sentence letter or a negated single sentence letter, you might plan on using indirect proof.

Work forwards from what you have. When you are starting a proof, look at the premises; later, look at the sentences that you have derived so far. Think about the elimination rules for the main operators of these sentences. These will tell you what your options are. For example: If you have A&B use &E to get A and B separately. If you have A _B see if you can find the negation of either A or B and use _E.

Repeat as necessary. Once you have decided how you might be able to get to the conclusion, ask what you might be able to do with the premises. Then consider the target sentences again and ask how you might reach them. Remember, a long proof is formally just a number of short proofs linked together, so you can fill the gap by alternately working back from the conclusion and forward from the premises.

Change what you are looking at. Replacement rules can often make your life easier. If a proof seems impossible, try out some di↵erent substitutions.For example: It is often di�cult to prove a disjunction using the basic rules. If you want to show A _ B , it is often easier to show ⇠A ! B and use the MC rule. Some replacement rules should become second nature. If you see a negated disjunction, for instance, you should immediately think of DeMorgan’s rule.

When all else fails, try indirect proof. If you cannot find a way to show something directly, try assuming its negation. Remember that most proofs can be done either indirectly or directly. One way might be easier—or perhaps one sparks your imagination more than the other—but either one is formally legitimate.

Take a break If you are completely stuck, put down your pen and paper, get up from your

6.9. SOUNDNESS AND COMPLETENESS 153

computer, and do something completely di↵erent for a while. Walk the dog. Do the dishes. Take a shower. I find it especially helpful to do something physically active. Doing other desk work or watching TV doesn’t have the same e↵ect. When you come back to the problem, everything will seem clearer. Of course, if you are in a testing situation, taking a break to walk around might not be advisible. Instead, switch to another problem.

A lot of times, when you are stuck, your mind keeps trying the same solution again and again, even though you know it won’t work. “If I only knew Q ! R,” you say to yourself, “it would all work. Why can’t I derive Q ! R!” If you go away from a problem and then come back, you might not be as focused on That One Thing that you were sure you needed, and you can find a di↵erent approach.

Practice Exercises

Part A

Show the following theorems are valid. Feel free to use T1 through T17

(1) T18 (Associativity of & , Ass& ): (A &B)&C A &(B &C )

(2) T19 (Associativity of _, Ass_): (A _ B) _ C A _ (B _ C )

(3) T20 (Associativity of $, Ass$): (A $ B) $ C A $ (B $ C )

6.9 Soundness and completeness

In section 4.6, we saw that we could use derivations to test for the same concepts we used truth tables to test for. Not only could we use derivations to prove that an argument is valid, we could also use them to test if a statement is a tautology or a pair of statements are equivalent. We also started using the single turnstile the same way we used the double turnstile. If we could prove that A was a tautology with a truth table, we wrote A , and if we could prove it using a derivation, we wrote A .

You may have wondered at that point if the two kinds of turnstiles always worked the same way. If you can show that A is a tautology using truth tables, can you also always show that it is true using a derivation? Is the reverse true? Are these things also true for tautologies and pairs of equivalent sentences? As it turns out, the answer to all these questions and many more like them is yes. We can show this by defining all these concepts separately and then proving them equivalent. That is, we imagine that we actually have two notions of validity, valid|= and valid` and then show that the two concepts always work the same way.

To begin with, we need to define all of our logical concepts separately for truth tables and derivations. A lot of this work has already been done. We handled all of the truth table

154 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

definitions in Chapter 5. We have also already given syntactic definitions for a tautologies and pairs of logically equivalent sentences. The other definitions follow naturally. For most logical properties we can devise a test using derivations, and those that we cannot test for directly can be defined in terms of the concepts that we can define.

For instance, we defined a syntactic tautology as a statement that can be derived without any premises (p. 145). Since the negation of a contradiction is a tautology, we can define a syntactic contradiction in SL as a sentence whose negation can be derived without any premises. The syntactic definition of a contingent sentence is a little di↵erent. We don’t have any practical, finite method for proving that a sentence is contingent using derivations, the way we did using truth tables. So we have to content ourselves with defining “contingent sentence” negatively. A sentence is syntactically contingent in SL if it is not a syntactic tautology or contradiction.

A set of sentences is syntactically inconsistent in SL if and only if one can derive a contradiction from them. Consistency, on the other hand, is like contingency, in that we do not have a practical finite method to test for it directly. So again, we have to define a term negatively. A set of set of sentences is syntactically consistent in SL if and only if they are not syntactically inconsistent.

Finally, an argument is syntactically valid in SL if and only if there is a derivation of it. All of these definitions are given in Table 6.15.

All of our concepts have now been defined both semantically and syntactically. How can we prove that these definitions always work the same way? A full proof here goes well beyond the scope of this book. However, we can sketch what it would be like. We will focus on showing the two notions of validity to be equivalent. From that the other concepts will follow quickly. The proof will have to go in two directions. First we will have to show that things which are syntactically valid will also be semantically valid. In other words, everything that we can prove using derivations could also be proven using truth tables. Put symbolically, we want to show that valid` implies valid|=. Afterwards, we will need to show things in the other directions, valid|= implies valid`

This argument from to is the problem of soundness. A proof system is sound if there are no derivations of arguments that can be shown invalid by truth tables. Demonstrating that the proof system is sound would require showing that any possible proof is the proof of a valid argument. It would not be enough simply to succeed when trying to prove many valid arguments and to fail when trying to prove invalid ones.

The proof that we will sketch depends on the fact that we initially defined a sentence of SL using a recursive definition (see p. 83). We could have also used recursive definitions to define a proper proof in SL and a proper truth table. (Although we didn’t.) If we had these definitions, we could then use a recursive proof to show the soundness of SL. A recursive proof works the same way a recursive definition does.With the recursive definition, we identified a group of base elements that were stipulated to be examples of the thing we were trying to define. In the case of a well formed formula, the base class was the set of sentence letters A, B, C . . . . We just announced that these were sentences. The second step of a recursive definition is to say that anything that is built up from your base class using certain rules also counts as an example of the

6.9. SOUNDNESS AND COMPLETENESS 155

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156 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

thing you are defining. In the case of a definition of a sentence, the rules corresponded to the five sentential connectives (see p. 82). Once you have established a recursive definition, you can use that definition to show that all the members of the class you have defined have a certain property. You simply prove that the property is true of the members of the base class, and then you prove that the rules for extending the base class don’t change the property. This is what it means to give a recursive proof.

Even though we don’t have a recursive definition of a proof in SL, we can sketch how a recursive proof of the soundness of SL would go. Imagine a base class of one-line proofs, one for each of our eleven rules of inference. The members of this class would look like this {A ,B} A &B ; A &B A ; {A _ B ,⇠A} B . . . etc. Since some rules have a couple di↵erent forms, we would have to have add some members to this base class, for instance A &B B Notice that these are all statements in the metalanguage. The proof that SL is sound is not a part of SL, because SL does not have the power to talk about itself.

You can use truth tables to prove to yourself that each of these one-line proofs in this base class is valid|=. For instance the proof {A ,B} A &B corresponds to a truth table that shows {A ,B} A &B This establishes the first part of our recursive proof.

The next step is to show that adding lines to any proof will never change a valid|= proof into an invalid|= one. We would need to this for each of our eleven basic rules of inference. So, for

instance, for & I we need to show that for any proof A1 . . .An B adding a line where we use & I to infer C &D, where C &D can be legitimately inferred from {A1 . . .An,B}, would not change a valid proof into an invalid proof. But wait, if we can legitimately derive C &D from these premises, then CandD must be already available in the proof. They are either members of {A1 . . .An,B} or can be legitimately derived from them. As such, any truth table line in which the premises are true must be a truth table line in which C and D are true. According to the characteristic truth table for & , this means that C &D is also true on that line. Therefore, C &D validly follows from the premises. This means that using the &E rule to extend a valid proof produces another valid proof.

In order to show that the proof system is sound, we would need to show this for the other inference rules. Since the derived rules are consequences of the basic rules, it would su�ce to provide similar arguments for the 11 other basic rules. This tedious exercise falls beyond the scope of this book.

So we have shown that A B implies A B . What about the other direction, that is why think that every argument that can be shown valid using truth tables can also be proven using a derivation.

This is the problem of completeness. A proof system has the property of completeness if and only if there is a derivation of every semantically valid argument. Proving that a system is complete is generally harder than proving that it is sound. Proving that a system is sound amounts to showing that all of the rules of your proof system work the way they are supposed to. Showing that a system is complete means showing that you have included all the rules you need, that you haven’t left any out. Showing this is beyond the scope of this book. The important point

6.9. SOUNDNESS AND COMPLETENESS 157

Property To prove it present To prove it absent

Being a tautology Derive the statement

Find the false line in the truth table for the sentence

Being a contradiction

Derive the negation of the statement

Find the true line in the truth table for the sentence

Contingency Find a false line and a true line in the truth table for the statement

Prove the statement or its negation

Equivalence Derive each statement from the other

Find a line in the truth tables for the statements where they have di↵erent values

Consistency Find a line in truth table for the sentence where they all are true

Derive a contradiction from the sentences

Validity Derive the conclusion form the premises

Find a line in the truth table where the premises are true and the conclusion false.

Table 6.16: When to provide a truth table and when to provide a proof.

is that, happily, the proof system for SL is both sound and complete. This is not the case for all proof systems and all formal languages. Because it is true of SL, we can choose to give proofs or give truth tables—whichever is easier for the task at hand.

Now that we know that the truth table method is interchangeable with the method of derivation, you can chose which method you want to use for any given problem. Students often prefer to use truth tables, because a person can produce them purely mechanically, and that seems ‘easier’. However, we have already seen that truth tables become impossibly large after just a few sentence letters. On the other hand, there are a couple situations where using derivations simply isn’t possible. We syntactically defined a contingent sentence as a sentence that couldn’t be proven to be a tautology or a contradiction. There is no practical way to prove this kind of negative statement. We will never know if there isn’t some proof out there that a statement is a contradiction and we just haven’t found it yet. We have nothing to do in this situation but resort to truth tables. Similarly, we can use derivations to prove two sentences equivalent, but what if we want to prove that they are not equivalent? We have no way of proving that we will never find the relevant proof. So we have to fall back on truth tables again.

Table 6.16 summarizes when it is best to give proofs and when it is best to give truth tables.

Practice Exercises

Part A Use either a derivation or a truth table for each of the following.

158 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

(1) Show that A ! [((B&C) _D) ! A] is a tautology.

(2) Show that A ! (A ! B) is not a tautology

(3) Show that the sentence A ! ⇠A is not a contradiction.

(4) Show that the sentence A $ ⇠A is a contradiction.

(5) Show that the sentence ⇠(W ! (J _ J)) is contingent

(6) Show that the sentence ⇠(X _ (Y _ Z)) _ (X _ (Y _ Z)) is not contingent

(7) Show that the sentence B ! ⇠S is equivalent to the sentence ⇠⇠B ! ⇠S

(8) Show that the sentence ⇠(X _O) is not equivalent to the sentence X &O

(9) Show that the set {⇠(A _B), C, C ! A} is inconsistent.

(10) Show that the set {⇠(A _B), ⇠B, B !A} is consistent

(11) Show that ⇠(A _ (B _ C)) .˙.⇠C is valid.

(12) Show that ⇠(A&(B _ C)) .˙.⇠C is invalid.

Part B Use either a derivation or a truth table for each of the following.

(1) Show that A ! (B ! A) is a tautology

(2) Show that ⇠(((N $ Q) _Q) _N) is not a tautology

(3) Show that Z _ (⇠Z $ Z) is contingent

(4) show that (L $ ((N ! N) ! L)) _H is not contingent

(5) Show that (A $ A)& (B&⇠B) is a contradiction

(6) Show that (B $ (C _B)) is not a contradiction.

(7) Show that ((⇠X $ X) _X) is equivalent to X

(8) Show that F &(K &R) is not equivalent to (F $ (K $ R))

(9) Show that the set {⇠(W ! W ), (W $ W )&W , E _ (W ! ⇠(E&W ))} is inconsistent.

(10) Show that the set {⇠R _ C, (C &R) !6 R, (⇠(R _R) ! R)} is consistent.

(11) Show that ⇠⇠(C $ ⇠C), ((G _ C) _G).˙. ((G ! C)&G) is valid.

(12) Show that ⇠⇠L, (C ! ⇠L) ! C).˙.⇠C is invalid.

Key Terms

6.9. SOUNDNESS AND COMPLETENESS 159

Completeness

Soundness

Syntactically consistent in SL

Syntactically contingent in SL

Syntactically inconsistent in SL

Syntactically logically equivalent in SL

Syntactically valid in SL

Syntactic contradiction in SL

Syntactic tautology in SL

160 CHAPTER 6. PROOFS IN SENTENTIAL LOGIC

Part III

Quantificational Logic

Chapter 7

Quantified Logic

This chapter introduces a logical language called QL. It is a version of quantified logic, because it introduces words like all and some, which tell us about quantities. Quantified logic is also sometimes called predicate logic, because the basic units of the language are predicates and terms.

7.1 From Sentences to Predicates

Consider the following argument, which is obviously valid in English:

If everyone knows logic, then either no one will be confused or everyone will. Everyone will be confused only if we try to believe a contradiction. Everyone knows logic. Therefore, if we don’t try to believe a contradiction, then no one will be confused.

In order to symbolize this in SL, we will need a symbolization key.

L: Everyone knows logic. N: No one will be confused. E: Everyone will be confused. B: We try to believe a contradiction.

Notice that N and E are both about people being confused, but they are two separate sentence letters. We could not replace E with ⇠N . Why not? ⇠N means “It is not the case that no one will be confused.” This would be the case if even one person were confused, so it is a long way from saying that everyone will be confused.

Once we have separate sentence letters for N and E, however, we erase any connection between the two. They are just two atomic sentences which might be true or false independently. In

163

164 CHAPTER 7. QUANTIFIED LOGIC

English, it could never be the case that both no one and everyone was confused. As sentences of SL, however, there is a truth value assignment for which N and E are both true.

Expressions like “no one”, “everyone”, and “anyone” are called quantifiers. By translating N and E as separate atomic sentences, we leave out the quantifier structure of the sentences. Fortunately, the quantifier structure is not what makes this argument valid. As such, we can safely ignore it. To see this, we translate the argument to SL:

1. L ! (N _ E) 2. E ! B 3. L

. . .˙. ⇠B ! N

This is a valid argument in SL. (You can do a truth table to check this.)

Now consider another argument. This one is also valid in English.

Willard is a logician. All logicians wear funny hats. Therefore, Willard wears a funny hat.

To symbolize it in SL, we define a symbolization key:

L: Willard is a logician. A: All logicians wear funny hats. F: Willard wears a funny hat.

Now we symbolize the argument:

1. L 2. A

. . .˙. F

This is invalid in SL. (Again, you can confirm this with a truth table.) There is something very wrong here, because this is clearly a valid argument in English. The symbolization in SL leaves out all the important structure. Once again, the translation to SL overlooks quantifier structure: The sentence “All logicians wear funny hats” is about both logicians and hat-wearing. By not translating this structure, we lose the connection between Willard’s being a logician and Willard’s wearing a hat.

Some arguments with quantifier structure can be captured in SL, like the first example, even though SL ignores the quantifier structure. Other arguments are completely botched in SL, like

7.2. BUILDING BLOCKS OF QUANTIFIED LOGIC 165

the second example. Notice that the problem is not that we have made a mistake while symbolizing the second argument. These are the best symbolizations we can give for these arguments in SL.

Generally, if an argument containing quantifiers comes out valid in SL, then the English language argument is valid. If it comes out invalid in SL, then we cannot say the English language argument is invalid. The argument might be valid because of quantifier structure which the natural language argument has and which the argument in SL lacks.

Similarly, if a sentence with quantifiers comes out as a tautology in SL, then the English sentence is logically true. If comes out as contingent in SL, then this might be because of the structure of the quantifiers that gets removed when we translate into the formal language.

In order to symbolize arguments that rely on quantifier structure, we need to develop a di↵erent logical language. We will call this language quantified logic, QL.

7.2 Building Blocks of Quantified Logic

Just as sentences were the basic unit of sentential logic, predicates will be the basic unit of quantified logic. A predicate is an expression like “is a dog.” This is not a sentence on its own. It is neither true nor false. In order to be true or false, we need to specify something: Who or what is it that is a dog?

The details of this will be explained in the rest of the chapter, but here is the basic idea: In QL, we will represent predicates with capital letters. For instance, we might let D stand for “ is a dog.” We will use lower-case letters as the names of specific things. For instance, we might let b stand for Bertie. The expression Db will be a sentence in QL. It is a translation of the sentence “Bertie is a dog.”

In order to represent quantifier structure, we will also have symbols that represent quantifiers. For instance, “9” will mean “There is some .” So to say that there is a dog, we can write 9xDx; that is: There is some x such that x is a dog.

That will come later. We start by defining singular terms and predicates.

Singular Terms

In English, a singular term is a word or phrase that refers to a specific person, place, or thing. The word “dog” is not a singular term, because there are a great many dogs. The phrase “Philip’s dog Bertie” is a singular term, because it refers to a specific little terrier.

A proper name is a singular term that picks out an individual without describing it. The name “Emerson” is a proper name, and the name alone does not tell you anything about Emerson. Of course, some names are traditionally given to boys and other are traditionally given

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to girls. If “Jack Hathaway” is used as a singular term, you might guess that it refers to a man. However, the name does not necessarily mean that the person referred to is a man—or even that the creature referred to is a person. Jack might be a gira↵e for all you could tell just from the name. There is a great deal of philosophical action surrounding this issue, but the important point here is that a name is a singular term because it picks out a single, specific individual.

Other singular terms more obviously convey information about the thing to which they refer. For instance, you can tell without being told anything further that “Philip’s dog Bertie” is a singular term that refers to a dog. A definite description picks out an individual by means of a unique description. In English, definite descriptions are often phrases of the form “the such-and-so.” They refer to the specific thing that matches the given description. For example, “the tallest member of Monty Python” and “the first emperor of China” are definite descriptions. A description that does not pick out a specific individual is not a definite description. “A member of Monty Python” and “an emperor of China” are not definite descriptions.

We can use proper names and definite descriptions to pick out the same thing. The proper name “Mount Rainier” names the location picked out by the definite description “the highest peak in Washington state.” The expressions refer to the same place in di↵erent ways. You learn nothing from my saying that I am going to Mount Rainier, unless you already know some geography. You could guess that it is a mountain, perhaps, but even this is not a sure thing; for all you know it might be a college, like Mount Holyoke. Yet if I were to say that I was going to the highest peak in Washington state, you would know immediately that I was going to a mountain in Washington state.

In English, the specification of a singular term may depend on context; “Willard” means a specific person and not just someone named Willard; “P.D. Magnus” as a logical singular term means me and not the other P.D. Magnus. We live with this kind of ambiguity in English, but it is important to keep in mind that singular terms in QL must refer to just one specific thing.

In QL, we will symbolize singular terms with lower-case letters a through w. We can add subscripts if we want to use some letter more than once. So a, b, c, . . . w, a1, f32, j390, and m12 are all terms in QL.

Singular terms are called constants because they pick out specific individuals. Note that x, y, and z are not constants in QL. They will be variables, letters which do not stand for any specific thing. We will need them when we introduce quantifiers.

Predicates

The simplest predicates are properties of individuals. They are things you can say about an object. “ is a dog” and “ is a member of Monty Python” are both predicates. In translating English sentences, the term will not always come at the beginning of the sentence: “A piano fell on ” is also a predicate. Predicates like these are called one-place or monadic, because there is only one blank to fill in. A one-place predicate and a singular term combine to make a sentence.

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Other predicates are about the relation between two things. For instance, “ is bigger than ”, “ is to the left of ”, and “ owes money to .” These are two-place

or dyadic predicates, because they need to be filled in with two terms in order to make a sentence.

In general, you can think about predicates as schematic sentences that need to be filled out with some number of terms. Conversely, you can start with sentences and make predicates out of them by removing terms. Consider the sentence, “Vinnie borrowed the family car from Nunzio.” By removing a singular term, we can recognize this sentence as using any of three di↵erent monadic predicates:

borrowed the family car from Nunzio. Vinnie borrowed from Nunzio.

Vinnie borrowed the family car from .

By removing two singular terms, we can recognize three di↵erent dyadic predicates:

Vinnie borrowed from . borrowed the family car from .

borrowed from Nunzio.

By removing all three singular terms, we can recognize one three-place or triadic predicate:

borrowed from .

If we are translating this sentence into QL, should we translate it with a one-, two-, or three-place predicate? It depends on what we want to be able to say. If the only thing that we will discuss being borrowed is the family car, then the generality of the three-place predicate is unnecessary. If the only borrowing we need to symbolize is di↵erent people borrowing the family car from Nunzio, then a one-place predicate will be enough.

In general, we can have predicates with as many places as we need. Predicates with more than one place are called polyadic. Predicates with n places, for some number n, are called n-place or n-adic.

In QL, we symbolize predicates with capital letters A through Z, with or without subscripts. When we give a symbolization key for predicates, we will not use blanks; instead, we will use variables. By convention, constants are listed at the end of the key. So we might write a key that looks like this:

Ax: x is angry. Hx: x is happy.

T1xy: x is as tall or taller than y. T2xy: x is as tough or tougher than y. Bxyz: y is between x and z.

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d: Donald g: Gregor m: Marybeth

We can symbolize sentences that use any combination of these predicates and terms. For example:

1. Donald is angry. 2. If Donald is angry, then so are Gregor and Marybeth. 3. Marybeth is at least as tall and as tough as Gregor. 4. Donald is shorter than Gregor. 5. Gregor is between Donald and Marybeth.

Sentence 1 is straightforward: Ad. The “x” in the key entry “Ax” is just a placeholder; we can replace it with other terms when translating.

Sentence 2 can be paraphrased as, “If Ad, then Ag and Am.” QL has all the truth-functional connectives of SL, so we translate this as Ad ! (Ag&Am).

Sentence 3 can be translated as T1mg&T2mg.

Sentence 4 might seem as if it requires a new predicate. If we only needed to symbolize this sentence, we could define a predicate like Sxy to mean “x is shorter than y.” However, this would ignore the logical connection between “shorter”’ and “taller.” Considered only as symbols of QL, there is no connection between S and T1. They might mean anything at all. Instead of introducing a new predicate, we paraphrase sentence 4 using predicates already in our key: “It is not the case that Donald is as tall or taller than Gregor.” We can translate it as ⇠T1dg.

Sentence 5 requires that we pay careful attention to the order of terms in the key. It becomes Bdgm.

Practice Exercises

Part A Which of the following are singular terms in English?

1) Bob Barker

2) some bees

3) the oldest person alive now

4) my dog Edie

5) numbers

6) the person with the highest grade in this class

7) Barak Obama

8) the fast lane on I-90 eastbound

9) a herd of elephants

10) a herd of Asian elephants.

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Part B Of the singular terms in Part A, which were proper names and which were definite descriptions?

Part C Given the following translation key, translate the sentences below.

Fxy: x is a friend of y Sxy: x is a sibling of y. Oxy: x is older than y Txy: x is taller than y. Cx: x is a child Ax: x is an adult. c: Caroline j: Joey a: Allison o: Jane m: Molly r: Rob

1) Joey is a child

2) Rob is taller than Joey

3) Joey is shorter or the same height as Rob

4) Joey and Alison are friends.

5) Caroline and Joey are siblings

6) Caroline and Joey are not friends

7) Alison and Jane are friends and siblings.

8) Molly is an adult

9) Molly is not a child

10) Molly is neither older nor taller than Rob.

7.3 Quantifiers

We are now ready to introduce quantifiers. Consider the symbolization key on page 167 with Donald, Gregor and Marybeth and the sample sentences that came with it. Let’s add these sentences to that list:

6. Everyone is happy. 7. Everyone is at least as tough as Donald. 8. Someone is angry.

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It might be tempting to translate sentence 6 as Hd&Hg&Hm. Yet this would only say that Donald, Gregor, and Marybeth are happy. We want to say that everyone is happy, even if we have not defined a constant to name them. In order to do this, we introduce the 8 symbol. This is called the universal quantifier.

A quantifier must always be followed by a variable and a formula that includes that variable. We can translate sentence 6 as 8xHx. Paraphrased in English, this means “For all x, x is happy.” We call 8x an x-quantifier. The formula that follows the quantifier is called the scope of the quantifier. We will give a formal definition of scope later, but intuitively it is the part of the sentence that the quantifier quantifies over. In 8xHx, the scope of the universal quantifier is Hx.

Sentence 7 can be paraphrased as, “For all x, x is at least as tough as Donald.” This translates as 8xT2xd.

In these quantified sentences, the variable x is serving as a kind of placeholder. The expression 8x means that you can pick anyone and put them in as x. There is no special reason to use x rather than some other variable. The sentence 8xHx means exactly the same thing as 8yHy, 8zHz, and 8x5Hx5.

To translate sentence 8, we introduce another new symbol: the existential quantifier, 9. Like the universal quantifier, the existential quantifier requires a variable. Sentence 8 can be translated as 9xAx. This means that there is some x which is angry. More precisely, it means that there is at least one angry person. Once again, the variable is a kind of placeholder; we could just as easily have translated sentence 8 as 9zAz.

Consider these further sentences:

9. No one is angry. 10. There is someone who is not happy. 11. Not everyone is happy.

Sentence 9 can be paraphrased as, “It is not the case that someone is angry.” This can be translated using negation and an existential quantifier: ⇠9xAx. Yet sentence 9 could also be paraphrased as, “Everyone is not angry.” With this in mind, it can be translated using negation and a universal quantifier: 8x⇠Ax. Both of these are acceptable translations, because they are logically equivalent. The critical thing is whether the negation comes before or after the quantifier.

In general, 8xA is logically equivalent to ⇠9x⇠A . This means that any sentence which can be symbolized with a universal quantifier can be symbolized with an existential quantifier, and vice versa. One translation might seem more natural than the other, but there is no logical di↵erence in translating with one quantifier rather than the other. For some sentences, it will simply be a matter of taste.

Sentence 10 is most naturally paraphrased as, “There is some x such that x is not happy.” This becomes 9x⇠Hx. Equivalently, we could write ⇠8xHx.

Sentence 11 is most naturally translated as ⇠8xHx. This is logically equivalent to sentence 10

7.3. QUANTIFIERS 171

and so could also be translated as 9x⇠Hx.

Although we have two quantifiers in QL, we could have an equivalent formal language with only one quantifier. We could proceed with only the universal quantifier, for instance, and treat the existential quantifier as a notational convention. We use square brackets [ ] to make some sentences more readable, but we know that these are really just parentheses ( ). In the same way, we could write “9x” knowing that this is just shorthand for “⇠8x⇠.” There is a choice between making logic formally simple and making it expressively simple. With QL, we opt for expressive simplicity. Both 8 and 9 will be symbols of QL.

Universe of Discourse

Given the symbolization key we have been using, 8xHx means “Everyone is happy.” Who is included in this everyone? When we use sentences like this in English, we usually do not mean everyone now alive on the Earth. We certainly do not mean everyone who was ever alive or who will ever live. We mean something more modest: everyone in the building, everyone in the class, or everyone in the room.

In order to eliminate this ambiguity, we will need to specify a universe of discourse—abbreviated UD. The UD is the set of things that we are talking about. So if we want to talk about people in Chicago, we define the UD to be people in the Chicago. We write this at the beginning of the symbolization key, like this:

UD: people in Chicago

The quantifiers range over the universe of discourse. Given this UD, 8x means “Everyone in Chicago” and 9x means “Someone in Chicago.” Each constant names some member of the UD, so we can only use this UD with the symbolization key above if Donald, Gregor, and Marybeth are all in Chicago. If we want to talk about people in places besides Chicago, then we need to include those people in the UD.

In QL, the UD must be non-empty ; that is, it must include at least one thing. It is possible to construct formal languages that allow for empty UDs, but this introduces complications.

Even allowing for a UD with just one member can produce some strange results. Suppose we have this as a symbolization key:

UD: the Ei↵el Tower Px: x is in Paris.

The sentence 8xPx might be paraphrased in English as “Everything is in Paris.” Yet that would be misleading. It means that everything in the UD is in Paris. This UD contains only the Ei↵el Tower, so with this symbolization key 8xPx just means that the Ei↵el Tower is in Paris.

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Non-referring terms

In QL, each constant must pick out exactly one member of the UD. A constant cannot refer to more than one thing—it is a singular term. Each constant must still pick out something. This is connected to a classic philosophical problem: the so-called problem of non-referring terms.

Medieval philosophers typically used sentences about the chimera to exemplify this problem. Chimera is a mythological creature; it does not really exist. Consider these two sentences:

12. Chimera is angry.

13. Chimera is not angry.

It is tempting just to define a constant to mean “chimera.” The symbolization key would look like this:

UD: creatures on Earth

Ax: x is angry.

c: chimera

We could then translate sentence 12 as Ac and sentence 13 as ⇠Ac.

Problems will arise when we ask whether these sentences are true or false.

One option is to say that sentence 12 is not true, because there is no chimera. If sentence 12 is false because it talks about a non-existent thing, then sentence 13 is false for the same reason. Yet this would mean that Ac and ⇠Ac would both be false. Given the truth conditions for negation, this cannot be the case.

Since we cannot say that they are both false, what should we do? Another option is to say that sentence 12 is meaningless because it talks about a non-existent thing. So Ac would be a meaningful expression in QL for some interpretations but not for others. Yet this would make our formal language hostage to particular interpretations. Since we are interested in logical form, we want to consider the logical force of a sentence like Ac apart from any particular interpretation. If Ac were sometimes meaningful and sometimes meaningless, we could not do that.

This is the problem of non-referring terms, and we will return to it later (see p. 193.) The important point for now is that each constant of QL must refer to something in the UD, although the UD can be any set of things that we like. If we want to symbolize arguments about mythological creatures, then we must define a UD that includes them. This option is important if we want to consider the logic of stories. We can translate a sentence like “Sherlock Holmes lived at 221B Baker Street” by including fictional characters like Sherlock Holmes in our UD.

7.4. TRANSLATING TO QUANTIFIED LOGIC 173

7.4 Translating to Quantified Logic

We now have all of the pieces of QL. Translating more complicated sentences will only be a matter of knowing the right way to combine predicates, constants, quantifiers, and connectives. Consider these sentences:

14. Every coin in my pocket is a quarter. 15. Some coin on the table is a dime. 16. Not all the coins on the table are dimes. 17. None of the coins in my pocket are dimes.

In providing a symbolization key, we need to specify a UD. Since we are talking about coins in my pocket and on the table, the UD must at least contain all of those coins. Since we are not talking about anything besides coins, we let the UD be all coins. Since we are not talking about any specific coins, we do not need to define any constants. So we define this key:

UD: all coins Px: x is in my pocket. Tx: x is on the table. Qx: x is a quarter. Dx: x is a dime.

Sentence 14 is most naturally translated with a universal quantifier. The universal quantifier says something about everything in the UD, not just about the coins in my pocket. Sentence 14 means that, for any coin, if that coin is in my pocket then it is a quarter. So we can translate it as 8x(Px ! Qx).

Since sentence 14 is about coins that are both in my pocket and that are quarters, it might be tempting to translate it using a conjunction. However, the sentence 8x(Px&Qx) would mean that everything in the UD is both in my pocket and a quarter: All the coins that exist are quarters in my pocket. This is would be a crazy thing to say, and it means something very di↵erent than sentence 14.

Sentence 15 is most naturally translated with an existential quantifier. It says that there is some coin which is both on the table and which is a dime. So we can translate it as 9x(Tx&Dx).

Notice that we needed to use a conditional with the universal quantifier, but we used a conjunction with the existential quantifier. What would it mean to write 9x(Tx ! Dx)? Probably not what you think. It means that there is some member of the UD which would satisfy the subformula; roughly speaking, there is some a such that (Ta ! Da) is true. In SL, A ! B is logically equivalent to ⇠A _ B , and this will also hold in QL. So 9x(Tx ! Dx) is true if there is some a such that (⇠Ta _Da); i.e., it is true if some coin is either not on the table or is a dime. Of course there is a coin that is not the table—there are coins lots of other places. So 9x(Tx ! Dx) is trivially true. A conditional will usually be the natural connective to use with a universal quantifier, but a conditional within the scope of an existential quantifier can do very

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strange things. As a general rule, do not put conditionals in the scope of existential quantifiers unless you are sure that you need one.

Sentence 16 can be paraphrased as, “It is not the case that every coin on the table is a dime.” So we can translate it as ⇠8x(Tx ! Dx). You might look at sentence 16 and paraphrase it instead as, “Some coin on the table is not a dime.” You would then translate it as 9x(Tx&⇠Dx). Although it is probably not obvious, these two translations are logically equivalent. (This is due to the logical equivalence between ⇠8xA and 9x⇠A , along with the equivalence between ⇠(A ! B) and A &⇠B .)

Sentence 17 can be paraphrased as, “It is not the case that there is some dime in my pocket.” This can be translated as ⇠9x(Px&Dx). It might also be paraphrased as, “Everything in my pocket is a non-dime,” and then could be translated as 8x(Px ! ⇠Dx). Again the two translations are logically equivalent. Both are correct translations of sentence 17.

We can now translate the argument from p. 164, the one that motivated the need for quantifiers:

Willard is a logician. All logicians wear funny hats. .˙. Willard wears a funny hat.

UD: people Lx: x is a logician. Fx: x wears a funny hat. w: Willard

Translating, we get:

1. Lw 2. 8x(Lx ! Fx)

. . .˙. Fw

This captures the structure that was left out of the SL translation of this argument, and this is a valid argument in QL.

Empty predicates

A predicate need not apply to anything in the UD. A predicate that applies to nothing in the UD is called an empty predicate.

Suppose we want to symbolize these two sentences:

18. Every monkey knows sign language.

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19. Some monkey knows sign language.

It is possible to write the symbolization key for these sentences in this way:

UD: animals Mx: x is a monkey. Sx: x knows sign language.

Sentence 18 can now be translated as 8x(Mx ! Sx).

Sentence 19 becomes 9x(Mx&Sx).

It is tempting to say that sentence 18 entails sentence 19; that is: if every monkey knows sign language, then it must be that some monkey knows sign language. This is a valid inference in Aristotelean logic: All Ms are S, .˙. some M is S. However, the entailment does not hold in QL. It is possible for the sentence 8x(Mx ! Sx) to be true even though the sentence 9x(Mx&Sx) is false.

How can this be? The answer comes from considering whether these sentences would be true or false if there were no monkeys.

We have defined 8 and 9 in such a way that 8A is equivalent to ⇠9⇠A . As such, the universal quantifier doesn’t involve the existence of anything—only non-existence. If sentence 18 is true, then there are no monkeys who don’t know sign language. If there were no monkeys, then 8x(Mx ! Sx) would be true and 9x(Mx&Sx) would be false.

We allow empty predicates because we want to be able to say things like, “I do not know if there are any monkeys, but any monkeys that there are know sign language.” That is, we want to be able to have predicates that do not (or might not) refer to anything.

What happens if we add an empty predicate R to the interpretation above? For example, we might define Rx to mean “x is a refrigerator.” Now the sentence 8x(Rx ! Mx) will be true. This is counterintuitive, since we do not want to say that there are a whole bunch of refrigerator monkeys. It is important to remember, though, that 8x(Rx ! Mx) means that any member of the UD that is a refrigerator is a monkey. Since the UD is animals, there are no refrigerators in the UD and so the setence is trivially true.

If you were actually translating the sentence “All refrigerators are monkeys”, then you would want to include appliances in the UD. Then the predicate R would not be empty and the sentence 8x(Rx ! Mx) would be false.

Picking a Universe of Discourse

The appropriate symbolization of an English language sentence in QL will depend on the symbolization key. In some ways, this is obvious: It matters whether Dx means “x is dainty” or “x is dangerous.” The meaning of sentences in QL also depends on the UD.

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. A UD must have at least one member.

. A predicate may apply to some, all, or no members of the UD.

. A constant must pick out exactly one member of the UD.

A member of the UD may be picked out by one constant, many constants, or none at all.

Let Rx mean “x is a rose,” let Tx mean “x has a thorn,” and consider this sentence:

20. Every rose has a thorn.

It is tempting to say that sentence 20 should be translated as 8x(Rx ! Tx). If the UD contains all roses, that would be correct. Yet if the UD is merely things on my kitchen table, then 8x(Rx ! Tx) would only mean that every rose on my kitchen table has a thorn. If there are no roses on my kitchen table, the sentence would be trivially true.

The universal quantifier only ranges over members of the UD, so we need to include all roses in the UD in order to translate sentence 20. We have two options. First, we can restrict the UD to include all roses but only roses. Then sentence 20 becomes 8xTx. This means that everything in the UD has a thorn; since the UD just is the set of roses, this means that every rose has a thorn. This option can save us trouble if every sentence that we want to translate using the symbolization key is about roses.

Second, we can let the UD contain things besides roses: rhododendrons, rats, rifles, and whatall else. Then sentence 20 must be 8x(Rx ! Tx).

If we wanted the universal quantifier to mean every thing, without restriction, then we might try to specify a UD that contains everything. This would lead to problems. Does “everything” include things that have only been imagined, like fictional characters? On the one hand, we want to be able to symbolize arguments about Hamlet or Sherlock Holmes. So we need to have the option of including fictional characters in the UD. On the other hand, we never need to talk about every thing that does not exist. That might not even make sense. There are philosophical issues here that we will not try to address. We can avoid these di�culties by always specifying the UD. For example, if we mean to talk about plants, people, and cities, then the UD might be “living things and places.”

Suppose that we want to translate sentence 20 and, with the same symbolization key, translate these sentences:

21. Esmerelda has a rose in her hair. 22. Everyone is cross with Esmerelda.

We need a UD that includes roses (so that we can symbolize sentence 20) and a UD that

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includes people (so we can translate sentence 21–22.) Here is a suitable key:

UD: people and plants Px: x is a person. Rx: x is a rose. Tx: x has a thorn.

Cxy: x is cross with y. Hxy: x has y in their hair.

e: Esmerelda

Since we do not have a predicate that means “. . . has a rose in her hair”, translating sentence 21 will require paraphrasing. The sentence says that there is a rose in Esmerelda’s hair; that is, there is something which is both a rose and is in Esmerelda’s hair. So we get: 9x(Rx&Hex).

It is tempting to translate sentence 22 as 8xCxe. Unfortunately, this would mean that every member of the UD is cross with Esmerelda—both people and plants. It would mean, for instance, that the rose in Esmerelda’s hair is cross with her. Of course, sentence 22 does not mean that.

“Everyone” means every person, not every member of the UD. So we can paraphrase sentence 22 as, “Every person is cross with Esmerelda.” We know how to translate sentences like this: 8x(Px ! Cxe)

In general, the universal quantifier can be used to mean “everyone” if the UD contains only people. If there are people and other things in the UD, then “everyone” must be treated as “every person.”

Practice Exercises

Part A Using the symbolization key given, translate each English-language sentence into QL.

UD: all animals Ax: x is an alligator. Mx: x is a monkey. Rx: x is a reptile. Zx: x lives at the zoo.

Lxy: x loves y. a: Amos b: Bouncer c: Cleo

1. Amos, Bouncer, and Cleo all live at the zoo. 2. Bouncer is a reptile, but not an alligator. 3. If Cleo loves Bouncer, then Bouncer is a monkey. 4. If both Bouncer and Cleo are alligators, then Amos loves them both.

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5. Some reptile lives at the zoo. 6. Every alligator is a reptile. 7. Any animal that lives at the zoo is either a monkey or an alligator. 8. There are reptiles which are not alligators. 9. Cleo loves a reptile.

10. Bouncer loves all the monkeys that live at the zoo. 11. All the monkeys that Amos loves love him back. 12. If any animal is an alligator, then it is a reptile. 13. Every monkey that Cleo loves is also loved by Amos. 14. There is a monkey that loves Bouncer, but sadly Bouncer does not reciprocate this love.

Part B Our language QL is an expanded and more powerful way of dealing with a class of arguments that were originally studied by Aristotle known as categorical syllogisms. Aristotle’s followers in the Latin medieval world developed a system of naming for the valid syllogisms that were known to them. These arguments and their names are listed below. Translate each into QL.

1) Barbara: All Ms are P s. All Ss are Ms. .˙. All Ss are P s.

2) Cesare: No P s are Ms. All Ss are Ms. .˙. NoSs are P s.

3) Datisi: All Ms are P s. Some Ms are Ss. .˙. Some Ss are P s.

4) Calemes: All P s are Ms. No Ms are Ss. .˙. NoSs are P s.

5) Celarent: No Ms are P s. All Ss are Ms. .˙. No Ss are P s.

6) Camestres: All P s are Ms. No Ss are Ms. .˙. No Ss are P s.

7) Disamis: Some Ms are P s. All Ms are Ss .˙. Some Ss are P s.

8) Dimatis: Some P s are Ms. All Ms are Ss. .˙. Some Ss are P s.

9) Darii: All Ms are P s. Some Ss are Ms. .˙. Some Ss are P s.

10) Festino: No P s are Ms. Some Ss are Ms. .˙. Some Ss are not P s.

11) Ferison No Ms are P s. Some Ms are Ss. .˙. Some Ss are not P s.

12) Fresison No P s are Ms Some Ms are Ss. .˙. Some Ss are not P s.

Part C Here are the remaining 12 syllogisms studied by Aristotle and his followers.Translate each into QL.

1) Ferio: No Ms are P s. Some Ss are Ms .˙. Some Ss are not P s.

2) Baroco: All P s are Ms. Some Ss are not Ms. .˙. Some Ss are not P s.

3) Bocardo: Some Ms are not P s. All Ms are Ss. .˙. Some Ss are not P s.

4) Calemos: All P s are Ms. And no Ms are Ss. .˙. Some Ss are not P s.

5) Barbar:i All Ms are P s. All Ss are Ms. .˙. Some Ss are P s.

6) Camestros: All P s are Ms. No Ss are Ms..˙. Some Ss are not P s.

7) Darapti: All Ms are P s. All Ms are Ss. .˙. Some Ss are P s.

8) Fesapo: No P s are Ms. All Ms are Ss. .˙. Some Ss are not P s.

9) Celaront: No Ms are P s. All Ss are Ms. .˙. Some Ss are not P s.

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10) Cesaro: No P s are Ms. All Ss are Ms .˙. Some Ss are not P s.

11) Felapton: No Ms are P s. All Ms are Ss. .˙. Some Ss are not P s.

12) Bamalip: All P s are Ms. All Ms are Ss. .˙. Some Ss are P s.

7.5 Recursive Syntax for QL

Up to this point, we have been working informally while translating things in to QL. Now we will introduce a more rigorous syntax for the language we are using. As in section 4 of chapter 2, we will be using recursive definitions. This time, however, we will need to define two kinds of grammatical units. In addition to defining a sentence in QL, we will need to define something called a well-formed formula or WFF. A well-formed formula will be like a sentence, but it will have some elements missing, and it won’t have a truth value until those elements are provided.

Expressions

There are six kinds of symbols in QL:

predicates A,B,C, . . . , Z with subscripts, as needed A1, B1, Z1, A2, A25, J375, . . .

constants a, b, c, . . . , w with subscripts, as needed a1, w4, h7,m32, . . .

variables x, y, z with subscripts, as needed x1, y1, z1, x2, . . .

connectives ⇠, & ,_,!,$ parentheses ( , ) quantifiers 8, 9

We define an expression of ql as any string of symbols of QL. Take any of the symbols of QL and write them down, in any order, and you have an expression.

Well-formed formulae

By definition, a term of ql is either a constant or a variable.

An atomic formula of ql is an n-place predicate followed by n terms.

At this point in Chapter 4, we gave a recursive definition for a sentence in SL (p.82). We said that the atomic sentences were sentences of SL, and then gave rules for building new sentences in SL. We are going to do something similar here, except this time we will just be defining a

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well-formed formula (w↵) in QL. Every atomic formula—every formula like Fx or Gab—is a w↵. And you can build new w↵s by applying the sentential connectives, just as before

We could just add a rule for each of the quantifiers and be done with it. For instance: If A is a w↵, then 8xA and 9xA are w↵s. However, this would allow for bizarre sentences like 8x9xDx and 8xDw. What could these possibly mean? We could adopt some interpretation of such sentences, but instead we will write the definition of a w↵ so that such abominations do not even count as well-formed.

In order for 8xA to be a w↵, A must contain the variable x and must not already contain an x-quantifier. 8xDw will not count as a w↵ because “x” does not occur in Dw, and 8x9xDx will not count as a w↵ because 9xDx contains an x-quantifier

1. Every atomic formula is a w↵.

2. If A is a w↵, then ⇠A is a w↵.

3. If A and B are w↵s, then (A &B), is a w↵.

4. If A and B are w↵s, (A _ B) is a w↵.

5. If A and B are w↵s, then (A ! B) is a w↵.

6. If A and B are w↵s, then (A $ B) is a w↵.

7. If A is a w↵, x is a variable, A contains at least one occurrence of x , and A contains no x -quantifiers, then 8xA is a w↵.

8. If A is a w↵, x is a variable, A contains at least one occurrence of x , and A contains no x -quantifiers, then 9xA is a w↵.

9. All and only w↵s of QL can be generated by applications of these rules.

Notice that the “x ” that appears in the definition above is not the variable x. It is a meta-variable that stands in for any variable of QL. So 8xAx is a w↵, but so are 8yAy, 8zAz, 8x4Ax4, and 8z9Az9.

We can now give a formal definition for scope: The scope of a quantifier is the subformula for which the quantifier is the main logical operator.

Sentences

A sentence is something that can be either true or false. In SL, every expression that satisfied the recursive syntax was a sentence. You could tell, because we could decide whether expression was true or false. This will not be the case for w↵s in QL. Consider the following symbolization key:

UD: people.

7.5. RECURSIVE SYNTAX FOR QL 181

Lxy: x loves y. b: Boris.

Consider the expression Lzz. It is an atomic formula: a two-place predicate followed by two terms. All atomic formula are w↵s, so Lzz is a w↵. Does it mean anything? You might think that it means that z loves himself, in the same way that Lbb means that Boris loves himself. Yet z is a variable; it does not name some person the way a constant would. The w↵ Lzz does not tell us how to interpret z. Does it mean everyone? anyone? someone? If we had a z-quantifier, it would tell us how to interpret z. For instance, 9zLzz would mean that someone loves themselves.

Some formal languages treat a w↵ like Lzz as implicitly having a universal quantifier in front. We will not do this for QL. If you mean to say that everyone loves themselves, then you need to write the quantifier: 8zLzz

In order to make sense of a variable, we need a quantifier to tell us how to interpret that variable. The scope of an x-quantifier, for instance, is the part of the formula where quantifier tells how to interpret x.

In order to be precise about this, we define a bound variable to be an occurrence of a variable x that is within the scope of an x -quantifier. A free variable is an occurrence of a variable that is not bound.

For example, consider the w↵ 8x(Ex _Dy) ! 9z(Ex ! Lzx). The scope of the universal quantifier 8x is (Ex _Dy), so the first x is bound by the universal quantifier but the second and third xs are free. There is not y-quantifier, so the y is free. The scope of the existential quantifier 9z is (Ex ! Lzx), so the occurrence of z is bound by it.

We define a sentence of QL as a w↵ of QL that contains no free variables.

Notational conventions

We will adopt the same notational conventions that we did for SL (p. 84.) First, we may leave o↵ the outermost parentheses of a formula. Second, we will use square brackets [ and ] in place of parentheses to increase the readability of formulae. Third, we will leave out parentheses between each pair of conjuncts when writing long series of conjunctions. Fourth, we will leave out parentheses between each pair of disjuncts when writing long series of disjunctions.

Substitution instance

If A is a w↵, c a constant, and x a variable, then A [c|x] is the w↵ made by replacing each occurance of x in A with c. This is called a substitution instance of 8xA and 9xA ; c is called the instantiating constant.

For example: Aa ! Ba, Af ! Bf , and Ak ! Bk are all substitution instances of

182 CHAPTER 7. QUANTIFIED LOGIC

8x(Ax ! Bx); the instantiating constants are a, f , and k, respectively. Raj, Rdj, and Rjj are substitution instances of 9zRzj; the instantiating constants are a, d, and j, respectively.

This definition will be useful later, when we define truth and derivability in QL. If 8xPx is true, then every substitution instance Pa, Pb, Pc. . . is true. To put the point informally, if everything is a P , then a is a P , b is a P , c is a P , and so on. Conversely, if some substitution instance of 9xPx such as Pa is true, then 9xPx must be true. Informally, if some specific a is a P , then there is some P .

On this definition, a substitution instance is formed by replacing just one variable with a constant. Also, quantifiers must be removed starting from the left. So substitutions instances of the sentence 8x8yPxy would include 8yPay and 8yPgy, but a sentence like Pab would actually be a substitution instance of a substitution instance of 8x8yPxy. To form it, you would first have to create the substitution instance 8yPay and then take a substitution instance of that newer sentence. The sentence 8xPxb does not count as a substitution instance of 8x8yPxy , although it would follow from it. Setting things up this way will make this rule consistent with the rules of existential and universal elimination in Chapter 7.

Practice Exercises

Part A Identify which variables are bound and which are free.

1. 9xLxy& 8yLyx 2. 8xAx&Bx 3. 8x(Ax&Bx)& 8y(Cx&Dy) 4. 8x9y[Rxy ! (Jz&Kx)] _Ryx 5. 8x1(Mx2 $ Lx2x1)& 9x2Lx3x2

Part B

1. Identify which of the following are substitution instances of 8xRcx: Rac, Rca, Raa, Rcb, Rbc, Rcc, Rcd, Rcx

2. Identify which of the following are substitution instances of 9x8yLxy: 8yLby, 8xLbx, Lab, 9xLxa

7.6 Tricky Translations

Ambiguous predicates

Suppose we just want to translate this sentence:

7.6. TRICKY TRANSLATIONS 183

23. Adina is a skilled surgeon.

Let the UD be people, let Kx mean “x is a skilled surgeon”, and let a mean Adina. Sentence 23 is simply Ka.

Suppose instead that we want to translate this argument:

The hospital will only hire a skilled surgeon. All surgeons are greedy. Billy is a surgeon, but is not skilled. Therefore, Billy is greedy, but the hospital will not hire him.

We need to distinguish being a skilled surgeon from merely being a surgeon. So we define this symbolization key:

UD: people Gx: x is greedy. Hx: The hospital will hire x. Rx: x is a surgeon. Kx: x is skilled. b: Billy

Now the argument can be translated in this way:

8x ⇥ ⇠(Rx&Kx) ! ⇠Hx

⇤

8x(Rx ! Gx) Rb&⇠Kb

.˙. Gb&⇠Hb

Next suppose that we want to translate this argument:

Carol is a skilled surgeon and a tennis player. Therefore, Carol is a surgeon and a skilled tennis player.

If we start with the symbolization key we used for the previous argument, we could add a predicate (let Tx mean “x is a tennis player”) and a constant (let c mean Carol). Then the argument becomes:

(Rc&Kc)&Tc .˙. T c&Kc

This translation is a disaster! It takes what in English is a terrible argument and translates it as a valid argument in QL. The problem is that there is a di↵erence between being skilled as a surgeon and skilled as a tennis player. Translating this argument correctly requires two separate predicates, one for each type of skill. If we let K1x mean “x is skilled as a surgeon” and K2x mean “x is skilled as a tennis player,” then we can symbolized the argument in this way:

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(Rc&K1c)&Tc .˙. T c&K2c

Like the English language argument it translates, this is invalid.

The moral of these examples is that you need to be careful of symbolizing predicates in an ambiguous way. Similar problems can arise with predicates like good, bad, big, and small. Just as skilled surgeons and skilled tennis players have di↵erent skills, big dogs, big mice, and big problems are big in di↵erent ways.

Is it enough to have a predicate that means “x is a skilled surgeon”, rather than two predicates “x is skilled” and ‘’x is a surgeon”? Sometimes. As sentence 23 shows, sometimes we do not need to distinguish between skilled surgeons and other surgeons.

Must we always distinguish between di↵erent ways of being skilled, good, bad, or big? No. As the argument about Billy shows, sometimes we only need to talk about one kind of skill. If you are translating an argument that is just about dogs, it is fine to define a predicate that means “x is big.” If the UD includes dogs and mice, however, it is probably best to make the predicate mean “x is big for a dog.”

Multiple quantifiers

Consider this following symbolization key and the sentences that follow it:

UD: People and dogs Dx: x is a dog. Fxy: x is a friend of y. Oxy: x owns y. f: Fifi g: Gerald

24. Fifi is a dog. 25. Gerald is a dog owner. 26. Someone is a dog owner. 27. All of Gerald’s friends are dog owners. 28. Every dog owner is the friend of a dog owner.

Sentence 24 is easy: Df .

Sentence 25 can be paraphrased as, “There is a dog that Gerald owns.” This can be translated as 9x(Dx&Ogx).

Sentence 26 can be paraphrased as, “There is some y such that y is a dog owner.” The subsentence “y is a dog owner” is just like sentence 25, except that it is about y rather than being about Gerald. So we can translate sentence 26 as 9y9x(Dx&Oyx).

7.6. TRICKY TRANSLATIONS 185

Sentence 27 can be paraphrased as, “Every friend of Gerald is a dog owner.” Translating part of this sentence, we get 8x(Fxg ! “x is a dog owner’). Again, it is important to recognize that “x is a dog owner” is structurally just like sentence 25. Since we already have an x-quantifier, we will need a di↵erent variable for the existential quantifier. Any other variable will do. Using z, sentence 27 can be translated as 8x

⇥ Fxg ! 9z(Dz&Oxz)

⇤ .

Sentence 28 can be paraphrased as “For any x that is a dog owner, there is a dog owner who is x’s friend.” Partially translated, this becomes

8x ⇥ x is a dog owner ! 9y(y is a dog owner&Fxy

⇤ .

Completing the translation, sentence 28 becomes

8x ⇥ 9z(Dz&Oxz) ! 9y(9z(Dz&Oyz)&Fxy

⇤ .

Consider this symbolization key and these sentences:

UD: people Lxy: x likes y.

i: Imre. k: Karl.

29. Imre likes everyone that Karl likes. 30. There is someone who likes everyone who likes everyone that he likes.

Sentence 29 can be partially translated as 8x(Karl likes x ! Imre likes x). This becomes 8x(Lkx ! Lix).

Sentence 30 is almost a tongue-twister. There is little hope of writing down the whole translation immediately, but we can proceed by small steps. An initial, partial translation might look like this:

9x everyone who likes everyone that x likes is liked by x

The part that remains in English is a universal sentence, so we translate further:

9x8y(y likes everyone that x likes ! x likes y).

The antecedent of the conditional is structurally just like sentence 29, with y and x in place of Imre and Karl. So sentence 30 can be completely translated in this way

9x8y ⇥ 8z(Lxz ! Lyz) ! Lxy

⇤

When symbolizing sentences with multiple quantifiers, it is best to proceed by small steps. Paraphrase the English sentence so that the logical structure is readily symbolized in QL. Then translate piecemeal, replacing the daunting task of translating a long sentence with the simpler task of translating shorter formulae.

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Translating pronouns

Let’s look at another situation where it helps to translate in small steps. For the next several examples, we will use this symbolization key:

UD: people Gx: x can play bass guitar. Rx: x is a rock star.

l: Lemmy

Now consider these sentences:

31. If Lemmy can play bass guitar, then he is a rock star. 32. If a person can play bass guitar, then he is a rock star.

Sentence 31 and sentence 32 have the same consequent (“. . . he is a rock star”), but they cannot be translated in the same way. It helps to paraphrase the original sentences, replacing pronouns with explicit references.

Sentence 31 can be paraphrased as, “If Lemmy can play bass guitar, then Lemmy is a rockstar.” This can obviously be translated as Gl ! Rl.

Sentence 32 must be paraphrased di↵erently: “If a person can play bass guitar, then that person is a rock star.” This sentence is not about any particular person, so we need a variable. Translating halfway, we can paraphrase the sentence as, “For any person x, if x can play bass guitar, then x is a rockstar.” Now this can be translated as 8x(Gx ! Rx). This is the same as, ‘’Everyone who can play bass guitar is a rock star.”

Consider these further sentences:

33. If anyone can play bass guitar, then Lemmy can. 34. If anyone can play bass guitar, then he or she is a rock star.

These two sentences have the same antecedent (“If anyone can play guitar. . .”), but they have di↵erent logical structures.

Sentence 33 can be paraphrased, “If someone can play bass guitar, then Lemmy can play bass guitar.” The antecedent and consequent are separate sentences, so it can be symbolized with a conditional as the main logical operator: 9xGx ! Gl.

Sentence 34 can be paraphrased, “For anyone, if that one can play bass guitar, then that one is a rock star.” It would be a mistake to symbolize this with an existential quantifier, because it is talking about everybody. The sentence is equivalent to “All bass guitar players are rock stars.” It is best translated as 8x(Gx ! Rx).

7.6. TRICKY TRANSLATIONS 187

The English words “any” and “anyone” should typically be translated using quantifiers. As these two examples show, they sometimes call for an existential quantifier (as in sentence 33) and sometimes for a universal quantifier (as in sentence 34). If you have a hard time determining which is required, paraphrase the sentence with an English language sentence that uses words besides “any” or “anyone.”

Quantifiers and scope

In the sentence 9xGx ! Gl, the scope of the existential quantifier is the expression Gx. Would it matter if the scope of the quantifier were the whole sentence? That is, does the sentence 9x(Gx ! Gl) mean something di↵erent?

With the key given above, 9xGx ! Gl means that if there is some bass guitarist, then Lemmy is a bass guitarist. 9x(Gx ! Gl) would mean that there is some person such that if that person were a bass guitarist, then Lemmy would be a bass guitarist. Recall that the conditional here is a material conditional; the conditional is true if the antecedent is false. Let the constant p denote the author of this book, someone who is certainly not a guitarist. The sentence Gp ! Gl is true because Gp is false. Since someone (namely p) satisfies the sentence, then 9x(Gx ! Gl) is true. The sentence is true because there is a non-guitarist, regardless of Lemmy’s skill with the bass guitar.

Something strange happened when we changed the scope of the quantifier, because the conditional in QL is a material conditional. In order to keep the meaning the same, we would have to change the quantifier: 9xGx ! Gl means the same thing as 8x(Gx ! Gl), and 9x(Gx ! Gl) means the same thing as 8xGx ! Gl.

This oddity does not arise with other connectives or if the variable is in the consequent of the conditional. For example, 9xGx&Gl means the same thing as 9x(Gx&Gl), and Gl ! 9xGx means the same things as 9x(Gl ! Gx).

Practice Exercises

Part A Using the symbolization key given, translate each English-language sentence into QL.

UD: all animals Dx: x is a dog. Sx: x likes samurai movies.

Lxy: x is larger than y. b: Bertie e: Emerson f: Fergis

1. Bertie is a dog who likes samurai movies.

188 CHAPTER 7. QUANTIFIED LOGIC

2. Bertie, Emerson, and Fergis are all dogs. 3. Emerson is larger than Bertie, and Fergis is larger than Emerson. 4. All dogs like samurai movies. 5. Only dogs like samurai movies. 6. There is a dog that is larger than Emerson. 7. If there is a dog larger than Fergis, then there is a dog larger than Emerson. 8. No animal that likes samurai movies is larger than Emerson. 9. No dog is larger than Fergis.

10. Any animal that dislikes samurai movies is larger than Bertie. 11. There is an animal that is between Bertie and Emerson in size. 12. There is no dog that is between Bertie and Emerson in size. 13. No dog is larger than itself. 14. Every dog is larger than some dog. 15. There is an animal that is smaller than every dog. 16. If there is an animal that is larger than any dog, then that animal does not like samurai

movies.

Part B For each argument, write a symbolization key and translate the argument into QL.

1. Nothing on my desk escapes my attention. There is a computer on my desk. As such, there is a computer that does not escape my attention.

2. All my dreams are black and white. Old TV shows are in black and white. Therefore, some of my dreams are old TV shows.

3. Neither Holmes nor Watson has been to Australia. A person could see a kangaroo only if they had been to Australia or to a zoo. Although Watson has not seen a kangaroo, Holmes has. Therefore, Holmes has been to a zoo.

4. No one expects the Spanish Inquisition. No one knows the troubles I’ve seen. Therefore, anyone who expects the Spanish Inquisition knows the troubles I’ve seen.

5. An antelope is bigger than a bread box. I am thinking of something that is no bigger than a bread box, and it is either an antelope or a cantaloupe. As such, I am thinking of a cantaloupe.

6. All babies are illogical. Nobody who is illogical can manage a crocodile. Berthold is a baby. Therefore, Berthold is unable to manage a crocodile.

Part C Using the symbolization key given, translate each English-language sentence into QL.

UD: candies Cx: x has chocolate in it. Mx: x has marzipan in it. Sx: x has sugar in it. Tx: Boris has tried x.

Bxy: x is better than y.

1. Boris has never tried any candy.

7.6. TRICKY TRANSLATIONS 189

2. Marzipan is always made with sugar. 3. Some candy is sugar-free. 4. The very best candy is chocolate. 5. No candy is better than itself. 6. Boris has never tried sugar-free chocolate. 7. Boris has tried marzipan and chocolate, but never together. 8. Any candy with chocolate is better than any candy without it. 9. Any candy with chocolate and marzipan is better than any candy that lacks both.

Part D Using the symbolization key given, translate each English-language sentence into QL.

UD: people and dishes at a potluck Rx: x has run out. Tx: x is on the table. Fx: x is food. Px: x is a person.

Lxy: x likes y. e: Eli f: Francesca g: the guacamole

1. All the food is on the table. 2. If the guacamole has not run out, then it is on the table. 3. Everyone likes the guacamole. 4. If anyone likes the guacamole, then Eli does. 5. Francesca only likes the dishes that have run out. 6. Francesca likes no one, and no one likes Francesca. 7. Eli likes anyone who likes the guacamole. 8. Eli likes anyone who likes the people that he likes. 9. If there is a person on the table already, then all of the food must have run out.

Part E Using the symbolization key given, translate each English-language sentence into QL.

UD: people Dx: x dances ballet. Fx: x is female. Mx: x is male. Cxy: x is a child of y. Sxy: x is a sibling of y.

e: Elmer j: Jane p: Patrick

1. All of Patrick’s children are ballet dancers. 2. Jane is Patrick’s daughter.

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3. Patrick has a daughter. 4. Jane is an only child. 5. All of Patrick’s daughters dance ballet. 6. Patrick has no sons. 7. Jane is Elmer’s niece. 8. Patrick is Elmer’s brother. 9. Patrick’s brothers have no children.

10. Jane is an aunt. 11. Everyone who dances ballet has a sister who also dances ballet. 12. Every man who dances ballet is the child of someone who dances ballet.

7.7 Identity

Consider this sentence:

35. Pavel owes money to everyone else.

Let the UD be people; this will allow us to translate “everyone” as a universal quantifier. Let Oxy mean “x owes money to y,” and let p mean Pavel. Now we can symbolize sentence 35 as 8xOpx. Unfortunately, this translation has some odd consequences. It says that Pavel owes money to every member of the UD, including Pavel; it entails that Pavel owes money to himself. However, sentence 35 does not say that Pavel owes money to himself; he owes money to everyone else. This is a problem, because 8xOpx is the best translation we can give of this sentence into QL.

The solution is to add another symbol to QL. The symbol = is a two-place predicate. Since it has a special logical meaning, we write it a bit di↵erently: For two terms t1 and t2, t1 = t2 is an atomic formula.

The predicate x = y means “x is identical to y.” This does not mean merely that x and y are indistinguishable or that all of the same predicates are true of them. Rather, it means that x and y are the very same thing.

When we write x 6= y, we mean that x and y are not identical. There is no reason to introduce this as an additional predicate. Instead, x 6= y is an abbreviation of ⇠(x = y).

Now suppose we want to symbolize this sentence:

36. Pavel is Mister Checkov.

Let the constant c mean Mister Checkov. Sentence 36 can be symbolized as p = c. This means that the constants p and c both refer to the same guy.

This is all well and good, but how does it help with sentence 35? That sentence can be paraphrased as, “Everyone who is not Pavel is owed money by Pavel.” This is a sentence

7.7. IDENTITY 191

structure we already know how to symbolize: “For all x, if x is not Pavel, then x is owed money by Pavel.” In QL with identity, this becomes 8x(x 6= p ! Opx).

In addition to sentences that use the word “else,” identity will be helpful when symbolizing some sentences that contain the words “besides” and “only.” Consider these examples:

37. No one besides Pavel owes money to Hikaru. 38. Only Pavel owes Hikaru money.

We add the constant h, which means Hikaru.

Sentence 37 can be paraphrased as, “No one who is not Pavel owes money to Hikaru.” This can be translated as ⇠9x(x 6= p&Oxh).

Sentence 38 can be paraphrased as, “Pavel owes Hikaru and no one besides Pavel owes Hikaru money.” We have already translated one of the conjuncts, and the other is straightforward. Sentence 38 becomes Oph&⇠9x(x 6= p&Oxh).

Expressions of quantity

We can also use identity to say how many things there are of a particular kind. For example, consider these sentences:

39. There is at least one apple on the table. 40. There are at least two apples on the table. 41. There are at least three apples on the table.

Let the UD be things on the table, and let Ax mean “x is an apple.”

Sentence 39 does not require identity. It can be translated adequately as 9xAx: There is some apple on the table—perhaps many, but at least one.

It might be tempting to also translate sentence 40 without identity. Yet consider the sentence 9x9y(Ax&Ay). It means that there is some apple x in the UD and some apple y in the UD. Since nothing precludes x and y from picking out the same member of the UD, this would be true even if there were only one apple. In order to make sure that there are two di↵erent apples, we need an identity predicate. Sentence 40 needs to say that the two apples that exist are not identical, so it can be translated as 9x9y(Ax&Ay&x 6= y).

Sentence 41 requires talking about three di↵erent apples. It can be translated as 9x9y9z(Ax&Ay&Az&x 6= y& y 6= z&x 6= z).

Continuing in this way, we could translate “There are at least n apples on the table.” There is a summary of how to symbolize sentences like these on p. 236.

192 CHAPTER 7. QUANTIFIED LOGIC

Now consider these sentences:

42. There is at most one apple on the table.

43. There are at most two apples on the table.

Sentence 42 can be paraphrased as, “It is not the case that there are at least two apples on the table.” This is just the negation of sentence 40:

⇠9x9y(Ax&Ay&x 6= y)

Sentence 42 can also be approached in another way. It means that any apples that there are on the table must be the selfsame apple, so it can be translated as 8x8y

⇥ (Ax&Ay) ! x = y

⇤ . The

two translations are logically equivalent, so both are correct.

In a similar way, sentence 43 can be translated in two equivalent ways. It can be paraphrased as, “It is not the case that there are three or more distinct apples”, so it can be translated as the negation of sentence 41. Using universal quantifiers, it can also be translated as

8x8y8z ⇥ (Ax&Ay&Az) ! (x = y _ x = z _ y = z)

⇤ .

See p. 236 for the general case.

The examples above are sentences about apples, but the logical structure of the sentences translates mathematical inequalities like a � 3, a  2, and so on. We also want to be able to translate statements of equality which say exactly how many things there are. For example:

44. There is exactly one apple on the table.

45. There are exactly two apples on the table.

Sentence 44 can be paraphrased as, “There is at least one apple on the table, and there is at most one apple on the table.” This is just the conjunction of sentence 39 and sentence 42: 9xAx& 8x8y

⇥ (Ax&Ay) ! x = y

⇤ . This is a somewhat complicated way of going about it. It is

perhaps more straightforward to paraphrase sentence 44 as “There is a thing which is the only apple on the table.” Thought of in this way, the sentence can be translated 9x

⇥ Ax&⇠9y(Ay&x 6= y)

⇤ .

Similarly, sentence 45 may be paraphrased as, “There are two di↵erent apples on the table, and these are the only apples on the table.” This can be translated as 9x9y

⇥ Ax&Ay&x 6= y&⇠9z(Az&x 6= z& y 6= z)

⇤ .

Finally, consider this sentence:

46. There are at most two things on the table.

7.7. IDENTITY 193

It might be tempting to add a predicate so that Tx would mean “x is a thing on the table.” However, this is unnecessary. Since the UD is the set of things on the table, all members of the UD are on the table. If we want to talk about a thing on the table, we need only use a quantifier. Sentence 46 can be symbolized like sentence 43 (which said that there were at most two apples), but leaving out the predicate entirely. That is, sentence 46 can be translated as 8x8y8z(x = y _ x = z _ y = z).

Techniques for symbolizing expressions of quantity (“at most”, “at least”, and “exactly”) are summarized on p. 236.

Definite descriptions

Recall that a constant of QL must refer to some member of the UD. This constraint allows us to avoid the problem of non-referring terms. Given a UD that included only actually existing creatures but a constant c that meant “chimera” (a mythical creature), sentences containing c would become impossible to evaluate.

The most widely influential solution to this problem was introduced by Bertrand Russell in 1905. Russell asked how we should understand this sentence:

47. The present king of France is bald.

The phrase “the present king of France” is supposed to pick out an individual by means of a definite description. However, there was no king of France in 1905 and there is none now. Since the description is a non-referring term, we cannot just define a constant to mean “the present king of France” and translate the sentence as Kf .

Russell’s idea was that sentences that contain definite descriptions have a di↵erent logical structure than sentences that contain proper names, even though they share the same grammatical form. What do we mean when we use an unproblematic, referring description, like “the highest peak in Washington state”? We mean that there is such a peak, because we could not talk about it otherwise. We also mean that it is the only such peak. If there was another peak in Washington state of exactly the same height as Mount Rainier, then Mount Rainier would not be the highest peak.

According to this analysis, sentence 47 is saying three things. First, it makes an existence claim: There is some present king of France. Second, it makes a uniqueness claim: This guy is the only present king of France. Third, it makes a claim of predication: This guy is bald.

In order to symbolize definite descriptions in this way, we need the identity predicate. Without it, we could not translate the uniqueness claim which (according to Russell) is implicit in the definite description.

Let the UD be people actually living, let Fx mean “x is the present king of France,” and let Bx mean “x is bald.” Sentence 47 can then be translated as 9x

⇥ Fx&⇠9y(Fy&x 6= y)&Bx

⇤ . This

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says that there is some guy who is the present king of France, he is the only present king of France, and he is bald.

Understood in this way, sentence 47 is meaningful but false. It says that this guy exists, but he does not.

The problem of non-referring terms is most vexing when we try to translate negations. So consider this sentence:

48. The present king of France is not bald.

According to Russell, this sentence is ambiguous in English. It could mean either of two things:

48a. It is not the case that the present king of France is bald. 48b. The present king of France is non-bald.

Both possible meanings negate sentence 47, but they put the negation in di↵erent places.

Sentence 48a is called a wide-scope negation, because it negates the entire sentence. It can be translated as ⇠9x

⇥ Fx&⇠9y(Fy&x 6= y)&Bx

⇤ . This does not say anything about the present

king of France, but rather says that some sentence about the present king of France is false. Since sentence 47 if false, sentence 48a is true.

Sentence 48b says something about the present king of France. It says that he lacks the property of baldness. Like sentence 47, it makes an existence claim and a uniqueness claim; it just denies the claim of predication. This is called narrow-scope negation. It can be translated as 9x

⇥ Fx&⇠9y(Fy&x 6= y)&⇠Bx

⇤ . Since there is no present king of France, this sentence is false.

Russell’s theory of definite descriptions resolves the problem of non-referring terms and also explains why it seemed so paradoxical. Before we distinguished between the wide-scope and narrow-scope negations, it seemed that sentences like 48 should be both true and false. By showing that such sentences are ambiguous, Russell showed that they are true understood one way but false understood another way.

Practice Exercises

Part A Using the symbolization key given, translate each English-language sentence into QL with identity. The last sentence is ambiguous and can be translated two ways; you should provide both translations. (Hint: Identity is only required for the last four sentences.)

UD: people Kx: x knows the combination to the safe. Sx: x is a spy.

7.7. IDENTITY 195

Vx: x is a vegetarian. Txy: x trusts y.

h: Hofthor i: Ingmar

1. Hofthor is a spy, but no vegetarian is a spy. 2. No one knows the combination to the safe unless Ingmar does. 3. No spy knows the combination to the safe. 4. Neither Hofthor nor Ingmar is a vegetarian. 5. Hofthor trusts a vegetarian. 6. Everyone who trusts Ingmar trusts a vegetarian. 7. Everyone who trusts Ingmar trusts someone who trusts a vegetarian. 8. Only Ingmar knows the combination to the safe. 9. Ingmar trusts Hofthor, but no one else. 10. The person who knows the combination to the safe is a vegetarian. 11. The person who knows the combination to the safe is not a spy.

Part B Using the symbolization key given, translate each English-language sentence into QL with identity. The last two sentences are ambiguous and can be translated two ways; you should provide both translations for each.

UD: cards in a standard deck Bx: x is black. Cx: x is a club. Dx: x is a deuce. Jx: x is a jack. Mx: x is a man with an axe. Ox: x is one-eyed. Wx: x is wild.

1. All clubs are black cards. 2. There are no wild cards. 3. There are at least two clubs. 4. There is more than one one-eyed jack. 5. There are at most two one-eyed jacks. 6. There are two black jacks. 7. There are four deuces. 8. The deuce of clubs is a black card. 9. One-eyed jacks and the man with the axe are wild. 10. If the deuce of clubs is wild, then there is exactly one wild card. 11. The man with the axe is not a jack. 12. The deuce of clubs is not the man with the axe.

Part C Using the symbolization key given, translate each English-language sentence into QL with identity. The last two sentences are ambiguous and can be translated two ways; you should provide both translations for each.

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UD: animals in the world Bx: x is in Farmer Brown’s field. Hx: x is a horse. Px: x is a Pegasus. Wx: x has wings.

1. There are at least three horses in the world. 2. There are at least three animals in the world. 3. There is more than one horse in Farmer Brown’s field. 4. There are three horses in Farmer Brown’s field. 5. There is a single winged creature in Farmer Brown’s field; any other creatures in the field

must be wingless. 6. The Pegasus is a winged horse. 7. The animal in Farmer Brown’s field is not a horse. 8. The horse in Farmer Brown’s field does not have wings.

Chapter 8

Semantics for Quantified Logic

8.1 Creating models in Quantified Logic

In Chapter 3 we developed the truth table system, a semantic method for evaluating sentences and arguments in SL. The method was called semantic, because it involved establishing meanings for all the non-logical parts of a sentence, in this case, the sentence letters. These were called non-logical because their meaning was not fixed by the rules of SL. The logical part of the language was the system of connectives and parentheses. As it turned out, the logical parts of our language were truth functional: the meaning of the larger sentences built up with the connectives and parentheses was simply a function of the truth of the smaller parts. Because the system of sentential connectives and parentheses is truth functional, our semantic system only needed to look at one aspect of the meaning of the sentence letters, their truth value. Thus an interpretation of a sentence in SL simply turned out to be a truth assignment on the sentence letters. That’s why you spent Chapter 3 merrily writing Ts and Fs under sentence letters. Each little row of Ts and Fs represented one way of assigning meaning to a sentence or argument, and all the little rows in the truth table represented all the possible ways of assigning meaning to the sentence or argument (at least as far as truth was concerned.)

In this chapter we are going to develop similar methods for QL. Because QL is a more complicated system than SL developing semantics for QL will be much more complicated, and there will be greater limitations for what we can do with these methods.

Basic Models

The first thing we need to do is to find an equivalent in QL of a little row of Ts and Fs beneath a sentence in SL. We called this little row of Ts and Fs an interpretation, because it gave one aspect of the meaning of the parts of the sentence, namely their truth values. To come up with an

197

198 CHAPTER 8. SEMANTICS FOR QUANTIFIED LOGIC

interpretation of a sentence in QL, we will need to look at more than the truth or falsity of sentences, though. We have, after all, broken open the atomic sentence to look at its subatomic parts, and now we will need to assign meaning to these parts. The way to start is to look at the symbolization keys we created when we were translating sentences in and out of QL. The symbolization key contained a universe of discourse (UD), a meaning for each predicate, and an object picked out by each term. For example:

UD: comic book characters Fx: x fights crime. b: Batman w: Bruce Wayne

Given this symbolization key, we can translate sentences like this

Fb: Batman fights crime. Fw: Bruce Wayne fights crime. Fb ! Fw: If Batman fights crime, then Bruce Wayne fights crime.

This symbolization key, along with some basic knowledge of superhero stories, gives us enough information to figure out that the sentence Fb is true. (Notice that the sentence Fb is not true just because of the interpretation. The way we interpret the sentence, plus the way the world is, makes the sentence true.)

This is the information we need to develop an interpretation of sentences in QL. We are not just assigning truth values anymore. We need to get deeper into meaning here, and look at the reference of the parts of the sentence. reference is an aspect of meaning that deals with the way bits of language pick out or identify objects in the real world. For our purposes the real world is the universe of discourse, the set of objects we are talking about. To explain how a singular term like b refers, we need only point out the member of the UD it refers to, in this case, Batman. In more technical terms, Batman is the referent of B. For our purposes it is enough to define a referent like this: the referent of a term in QL is the unique object picked out by that term.

(Notice that here to explain what b refers to, I have been forced to simply use another singular term in another language, the name “Batman.” To really pick out the object referred to, I would need to draw your attention to something in the actual world, for instance by holding up a comic book and pointing at the character on the cover.)

We want to use similar methods to talk about the meaning of predicates. To do this we will talk about the extension of a predicate. The extension of a predicate is the set of objects in the UD that the predicate applies to. So if b is in the extension of the predicate F then the sentence Fb is true.

(Identifying the extension of a predicate like F again forces us to rely on another language, English. We can say that the extension of F includes Batman, Superman, Green Lantern, etc. The situation is worse because F has an indefinitely large extension, so we have relied on the English predicate “fights crime” to describe it.)

8.1. CREATING MODELS IN QUANTIFIED LOGIC 199

All this means that we are able to talk about the meaning of sentences in QL entirely in terms of sets. We use curly brackets { and } to denote sets. The members of the set can be listed in any order, separated by commas. The fact that sets can be in any order is important, because it means that {foo, bar} and {bar, foo} are the same set. It is possible to have a set with no members in it. This is called the empty set. The empty set is sometimes written as {}, but usually it is written as the single symbol ;.

We are now able to give the equivalent of a line in a truth table for QL. An interpretation in QL will be called a model. A model of sentences or arguments in QL consists of a set that is the universe of discourse, individual members of that set that are the referents of the singular terms in the sentences or arguments, and subsets of the universe of discourse which are the extensions of the predicates used in the sentences or arguments.

To see how this works imagine I have a bunch of sentences in QL, which include the predicate H and the singular term f . Now consider this symbolization key:

UD: People who played as part of the Three Stooges Hx: x had head hair.

f: Mister Fine

What is the model that corresponds to this interpretation? There were six people who played as part of the Three Stooges over the years, so the UD will have six members: Larry Fine, Moe Howard, Curly Howard, Shemp Howard, Joe Besser, and Curly Joe DeRita. Curly, Joe, and Curly Joe were the only completely bald stooges. The result is this model:

UD = {Larry, Curly, Moe, Shemp, Joe, Curly Joe} extension(H) = {Larry, Moe, Shemp}

referent(f) = Larry

You do not need to know anything about the Three Stooges in order to evaluate whether sentences are true or false in this model. Hf is true, since the referent of f (Larry) is in the extension of H. Both 9xHx and 9x⇠Hx are true, since there is at least one member of the UD that is in the extension of H and at least one member that is not in the extension of H. In this way, the model captures all of the formal significance of the interpretation.

Models for multiplace predicates

Now consider this interpretation:

UD: whole numbers less than 10 Ex: x is even. Nx: x is negative. Lxy: x is less than y. Txyz: x times y equals z.

200 CHAPTER 8. SEMANTICS FOR QUANTIFIED LOGIC

What is the model that goes with this interpretation? The UD is the set {1, 2, 3, 4, 5, 6, 7, 8, 9}.

The extension of a one-place predicate like E or N is just the subset of the UD of which the predicate is true. The extension of E is the subset {2, 4, 6, 8}. There are many even numbers besides these four, but these are the only members of the UD that are even. There are no negative numbers in the UD, so N has an empty extension; i.e. extension(N) = ;.

The extension of a two-place predicate like L is somewhat vexing. It seems as if the extension of L ought to contain 1, since 1 is less than all the other numbers; it ought to contain 2, since 2 is less than all of the other numbers besides 1; and so on. Every member of the UD besides 9 is less than some member of the UD. What would happen if we just wrote extension(L) = {1, 2, 3, 4, 5, 6, 7, 8}?

The problem is that sets can be written in any order, so this would be the same as writing extension(L) = {8, 7, 6, 5, 4, 3, 2, 1}. This does not tell us which of the members of the set are less than which other members.

We need some way of showing that 1 is less than 8 but that 8 is not less than 1. The solution is to have the extension of L consist of pairs of numbers. An ordered pair is like a set with two members, except that the order does matter. We write ordered pairs with angle brackets < and >. The ordered pair <foo, bar> is di↵erent than the ordered pair <bar, foo>. The extension of L is a collection of ordered pairs, all of the pairs of numbers in the UD such that the first number is less than the second. Writing this out completely:

extension(L) = {<1,2>, <1,3>, <1,4>, <1,5>, <1,6>, <1,7>, <1,8>, <1,9>, <2,3>, <2,4>, <2,5>, <2,6>, <2,7>, <2,8>, <2,9>, <3,4>, <3,5>, <3,6>, <3,7>, <3,8>, <3,9>, <4,5>, <4,6>, <4,7>, <4,8>, <4,9>, <5,6>, <5,7>, <5,8>, <5,9>, <6,7>, <6,8>, <6,9>, <7,8>, <7,9>, <8,9>}

Three-place predicates will work similarly; the extension of a three-place predicate is a set of ordered triples where the predicate is true of those three things in that order. So the extension of T in this model will contain ordered triples like <2,4,8>, because 2⇥ 4 = 8.

Generally, the extension of an n-place predicate is a set of all ordered n-tuples <a1, a2, . . . , an> such that a1–an are members of the UD and the predicate is true of a1–an in that order.

Models for identity

Identity is a special predicate of QL. We write it a bit di↵erently than other two-place predicates: x = y instead of Ixy. We also do not need to include it in a symbolization key. The sentence x = y always means “x is identical to y,” and it cannot be interpreted to mean anything else. In the same way, when you construct a model, you do not get to pick and choose which ordered pairs go into the extension of the identity predicate. It always contains just the ordered pair of each object in the UD with itself.

The sentence 8xIxx, which contains an ordinary two-place predicate, is contingent. Whether it

8.1. CREATING MODELS IN QUANTIFIED LOGIC 201

is true for an interpretation depends on how you interpret I, and whether it is true in a model depends on the extension of I.

The sentence 8x x = x is a tautology. The extension of identity will always make it true.

Notice that although identity always has the same interpretation, it does not always have the same extension. The extension of identity depends on the UD. If the UD in a model is the set {Doug}, then extension(=) in that model is {<Doug, Doug>}. If the UD is the set {Doug, Omar}, then extension(=) in that model is {<Doug, Doug>, <Omar, Omar>}. And so on.

If the referent of two constants is the same, then anything which is true of one is true of the other. For example, if referent(a) = referent(b), then Aa $ Ab, Ba $ Bb, Ca $ Cb, Rca $ Rcb, 8xRxa $ 8xRxb, and so on for any two sentences containing a and b. In metaphysics, this is called principle of the indiscernibility of identicals

In our system, the reverse of this principle is not true. It is possible that anything which is true of a is also true of b, yet for a and b still to have di↵erent referents. This may seem puzzling, but it is easy to construct a model that shows this. Consider this model:

UD = {Rosencrantz, Guildenstern} referent(a) = Rosencrantz referent(b) = Guildenstern

for all predicates P , extension(P ) = ; extension(=) = {<Rosencrantz, Rosencrantz>,

<Guildenstern, Guildenstern>}

This specifies an extension for every predicate of QL: All the infinitely-many predicates are empty. This means that both Aa and Ab are false, and they are equivalent; both Ba and Bb are false; and so on for any two sentences that contain a and b. Yet a and b refer to di↵erent things. We have written out the extension of identity to make this clear: The ordered pair < referent(a), referent(b) > is not in it. In this model, a = b is false and a 6= b is true.

Practice Exercises

Part A Determine whether each sentence is true or false in the model given.

UD = {Corwin, Benedict} extension(A) = {Corwin, Benedict} extension(B) = {Benedict} extension(N) = ;

referent(c) = Corwin

1. Bc 2. Ac $ ⇠Nc 3. Nc ! (Ac _Bc)

202 CHAPTER 8. SEMANTICS FOR QUANTIFIED LOGIC

4. 8xAx 5. 8x⇠Bx 6. 9x(Ax&Bx) 7. 9x(Ax ! Nx) 8. 8x(Nx _ ⇠Nx) 9. 9xBx ! 8xAx

Part B Determine whether each sentence is true or false in the model given.

UD = {Waylan, Willy, Johnny} extension(H) = {Waylan, Willy, Johnny} extension(W ) = {Waylan, Willy} extension(R) = {<Waylan, Willy>,<Willy, Johnny>,<Johnny, Waylan>} referent(m) = Johnny

1. 9x(Rxm&Rmx) 2. 8x(Rxm _Rmx) 3. 8x(Hx $ Wx) 4. 8x(Rxm ! Wx) 5. 8x

⇥ Wx ! (Hx&Wx)

⇤

6. 9xRxx 7. 9x9yRxy 8. 8x8yRxy 9. 8x8y(Rxy _Ryx) 10. 8x8y8z

⇥ (Rxy&Ryz) ! Rxz

⇤

Part C Determine whether each sentence is true or false in the model given.

UD = {Lemmy, Courtney, Eddy} extension(G) = {Lemmy, Courtney, Eddy} extension(H) = {Courtney} extension(M) = {Lemmy, Eddy}

referent(c) = Courtney referent(e) = Eddy

1. Hc 2. He 3. Mc _Me 4. Gc _ ⇠Gc 5. Mc ! Gc 6. 9xHx 7. 8xHx 8. 9x⇠Mx 9. 9x(Hx&Gx)

8.2. WORKING WITH MODELS 203

10. 9x(Mx&Gx) 11. 8x(Hx _Mx) 12. 9xHx& 9xMx 13. 8x(Hx $ ⇠Mx) 14. 9xGx& 9x⇠Gx 15. 8x9y(Gx&Hy)

Part D Write out the model that corresponds to the interpretation given.

UD: natural numbers from 10 to 13 Ox: x is odd. Sx: x is less than 7. Tx: x is a two-digit number. Ux: x is thought to be unlucky. Nxy: x is the next number after y.

8.2 Working with Models

Working with models is in some ways like working with truth tables and in some ways not. With truth tables, we could conclusively show that a sentence was a tautology or a contradiction, because the truth table would always have a finite number of lines. We cannot, however, use models to show that a sentence is a tautology or a contradiction, because there are infinitely many ways to model a sentence, and and no single way to talk about all of them, the way we talked about all the lines in a truth table. One thing we can do is show conclusively that a sentence is neither a tautology nor a contradiction, and is instead contingent. A contingent sentence will have at least one model where it is false and one model where it is true.

As we shall see, this pattern plays itself out with the other logical properties we have covered. Because there are infinitely many ways to model a sentence, we cannot construct models to prove two sentences equivalent, but we can use them to show that two sentences are not equivalent. We cannot use models to show that a set of sentences is inconsistent, but we can use models to show that a set of sentences is consistent. Finally, we cannot construct a finite number of models to show that an argument is valid, but we can use one to show that an argument is invalid.

We will use the double turnstile symbol for QL much as we did for SL. “A B”’ means that an argument from A to B is semantically valid. A means that A is a semantic tautology. A B means A and B are semantically equivalent.

In 5.3 on page 98 we stipulated semantic definitions for various logical concepts in SL that matched our truth table method for determining these concepts. So a sentence was said to be a tautology in SL if the column under its main connective contained only Ts. This was an alternative to saying the the truth table method was an imprecise way of getting at the ordinary language versions of these concepts. We will do something similar for the semantic definitions of logical notions in QL.

204 CHAPTER 8. SEMANTICS FOR QUANTIFIED LOGIC

1. A semantic tautology in ql is a sentence A that is true in every model; i.e., A .

2. A semantic contradiction in ql is a sentence A that is false in every model; i.e., ⇠A .

3. A sentence is semantically contingent in ql if and only if it is neither a tautology nor a contradiction.

4. Two sentences A and B are semantically equivalent in ql if and only if they have the same truth value in every model.

5. The set {A1,A2,A3, · · · } is semantically consistent in ql if and only if there is at least one model in which all of the sentences are true. The set is semantically inconsistent in ql if and if only there is no such model.

6. An argument “ P1,P2, · · · , .˙. C ” is semantically valid in ql if and only if there is no model in which all of the premises are true and the conclusion is false; i.e., {P1,P2, · · · } C . It is semantically invalid in ql otherwise.

Models to show contingency

Suppose we want to show that 8xAxx ! Bd is not a tautology. This requires showing that the sentence is not true in every model; i.e., that it is false in some model. If we can provide just one model in which the sentence false, then we will have shown that the sentence is not a tautology.

What would such a model look like? In order for 8xAxx ! Bd to be false, the antecedent (8xAxx) must be true, and the consequent (Bd) must be false.

To construct such a model, we start with a UD. It will be easier to specify extensions for predicates if we have a small UD, so start with a UD that has just one member. Formally, this single member might be anything, so let’s just call it ↵.

We want 8xAxx to be true, so we want all members of the UD to be paired with themselves in the extension of A; this means that the extension of A must be {< ↵,↵ >}.

We want Bd to be false, so the referent of d must not be in the extension of B. We give B an empty extension.

Since ↵ is the only member of the UD, it must be the referent of d. The model we have constructed looks like this:

UD = {↵} extension(A) = {< ↵,↵ >} extension(B) = ;

referent(d) = ↵

Strictly speaking, a model specifies an extension for every predicate of QL and a referent for every constant. As such, it is generally impossible to write down a complete model. That would require writing down infinitely many extensions and infinitely many referents. However, we do not

8.2. WORKING WITH MODELS 205

need to consider every predicate in order to show that there are models in which 8xAxx ! Bd is false. Predicates like H and constants like f13 make no di↵erence to the truth or falsity of this sentence. It is enough to specify extensions for A and B and a referent for d, as we have done. This provides a partial model in which the sentence is false.

Perhaps you are wondering: What is ↵? What does the predicate A mean in English? The partial model could correspond to an interpretation like this one:

UD: Paris Axy: x is in the same country as y. Bx: x was founded in the 20th century. d: the City of Lights

However, we don’t have to say that this model corresponds to any particular interpretation of the sentence in English in order to know that the sentence 8xAxx ! Bd is not a tautology. We could have made our one-object universe of discourse contain only Mahatma Gandhi, or a pebble on a beach in Africa, or the number 2. As long as the predicate and terms were given the right reference, the sentence would come out false. Thus in the future we can evalaute sentences and arguments using UDs with arbitrarily named elements, like ↵, �, �, etc.

We use the same method to show that 8xAxx ! Bd is not a contradiction. We need only specify a model in which 8xAxx ! Bd is true; i.e., a model in which either 8xAxx is false or Bd is true. Here is one such partial model:

UD = {↵} extension(A) = {< ↵,↵ >} extension(B) = {↵}

referent(d) = ↵

We have now shown that 8xAxx ! Bd is neither a tautology nor a contradiction. By the definition of “contingent in QL,” this means that 8xAxx ! Bd is contingent. In general, showing that a sentence is contingent will require two models: one in which the sentence is true and another in which the sentence is false.

Notice, however, that we cannot show that a sentence actually is a tautology or a contradiction using one or two models this way. For a sentence to be a tautology, it must be true in every possible model. Similarly, if a sentence is a contradiction, it is false in all possible models. But there are infinitely many possible models for any sentence, and we don’t have any tools in this text that will let us reason about all of them at once.

Models to show non-equivalence

Suppose we want to show that 8xSx and 9xSx are not logically equivalent. We need to construct a model in which the two sentences have di↵erent truth values; we want one of them to be true and the other to be false. We start by specifying a UD. Again, we make the UD small so that we

206 CHAPTER 8. SEMANTICS FOR QUANTIFIED LOGIC

can specify extensions easily. We will need at least two members. Let the UD be {↵, �}. (If we chose a UD with only one member, the two sentences would end up with the same truth value. In order to see why, try constructing some partial models with one-member UDs.)

We can make 9xSx true by including something in the extension of S, and we can make 8xSx false by leaving something out of the extension of S. It does not matter which one we include and which one we leave out. Making ↵ the only S, we get a partial model that looks like this:

UD = {↵, �} extension(S) = {↵}

This partial model shows that the two sentences are not logically equivalent.

Notice, though, that we cannot show that two sentences are logically equivalent by simply producing a model. If we claim that two sentences are logically equivalent, we are once again making a claim about every possible model.

Models to show consistency

Suppose I wanted to show that the set of sentences {9xFx, 9x⇠Fx, 8xGx} is consistent. For this to be the case, we need at least one model where all three sentences are true. In this case that means having one object in our UD that is F and one that is not F , and we need both of them to be G. This would do the trick.

UD = {↵, �} extension(F ) = {↵} extension(G) = {↵, �}

We cannot show a set of sentences to be inconsistent this way, because as before, that would mean creating an infinite number of models.

Models to show invalidity

Back on p. 183, we said that this argument would be invalid in QL:

(Rc&K1c)&Tc

.˙. T c&K2c

In order to show that it is invalid, we need to show that there is some model in which the premises are true and the conclusion is false. We can construct such a model deliberately. Here is one way to do it:

8.2. WORKING WITH MODELS 207

UD = {↵} extension(T ) = {↵}

extension(K1) = {↵} extension(K2) = ; extension(R) = {↵}

referent(c) = ↵

As you have probably guessed, we cannot show an argument valid with models so simply.

Practice Exercises

Part A Show that each of the following is contingent.

1.? Da&Db 2.? 9xTxh 3.? Pm&⇠8xPx 4. 8zJz $ 9yJy 5. 8x(Wxmn _ 9yLxy) 6. 9x(Gx ! 8yMy)

Part B Show that the following pairs of sentences are not logically equivalent.

1. Ja, Ka 2. 9xJx, Jm 3. 8xRxx, 9xRxx 4. 9xPx ! Qc, 9x(Px ! Qc) 5. 8x(Px ! ⇠Qx), 9x(Px&⇠Qx) 6. 9x(Px&Qx), 9x(Px ! Qx) 7. 8x(Px ! Qx), 8x(Px&Qx) 8. 8x9yRxy, 9x8yRxy 9. 8x9yRxy, 8x9yRyx

Part C Show that the following sets of sentences are consistent.

1. {Ma, ⇠Na, Pa, ⇠Qa} 2. {Lee, Lef , ⇠Lfe, ⇠Lff} 3. {⇠(Ma& 9xAx), Ma _ Fa, 8x(Fx ! Ax)} 4. {Ma _Mb, Ma ! 8x⇠Mx} 5. {8yGy, 8x(Gx ! Hx), 9y⇠Iy} 6. {9x(Bx _Ax), 8x⇠Cx, 8x

⇥ (Ax&Bx) ! Cx

⇤ }

7. {9xXx, 9xY x, 8x(Xx $ ⇠Y x)} 8. {8x(Px _Qx), 9x⇠(Qx&Px)}

208 CHAPTER 8. SEMANTICS FOR QUANTIFIED LOGIC

9. {9z(Nz&Ozz), 8x8y(Oxy ! Oyx)} 10. {⇠9x8yRxy, 8x9yRxy}

Part D Construct models to show that the following arguments are invalid.

1) 8x(Ax ! Bx) 9xBx 2) {8x(Rx ! Dx), 8x(Rx ! Fx)} 9x(Dx&Fx) 3) 9x(Px ! Qx) 9xPx 4) Na&Nb&Nc 8xNx 5) {Rde, 9xRxd} Red 6) {9x(Ex&Fx), 9xFx ! 9xGx} 9x(Ex&Gx) 7) {8xOxc, 8xOcx} 8xOxx 8) {9x(Jx&Kx), 9x⇠Kx, 9x⇠Jx} 9x(⇠Jx&⇠Kx) 9) {Lab ! 8xLxb, 9xLxb} Lbb

Part E

1.? Show that {⇠Raa, 8x(x = a _Rxa)} is consistent. 2.? Show that {8x8y8z(x = y _ y = z _ x = z), 9x9y x 6= y} is consistent. 3.? Show that {8x8y x = y, 9x x 6= a} is inconsistent. 4. Show that 9x(x = h&x = i) is contingent. 5. Show that {9x9y(Zx&Zy&x = y), ⇠Zd, d = s} is consistent. 6. Show that 8x(Dx ! 9yTyx) .˙. 9y9z y 6= z is invalid.

Chapter 9

Proofs in Quantified Logic

9.1 Rules for Quantifiers

For proofs in QL, we use all of the basic rules of SL plus four new basic rules: both introduction and elimination rules for each of the quantifiers.

Since all of the derived rules of SL are derived from the basic rules, they will also hold in QL. We will add another derived rule, a replacement rule called quantifier negation.

Universal elimination

If you have 8xAx, it is legitimate to infer that anything is an A. You can infer Aa, Ab, Az, Ad3— in short, you can infer Ac for any constant c. This is the general form of the universal elimination rule (8E):

m 8xA

A [c|x ] 8E m

A [c|x ] is a substitution instance of 8xA . The symbols for a substitution instance are not symbols of QL, so you cannot write them in a proof. Instead, you write the subsituted sentence with the constant c replacing all occurrences of the variable x in A . For example:

1 8x(Mx ! Rxd)

2 Ma ! Rad 8E 1

3 Md ! Rdd 8E 1

209

210 CHAPTER 9. PROOFS IN QUANTIFIED LOGIC

Existential introduction

When is it legitimate to infer 9xAx? If you know that something is an A— for instance, if you have Aa available in the proof.

This is the existential introduction rule (9I):

m A

9xA [x ||c] 9I m

It is important to notice that A [x ||c] is not the same as a substitution instance. We write it with two bars to show that the variable x does not need to replace all occurrences of the constant c. You can decide which occurrences to replace and which to leave in place. For example:

1 Ma ! Rad

2 9x(Ma ! Rax) 9I 1

3 9x(Mx ! Rxd) 9I 1

4 9x(Mx ! Rad) 9I 1

5 9y9x(Mx ! Ryd) 9I 4

6 9z9y9x(Mx ! Ryz) 9I 5

Universal introduction

A universal claim like 8xPx would be proven if every substitution instance of it had been proven, if every sentence Pa, Pb, . . . were available in a proof. Alas, there is no hope of proving every substitution instance. That would require proving Pa, Pb, . . ., Pj2, . . ., Ps7, . . ., and so on to infinity. There are infinitely many constants in QL, and so this process would never come to an end.

Consider a simple argument: 8xMx, .˙. 8yMy

It makes no di↵erence to the meaning of the sentence whether we use the variable x or the variable y, so this argument is obviously valid. Suppose we begin in this way:

1 8xMx want 8yMy

2 Ma 8E 1

We have derived Ma. Nothing stops us from using the same justification to derive Mb, . . ., Mj2, . . ., Ms7, . . ., and so on until we run out of space or patience. We have e↵ectively shown the way to prove Mc for any constant c. From this, 8xMx follows.

9.1. RULES FOR QUANTIFIERS 211

1 8xMx

2 Ma 8E 1

3 8yMy 8I 2

It is important here that a was just some arbitrary constant. We had not made any special assumptions about it. If Ma were a premise of the argument, then this would not show anything about all y. For example:

1 8xRxa

2 Raa 8E 1

3 8yRyy not allowed!

This is the schematic form of the universal introduction rule (8I):

m A

8xA [x |c]⇤ 8I m

⇤ c must not occur in any undischarged assumptions.

Note that we can do this for any constant that does not occur in an undischarged assumption and for any variable.

Note also that the constant may not occur in any undischarged assumption, but it may occur as the assumption of a subproof that we have already closed. For example, we can prove 8z(Dz ! Dz) without any premises.

1 Df want Df

2 Df R 1

3 Df ! Df !I 1–2

4 8z(Dz ! Dz) 8I 3

Existential elimination

A sentence with an existential quantifier tells us that there is some member of the UD that satisfies a formula. For example, 9xSx tells us (roughly) that there is at least one S. It does not tell us which member of the UD satisfies S, however. We cannot immediately conclude Sa, Sf23, or any other substitution instance of the sentence. What can we do?

Suppose that we knew both 9xSx and 8x(Sx ! Tx). We could reason in this way:

212 CHAPTER 9. PROOFS IN QUANTIFIED LOGIC

Since 9xSx, there is something that is an S. We do not know which constants refer to this thing, if any do, so call this thing ⌦. From 8x(Sx ! Tx), it follows that if ⌦ is an S, then it is a T . Therefore ⌦ is a T . Because ⌦ is a T , we know that 9xTx.

In this paragraph, we introduced a name for the thing that is an S. We called it ⌦, so that we could reason about it and derive some consequences from there being an S. Since ⌦ is just a bogus name introduced for the purpose of the proof and not a genuine constant, we could not mention it in the conclusion. Yet we could derive a sentence that does not mention ⌦; namely, 9xTx. This sentence does follow from the two premises.

We want the existential elimination rule to work in a similar way. Yet since Greek letters like ⌦ are not symbols of QL, we cannot use them in formal proofs. Instead, we will use constants of QL which do not otherwise appear in the proof. A constant that is used to stand in for whatever it is that satisfies an existential claim is called a proxy. Reasoning with the proxy must all occur inside a subproof, and the proxy cannot be a constant that is doing work elsewhere in the proof.

This is the schematic form of the existential elimination rule (9E):

m 9xA

n A [c|x ]⇤

p B

B 9E m, n–p

⇤ The constant c must not appear in 9xA , in B , or in any undischarged assumption.

With this rule, we can give a formal proof that 9xSx and 8x(Sx ! Tx) together entail 9xTx. The structure of the proof is e↵ectively the same as the English-language argument with which we began, except that the subproof uses the constant “a” rather than the bogus name ⌦.

1 9xSx

2 8x(Sx ! Tx) want 9xTx

3 Sa

4 Sa ! Ta 8E 2

5 Ta !E 3, 4

6 9xTx 9I 5

7 9xTx 9E 1, 3–6

9.1. RULES FOR QUANTIFIERS 213

Quantifier negation

When translating from English to QL, we noted that ⇠9x⇠A is logically equivalent to 8xA . In QL, they are provably equivalent. We can prove one half of the equivalence with a rather gruesome proof:

1 8xAx want ⇠9x⇠Ax

2 9x⇠Ax for reductio

3 ⇠Ac for 9E

4 8xAx for reductio

5 Ac 8E 1

6 ⇠Ac R 3

7 ⇠8xAx ⇠I 4–6

8 ⇠8xAx 9E 3–7

9 8xAx R 1

10 ⇠9x⇠Ax ⇠I 2–8

In order to show that the two sentences are genuinely equivalent, we need a second proof that assumes ⇠9x⇠A and derives 8xA . We leave that proof as an exercise for the reader.

It will often be useful to translate between quantifiers by adding or subtracting negations in this way, so we add two derived rules for this purpose. These rules are called quantifier negation (QN):

⇠8xA 9x ⇠A ⇠9xA 8x ⇠A QN

Since QN is a replacement rule, it can be used on whole sentences or on subformulae.

Practice Exercises

Part A Provide a justification (rule and line numbers) for each line of proof that requires one.

1) . 1 ⇠(9xMx _ 8x⇠Mx)

2 ⇠9xMx&⇠8x⇠Mx

3 ⇠9xMx

4 8x⇠Mx

5 ⇠8x⇠Mx

6 9xMx _ 8x⇠Mx

214 CHAPTER 9. PROOFS IN QUANTIFIED LOGIC

2) . 1 8x9y(Rxy _Ryx)

2 8x⇠Rmx

3 9y(Rmy _Rym)

4 Rma _Ram

5 ⇠Rma

6 Ram

7 9xRxm

8 9xRxm

3) . 1 8x(9yLxy ! 8zLzx)

2 Lab

3 9yLay ! 8zLza

4 9yLay

5 8zLza

6 Lca

7 9yLcy ! 8zLzc

8 9yLcy

9 8zLzc

10 Lcc

11 8xLxx

4) . 1 8x(Jx ! Kx)

2 9x8yLxy

3 8xJx

4 Ja

5 Ja ! Ka

6 Ka

7 8yLay

8 Laa

9 Ka&Laa

10 9x(Kx&Lxx)

11 9x(Kx&Lxx)

Part B Without using the QN rule, prove ⇠9x⇠A 8xA

Part C Provide a proof of each claim.

1. 8xFx _ ⇠8xFx 2. {8x(Mx $ Nx),Ma& 9xRxa} 9xNx 3. {8x(⇠Mx _ Ljx), 8x(Bx ! Ljx), 8x(Mx _Bx)} 8xLjx 4. 8x(Cx&Dt) 8xCx&Dt 5. 9x(Cx _Dt) 9xCx _Dt

Part D

In the previous chapter (p. 215), we gave the following example

9.1. RULES FOR QUANTIFIERS 215

The hospital will only hire a skilled surgeon. All surgeons are greedy. Billy is a surgeon, but is not skilled. Therefore, Billy is greedy, but the hospital will not hire him.

UD: people Gx: x is greedy. Hx: The hospital will hire x. Rx: x is a surgeon. Kx: x is skilled. b: Billy

8x ⇥ ⇠(Rx&Kx) ! ⇠Hx

⇤

8x(Rx ! Gx) Rb&⇠Kb

.˙. Gb&⇠Hb

Prove the symbolized argument.

Part E On page 178 you translated the basic categorical syllogisms studied by Aristotle and his followers into QL. Now you need to provide derivations for some of them.

1) Barbara: All Bs are Cs. All As are Bs. .˙. All As are Cs.

2) Baroco: All Cs are Bs. Some A is not B. .˙. Some A is not C.

3) Bocardo: Some B is not C. All Bs are As. .˙. Some A is not C.

4) Celantes: No Bs are Cs. All As are Bs. .˙. No Cs are As.

5) Celarent: No Bs are Cs. All As are Bs. .˙. No As are Cs.

6) Campestres: All Cs are Bs. No As are Bs. .˙. No As are Cs.

7) Cesare: No Cs are Bs. All As are Bs. .˙. No As are Cs.

8) Dabitis: All Bs are Cs. Some A is B. .˙. Some C is A.

9) Darii: All Bs are Cs. Some A is B. .˙. Some A is C.

10) Disamis: Some B is C. All Bs are As. .˙. Some A is C.

11) Ferison: No Bs are Cs. Some B is A. .˙. Some A is not C.

12) Festino: No Cs are Bs. Some A is B. .˙. Some A is not C.

13) Frisesomorum: Some B is C. No As are Bs. .˙. Some C is not A.

Part F Now prove the conditionally valid syllogisms using QL. Symbolize each of the following and add the additional assumptions “There is an A” and “There is a B.” Then prove that the supplemented arguments forms are valid in QL. Calemos and Cesaro have been skipped because they are trivial variations of Camestros and Celaront.

1) Barbari: All Bs are Cs. All As are Bs. .˙. Some A is C.

2) Celaront: No Bs are Cs. All As are Bs. .˙. Some A is not C.

3) Camestros: All Cs are Bs. No As are Bs. .˙. Some A is not C.

4) Darapti: All As are Bs. All As are Cs. .˙. Some B is C.

216 CHAPTER 9. PROOFS IN QUANTIFIED LOGIC

5) Felapton: No Bs are Cs. All As are Bs. .˙. Some A is not C.

6) Baralipton: All Bs are Cs. All As are Bs. .˙. Some C is A.

7) Fapesmo: All Bs are Cs. No As are Bs. .˙. Some C is not A.

Part G Provide a proof of each claim.

1) 8x8yGxy 9xGxx 2) 8x8y(Gxy ! Gyx) 8x8y(Gxy $ Gyx) 3) {8x(Ax ! Bx), 9xAx} 9xBx 4) {Na ! 8x(Mx $ Ma),Ma,⇠Mb} ⇠Na 5) 8z(Pz _ ⇠Pz) 6) 8xRxx ! 9x9yRxy 7) 8y9x(Qy ! Qx)

Part H Show that each pair of sentences is provably equivalent.

1) 8x(Ax ! ⇠Bx) ⇠9x(Ax&Bx) 2) 8x(⇠Ax ! Bd) 8xAx _Bd 3) 9xPx ! Qc 8x(Px ! Qc)

Part I Show that each of the following is provably inconsistent.

1) {Sa ! Tm, Tm ! Sa, Tm&⇠Sa} 2) {⇠9x9yLxy, Laa} 3) {8x(Px ! Qx), 8z(Pz ! Rz), 8yPy, ⇠Qa&⇠Rb}

Part J Write a symbolization key for the following argument, translate it, and prove it:

There is someone who likes everyone who likes everyone that he likes. Therefore, there is someone who likes himself.

Part K For each of the following pairs of sentences: If they are logically equivalent in QL, give proofs to show this. If they are not, construct a model to show this.

1) 8xPx ! Qc 8x(Px ! Qc) 2) 8xPx&Qc 8x(Px&Qc) 3) Qc _ 9xQx 9x(Qc _Qx) 4) 8x8y8zBxyz 8xBxxx 5) 8x8yDxy 8y8xDxy 6) 9x8yDxy 8y9xDxy

9.2. RULES FOR IDENTITY 217

Part L For each of the following arguments: If it is valid in QL, give a proof. If it is invalid, construct a model to show that it is invalid.

1) 8x9yRxy 9y8xRxy 2) 9y8xRxy 8x9yRxy 3) 9x(Px&⇠Qx) 8x(Px ! ⇠Qx) 4) {8x(Sx ! Ta), Sd} Ta 5) {8x(Ax ! Bx), 8x(Bx ! Cx)} 8x(Ax ! Cx) 6) {9x(Dx _ Ex), 8x(Dx ! Fx)} 9x(Dx&Fx) 7) 8x8y(Rxy _Ryx) Rjj 8) 9x9y(Rxy _Ryx) Rjj 9) {8xPx ! 8xQx, 9x⇠Px} 9x⇠Qx 10) {9xMx ! 9xNx, ⇠9xNx} 8x⇠Mx

9.2 Rules for Identity

The identity predicate is not part of QL, but we add it when we need to symbolize certain sentences. For proofs involving identity, we add two rules of proof.

Suppose you know that many things that are true of a are also true of b. For example: Aa&Ab, Ba&Bb, ⇠Ca&⇠Cb, Da&Db, ⇠Ea&⇠Eb, and so on. This would not be enough to justify the conclusion a = b. (See p. 201.) In general, there are no sentences that do not already contain the identity predicate that could justify the conclusion a = b. This means that the identity introduction rule will not justify a = b or any other identity claim containing two di↵erent constants.

However, it is always true that a = a. In general, no premises are required in order to conclude that something is identical to itself. So this will be the identity introduction rule, abbreviated =I:

c = c =I

Notice that the =I rule does not require referring to any prior lines of the proof. For any constant c, you can write c = c on any point with only the =I rule as justification.

If you have shown that a = b, then anything that is true of a must also be true of b. For any sentence with a in it, you can replace some or all of the occurrences of a with b and produce an equivalent sentence. For example, if you already know Raa, then you are justified in concluding Rab, Rba, Rbb. Recall that A [a||b ] is the sentence produced by replacing a in A with b . This is not the same as a substitution instance, because b may replace some or all occurrences of a. The identity elimination rule (=E) justifies replacing terms with other terms that are identical to it:

218 CHAPTER 9. PROOFS IN QUANTIFIED LOGIC

m a = b

n A

A [a||b ] =E m, n

A [b ||a] =E m, n

To see the rules in action, consider this proof:

1 8x8y x = y

2 9xBx

3 8x(Bx ! ⇠Cx) want ⇠9xCx

4 Be

5 8y e = y 8E 1

6 e = f 8E 5

7 Bf =E 6, 4

8 Bf ! ⇠Cf 8E 3

9 ⇠Cf !E 8, 7

10 ⇠Cf 9E 2, 4–9

11 8x⇠Cx 8I 10

12 ⇠9xCx QN 11

Practice Exercises

Part A Provide a proof of each claim.

1) {Pa _Qb,Qb ! b = c,⇠Pa} Qc 2) {m = n _ n = o,An} Am _Ao 3) {8xx = m,Rma} 9xRxx 4) ⇠9xx 6= m 8x8y(Px ! Py) 5) 8x8y(Rxy ! x = y) Rab ! Rba 6) {9xJx, 9x⇠Jx} 9x9yx 6= y 7) {9xDx, 8x(x = p $ Dx)} Dp 8) {9x[(Kx&Bx)& 8y(Ky ! x = y)],Kd} Bd

9.2. RULES FOR IDENTITY 219

9) Pa ! 8x(Px _ x 6= a)

220 CHAPTER 9. PROOFS IN QUANTIFIED LOGIC

Appendix A

Other Symbolic Notation

In the history of formal logic, di↵erent symbols have been used at di↵erent times and by di↵erent authors. Often, authors were forced to use notation that their printers could typeset.

In one sense, the symbols used for various logical constants is arbitrary. There is nothing written in heaven that says that ‘⇠’ must be the symbol for truth-functional negation. We might have specified a di↵erent symbol to play that part. Once we have given definitions for well-formed formulae (w↵) and for truth in our logic languages, however, using ‘⇠’ is no longer arbitrary. That is the symbol for negation in this textbook, and so it is the symbol for negation when writing sentences in our languages SL or QL.

This appendix presents some common symbols, so that you can recognize them if you encounter them in an article or in another book.

summary of symbols negation ¬, ⇠

conjunction &, ^, disjunction _ conditional !, �

biconditional $, ⌘ Negation Two commonly used symbols are the hoe, ‘¬’, and the swung dash, ‘⇠.’ In some more advanced formal systems it is necessary to distinguish between two kinds of negation; the distinction is sometimes represented by using both ‘¬’ and ‘⇠.’

Disjunction The symbol ‘_’ is typically used to symbolize inclusive disjunction.

Conjunction Conjunction is often symbolized with the ampersand, ‘&.’ The ampersand is actually a decorative form of the Latin word ‘et’ which means ‘and’; it is commonly used in English writing. As a symbol in a formal system, the ampersand is not the word ‘and’; its meaning is given by the formal semantics for the language. Perhaps to avoid this confusion, some systems use a di↵erent symbol for conjunction. For example, ‘^’ is a counterpart to the symbol used for disjunction. Sometimes a single dot, ‘ ’, is used. In some older texts, there is no symbol for conjunction at all; ‘A and B’ is simply written ‘AB.’

221

222 APPENDIX A. OTHER SYMBOLIC NOTATION

Material Conditional There are two common symbols for the material conditional: the arrow, ‘!’, and the hook, ‘�.’

Material Biconditional The double-headed arrow, ‘$’, is used in systems that use the arrow to represent the material conditional. Systems that use the hook for the conditional typically use the triple bar, ‘⌘’, for the biconditional.

Quantifiers The universal quantifier is typically symbolized as an upside-down A, ‘8’, and the existential quantifier as a backwards E, ‘9.’ In some texts, there is no separate symbol for the universal quantifier. Instead, the variable is just written in parentheses in front of the formula that it binds. For example, ‘all x are P ’ is written (x)Px.

In some systems, the quantifiers are symbolized with larger versions of the symbols used for conjunction and disjunction. Although quantified expressions cannot be translated into expressions without quantifiers, there is a conceptual connection between the universal quantifier and conjunction and between the existential quantifier and disjunction. Consider the sentence 9xPx, for example. It means that either the first member of the UD is a P , or the second one is, or the third one is, . . . . Such a system uses the symbol ‘

W ’ instead of ‘9.’

Polish notation

This section briefly discusses sentential logic in Polish notation, a system of notation introduced in the late 1920s by the Polish logician Jan Lukasiewicz.

Lower case letters are used as sentence letters. The capital letter N is used for negation. A is used for disjunction, K for conjunction, C for the conditional, E for the biconditional. (‘A’ is for alternation, another name for logical disjunction. ‘E’ is for equivalence.)

notation Polish of SL notation ⇠ N & K _ A ! C $ E

In Polish notation, a binary connective is written before the two sentences that it connects. For example, the sentence A&B of SL would be written Kab in Polish notation.

The sentences ⇠A ! B and ⇠(A ! B) are very di↵erent; the main logical operator of the first is the conditional, but the main connective of the second is negation. In SL, we show this by putting parentheses around the conditional in the second sentence. In Polish notation, parentheses are never required. The left-most connective is always the main connective. The first sentence would simply be written CNab and the second NCab.

This feature of Polish notation means that it is possible to evaluate sentences simply by working through the symbols from right to left. If you were constructing a truth table for NKab, for example, you would first consider the truth-values assigned to b and a, then consider their conjunction, and then negate the result. The general rule for what to evaluate next in SL is not nearly so simple. In SL, the truth table for ⇠(A&B) requires looking at A and B, then looking in the middle of the sentence at the conjunction, and then at the beginning of the sentence at the

Appendix A: Other Symbolic Notation 223

negation. Because the order of operations can be specified more mechanically in Polish notation, variants of Polish notation are used as the internal structure for many computer programming languages.

224 APPENDIX A. OTHER SYMBOLIC NOTATION

Appendix B

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228 APPENDIX B. BIBLIOGRAPHY

Appendix C

Glossary

Antecedent The sentence to the left of a conditional.. 73

Argument a connected series of statements designed to convince an audience of another statement. 9

Artificial language A language that was consciously developed by identifiable individuals for some purpose. 51

Atomic sentence A sentence that does not have any sentences as proper parts. 67

Biconditional A sentential connective, written as double headed arrow, $, used to represent a situation where A implies B and B implies A. This is also the situation where A and B are logically equivalent. This is often expressed by the English phrase “if and only if.” 75

Bivalent A property of logical systems which is present when the system only has two truth values, generally “true” and “false.” 54

Canonical form a method for representing arguments where each premise is written on a separate, numbered, line, followed by a horizontal bar and then the conclusion. Statements in the argument might be paraphrased for brevity and indicator words are removed. 10

Cogent A property of arguments that holds when the argument is strong and the premises are true. 38

Cognitive bias a habit of reasoning that can become dysfunctional in certain circumstances. Often these biases are not a matter of explicit belief. See also fallacy 31

Complete truth table A table that gives all the possible interpretations for a sentence or set of sentences in SL. 94

Completeness A property held by logical systems if and only if implies 156

229

230 Glossary

Conclusion the statement that an argument is trying to convince an audience of. 9

Conclusion indicator a word or phrase such as “therefore” used to indicate that what follows is the conclusion of an argument. 10

Conditional The symbol !, used to represent words and phrases that function like the English phrase “if . . . then.” 73

Confirmation bias The tendency to discount or ignore evidence and arguments that contradict one’s current beliefs. 31

Conjunct A sentences joined to another by a conjunction. 70

Conjunction The symbol & , used to represent words and phrases that function like the English word “and.” 70

Consequent The sentence to the right of a conditional. 73

Consistency A property possessed by a set of sentences when they can all be true at the same time, but are not necessarily so. 56

Content neutrality the feature of the study of logic that makes it indi↵erent to the topic being argued about. If a method of argument is considered rational in one domain, it should be considered rational in any other domain, all other things being equal. 4

Contingent statement A statement that is neither a tautology nor a contradiction. 54

Contradiction A statement that must be false, as a matter of logic. 54

Contradictories Two statements that must have opposite truth values, so that one must true and the other false. 56

Critical thinker A person who has both sharpened their reasoning abilities using metareasoning and deploys those sharpened abilities in real world situations.. 5

Critical thinking The use of metareasoning to improve our reasoning in practical situations. Sometimes the term is also used to refer to the results of this e↵ort at self improvement, that is, reasoning in practical situations that has been sharpened by reflection and metareasoning. 5

Deductive A style of arguing where one attempts to use valid arguments. 38

Disjunct A sentences joined to another by a disjunction. 72

Disjunction The symbol _, used to represent words and phrases that function like the English word “or” in its inclusive sense. 72

Exclusive or A kind of disjunction that excludes the possibility that both disjuncts are true. The exclusive or says “This or that, but not both.” 73

Explainee The part of an explanation that one gains a greater understanding of as a result of the explainer. 20

Glossary 231

Explainer The part of an explanation that provides greater understanding of the explainee. 20

Explanation A kind of reasoning where reasons are used to provide a greater understanding of something that is already known. 20

Expository passage A nonargumentative passage that organizes statements around a central theme or topic statement. 16

Fallacy A common mistake in reasoning. Fallacies are generally conceived of as mistake forms of inference and are generally explained by arguments represented in canonical form. See also cognitive bias. 33

Formal language An artificial language designed to bring out the logical structure of ideas and remove all the ambiguity and vagueness that plague natural languages like English. Sometimes, formal languages are also said to be languages that can be implemented by a machine. 52

Formal logic A way of studying logic that achieves content neutrality by replacing parts of the arguments being studied with abstract symbols. Often this will involve the construction of full formal languages. 4

Inclusive or A kind of disjunction that allows for the possibility that both disjuncts are true. The inclusive or says “This or that, or both.” 73

Inconsistency A property possessed by a set of sentences when they cannot all be true at the same time, but they may all be false at the same time. 56

Inductive A style of arguing where one attempts to use strong arguments. 38

Inference the act of coming to believe a conclusion on the basis of some set of premises. 13

Informal logic The study of arguments given in ordinary language. 5

Interpretation A correspondence between nonlogical symbols of the object language and elements of some other language or logical structure. 91

Invalid A property of arguments that holds when the premises do not force the truth of the conclusion. The opposite of valid. 32

Logic the part of the study of reasoning that focuses on argument. 4

Logical constant A symbol whose meaning is fixed by a formal language. Sometimes these are just called “logical symbols.” They are contrasted with non-logical symbols. 68, 91

Logical equivalence A property held by a pair of sentences that must always have the same truth value. 56

Main connective The last connective that you add when you assemble a sentence using the recursive definition. 83

232 Glossary

Metacognition Thought processes that are applied to other thought processes See also metareasoning. 4

Metalanguage The language logicians use to talk about the object language. In this textbook, the metalanguage is English, supplemented by certain symbols like metavariables and technical terms like “valid.” 81

Metareasoning Using reasoning to study reasoning. See also metacognition. 4

Metavariables A variable in the metalanguage that can represent any sentence in the object language. 82

Narrative A nonargumentative passage that describes a sequence of events or actions. 17

Natural language A language that develops spontaneously and learned by infants as their first language. 51

Necessary condition A condition that must be true in order for something else to be, generally contrasted with a su�cient condition. 78

Negation The symbol ⇠, used to represent words and phrases that function like the English word “not”. 69

Nonlogical symbol A symbol whose meaning is not fixed by a formal language. 68, 91

Object language A language that is constructed and studied by logicians. In this textbook, the object languages are SL and QL. 81

Practical argument An argument whose conclusion is a statement that someone should do something. 9

Premise a statement in an argument that provides evidence for the conclusion 9

Premise indicator a word or phrase such as “because” used to indicate that what follows is the premise of an argument. 10

Reason The premise of an argument or the explainer in an explanation; the part of reasoning that provides logical support for the target proposition. 20

Recursive definition A definition that defines a term by identifying base class and rules for extending that class. Also called an “inductive definition.” 83

Rhetoric The study of e↵ective persuasion. 6

Scope The sentences that are joined by a connective. These are the sentences the connective was applied to when the sentence was assembled using a recursive definition. 83

Semantic contradiction in SL A statement that has only Fs in the column under the main connective of its complete truth table. 98

Glossary 233

Semantic tautology in SL A statement that has only Ts in the column under the main connective of its complete truth table. 98

Semantically consistent in SL A property held by sets of sentences if and only if the complete truth table for that set contains one line on which all the sentences are true 99

Semantically contingent in SL A property held by a sentence in SL if and only if the complete truth table for that sentence has both Ts and Fs under its main connective. 98

Semantically logically equivalent in SL A property held by pairs of sentences if and only if the complete truth table for those sentences has identical columns under the two main connectives. 98

Semantically valid in SL A property held by arguments if and only if the complete truth table for the argument contains no rows where the premises are all true and the conclusion false. 99

Semantics The meaning of a bit of language is its meaning, including truth and falsity. 81

Sentence letter A single capital letter, used in SL to represent a statement. 65

Sentence of SL A string of symbols in SL that can be built up using according to the recursive rules given on page 82

Sentential connective A logical operator in SL used to combine sentence letters into larger sentences. 67

Sentential logic A system of logic in which statements can be defined using a recursive definition with only sentences in the base class. 83

Simple statement of belief A kind of nonargumentative passage where the speaker simply asserts what they believe without giving reasons. 15

Sound A property of arguments that holds if the argument is valid and has all true premises. 34

Soundness A property held by logical systems if and only if implies 154

Statement A unit of language that can be true or false. 6

Strong A property of arguments which holds when the premises, if true, mean the conclusion must be likely to be true. 38

Su�cient condition A condition that is all you need for something to be true, generally contrasted with a necessary condition. 78

Syntactic contradiction in SL A statement in SL whose negation can be derived without any premises. 154

Syntactic tautology in SL A statement in SL that can be derived without any premises 145

Syntactically consistent in SL A property held by sets of sentences in SL if and only if they are not syntactically inconsistent. 154

234 Glossary

Syntactically contingent in SL A property held by a statement in SL if and only if it is not a syntactic tautology or a syntactic contradiction. 154

Syntactically inconsistent in SL A property held by sets of sentences in SL if and only if one can derive a contradiction from them. 154

Syntactically logically equivalent in SL A property held by pairs of statements in SL if and only if there is a derivation which takes you from each one to the other one. 143

Syntactically valid in SL A property held by arguments in SL if and only if there is a derivation that goes from the premises to the conclusion. 154

Syntax The structure of a bit of language, considered without reference to truth, falsity, or meaning. 81

Target proposition The conclusion of an argument or the explainee in an explanation; the part of reasoning that is logically supported by the reasons. 20

Tautology A statement that must be true, as a matter of logic. 54

Translation key A list that assigns English phrases or sentences to variable names. Also called a “symbolization key” or simply a “dictionary.” 65

Truth assignment A function that maps the sentence letters in SL onto truth values. 92

Truth value The status of a statement with relationship to truth. For this textbook, this means the status of a statement as true or false 53, 91

Truth-functional connective an operator that builds larger sentences out of smaller ones and fixes the truth value of the resulting sentence based only on the truth value of the component sentences. 92

Unique readability A property of formal languages which is present when each well formed formula is the product of a unique process of recursive construction. 83

Valid A property of arguments where it is impossible for the premises to be true and the conclusion false. 30

Weak A property of arguments that are neither valid nor strong. In a weak argument, the premises would not even make the conclusion likely, even if they were true. 38

Appendix D

Quick Reference

Characteristic Truth Tables

A ⇠A T F F T

A B A &B A_B A!B A$B T T T T T T T F F T F F F T F T T F F F F F T T

Symbolization

Sentential Connectives (chapter 4)

It is not the case that P . ⇠P Either P , or Q. (P _Q)

Neither P , nor Q. ⇠(P _Q) or (⇠P &⇠Q) Both P , and Q. (P &Q)

If P , then Q. (P ! Q) P only if Q. (P ! Q)

P if and only if Q. (P $ Q) Unless P , Q. P unless Q. (P _Q)

235

236 APPENDIX D. QUICK REFERENCE

Predicates (chapter 7)

All F s are Gs. 8x(Fx ! Gx)

Some F s are Gs. 9x(Fx&Gx)

Not all F s are Gs. ⇠8x(Fx ! Gx) or 9x(Fx&⇠Gx)

No F s are Gs. 8x(Fx ! ⇠Gx) or ⇠9x(Fx&Gx)

Identity (section 7.7)

Only j is G. 8x(Gx $ x = j) Everything besides j is G. 8x(x 6= j ! Gx)

j is more R than anyone else. 8x(x 6= j ! Rjx)

The F is G. 9x(Fx& 8y(Fy ! x = y)&Gx)

‘The F is not G’ can be translated two ways:

It is not the case that the F is G. (wide) ⇠9x(Fx& 8y(Fy ! x = y)&Gx) The F is non-G. (narrow) 9x(Fx& 8y(Fy ! x = y)&⇠Gx)

Using identity to symbolize quantities

There are at least F s.

one 9xFx two 9x19x2(Fx1 &Fx2 &x1 6= x2)

three 9x19x29x3(Fx1 &Fx2 &Fx3 &x1 6= x2 &x1 6= x3 &x2 6= x3) four 9x19x29x39x4(Fx1 &Fx2 &Fx3 &Fx4 &x1 6= x2 &x1 6= x3 &x1 6= x4 &x2 6=

x3 &x2 6= x4 &x3 6= x4) n 9x1 · · · 9xn(Fx1 & · · · &Fxn &x1 6= x2 & · · · &xn�1 6= xn)

There are at most F s.

One way to say ‘at most n things are F ’ is to put a negation sign in front of one of the symbolizations above and say ⇠ ‘at least n+ 1 things are F .’ Equivalently:

one 8x18x2 ⇥ (Fx1 &Fx2) ! x1 = x2

⇤

237

two 8x18x28x3 ⇥ (Fx1 &Fx2 &Fx3) ! (x1 = x2 _ x1 = x3 _ x2 = x3)

⇤

three 8x18x28x38x4 ⇥ (Fx1 &Fx2 &Fx3 &Fx4) ! (x1 = x2 _ x1 = x3 _ x1 = x4 _ x2 =

x3 _ x2 = x4 _ x3 = x4) ⇤

n 8x1 · · · 8xn+1 ⇥ (Fx1 & · · · &Fxn+1) ! (x1 = x2 _ · · · _ xn = xn+1)

⇤

There are exactly F s.

One way to say ‘exactly n things are F ’ is to conjoin two of the symbolizations above and say ‘at least n things are F ’ & ‘at most n things are F .’ The following equivalent formulae are shorter:

zero 8x⇠Fx one 9x

⇥ Fx&⇠9y(Fy&x 6= y)

⇤

two 9x19x2 ⇥ Fx1 &Fx2 &x1 6= x2 &⇠9y

� Fy& y 6= x1 & y 6= x2

�⇤

three 9x19x29x3 ⇥ Fx1 &Fx2 &Fx3 &x1 6= x2 &x1 6= x3 &x2 6= x3 &⇠9y(Fy& y 6=

x1 & y 6= x2 & y 6= x3) ⇤

n 9x1 · · · 9xn ⇥ Fx1 & · · · &Fxn &x1 6= x2 & · · · &xn�1 6= xn &⇠9y(Fy& y 6=

x1 & · · · & y 6= xn) ⇤

Specifying the size of the UD

Removing F from the symbolizations above produces sentences that talk about the size of the UD. For instance, ‘there are at least 2 things (in the UD)’ may be symbolized as 9x9y(x 6= y).

Basic Rules of Proof

Reiteration

m A

A R m

Conjunction Introduction

238 APPENDIX D. QUICK REFERENCE

m A

n B

A &B &I m, n

m A

n B

B &A &I m, n

Conjunction Elimination

m A &B

A &E m

m A &B

B &E m

Disjunction Introduction

m A

A _ B _I m

m A

B _ A _I m

Disjunction Elimination

m A _ B

n ⇠B

A _E m, n

m A _ B

n ⇠A

B _E m, n

Conditional Introduction

m A want B

n B

A ! B !I m–n

Conditional Elimination

m A ! B

n A

B !E m, n

Biconditional Introduction

239

m A want B

n B

p B want A

q A

A $ B $I m–n, p–q

Biconditional Elimination

m A $ B

n B

A $E m, n

m A $ B

n A

B $E m, n

Negation Introduction

m A for reductio

n� 1 B

n ⇠B

⇠A ⇠I m–n

Negation Elimination

m ⇠A for reductio

n� 1 B

n ⇠B

A ⇠E m–n

240 APPENDIX D. QUICK REFERENCE

[4]

Quantifier Rules

Existential Introduction

m A

9xA [x ||c] 9I m

x may replace some or all occurrences of c in A .

Existential Elimination

m 9xA

n A [c|x ]

p B

B 9E m, n–p

The constant c must not appear in 9xA , in B , or in any undischarged assumption.

Universal Introduction

m A

8xA [x |c] 8I m

c must not occur in any undischarged assumptions.

Universal Elimination

m 8xA

A [c|x ] 8E m

Identity Rules

241

Identity Introduction

c = c =I Identity Elimination

m c = d

n A

A [c||d ] =E m, n

A [d ||c] =E m, n

One constant may replace some or all occurrences of the other.

Derived Rules

Constructive Dilemma (CD)

m A _ B

n A ! C

p B ! C

C _⇤ m, n, p

Modus Tollens (MT)

m A ! B

n ⇠B

⇠A MT m, n

Hypothetical Syllogism (HS)

m A ! B

n B ! C

A ! C HS m, n

Replacement Rules

Commutivity (Comm)

(A &B) () (B &A) (A _ B) () (B _ A) (A $ B) () (B $ A)

DeMorgan (DeM)

⇠(A _ B) () (⇠A &⇠B) ⇠(A &B) () (⇠A _ ⇠B)

Double Negation (DN)

⇠⇠A () A

Material Conditional (MC)

(A ! B) () (⇠A _ B) (A _ B) () (⇠A ! B)

Biconditional Exchange ($ex)

[(A ! B)& (B ! A)] () (A $ B)

Quantifier Negation (QN)

⇠8xA () 9x ⇠A ⇠9xA () 8x ⇠A

About the authors:

P.D. Magnus is an associate professor of philosophy in Albany, New York. His primary research is in the philosophy of science, concerned especially with the underdetermination of theory by data.

J. Robert Loftis is an associate professor of philosophy at Lorain County Community College in Elyria, Ohio. He received his Ph.D. in philoso- phy from Northwestern University.

Cathal Woods is Batten Associate Professor of Philosophy at Virginia Wesleyan University. He received his Ph.D. in philosophy from The Ohio State University.