PHI 1101: Reasoning and Critical Thinking: Course Notes, Units 1-8 and Test 1 Review

Table of contents :
Introduction
1.1 What this course is about
1.2 Taking the online version of this course
1.3 The Art of Thinking
1.4 Truth
1.5 Belief
1.6 Knowledge
1.7 Knowledge and Reasonable Belief

2 Arguments
2.1 Introduction
2.2 Explanations and arguments
2.3 Simple and complex arguments

3 Arguments in standard form
3.1 Real life is messy
3.2 The principle of charity
3.3 Principles or forms of inference
3.4 Unstated premises and conclusions
3.5 Diagramming arguments

4 Evaluating Arguments
4.1 Introduction
4.2 Evaluating Premises: General Remarks
4.3 Evaluating reasoning

5 Validity and Soundness (1)
5.1 Deductive and non-deductive reasoning
5.2 Validity
5.3 Developing a sense of possibility
5.4 Implication and equivalence
5.5 Validity, truth, and soundness

6 Validity and Soundness (2)
6.1 Truth preservation and formal validity
6.2 Proving Validity: A pair of examples
6.3 On Proofs and Refutations

7 Basic Syllogistic and Venn Diagrams
7.1 Introduction
7.2 Syllogisms
7.3 Venn diagrams: the basics
7.4 Two terms; categorical propositional forms

8 Venn Diagrams (2): Syllogisms
8.1 Reminder
8.2 Evaluating Reasoning, Once Again

9 Overview of Units 1-8
10 Exercises for Review
11 Solutions to Exercises

Citation preview

PHI 1101: Reasoning and Critical Thinking: Course Notes, Units 1-8 and Test 1 Review P Rusnock

Copyright © 2020 Paul Rusnock

C ONTENTS 1

Introduction

1.1 1.2 1.3 1.4 1.5 1.6 1.7 2

3

4

5

6

7

What this course is a b o u t................................................................. Taking the on-line version of this c o u r s e ...................................... The Art of T hin king.......................................................................... Truth..................................................................................................... Belief..................................................................................................... Know ledge......................................................................................... Knowledge and Reasonable B elief..................................................

5

5 5 6 7 8 9 10

Arguments

11

2.1 Introduction ...................................................................................... 2.2 Explanations and arguments........................................................... 2.3 Simple and complex argu m ents.....................................................

11 13 14

Arguments in standard form

16

3.1 3.2 3.3 3.4 3.5

18 19 20 21 22

Real life is m essy................................................................................ The principle of ch a rity .................................................................... Principles or forms of inference ..................................................... Unstated premises and conclusions............................................... Diagramming argum ents.................................................................

Evaluating Arguments

26

4.1 Introduction ...................................................................................... 4.2 Evaluating Premises: General Rem arks......................................... 4.3 Evaluating reaso n in g .......................................................................

26 27 32

Validity and Soundness (1)

34

5.1 5.2 5.3 5.4 5.5

34 34 36 37 38

Deductive and non-deductive reasoning...................................... V a lid ity ............................................................................................... Developing a sense of possibility .................................................. Implication and equivalence........................................................... Validity truth, and sou nd n ess........................................................

Validity and Soundness (2)

39

6.1 Truth preservation and formal v a lid ity ......................................... 6.2 Proving Validity: A pair of ex am p les............................................ 6.3 On Proofs and R efutations..............................................................

39 42 43

Basic Syllogistic and Venn Diagrams

45

7.1 Introduction ...................................................................................... 7.2 Syllogisms .........................................................................................

45 45

7 3 Venn diagrams: the basics.................................................................. 7.4 Two terms; categorical propositional form s....................................

47 50

8 Venn Diagrams (2): Syllogisms 8.1 R em inder............................................................................................ 8.2 Evaluating Reasoning, Once A g a in ...............................................

55 55 63

9 Overview of Units 1-8

65

10 Exercises for Review

67

11 Solutions to Exercises

77

C HAPTER 1

I N TRO D U CTIO N 1.1

W HAT

THIS COURSE IS ABOUT

This course, which is called Reasoning and Critical Thinking, presents a variety of techniques that can be used in our own reasoning and in evaluating the rea­ soning of others. The emphasis throughout is on kinds of reasoning that are common to most kinds of inquiry, with the hope and expectation that pretty much everything you learn in this course will be applicable in one way or an­ other throughout your studies, whatever their particular focus may be, and also in life beyond the walls of the academy. The presentation is also introduc­ tory in approach. Everything that is covered here can be covered in greater depth and detail. The goal is simply to give you a decent tool-kit to get you started, and to show you how to use it. Our hope is that you will continue to improve these skills by applying and refining them as you progress through your studies. We will study a number of different kinds of reasoning, including the deduc­ tive variety which is the main topic of courses on formal logic, as well as various sorts of non-deductive reasoning that are widely used in academic research as well as everyday life. Finally, we will point out a number of common fallacies, patterns of reasoning that should not convince people but nonetheless often succeed in doing just that.

1.2

TAKING

THE ON - LINE VERSION OF THIS COURSE

Skill in correct reasoning can only be developed through methodical practice. Accordingly, when I teach the in person version of this course, I spend a good deal of class time working on exercises with students. This will obviously be difficult to duplicate in an on-line course, but I believe we can achieve a similar result by working together. On your end, it is very important first of all to read the text for each unit carefully and, in many cases, repeatedly, until you have become well acquainted with the concepts presented. Second, listen carefully to the on-line lecture (with slides), taking notes as you do. These are usually fairly short, and you may find it useful to listen to them more than once. After following the lecture, I encourage you to read the text for the unit again, and then work through the exercises included in that text, comparing your answers with the solutions provided. For my part, I will also be including as part of the course a number of on­ line quizzes to give you more practice and immediate feedback. In addition, I, 5

Reasoning and Critical Thinking along with one or more teaching assistants, will be available on-line at a variety of times during the week. Please do not hesitate to get in touch with one of us if you have any questions.

1.3

THE A RT

OF

T H IN K IN G

Most mammals are able to move about and fend for themselves at least to some extent within hours of being bom. We are not like that: most of us can't walk until we are over a year old, and even then not so well at first. During that first year we are pretty much helpless, entirely dependent upon others to take care of us. The explanation for this is fairly simple: our heads, even as small children, grow so large that birth would be a physical impossibility at the time (around 18 months) when we would be more or less capable of functioning with some independence. Presumably, the extra space for brains gave our species an evolutionary advantage that outweighed the heavy costs involved in childrearing, allowing us to multiply and spread across the earth. We have done so well at least partly because we can think. All the same, complaints about our failure to do so are peppered throughout literature, history, and philosophy. Hamlet, for instance, wondered aloud: What is a man If his chief good and market of his time Be but to sleep and feed? A beast, no more. Sure, he that made us with such large discourse, Looking before and after, gave us not That capability and godlike reason To fust in us unused.1 While the English philosopher Bertrand Russell drily remarked: Most people would sooner die than think; in fact, they do so. There can be no doubt that the failure to think, or to think carefully enough, be­ fore acting has led to an impressive number of disasters in human history, and can be blamed for many of the ills that plague us. It is primarily for this reason that educators place so much emphasis upon the importance of inculcating the skills of reasoning and critical thinking in students. Thinking comes naturally to us, however, as naturally as breathing, walk­ ing, or running. So it is not obvious at first glance that there is anything to be gained from studying reasoning. A little reflection, however, suggests that there might be. For although everyone breathes and the vast majority of us run at one time or another, we also know from experience that attentive prac­ tice and training can make us better at these things. Singers, trumpet players, ^Hamlet, Act 4, Scene 4.

6

Introduction and swimmers have much to gain from learning how to breathe in the way that best serves their purposes, and champion sprinters have invariably spent a great deal of time on the mechanics of running. In both cases, high perfor­ mance is the result of practice, methodical observation, and adjustment. So too, it can be argued, with thinking. By slowing down and paying at­ tention to what we are doing when we are thinking, we can both reinforce the habits that direct us towards truth, and fight against the bad ones that lead us and others into error. If you have a computer, it can be an eye-opening experience to carefully go through the list of files on its hard drive. You will be astounded by the number and variety of things that have somehow or other made their way there. Often, you don't have to do much if anything to help the process along. Files just seem to appear, whether you want them to or not. Some, indeed, are downright harmful, like viruses, spy- and mal-ware, etc. It would not be much of a stretch to compare our minds to such a hard drive. All of us have things rattling around in our minds whose origin we would be at a loss to explain. Some of them can also be harmful, either to ourselves or others. At the same time, some of these thoughts may be among our most firmly held beliefs. And there is no doubt that large-scale efforts are constantly underway to get us to accept certain things, most notably, but not exclusively, through advertising. It would be foolish simply to assume that we are immune to these techniques, and that our minds are free from prejudices, fake-facts (factoids), and the like. One advantage of slowing down and examining our own arguments is that it can make us aware of such mind junk, and help us to get rid of at least some of it. By the very nature of the case, this is not a once in a lifetime business, as Descartes hoped it might be. Rather, like keeping house, it requires constant vigilance and the occasional spring cleaning.

1.4

TRUTH

We use the words 'true' and 'false' all the time, but, if are you are like most people, you will find it difficult to say precisely what you mean by them. It is like this with many of our basic concepts: though we use them quite fluently, we may find ourselves unable to define them. In the sense that we are concerned with here, truth and falsity are correctly applied to sentences, statements, or propositions. We say, for example, that it is true that whales are mammals, or false that whales can fly. But it wouldn't make any sense to say, for example, that a whale or a chair or the colour red is true or false. When is a statement or proposition true? Here is what Aristotle said: To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is 7

Reasoning and Critical Thinking true.2 If we say, for instance, that bats are mammals, we speak truly, since bats are mammals. Similarly, if we say that Elizabeth is the Queen of France, we speak falsely, because she is not Queen of that country. Logicians customarily use single quotation marks to form names of linguis­ tic expressions. For example, we write: 'Chicago' ends with a vowel. —to express a truth about the name of a certain midwestern US city. We say in such cases that the expression is mentioned, in contrast to cases where it is used, e.g.: Chicago sits on the shore of Lake Michigan. With this notational convention, we can, as Alfred Tarski noted, capture Aris­ totle's idea in individual cases by writing, e.g.: 'Bats are mammals' is true if and only if bats are mammals. —where the sentence in question is mentioned in its first occurrence and used in the second. Whether there is more to truth than this and, if so, what it is, are matters of some dispute, which we will not go into here. We have nonetheless seen enough to catch sight of some of the main features of truth. Truth involves what can be said, but also what can be talked about, and the relations between the two.

1.5

BELIEF

Like 'truth', the word 'belief' is used in several senses. The one that we will be concerned with here is that according to which to believe something (a proposi­ tion) is to take it to be true. You occasionally hear people conflate truth and belief, saying, for example, that if Smith genuinely believes that penguins can fly, then it is true fo r him that penguins can fly. I will not be using the word 'true' in this sense. Moreover, it seems to me that there is no need to use the word 'true' in this way, since the word 'believes' already exists. We can easily see that truth and belief are different things by noting the following: 1. Something can be believed without being true: this is obvious, since we all make mistakes. 2. Something can be true without being believed: did anyone in the thir­ teenth century, for example, know that there are 78 protons in an atom of 2M

etaphysics, tr. W. D. Ross (Oxford University Press, 1924), IV, 7, 1101 (>25.

8

Introduction platinum? Similarly, we rightly expect that in the future new truths will be discovered that were previously unknown to anyone alive either now or at any earlier time. 3. Belief is relative to individuals but truth isn't. For there to be a belief, there has to be someone who believes something. But a proposition is simply true or false. Finally, there is a further important difference between truth and belief: with truth, there are just two possibilities: a proposition is either true or its false. With belief, on the other hand, there are three options: 1. Someone can accept a proposition P (believe it); 2. Reject P (believe that P is false); 3. Or remain neutral (not make a judgment either way). Often, the third option is overlooked, even when it is the most advisable under the circumstances. This is because we often feel pressured to make a judgment, even in cases where we don't have enough information to do so intelligently and there is no need to decide right away.

1.6

K NOW LEDGE

Knowledge involves both truth and belief. To know something, we must believe it, and our belief must also be true. As the old saying goes, you can't know what isn't so. But knowledge cannot be reduced to either truth or belief, or to the combination of the two. Consider Longshot Louie, who spends his days at one of the few remaining horse tracks. In every race, Louie picks the horse with the longest odds, the one that, in the judgment of all the bettors, is least likely to win. Louie is an optimistic sort, however, and genuinely believes that every one of his bets will be a winner. Now, around ninety-seven percent of the time, he turns out to be wrong, but once in a while his horse comes in. In such cases, he believed his horse would win, and that belief turned out to be true. Yet I think it fair to say that no one would say that he knew that it would win. Examples like this one seem to show quite clearly that knowledge requires more than just happening to form a true belief—we must have formed the belief in the right way, perhaps. But it is hard to say exactly what the right way is, or to recognize when we have formed a belief in the right way. It is also hard to find an example where we can be absolutely sure that we do know something—most of us can recall cases where we believed something very strongly, didn't think we could be mistaken, and yet turned out to be wrong all the same. For this reason, skepticism—the view that we can know very little if anything at all—has always been attractive to some people. 9

1.7

KNOWLEDGE

AND

R EASONABLE BELIEF

Even though it can be difficult to say whether we genuinely know something, almost everyone will agree that, among the beliefs that people hold, some are more reasonable than others, that some are quite reasonable, and some are quite unreasonable. In addition, most will agree that the more reasonable our beliefs are, the more likely they are to be true. Accordingly, we will set our sights on reasonable belief rather than knowledge, aiming to develop methods and habits of thought that will produce the beliefs that are most reasonable given the information available to us. If it turns out that some of these beliefs also qualify as knowledge, we may gratefully accept this as a bonus.

C HAPTER 2

A RGUMENTS 2.1

I NTRODUCTION

Reasonable beliefs, as we said, are often beliefs supported by good reasons. Arguments are the written or spoken expression of this relationship of support. In this part, we will look at the elements and structure of arguments. In the general sense we are concerned with here, an argument is a set of claims (called premises) along with a further claim, which is called the conclu­ sion, and an inference linking premises and conclusion. Here is an example: Socrates was the victim of injustice. No one, however, can act un­ justly towards himself. Therefore, someone else must have acted unjustly. Usually, when people propose arguments, the premises are supposed to provide at least some support for the conclusion, that is, to give anyone hear­ ing the argument at least some reason to accept it. Thus you will sometimes hear arguments described as attempts to persuade someone that some claim or other is true. The concept of an argument used here is broader, however, covering not only cases where there is an actual attempt to convince someone of the truth of a given claim (conclusion), but also merely possible cases where no one in fact makes the argument. Sometimes, too, people will say of a really bad argument that it is no argu­ ment at all. Again, we will take a broader view. For us, any actual or possible attempt to convince someone that something is true by giving reasons will be counted as an argument, no matter how inept the attempt may be. Thus, along­ side perfectly respectable bits of reasoning such as: Cavities hurt a lot and dental work is expensive. So you should brush your teeth. We will also count the following as an argument (just a horribly bad one): Dental work is expensive. So if your teeth hurt, you should just ignore it. Of course, not everything people say consists of arguments. Sometimes, we just describe things, like this: A Saturday afternoon in November was approaching the time of twilight, and the vast tract of unenclosed wild known as Egdon Heath embrowned itself moment by moment. Overhead the hollow stretch of whitish cloud shutting out the sky was as a tent which had the whole heath for its floor. 11

Reasoning and Critical Thinking Or this: It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season of Light, it was the season of Darkness, it was the spring of hope, it was the winter of despair, we had everything before us, we had nothing before us, we were all going direct to heaven, we were all going direct the other way—in short, the period was so far like the present period, that some of its noisiest authorities insisted on its being received, for good or for evil, in the superlative degree of comparison only. Or we may simply report facts, without drawing any conclusions from them: Yesterday, the high temperature in New York City was 23 degrees, with a low of 16. The sky was partly cloudy, but there was no rain. Thus arguments are far from the only sort of thing to be found in what we say. In fact, they are somewhat rare. English is rich in expressions which help to indicate that an argument is intended. When giving reasons in support of some claim, we often use words such as since, because, as, seeing that, given that (these and similar words are called premise indicators), while when drawing conclusions, we often use ex­ pressions such as so, thus, therefore, hence and so on (conclusion indicators).

Premise indicators as as shown by as we can see from because considering that for insofar as on account of since

Conclusion indicators as a consequence as a result for these reasons hence it follows that it must be that so therefore thus we may conclude/infer which proves/shows that

Here are a few examples: I think. Therefore, I am Since all people are born equal, hereditary rights and privileges can­ not be justified. [T]heir Lordships have come to the conclusion that the word 'per­ sons' in sec. 24 includes members both of the male and female sex 12

Arguments and that, therefore, ... women are eligible to be summoned to and become members of the Senate of Canada, and they will humbly advise His Majesty accordingly. While the even natural numbers can be mapped one to one onto the set of all natural numbers, the real numbers cannot. This shows that not all infinite sets are the same size. Corporations do not have the same rights to freedom of speech as individuals under the US constitution because they were not in­ tended to be covered under the opening formula 'We the people.' As you know, many words in English do double-duty, and the presence of the above words is not an infallible sign that an argument is present. Consider, for example, the following passage: Steve's been depressed ever since he lost his job. Clearly, there is no inference here; the word since just indicates a sequence of events.

2.2

E XPLANATIONS

AND ARGUMENTS

One particularly tricky case is that of explanations. In explanations, as in argu­ ments, reasons are given, and often enough the same indicator words are used (since, because, etc.). But whereas in the case of arguments the aim is to con­ vince someone that something is true, in an explanation, that something is true is usually taken for granted, and reasons are given to show why it is. By way of example, consider the following: Even though he finished first in the 100 metre race in the 1988 Olympics, Ben Johnson did not win the gold medal, because he was disquali­ fied for doping. In this passage, no attempt is made to convince us that Johnson did not win the gold medal. Rather, this is simply something we are told. When reasons are given here, it is simply to explain why Johnson did not win. Interestingly, some explanations can be converted into arguments, and vice versa, simply by changing some tenses. For instance, the explanation: The floor is wet because it was raining and the window was left open. corresponds to the argument: It's raining and the window has been left open, so the floor will get wet. 13

Reasoning and Critical Thinking It is also possible that the same considerations can serve at the same time as an argument and an explanation, showing simultaneously that something is true and why it is. Many scientific explanations and mathematical proofs are of this sort. The following partly visual argument, for example, seems to show both that and why the well-known formula for the difference of two squares is correct when a > b > 0: a 2 —&2 = (a — 6) (a + b)

For when we remove the square 3, whose area is b2 , from the larger square whose area is a 2 , we are left with the two rectangles 1 and 2, whose sides, respectively, have lengths a, a —b and b. a — b. Together, then, we have: a 2 —b2 = (a —b) •a + (a —b) •b = (a —b) (a + b)

2.3

S IMPLE

AND COMPLEX ARGUMENTS

In many cases, an argument involves only one inference. This is the case with most of the examples just given. But there are also more complex argu­ ments, involving several inferences, where one or more of the conclusions that is drawn is subsequently used as a premise in another part of the argument. Here is an example of such a complex argument: Either the chauffeur or the butler killed Thickson. But the butler was in London on the day of the murder. So he didn't do it. It must have been the chauffeur. Here, we see that the claim The butler didn't do it functions both as a premise and as a conclusion. It is a conclusion of the argument: The butler was in London the day of the murder. So: The butler didn't do it. 14

And a premise of the argument: Either the chauffeur or the butler killed Thickson. The butler didn't do it. So: It must have been the chauffeur. We will call such claims intermediate conclusions of complex arguments, and the two arguments above will be called sub-arguments of the original, complex argument. There is no limit to the complexity of arguments. In mathematics and sci­ ences such as physics and astronomy, for example, it is common for arguments to be based upon hundreds or even thousands of sub-arguments, some of which have been elaborated by many individuals over hundreds or even thou­ sands of years. With the help of computers, the boundaries have been pushed still farther.

C HAPTER 3

A RGUM ENTS

IN STANDARD FORM

Aiming as we do at truth, we are interested in arguments insofar as they lead us closer to it. Above all, we want to be able to tell the good arguments from the bad. But before we can evaluate an argument, we have to know precisely what the argument is. And this is not always obvious at first glance, even when the person proposing the argument has spoken or written clearly. It is even more difficult when the arguer is not very clear, something that is more often than not the case. A helpful, if slightly artificial, way of getting clearer on the content and structure of an argument is to express it in the form of a list, where each claim is numbered separately, premises and conclusions are clearly labelled, premises always appearing in the list before the conclusions they support, and inferences indicated by referring to the premises supporting a given conclusion. We will say that an argument presented in this way appears in standard form. The following examples will help to illustrate what I mean. Let us begin with the simplest case, where there is just one premise and one conclusion: You shouldn't drink and drive because the penalties are stiff if you are caught. To put this in standard form, we list the premise, followed by the conclusion. 1. The penalties are stiff if you are caught drinking and driving (P). 2. You shouldn't drink and drive. (C: from 1) The line separating the premise from the conclusion indicates an inference, and the number in parentheses indicates which premise is supposed to support the conclusion. Similarly, when there are two or more premises: You shouldn't drink and drive because the penalties are stiff if you are caught. Besides, you might cause an accident. 1. The penalties are stiff if you are caught drinking and driving. (P) 2. You might cause an accident if you drink and drive. (P) 3. You shouldn't drink and drive. (C: from 1,2) Consider now a complex argument: You shouldn't drink and drive because the penalties are stiff if you are caught. Besides, you might cause an accident. And you do have to drive tonight. So you shouldn't have anything to drink tonight. 16

3. Arguments in Standard Form Here, the indicator words because and so tell us that (at least) two inferences have been made, and thus that two conclusions have been drawn, namely: You shouldn't drink and drive. You shouldn't have anything to drink tonight. 1. 2. 3. 4. 5.

The penalties are stiff if you are caught drinking and driving. (P) You might cause an accident if you drink and drive. (P) You shouldn't drink and drive. (IC: from 1, 2) You have to drive tonight (P). You shouldn't have anything to drink tonight. (FC: from 3,4)

Here, we use the abbreviations 'IC' and 'FC' to indicate intermediate and final conclusions, respectively. Finally, here is a more complicated example: Unemployment won't increase because the economy will continue to grow, and employment usually increases when the economy is growing. But EI expenditures would only increase if unemploy­ ment worsened. So we can count on EI costing either the same or less in years ahead. And tax revenues tend to increase in a growing economy, so we should expect this as well. Hence if other expendi­ tures are held constant, we should expect the deficit to decrease. Here, the indicator words ('because', 'so', 'so', 'hence') help us pick out four separate conclusions, namely: 1. 2. 3. 4.

Unemployment won't increase. EI expenditures won't increase. Tax revenues will increase. The deficit will decrease.

They also help us determine for the most part which premises are used to sup­ port these conclusions. And the fact that the first conclusion is used to support the second, and that the second along with the third used to support the fourth helps us to sort out almost all the features of the argument's structure. One detail, however, is not clearly indicated from the way the passage is written, namely, that the premise stating that the economy will continue to grow is used to support two different conclusions (nos. 2 and 3). This has to be gathered from the context, rather than explicit indication. Finally, we arrive at something like this: 1. If the economy grows, employment usually increases. (Pr) 2. The economy will continue to grow. (Pr) 3. Unemployment won't increase. (IC, from 1, 2) 4. EI expenditures will increase only if unemployment does. (Pr) 5. EI expenditures won't increase. (IC, from 3,4) 6. Tax revenues usually increase in a growing economy. (Pr) 7. Tax revenues will increase. (IC, from 2,6) 8. If other expenditures are held constant, the deficit will likely decrease. (FC, from 5, 7) 17

Reasoning and Critical Thinking

3.1

R EAL

LIFE IS MESSY

The clear, direct expression of an argument in standard form often contrasts sharply with the arguments we encounter in daily life. Statements may not be entirely clear, words may be used in unusual senses, and there is often a lot of extraneous material. Parts can also be missing: premises or conclusions are sometimes simply hinted at, the expectation being that the audience will supply them on their own. Given all this, it can be a challenge at times to figure out exactly what argument was intended. Consider, by way of example, the following passage: Guns are dangerous, even in the hands of highly-trained police. We've all heard the story: a teenager had a psychotic episode on a Toronto streetcar. Waving a knife at the passengers and the driver, he managed to scare everyone out of their wits. By some miracle, however, everyone except the teenager got off the streetcar safely. Then the police showed up. The young man with a knife no longer posed any danger to the public, yet he was shot all the same. And then shot again, and again, and again, so that finally he died. This tragic result was completely unnecessary. Not because the po­ liceman didn't have to shoot the man so many times. For the police are human, and will always make mistakes. There's no way of pre­ venting that from happening. They're not perfect, and will some­ times do the wrong thing. It's the gun that was the real problem that night in Toronto. Once you fire a gun, there is always a real chance of killing someone. If only he had had a different sort of weapon, one that was capable of stopping a suspect without killing him. Then that young man would still be alive, and could have received the treatment that would allow him to get on with his life. But we do have such weapons: they're called tasers. Why didn't this policeman have a taser? Why don't all policemen carry tasers? Everyone would be so much safer then. But instead they carry guns, making tragedies like this one all but inevitable. This prob­ lem would be solved once and for all by a simple change: give all policemen tasers. As is often the case, there are many words here, but not a whole lot of argu­ ment. Notice the repetition for the sake of emphasis, the addition of narrative detail to add colour, the asking of questions with the expectation that the reader already knows the answer, and so on. From our point of view, these rhetori­ cal techniques just obscure the argument. We want to know what reasons are given in support of what conclusion. From this perspective, the main point of the above passage is to convey a fairly simple argument, which can be summa­ rized as follows:

18

3. Arguments in Standard Form 1. A teenager who had threatened people was shot and killed by a policeman in Toronto. 2. If he had been shot with a taser instead, he would not have been killed. 3. A death could have been prevented if that policeman had had a taser. (IC, from 1, 2) 4. Similar preventable deaths are almost certain to occur in the future. 5. All policemen should carry tasers. (FC, from 3,4)

3.2

THE PRINCIPLE OF CHARITY LORD POLONIUS: My lord, I will use them according to their desert. HAMLET: God's bodkin, man, much better: use every man after his desert, and who should 'scape whipping? Use them after your own honour and dignity: the less they deserve, the more merit is in your bounty. (Hamlet, Act II, scene ii)

As just noted, the actual expression of an argument is sometimes not clear enough to fix a single meaning, so that what is said can be interpreted in more than one way. In many cases, the interpretation we choose will affect the strength of the argument. If our main concern is to figure out exactly what the person who presented the argument meant, we may well settle on an inter­ pretation that makes for a weak argument (lots of arguments are weak). If, on the other hand, we want to figure out what we can learn about the topic un­ der discussion, we may do well to choose an interpretation that makes for the strongest possible argument compatible with what was actually said (and al­ lowing for occasional slips of the tongue or pen). This is done not merely to be nice to the person who proposed the argument (though there is nothing wrong with being nice), but also to make the best use of the argument in our search for truth. When we interpret arguments (or what people say more generally) in this way, we are said to be using the principle o f charity. Consider, for example, statements of the form: All A are not B. This grammatical form is used to make quite different sorts of claims. A wellknown proverb tells us, for example, that All that glitters is not gold. which amounts to saying that just because something is shiny doesn't mean it is gold, or: Not everything that glitters is gold. On the other hand, if someone said: Everyone is not honest. he might mean either that no one is honest or instead that not everyone is honest. So in the context of a given argument we might well have to decide which was meant, and the principle of charity would advise us to adopt the interpretation that makes for the stronger argument. Consider now this argument: 19

Reasoning and Critical Thinking Democratic decision-making could only be trusted in all cases if everyone were honest. But everyone is not honest. So it's clear that democratic decision-making cannot always be trusted. Here, it seems most reasonable to take the statement 'Everyone is not honest' to be making the claim that not everyone is honest. For the stronger and highly implausible claim that no one is honest is not only far more sweeping and hence more difficult to justify, but also stronger than the argument requires (since the reasoning would still be valid on the weaker interpretation). As noted above, it is not always appropriate to use the principle of charity. Indeed, something like the opposite seems to be called for on occasion. If, for example, we are in an adversarial situation (for example, in a dispute over a contract), we may be wise to ask what the worst possible interpretation of what was said might be. For we might well be faced with the consequences of that interpretation at some point.

3.3

P RINCIPLES

OR FORMS OF INFERENCE

One of the most important elements of any argument is normally invisible in its written or spoken presentation, even when the argument is presented in the standard form described above. I have in mind the inference or inferences used in drawing conclusions from premises. For while we can see that a conclusion has been drawn, it is rarely obvious from the written or spoken presentation exactly how it was drawn. Usually, if not always, reasoning is guided by general principles, in such a way that the kind of reasoning used in one case can also be applied in others. Consider, for example, the following simple argument: The hamster just had babies. So it must be a female. Here, it would be reasonable to suppose that the principle used to draw the conclusion is that only females have babies, a principle that can also be applied to cats, dogs, racoons, etc. But a narrower principle might also have been applied in this case, e.g., a hamster that has babies must be female. Instead of principles, we can also speak of forms or patterns of inference. To the above principles, for example would correspond the forms: X had babies. So X is female.

The hamster X had babies. So X is female.

The important points to note here are that in this and other arguments, more than one principle or form might have been used, and that quite often nothing in the statement of the argument indicates which one it was. It can easily happen that some of the principles that might have been used in a given argument are trustworthy but others are not. Consider, for example, the following argument: 20

3. Arguments in Standard Form Flipper is a dolphin. So Flipper is a mammal. We might suppose that the reasoning followed the reliable pattern: X is a dolphin. So X is a mammal. But we might also think that the conclusion was drawn according to the unre­ liable form: X is a Y. So X is a Z. and that the argument was consequently defective. Since our decision here can affect our estimate of how good an argument is, we need to be cautious in our assumptions about which form of inference is actually used in it. Here, as above, the principle of charity should normally guide our interpretations.

3.4

U NSTATED

PREMISES AND CONCLUSIONS

In some cases, people presenting arguments leave one or more premises, or even a conclusion, unstated, expecting that those they are addressing will be able to supply what is missing on their own. Usually, this is accomplished by using a familiar pattern or form of inference which the audience will im­ mediately recognize. Knowing the pattern, they are able to add whatever the speaker leaves out. In the example we just considered, for instance, many people would say that the form of inference was actually the trustworthy: X is a Y. All Y are Z. So X is a Z. And that the arguer left out an obvious premise, in this case, that all dolphins are mammals. For another example, consider this common form of inference: If P then Q. Not Q. Therefore, not P. which we see, for example, in the following argument: If it were snowing, the streets would be white. But they aren't white. So it's not snowing. Relying on us to recognize that this pattern is being used, someone might simply say: It's not snowing. If it were snowing, the streets would be white. 21

Reasoning and Critical Thinking And expect us to supply the premise "The streets aren't white" in order to complete the argument. Since our ultimate goal in clearly displaying the content and structure of arguments is to figure out whether they are any good, it is important to point out such unstated premises and conclusions whenever we are confident that the arguer intended them. For it might well turn out that everything that is ex­ plicitly stated is perfectly acceptable, but an unstated premise that is essential to the argument is not.

3.5

D IAGRAM M ING

ARGUMENTS

It can also be a useful exercise to draw a diagram of an argument's structure. Es­ pecially in the case of complex arguments, such diagrams often give us a better understanding of the roles played by various premises, the contributions of various sub-arguments, etc., than the simple lists we produce when presenting arguments in standard form. We'll use a popular system to represent argument structure, in which the premises directly supporting a given conclusion will appear immediately above it. Final conclusions will thus always appear at the bottom of our diagrams. Arrows, finally, will be used to indicate which premises support which conclu­ sions. Let's begin with the simplest case, where there is only a single premise sup­ porting the conclusion. [Pl] Joe hasn't eaten for three days. So [C] he must be hungry. Here, we simply draw an arrow from the premise to the conclusion, like this: Pi

When two or more premises support a given conclusion, it is common to indicate in the diagram whether they do so more or less independently of one another, or only when taken together. In the former case, if one of the premises were to be rejected, the argument might still be strong; in the latter however, rejecting a premise usually results in an argument with no force of conviction whatsoever. As an example of the first sort of argument, consider the following: P l: Smith has a violent temper. P2: Smith is frequently intoxicated. C: You shouldn't hire Smith as a babysitter. 22

3. Arguments in Standard Form In this example, each premise provides support to the conclusion, regardless of whether we consider the other. Consider, by contrast, the following argument: P l: If Archie doesn't go, Betty won't go either. P2: Archie's not going. C: Betty won't go. Here, dropping either premise results in completely unconvincing arguments: P: If Archie doesn't go, Betty won't go. P: Archie's not going. C: Betty won't go. C: Betty won't go. For the first sort of argument, we'll use a separate arrow to connect each premise with the conclusion it (independently) supports, while in the second kind, we'll draw a line under the premises that work together and connect them to the conclusion with a single arrow, like so: Pl: Smith has a violent temper. P2: Smith is frequently intoxicated. C: You shouldn't hire Smith as a babysitter.

P l: If Archie doesn't go, Betty won't go either. P2: Archie's not going. C: Betty won't go. P2

The same options are available when there are three or more premises, as in the following examples: P l: This apartment is filthy. P2: It's infested with fleas. P3: The landlord is a crook. C: You shouldn't rent it.

P l: Tom, Dick or Harry stole the bread. P2: Tom didn't do it. P3: Dick didn't do it. C: Harry did it.

It is in the case of complex arguments that diagrams are the most helpful. For example: 23

Reasoning and Critical Thinking The Honda is cheap, has low mileage, and is in good repair. So you should buy it. But you can't afford two cars, so that means the Chevy is out. P i: The Honda is cheap. P2: The Honda has low mileage. P 3 : The Honda is in good repair. IC4: You should buy the Honda (from 1,2 ,3). P5: You can only afford one car. F C s: You shouldn't buy the Chevy, (from 4, 5) FCs For a particular messy final example, let's diagram the argument we looked at above (p. 17): 1. If the economy grows, employment usually increases. (Pr) 2. The economy will continue to grow. (Pr) 3. Unemployment won't increase. (IC, from 1,2) 4. EI expenditures will increase only if unemployment does. (Pr) 5. EI expenditures won't increase. (IC, from 3,4) 6. Tax revenues usually increase in a growing economy. (Pr) 7. Tax revenues will increase. (IC, from 2, 6) 8. If other expenditures are held constant, the deficit will likely decrease. (FC, from 5, 7)

Pi

P2

IC 3

P4

P2

P6

FCS One thing that becomes especially clear from looking at the above diagram is just how fragile the argument is, in particular, with respect to its dependence 24

on premise 2 (i.e., that the economy will continue to grow). If this premise turned out to be unacceptable, we would lose support for the intermediate con­ clusion IC3, as only in conjunction with P? does P i support IC3. For similar reasons, we would lose support for IC7. But if IC 3 turned out to be unsup­ ported, we would also lose support for I C 5 , and thus for the final conclusion. All of this is apparent at a glance from the diagram. By contrast, even if premise Pi in the previous example turned out to be unacceptable, it is clear from the diagram that the intermediate and final con­ clusions would still enjoy some support.

C HAPTER 4

EVALUATING A RGUM ENTS 4.1 I NTRODUCTION This chapter deals in general terms with the question: how can you know whether or not a given argument is a good one? We will discuss several differ­ ent ways to answer this question, some of which will be worked out in greater detail in the following chapters. In a simple argument, we begin with premises and draw (or infer) a con­ clusion. Figuring out whether the argument is good thus involves asking two questions: 1.

Are the premises good?

2.

Is the reasoning (the inference) good?

Though most people tend to size up arguments and evaluate them as wholes, there is much to be gained from slowing down and looking at the two ques­ tions separately. The first question is sometimes easy to answer, sometimes not so easy. We can all agree, for example, that the following would be perfectly acceptable premises: • The earth is roughly spherical in shape. • 2+2=4 • Brazil is in the southern hemisphere. • It usually snows in the winter in the Yukon Territory. • Germany and the Soviet Union fought in World War II. It is just as easy to agree that the following would not be acceptable premises: • The Prime Minister is a little green man from outer space. • After winning the Second World War, Hitler performed in many Broad­ way musicals. • Sumo wrestlers are great marathon runners. • The United States spends far more on food stamps than on the military. • The water in Lake Superior is pleasantly warm in January. Then there are the not-so-obvious cases, where we either just don't know or else would have to do some research to find out, e.g.: 26

Argument Evaluation • Within the next ten years, scientists will discover a cure for AIDS. • The most common cause of accidental death in Canada is automobile accidents. • Koodo offers the cheapest cell-phone service in Ontario. And similar remarks apply to reasoning. Sometimes, it is obviously good, sometimes obviously bad, and sometimes hard to tell, as the following exam­ ples indicate: • Joe has two dollars and Samantha has three. So between them they have five. • Joanne is a Libra. So she must have a good job. • There are more than 60 people in this room. So it is close to certain that at least two of them have the same birthday. It is also important to recognize that the two questions are independent. That is, there are arguments with: • Good premises and good reasoning • Good premises and bad reasoning • Bad premises and good reasoning • Bad premises and bad reasoning Here are examples to prove the point: • Good premises and good reasoning: Paris is in France. France is in Europe. So Paris is in Europe. • Good premises and bad reasoning: Paris is in Europe. Spain is in Europe. So Paris is in Spain. • Bad premises and good reasoning: Paris is in Spain. Spain is in Asia. So Paris is in Asia. • Bad premises and bad reasoning: Paris is in Asia. Spain is in Asia. So Paris is in Spain.

4.2

E VALUATING P REMISES : G ENERAL R EMARKS

While it is a nice thing if the premises we use in an argument are not only true but known with certainty to be true, we often have to settle for less than this. If I buy a lottery ticket, for example, I may not know for certain that I will not win millions of dollars, yet it would be quite reasonable to proceed on the assumption that I won't, and correspondingly unreasonable to count on the prospective winnings to support me when I retire. Similarly, if I want to buy some milk, it is usually quite reasonable to assume that the local grocery store will have some for sale, even though once again I do not know for certain that 27

Reasoning and Critical Thinking this is so. For reasons of this sort, when evaluating premises, we usually set our sights not on certain truth, but on rational acceptability, asking ourselves: is it reasonable to accept this claim, even if we are not entirely sure that it is true? Of course, if we ever find ourselves in a situation where we can be certain, we may gratefully accept this as a bonus. Reasonableness being a matter of degree, we can also distinguish degrees o f acceptance of premises. The English language is rich in expressions that show the extent to which we are willing to accept claims. Compare, for example, the following statements, each of which indicates a different level of commitment or non-commitment to the claim that Smith will lose the election: • There can be no doubt that Smith will lose the election. • Smith will almost certainly lose the election. • It is very likely that Smith will lose the election. • Smith will probably lose the election. • Smith may lose the election. • It is at least possible that Smith will lose the election. (We can also look upon these as different claims of steadily decreasing strength.) Now circumstances vary, and the different purposes for which premises are used can give rise to different standards of what is reasonable. When I board an airplane to fly to Toronto, for example, it can be quite reasonable for me to proceed on the assumption that the airplane will not crash. An engineer or mechanic responsible for maintaining the airplane in good condition, by contrast, would not be acting reasonably in simply assuming this. Rather, it is their job to make sure that it is so, so far as this lies within their power. How certain is certain enough? Again this is something that varies accord­ ing to circumstances. If the risk involved in making a mistake is relatively small, we can afford to be somewhat less demanding when assessing premises. Where the risk involved with error is high, by contrast, we do well to be more careful. Think, for example, of the question whether a concrete structure will leak. The level of conviction reasonably required in the cases of a swimming pool and a nuclear power plant will differ by a wide margin. In many cases, of course, we can remain neutral, neither accepting nor re­ jecting a given claim. But there are other cases where this is not possible, where we have to decide. Suppose, for instance, that through some misfortune I find myself in the middle of the highway, with vehicles whizzing past me on all sides. While it is true that I cannot be absolutely certain that I will make it to the side of the road alive if I decide to move, it is still reasonable to act on this assumption, since I am even more likely to die if I stay where I am. (The question of when and how to move is of course a separate one.) People speak in such cases of moral certainty, a degree of conviction that is sufficient to guide action. Finally, there may be cases where we are not sure whether we should accept a premise, but where we are still interested in evaluating an argument in which 28

Argument Evaluation it is used. In such cases, we may consider the premise as a hypothesis, simply in order to see what would follow from it if it were true. This is sometimes called entertaining a premise for the sake of argument. To make it clear that this is what we are doing, it is best to use the subjunctive mood, like this: Suppose for the sake of argument that this were the case. Then so and so would also be the case, etc. Scope of claims Many claims state things about one or more objects of a cer­ tain kind. Someone might say, for example, that some politicians sooner or later compromise their principles, or that many or most or even all do. Obviously, such claims differ in strength, and the earlier ones are easier to establish than the later. We will say that they differ in scope. Customarily, we distinguish particular and general claims. A particular claim is one stating that one or more objects of a certain kind have a certain property, for example:

• At least one Canadian PM was female. • Some MPs are female. • Several senators have been charged with fraud. With general claims, by contrast, a statement is made about a more substantial number of objects or individuals of a certain kind, e.g.: • Many mammals live on land. • Most of the people on Earth live in Asia. • All crows are black. General claims can be either vague or precise. We might say, for instance, that many doctors prefer brand X, or that 75% of them do. When a precise propor­ tion is stated (with or without a margin of error), we have a statistical claim. There are two especially noteworthy special cases of statistical claims, namely, when either 0% (none) or 100% (all) of the things of a given kind are claimed to have a certain property. For example: • No crows are pink. • All people die sooner or later. We'll look at these more closely in the next section. A universal generalization is a claim made about all things of a certain kind. For example:

Universal generalizations and counterexamples

• All mammals are warm-blooded. • All politicians are crooks. • Every even number is equal to the difference of two odd numbers. 29

Reasoning and Critical Thinking There are also negative universal generalizations, such as: • No bird is a mammal. • No human being is perfect. • No twentieth-century American president was female. It is easy to see why such negative claims are also called universal generaliza­ tions, since they too can be taken to say something about all things of a certain kind. To say that no birds are mammals, for instance, amounts to saying that all birds are non-mammals. Universal generalizations, even if true, are often difficult to establish. How can you know, for instance, if all crows are black? Have you seen them all? Has anybody? Because of this, some people go so far as to say: No one should ever make a universal generalization.1 But this would be going too far. Clearly, there are some cases where it is possible to establish a universal generalization. If I say, for example: All twentieth-century US presidents were male. I can verify that this is so by looking through the historical record. Similarly, I can prove that: Every even number is equal to the difference of two odd numbers. by means of a simple argument, even though there are infinitely many even numbers. In many other cases, however, this is either not practical or flat out impos­ sible. Consider the following claims, for example: • Every asteroid in our galaxy contains some iron. • No one will ever run a mile in less than three and a half minutes. • Every robin that lives at least a year eats an earthworm. On the other hand, if a universal generalization is false, it can be very easy to show this. The claim: Every twentieth-century British Prime Minister was male. for example, is easily shown to be false by pointing to Margaret Thatcher. Thatcher serves as a so-called counterexample to the universal claim. False negative universals can also be refuted by counterexample. For in­ stance: There has never been a female Canadian PM. —can be refuted by the counterexample of Kim Campbell.

1 This

is interesting advice. Why?

Argument Evaluation Particular claims, by contrast, cannot be refuted by counterexample. Citing an example of a British Prime Minister who is not female, for instance, in no way shows the particular claim that there has been a female British PM to be false. On the other hand, one or more examples do suffice to show that a particular claim is true. Knowing that Zibanejad, Karlsson, and Alfredsson are Swedish hockey players, for instance, puts me in a position to accept the claim that some (or several) hockey players are Swedish, while knowing that Alberta is landlocked justifies me in stating that at least one Canadian province is not an island. Inconsistency We say that a set of claims is inconsistent if it is impossible for all of them to be true together. Put otherwise, a set of claims is inconsistent if there is no possible set of circumstances in which they are all true. The follow­ ing claims, for example, are inconsistent, since there is no way both of them could be true: Jack is older than Jill. Jill is older than Jack. An interesting feature about inconsistency is that it can often be recognized in cases where we do not know whether the individual claims are themselves true or false. In the above example, for instance, we have no idea who Jack and Jill are, and no idea of their ages, yet we can still see quite clearly that it is impossible for both claims to be true. No matter what the facts may be, then, at least one of the two claims must be false. If we recognize that a set of premises is inconsistent, then, we can imme­ diately conclude that at least one of them must be false. We may not have reason to reject any particular one of the premises, but we can at least say in such cases that the set containing all of them is unacceptable. If the argument depends crucially upon all of them, then, it will be a weak one. When a set of claims is inconsistent, people sometimes say that the claims contradict one another. This has to be interpreted carefully. In the strict sense, we say that one claim contradicts another if the former denies the truth of the latter. For example: • A: I didn't eat the last cookie. B: You did too! In this sense, contradiction is a relation between two claims or statements. But inconsistency doesn't require the presence of two contradictory claims, as the following examples show: • Jack is older than Jill. Jill is older than Tom. Tom is older than Jack. • Smith drove his car into a telephone pole. This accident occurred in Chicago last Wednesday. Smith was not in Chicago last Wednesday. • Jones isn't happy. Smith isn't happy. Either Smith or Jones is happy. In each of these examples, any two of the three statements are consistent, but there is no way that all three could be true. So there is inconsistency, even 31

Reasoning and Critical Thinking though there is no contradiction in the strict sense of the term. In order to avoid confusion, then, it is probably best not to speak of contradiction in such cases and instead to think of inconsistency as a relation between an arbitrary number of statements, consisting in the fact that there is no possible set of circumstances in which all of them would be true.

4.3

E VALUATING

REASONING

If we are satisfied that the premises of a given argument are acceptable at least to some degree, we can move on to evaluate the reasoning employed in the argument, the one or several inferences that take us from the premises to the final conclusion. It would be nice if all the reasoning we used were of the sort which cannot possibly go wrong, which gives us an ironclad guarantee that the conclusion will be true, provided that the premises are. As with premises, however, we often have to settle for less than this, and be happy if the infer­ ences are of the sort which make it reasonable to accept the conclusion provided that the premises are true. So, too, we may ask how certain is certain enough? when evaluating inferences, and reply equally truthfully that it depends upon our purposes and circumstances, that the more that is at stake, the more careful and precise we need to be, and so on. As noted above in Chapter 2, reasoning is usually guided by general prin­ ciples. Arguments based on arithmetic provide familiar examples of this. All of the following arguments, for instance, employ the same principle in arriving at the conclusion: • Sarah has two coconuts and Jillian has three. So together they have five. • The Yankees lost two games last week and three games the week before. So they lost five games in the last two weeks. • Josephine has two rings on her left hand and three on her right. So she's wearing five rings on her hands. If we wanted to state the principle used in the reasoning, we might try: Whenever there are two things and three other things of the same kind, then there are five things of that kind. Or, more simply: 2+ 3= 5 Since this is a known truth, we can be fully confident that the reasoning used in the above arguments is correct. Now consider the following arguments: • Senators Duffy, Brazeau, Wallin, and Harb have all been caught fudging their expenses. Obviously, all senators abuse their expense accounts. • My bus was late every day this week. It's never on time. 32

• Hitler, Stalin and Pol Pot started as dictators and finished as mass mur­ derers. It's clear that dictatorial power inevitably leads to killing. In these cases, the principle used seems to be something like: If a few things of a certain kind have a given property or feature, then all do. We can easily see that this principle is not to be trusted by considering some other obviously bad arguments where it is used, e.g.: • Reagan, Bush, and Trump were Republican US presidents. So all US pres­ idents are Republican. • The Red Sox won their first three games, so they won't lose this season. In effect, these two arguments serve as counterexamples to the claim that the above principle is a generally valid one, i.e., that it can be relied upon in every case. The main point to bear in mind is that if a principle leads you astray in some cases, it shouldn't be relied upon in any. For if we wanted to have complete confidence in using a principle that recognizably steers us wrong in some cases, we would need to know that the present case was not one of them. To do this, we would have to be able to say that if such and such conditions were satisfied, the principle could be used safely, and otherwise not; and that the case at hand is one of the ones where those conditions are met. But that would mean that we are actually using a different principle. Hence if we have correctly identified a general principle underlying a given inference, we can show that the reasoning is weak by pointing to a different case where the same principle is used, but produces obviously unacceptable results. As we see from the examples given above, principles of inference are some­ times easily stated in a single sentence. In other cases, it is more convenient to represent them by displaying an associated form of inference. Consider the following argument: If Sarah just got out of the swimming pool, her hair would be wet. Her hair is wet. She just got out of the swimming pool. It seems to embody a pattern or form of reasoning that we can represent as follows: If P then W. W.________ P. —a form we can show to be untrustworthy by citing an example like this: If Napoleon had been killed at Waterloo, he'd be dead now. Napoleon is dead now. Napoleon was killed at Waterloo. —where it is obvious that the conclusion doesn't follow.

C HAPTER 5

VALIDITY 5.1

D EDUCTIVE

AND

S OUNDNESS (1)

AND NON - DEDUCTIVE REASONING

It is customary to distinguish two types of reasoning, deductive and non-deductive. In good arguments of the former kind, the truth of the premises is said to guar­ antee the truth of the conclusion. We call such arguments deductively valid, or simply valid. Here is an example: Jack is younger than Jill and Jill is younger than John. So Jack is younger than John. Note that while we may not know in this case whether the premises are true, we can nevertheless recognize that the argument is valid: we see that if the premises were true, the conclusion would have to be true as well. With the second, non-deductive, kind of argument, there is no absolute guarantee even when the reasoning is good: the conclusion might still be false even if all the premises were true. Here is an example: I have walked past the store on the comer almost every day for the past fifteen years, and every time they have had cigarettes and lottery tickets. So if I go down there today, they will have cigarettes and lottery tickets. If we were to evaluate this second argument as a piece of deductive reasoning, we would have to say that it was a bad (or invalid) one, since the conclusion might be false even if the premise were true. But an argument that is bad when considered as a piece of deductive reasoning can nevertheless be quite good, in that the truth of the premises would make it quite likely or probable that the conclusion would be true. In this case, the principle of charity would tell us to interpret the argument as a non-deductive one. In this part, we focus on evaluating the reasoning in deductive arguments. When the reasoning in such an argument is good, we will say that the argu­ ment is valid and, when it is bad, invalid.

5.2

VALIDITY

When an argument is valid, we also say that the conclusion follows from the premises, that the conclusion is a consequence o f the premises, or that the premises imply or entail the conclusion. What we mean by all these expressions is that there is certain relation between the premises and the conclusion, such that if the premises were all true, the conclusion would also have to be true. 34

Validity and Soundness (1) For example, consider the argument: Jack is 21 years old and Jill is 25. So Jack is younger than Jill. Without knowing who Jack and Jill are, or what their ages are, we can nonethe­ less see that the conclusion follows: if the premises were true, and Jack and Jill 21 and 25 years old, respectively, the conclusion would b e guaranteed to be true. On the other hand, we can see that the conclusion of the following argu­ ment does not follow from its premises: Jill is 23 years old. Jack is not 16 years old. So Jack is older than Jill. —since the premises would be true and the conclusion false, for instance, if Jill were 23 and Jack 18. Notice that, in both cases, when asking about validity, we do not merely want to know if in fact the premises and conclusion are true or false, but also whether it is possible for the premises to be true and the conclusion false. It might well be, for example, that Jack is actually 63 years old and Jill 83; the fact that they could be 23 and 18 is enough to show that there is a possible situation in which the premises are true and the conclusion false, so that the second argument is invalid. In the first argument, by contrast, we can easily see that there is no possible situation in which the premises would be true and the conclusion false. Thinking in terms of necessity and possibility is often a very good way to assess deductive reasoning. For example, let's consider four closely related arguments: I. If Joe has $10, he has more than $5. Joe has $10. Joe has more than $5.

II. If Joe has $10, he has more than $5. Joe has more than $5. Joe has $10.

III. If Joe has $10, he has more than $5. Joe doesn't have $10. Joe doesn't have more than $5.

IV. If Joe has $10, he has more than $5. Joe doesn't have more than $5. Joe doesn't have $10.

What we want to know in each case is whether it is possible for the premises to be true and the conclusion nonetheless false. To do this, we consider various possibilities, and we quickly grasp that the relevant ones concern how much money Joe has. In the first argument, the second premise will only be true if Joe happens to have $10, so this is the only case we need to consider. We notice, too, that the first premise will be true no matter how much money Joe has. Finally, in this, the only possible case where both premises are true, we also see that the conclusion is true. Now consider the second argument. Again, the first premise will be true regardless of how much money Joe has, so the second one is the key. It will 35

Reasoning and Critical Thinking be true, we recognize, if Joe has $5.01 or any larger amount of money. Un­ like the first argument, then, there are many relevant possibilities in this case. But we easily see that there are possible cases where the conclusion would be false even though both premises were true, for instance, if Joe had $7. Similar reflections will permit you to assess the reasoning in the other two arguments. Let's now consider a second set of examples: I. Harry plays the bassoon. Hermione plays the Harp. Harry and Hermione play different instruments. II. Harry plays the bassoon. Harry and Hermione play different instruments. Hermione plays the Harp. III. Hermione plays the Harp. Harry and Hermione play different instruments. Harry plays the bassoon. IV. Harry plays the bassoon. Hermione plays the bassoon. Harry and Hermione play the same instrument. V. Harry plays the bassoon. Harry and Hermione play the same instrument. Hermione plays the bassoon.

Here, by reflecting on what might be the case, you should be able to see that arguments II and III are invalid. For II, consider a situation where Harry plays the bassoon and Hermione plays the flute. For III, suppose Hermione plays the harp and Harry plays the piccolo. These are clearly possible situations in which the premises would be true and the conclusion false Similarly, you should be able to see that in the case of arguments I, IV, and V, it is impossible for the conclusion to be true if the premises are both true.

5.3

D EVELOPING

A SENSE OF POSSIBILITY

Things are the way they are, but we also recognize that they could be other­ wise. As I write this, for example, I am wearing a blue shirt, but I could easily 36

Validity and Soundness (1) have been wearing a white or a red one. When the evening rolls around, I ei­ ther will or I won't go for a walk, but both possibilities remain open. Yesterday, similarly, I did go for a walk, but I recognize that I might not have. The Liberals won a minority in the 2019 federal election, but they might have lost, or won a majority. And so on. On the other hand, some things are just not possible: I can't be taller or older than myself, for example, nor can I be my own father. Thinking about validity in terms of possibility and necessity is a useful tech­ nique, but it requires you to develop a good sense of possibility. In particular, we need to develop the ability to recognize which factors are relevant for as­ sessing the argument at hand. In the first set of examples above, we can see that the relevant factor is how much money Joe has. His age, height, weight, the state of the weather, etc., are not relevant. In the second set, what matters is what instruments, if any, Harry and Hermione play. Now while it is true that, in addition to Joe possibly having different amounts of money, he might also have been older or younger, lighter or heavier, taller or shorter, living here or there, etc., we don't need to consider any of these fea­ tures in assessing the above arguments. And similarly with the five arguments about musical instruments. In effect, we are not considering fully articulated possible situations, but rather entire kinds of possible situations, grouping them according to the features considered to be relevant to the validity of the argu­ ment. So, for example, as long as Hermione plays the harp and Harry the trumpet, we will have a possible situation in which the premises of argument III are true and its conclusion false, regardless of how the other details go.

5.4

I MPLICATION

AND EQUIVALENCE

When an argument is valid, we also say that its premises imply (or entail) its conclusion. Thus the following statements amount to the same thing: • The argument A, B, C

Z is valid.

• A. B.C imply Z If A, B, C imply Z , we also say that Z follows from A, B, C. In the special case where A and B imply each other, we say that A and B are equivalent. Equivalent claims have the same truth-value in every possible situation and in this sense convey the same information. Here are a couple of examples: A: Smith is taller than Jones.

A: Some apples are not red.

B: Jones is shorter than Smith.

B: Not all apples are red.

When we have two propositions, A and B, there are four possibilities with respect to implication: (1) They imply each other, (2) the first implies the second but not vice versa, (3) The second implies the first, but not vice vera, or (4) 37

neither implies the other. We've already called the first case equivalence. In the second and third cases, where one of the pair implies the other but not vice versa, we say that a one-sided, or unilateral, relation of implication holds between them. Finally, when neither implies the other, we sometimes say that they are (logically) independent. A implies B and B implies A A implies B but B doesn't imply A B implies A but A doesn't imply B Neither implies the other

Equivalence Unilateral implication Unilateral implication Independence

In a case of unilateral implication, e.g., if A implies B but not vice versa, we sometimes say that A makes a stronger claim than B , or that A claims more than B.

5.5

VALIDITY,

TRUTH , AND SOUNDNESS

As we have seen, the questions of whether an argument is valid and whether its premises and conclusion are true are independent. One's reasoning can be perfectly good, and yet one's premises and conclusions false—for instance, if in filling in our tax return we perform correct calculations but use the wrong numbers. Similarly, one's premises may be true, and the conclusion true as well, even though our reasoning is invalid, as in the following example: Some people like hockey. Some people like beer. So some people like both hockey and beer. The only condition that validity places on the truth of the premises and con­ clusion is that if (always a big if) all the premises happened to be true, the conclusion could not possibly be false. Thus there are no valid arguments with all true premises and a false conclusion (because whatever is the case is also possible). All the other combinations, however, can occur. That is, we may have a valid argument with: • All true premises, and a true conclusion. • Not all premises true, and the conclusion true. • Not all premises true, and the conclusion false. While invalid arguments may have any combination of truth values among their premises and conclusions. The best arguments, of course, have good reasoning and true premises. In the case of deductive arguments, we introduce a special term for this: Definition: An argument is called sound if and only if it is valid and all its premises are true.

C HAPTER 6

VALIDITY 6.1

TRUTH

AND

S OUNDNESS (2)

PRESERVATION AND FORMAL VALIDITY

As mentioned previously, the reasoning in individual arguments usually em­ bodies patterns or forms that can also be found in other arguments. Another common approach to assessing deductive reasoning accordingly focusses on the forms of arguments or inferences. Using this approach, we first identify the form or pattern of reasoning we believe to have been used in a given ar­ gument, and then determine whether this pattern is reliable. We do this by considering the original argument along with various others that embody the pattern, asking whether any of them leads from truth to falsity. Let us take the following argument as an example: Some swimmers do not skate. So some skaters do not swim. Noting that certain words are repeated, we can discern the following pattern: Some A are not B. So some B are not A. On the reasonable assumption that this is the form of inference used in the ar­ gument, we can then ask ourselves whether it is a trustworthy one, by consid­ ering various other arguments that result when different terms are substituted for 'A' and ‘B ’. For instance:

(T) Some lawyers don't golf. (T) Some golfers are not lawyers.

(F) Some who are rich are not wealthy. (F) Some who are wealthy are not rich.

(T) Some mammals are not dogs. (F) Some dogs are not mammals.

(F) Some rabbits are not mammals. (T) Some mammals are not rabbits.

For our purposes, the argument on the bottom left is the key: it shows us that there are arguments of the form we are interested in that take us from true premises to a false conclusion, and that accordingly the pattern or form of inference is not trustworthy. This argument serves as a counterexample to the claim that the above form of inference is truth-preserving, never leading from true premises to a false conclusion. 39

Reasoning and Critical Thinking This approach is quite different from the one we discussed in the previ­ ous section. Instead of thinking about what our premises say, and considering whether those very claims would be true or false in various possible circum­ stances, we instead look at arguments of the same form on all sorts of subjects, asking whether any of them has true premises and a false conclusion. Put otherwise, we consider what happens when we vary components rather than circumstances. Looking at reasoning in this way helps us to separate what we earlier recog­ nized as independent questions, namely: (1) are the premises and conclusion true or false? and (2) is the reasoning good? Consider, for example, the follow­ ing argument with the associated argument form: All lawyers are educated. Some rich people are educated. So some lawyers are rich.

All A are B. Some C are B . Some A are C.

When asked to evaluate this argument, it is natural to think about lawyers, education, and wealth, and since every one of the statements is true, we can easily be led by consideration of the subject-matter to deem the reasoning valid. But if, as is usually the case, the reasoning is formal, it is the pattern that matters, not the subject-matter that fills it in a particular instance. Removing the words 'lawyer', 'educated' and 'rich', and replacing them by letters like 'A', 'B', 'C", helps us to focus on the pattern, and thus to fight against the distraction created by the subject-matter. This technique is all the more important in that all of us have a natural tendency to think that any argument in support of a conclusion we think is true must be a good one. Thinking in terms of patterns or forms of reasoning, however, we can see that the reasoning used in above argument is the same as in this obviously bad one, and hence invalid: All A are B. Some C are B. Some A are C.

All whales live in the ocean. Some fish live in the ocean. Some whales are fish.

Let us now introduce some terminology. Beginning with a particular argu­ ment, for example: All mice are rodents. Not all mammals are rodents. So some mam­ mals are not mice. we can consider some of its parts variable, i.e., subject to replacement by other parts of the same kinds. In the above argument, for instance, we might consider 'mouse', 'rodent', and 'mammal' to be variable. Replacing these with the letters P , Q, and B , we obtain: All P are Q. Not all B are Q. So some B are not P. 40

Validity and Soundness (2) If we now stipulate that P , Q and B can be replaced by any general terms (common nouns) you like, we have an argument form. An instance of this form is the result of substituting general terms for P , Q, and B. Here are a few examples: • All lawyers are human. Not all animals are human. So some animals are not lawyers. • All cars are Volkswagens. Not all blenders are Volkswagens. So some blenders are not cars. • All parties are events. Not all meetings are events. So some meetings are not parties. • All violinists are musicians. Not all artists are musicians. So some artists are not violinists. Argument forms that never take us from true premises to a false conclusion are obviously important. We will call them truth-preserving: Definition: An argument form is said to be truth-preserving if and only if no instance of the form has true premises and a false conclusion. Among truth-preserving argument forms, we can make a further distinc­ tion, depending on whether the basis of the truth-preservation is or is not nec­ essary. By way of example, consider the following argument forms: Some P is Q. Some Q is P .

X was a twentieth-century US President. X was male.

While both forms are truth-preserving, we see that the second has this feature contingently—that is, if history had gone differently and a female US President had been elected in the twentieth century (as appears to be entirely possible), the second form would not be truth-preserving. With the first form, by con­ trast, we see that the connection between premises and conclusion is more ro­ bust, and would remain intact regardless of how the world had turned out. This distinction is the basis for two further definitions: Definition: An argument form is said to be valid if and only if is necessarily truth-preserving. Definition: An argument is said to be valid if and only if its form is valid. Finally, among valid forms, we can make a further distinction, based on whether this feature depends only on logic. If we accept, for instance, that an animal must be a mammal in order to be a cat, then the form: 41

Reasoning and Critical Thinking X is a cat. X is a mammal. will qualify as formally valid. To recognize this, however, we need to draw on biology—logic alone won't tell us whether an argument of this form is formally valid. By contrast, the form considered above: Some P is Q. Some Q is P. is formally valid for purely logical reasons. Understandably, logicians tend to focus on the second sort of forms, which we might call logically valid.

6.2

P ROVING VALIDITY : A

PAIR OF EXAMPLES

As we saw, a single counterexample—an argument of a given form that has all true premises and a false conclusion—is enough to show that that argument form is not truth-preserving and hence invalid. A single example, or even a thousand examples, however, aren't enough to show that an argument form is truth-preserving. For we need to show that there is no instance with true premises and a false conclusion—citing ten, even a hundred instances where this doesn't happen won't get us any closer to that goal. Instead, we have to focus on the general features of the form, and construct an argument that shows us that no instance of that form has true premises and a false conclusion. For example, consider the form of a refutation by counterexample: a is P but not Q. So not all P are Q. Why does this form preserve truth? Well, to make the conclusion false it would have to be true that all P are Q. But then, if the premise were true, a would have to be P , and hence Q (because all P are supposed to be Q). Yet the premise also states that a is not Q. But it isn't possible for a to be both Q and not Q. Hence it's impossible for the premise to be true and the conclusion false. For a second example, consider the argument form we have called modus tollens: If P then Q. Not Q. Not P. To see why this form is truth-preserving, notice that a conditional statement, that is, a statement of the form 'If P then Q', is false if P is true and Q is false. If I say, for example, that if I buy a ticket, I will win the lottery, my claim is revealed to be false when I buy a ticket and do not win. But if this is so, then, whenever 'If P then Q' is true and Q is false, P cannot be true. Hence, under these conditions, 'Not P' must be true. 42

Validity and Soundness (2) In the next couple of lectures, we will present a general method that can be used to decide in quite a few cases whether a given argument form preserves truth. More powerful methods are developed in courses on formal logic.

6.3

ON P ROOFS

AND

R EFUTATIONS

The problems at the end of this section ask you to determine whether certain statements are true or false, and to prove that your answer is correct. By a proof, I mean: construct a sound argument with the given proposition as conclusion. Before answering these questions, the following two points should be noted about the way logicians use the word 'some': 1. When we say 'Some A are B ', this should not be taken to imply that some A are not B . Thus 'Some A are B' still counts as true when all A are B. 2. We say that 'Some A are B' is true even in cases where there is only one such A. Thus, in order to avoid any possible misunderstanding, you could substitute 'At least one A is B' for the expression 'Some A are B'. Claims of the forms 'Some A are B' and 'Some A are not B' are called par­ ticular, in contrast to claims of the forms 'All A are B' and 'No A are B ', which, as we saw, are called universal. Note that in order to show that a statement of the form 'Some A are B ' is true, it suffices to give a single example of an A which is also B . This kind of argument is sometimes called proof by example. In order to show that such a statement is false, however, single examples are powerless: we have to appeal to the definitions of the terms involved, and draw general conclusions from them. That is, we must construct a general argu­ ment that covers all the relevant cases, showing why there can be no example. Similarly, we recall that a single counterexample is enough to prove that a universal claim is false. As we saw, this method of disproof is called refutation by counterexample. To prove a universal claim true, however, requires a general argument that covers all the cases. The following table sums up the kinds of arguments required in each case:

Kind of claim Universal Universal Particular Particular

To prove True False True False

43

Method General Argument Counterexample Example General Argument

Here are a couple of sample questions and answers to help get you started: (a) True or false? Some valid arguments have false premises. Answer: True. Proof by example: The following argument is valid and has a false premise: Ottawa is in Alberta. Alberta is in Canada. Therefore Ottawa is in Canada. (b) Some sound arguments are invalid. Answer: False. Proof by general argument: By definition, a sound argument is a valid argument with true premises. It follows that every sound argument is valid, and hence that no such argument is invalid.

C HAPTER

BASIC S YLLOGISTIC 7.1

AND

7

V ENN D IAGRAM S

I NTRODUCTION

In this and the next part, we will deal with the class of deductive argument forms called categorical syllogisms. We will see how to represent the premises and conclusions of such arguments pictorially, using so-called Venn diagrams, and how to use these diagrams to determine whether or not a given syllogistic form is valid. Categorical syllogisms and Venn diagrams are a standard topic in critical thinking textbooks, and there are some good reasons for this. To begin with, categorical syllogisms are fairly common argument forms, encountered in a va­ riety of different situations. Second, the method of Venn diagrams is relatively simple and self-contained, so that it can be presented fairly briefly. Finally, even though this method deals only with a limited class of argument forms, it can be used to illustrate a number of more general points about arguments and argument forms. The simplicity of the system we will study comes at a cost, however, since the validity of a great many important argument forms cannot be judged with the aid of Venn diagrams. More powerful methods are needed to deal with other cases; these are studied in courses devoted to formal or symbolic logic. 7.2

S YLLOGISMS

The study of syllogisms began a long time ago. Aristotle (384-322 BC), who wrote the earliest works on logic that survive, presented a remarkably good account of syllogisms in his book Prior Analytics. Syllogistic is thus one of the oldest parts of one of the oldest sciences (logic). Because of his influence, syl­ logisms were the primary topic of most logicians through the middle ages and even well into the modem period. What is a syllogism? Aristotle said this: A syllogism is a discourse in which, certain things being said, other things follow of necessity from their being so.1 As it stands, this sounds like a definition of 'valid argument'; in any case, it is both too broad and too narrow with respect to modem usage. It is too narrow because it is restricted to valid arguments, while modern usage recognizes both 1 Prior

analytics 1 ,1 (24 i ’ 18f).

45

Reasoning and Critical Thinking valid and invalid syllogisms. It is also too broad, because not all valid argu­ ments are of the form now called syllogistic. The form called modus tollens, for example: If P , then Q. Not Q. Therefore, not P. is not considered a syllogism, though it is a valid form, and seems to fit Aristo­ tle's definition. In practice, Aristotle only considers a limited class of argument forms, the so-called categorical syllogisms along with modal syllogisms. We will limit our­ selves even further, by ignoring the modal syllogisms and restricting our at­ tention to categorical syllogisms alone. These are argument forms with two premises and a conclusion, where premises and conclusion all belong to one of the four categorical propositional forms called A, E, I, and O: • A: All P are Q

• E: No P is Q

• I: Some P is Q

• O: Some P is not Q.

The A and E forms are called universal propositions, statements, or claims. This is because they speak about everything of a certain kind, saying either that all P are Q or (in the case of E-propositions) that all P are non-Q. I- and O-propositions, by contrast, are called particular, as they speak not of all things of a certain kind, but only of some of them. A- and I-propositions are called affirmative, while E- and O-propositions are called negative. Thus we have: • A: universal affirmative

• E: universal negative

• I: particular affirmative

• O: particular negative

This seems a good place to recall a peculiarity of the way modern logicians understand the words 'all' and 'some'. For us, the word 'some' means nothing more and nothing less than 'at least one'. Thus we count a statement such as: Some 20th-century Canadian Prime Ministers were women. as true, even though we only had one female PM in the last century. We also deem a statement such as: Some 20th-century US Presidents were men. to be true, even though all were. The way modem logicians understand claims of the form 'All P are Q' also takes some getting used to. The odd case is when there are no Ps. Strange as it may sound at first, we consider the claim 'All P are Q' to be true in this case. Perhaps the easiest way to get used to this is to think of this claim as meaning that there aren't any P s that aren't Q (i.e., no counterexamples), as the latter is clearly true when there are no Ps. So much for the basic propositional building-blocks of syllogisms. Here now is an example of a syllogistic form (an invalid one, as it turns out): 46

Venn Diagrams (1) All P are Q. Some Q is not R. •_Some P is not R. And here is an instance of the above form: All dogs are mammals. Some mammal is not brown. So some dog is not brown. Usually, only syllogisms involving three terms (P, Q, R) are considered, with one of the three occurring in both premises, but not in the conclusion. The term that occurs twice in the premises is called the middle term ('Q', 'mammal', in the above example).

7.3

V ENN

DIAGRAMS : THE BASICS

We now turn to the graphical representation of various propositional forms. To begin with, we will always assume that there is a so-called universe o f dis­ course in the background. This will be the set of all objects we are or might be talking about in a given situation. English, like other languages, has differ­ ent words to mark differences in the universe of discourse. For example, if we are only speaking about people, we use the words 'everyone', 'someone', 'no one', or 'everybody', 'somebody', 'nobody', while if we are speaking of things in general, we tend to use the words 'everything', 'something', etc. We also have words such as 'sometime', 'anytime', 'anywhere' and 'somewhere' that are used when the universe of discourse contains only times or places, and others besides. In the system of Venn diagrams, the universe of discourse (U) is depicted as a rectangle, like this:

The extensions of concepts or terms are then depicted as circles (or parts of circles) lying within the universe of discourse. The extension of a term A, for example, might be depicted like this: 47

Reasoning and Critical Thinking

The idea here is that all the objects to which the term 'A' applies are inside the circle. Thus the A-circle divides the universe into two parts: inside, where all the As are found, and outside, where the non-As are found. We use an 'x to indicate that an object occupies a given region. An ‘x inside the circle, for example, indicates that there is at least one A, or that something is A.

While an 'x outside the circle indicates that at least one thing is not A: Something is not A.

xs both inside and outside will then indicate that something is, and something else is not, A: Something is A and something is not A.

48

Venn Diagrams (1) Finally, if we simply wish to indicate that there is something, without spec­ ifying whether or not it is an A, we may put an 'x on the circumference of the /I-circle, like this: There is something.

Shading is used to indicate that a region is empty. If, for example, we shade in the entire interior of the .1-circle, this represents a situation where nothing whatsoever is an A:

This would be the case, for instance, if the universe of discourse contained only people and the term A was 'female 20th-century president of the USA'. If, by contrast, we shade in the entire region outside the circle, this indicates that everything in the universe of discourse is an A:

We would encounter this situation, for example, if the universe of discourse were again restricted to people, and the term A was 'mortal'.

49

Reasoning and Critical Thinking It is also entirely possible that the entire universe of discourse is empty:

There is nothing

This would be the case, for example, if the universe of discourse were re­ stricted to female twentieth-century US presidents or to free lunches. Aristotle, and many logicians after him, did not consider either empty terms or empty universes when formulating their theories. Because of this, modern logicians sometimes disagree with the ancients on the validity of certain infer­ ences. 7 .4

TW O TERMS; CATEGORICAL PROPOSITIONAL FORMS

When two terms are involved, we draw two overlapping circles, like this:

Two terms; the basic set-up

These circles divide the universe of discourse into four separate regions:

Regions

Venn Diagrams (1) Region 1 contains the things that are neither A nor B , region 2 the things that are A but not B , region 3 the things that are both A and B , and region 4 the things that are B but not A. There are a variety of possibilities for placing 'x's and shading; some of these represent categorical propositional forms. Suppose, for example, we place an ‘x in region 2, i.e., inside the ,1-circle but outside the B-circle. This indicates that there is an object that is A but not B , or, Some A is not B.

Similarly,

An x in the area common to the A- and B - circles (region 3) indicates that something is both A and B , or Some A is B .

Some A is B

An x outside of both circles indicates that there is something that is neither A nor B: 51

Reasoning and Critical Thinking Something is neither A nor B

Finally, multiple xs will express combinations of such propositions, e.g., Something is both A and B and something is neither A nor B

As was the case before, we also have the option of placing an ‘x on the circumference of a circle. If, for example, we wanted to indicate that something was A without specifying whether or not it was also B , we could place it within the /l-circle, but on the circumference of the B-circle, like this: Something is A

Similar possibilities exist for shading. If, to begin with, we were to shade in the part of the /l-circle that lies outside the B-circle (region 2), this would indicate that the only room for As is within the B-circle, or that there aren't any As that aren't also Bs. More familiarly: All A are B. All A are B

52

Venn Diagrams (1) Similarly:

Now if we shade in the area common to the A- and B-circles, this indicates that nothing is both A and B , or No A is B :

Finally, we have the option of shading in everything outside the two circles (i.e., region 1). This indicates that everything within the universe of discourse lies within the A-circle, the h-circle, or both. More familiarly: Everything is either A or B.

Here are a few other possibilities, including combinations of shading and xs:

All and only A are B.

Nothing is either A or B.

Nothing is A but something is B

Nothing is B and everything is A

C HAPTER 8

V ENN D IAGRAM S (2): SYLLOGISMS 8.1

R EMINDER

In the last part, we introduced the system of Venn diagrams, and showed how these diagrams could be used to represent commonly occurring kinds of propositions, among them the building blocks of categorical syllogisms (tradi­ tionally called A, E, I, and O propositions): A: All A are B

O: Some A is not B

We also noted that Venn diagrams could be used to represent various other propositional forms, for instance: Nothing is either A or B.

Nothing is A but something is B

Reasoning and Critical Thinking In this part, we will show how Venn diagrams can be constructed for cate­ gorical syllogisms, and used to determine whether the forms of these are argu­ ments are valid. Since the categorical syllogisms we are interested in involve three terms, we will use three overlapping circles to depict them:

These circles divide the universe of discourse into 8 distinct regions: Three circles: regions

The objects contained within each of these regions may be characterized as follows: 1. Neither A nor B nor C

5. A and B, but not C

2. A but neither B nor C

6. A and B and C

3. A and C, but not B

7. B and C but not A

4. C, but neither A nor B

8. B, but neither A nor C

56

Venn Diagrams (2)

In this case, there are hundreds of ways to shade regions and place xs. In­ stead of taking the time to consider some of these these separately, we will proceed directly to diagramming syllogisms. In brief, the procedure is as fol­ lows: we represent the premises, then inspect the diagram to see whether the conclusion is also represented. Consider, to begin with, the syllogism called AAA, or Barbara:

All A are B All B are C So All A are C. We begin by representing the first premise, All A are B, by shading in the part of the A circle that lies outside the / J -circle (regions 2 and 3): Barbara: first premise

Next, we represent the second premise, All B are C, by shading in the part of the B-circle that lies outside the C-circle (regions 5 and 8): Barbara: second premise

When we superimpose these two diagrams, we obtain this one: 57

Reasoning and Critical Thinking Barbara: both premises

We now have a faithful depiction of the information contained in the premises and look to see whether the conclusion, All A are C, is represented in the dia­ gram. That conclusion, recall, looks like this: Barbara: conclusion

In terms of the numbering given above, regions 2 and 5 are shaded in. And we find that this is indeed the case once we have represented the premises—there is no place left for any As except within the C-circle. The truth of the premises thus forces the truth of the conclusion; in other words, the argument is valid. The following pair of diagrams allows you to make a direct comparison: Barbara: conclusion

Barbara: both premises

By way of contrast, consider the related argument form EEE: 58

Venn Diagrams (2) No A is B No B is C So No A is C. As above, we begin by representing the first premise, No A is B, by shading in the area common to the A and B circles (regions 5 and 6): EEE: first premise

Next, we represent the second premise, No B is C, by shading in the area common to the B and C'-circles (regions 6 and 7): EEE: second premise

When these two are superimposed, we obtain this one: EEE: both premises

59

Reasoning and Critical Thinking We now look to see whether the conclusion, No A is C , is represented in the diagram. That conclusion looks like this, with regions 3 and 6 shaded in: EEE: conclusion

Clearly, this conclusion is not represented in the diagram depicting the two premises, since region 3 remains open. There is still room for As that are also Cs. The truth of the premises does not force the truth of the conclusion. In brief, the form is invalid. Again, a side-by side diagram makes this perfectly clear: EEE: conclusion

The next form we shall consider is traditionally called Baroco: All R are Q Some P is not Q So Some P is not R. Extra care is required in this case, since we have both a universal premise (which is depicted with shading) and a particular one (depicted with an x). We adopt the following rule for such cases: always diagram the universal premise first. The reason for this is that the shading required to depict the universal premise may restrict our options for placing an x. We shall see that this is the case with Baroco.

60

Venn Diagrams (2)

We begin by representing the universal premise 'All R are Q':

Baroco: universal premise

Next, we represent the particular premise 'Some P is not Q'. The region for things that are P but not Q is composed of two parts, those numbered 2 and 3 in our diagram. We see, however, that according to the first, universal premise, region 3 is empty. We are thus forced to place our x is region 2, like so:

Baroco: both premises

But now the conclusion, 'Some P is not R' is clearly represented in our dia­ gram: again, the truth of the premises forces the truth of the conclusion, and the form is valid. For a final example, consider the following form: All P are Q Some Q is R So Some P is R. As in the previous case, we begin by representing the universal premise 'All P are Q.': 61

Reasoning and Critical Thinking First premise: All P are Q

We now have to depict the second premise, 'Some Q is R'. In order to do this, recall, we need to place an x in the area common to the Q and R circles. Now this area is composed of two regions, numbered 6 and 7 in our diagram. Where should we put the x l The rule to be observed here is: never introduce any additional assumptions when constructing diagrams. Placing the ‘x in region 6, for example, would go beyond the information given in our premises, by assuming that there is something that is Q and R and also P ; on the other hand, placing it in region 7 would amount to assuming that there is something that is Q and R but not P . Now this additional information is in no way contained in the premise 'Some Q is R', which makes no mention of P. What should we do? Recall that we can remain noncommittal, making no assumption about whether or not an object which is both Q and R is P as well by placing an x on the part of the circumference of the P circle that separates regions 6 and 7, like this: Both premises

We now have a faithful depiction of the information contained in the premises. And we can see that they do not force the conclusion, 'Some P is R' to be true, because the only individual we know to exist based on the premises (our x) sits on the circumference of the P circle, x might be P , but equally well it might not. Nothing in the premises forces it to lie within the P circle, as is required to depict the conclusion 'Some P is R'. Thus the truth of the premises does not force the truth of the conclusion and this form is invalid. 62

Venn Diagrams (2)

Summing up, we can give the following instructions for constructing Venn diagrams to test argument forms for validity: 1. First, depict the premises. (a) Universal premises must be depicted before particular premises. (b) When placing 'x's, do not introduce any assumptions: if two sub­ regions are open within the required region, place the x on the line separating them. 2. Finally, inspect the diagram to see whether the conclusion is also de­ picted. (a) If so, the form is valid. (b) Otherwise, invalid.

8.2

E VALUATING R EASONING , O NCE A GAIN

Venn diagrams only work for a limited class of argument forms, but they do provide a striking illustration of some important points concerning the evalua­ tion of arguments, especially deductive ones. Try for a moment to forget what you've learned in this and previous chapters, and quickly read the following arguments, relying on your first impression to judge which of them seem valid, and which invalid. 1. All lawyers are educated. Some rich people are educated. So some lawyers are rich. 2. All mammals are warm-blooded. Some dolphins are warm-blooded. So some mammals are dolphins. 3. All mammals are warm-blooded. Some animals that live in the ocean are warm blooded. So some mammals live in the ocean. 4. All snakes are cold-blooded. Some reptiles are cold-blooded. So some snakes are reptiles. 5. All rabbits are cold-blooded. Some reptiles are cold-blooded. Some some rabbits are reptiles. 6. All Texans are North Americans. Some US residents are North Ameri­ cans. So some Texans are US residents. 7. All nudibranchs are coelacanths. Some chordates are coelacanths. So some nudibranchs are chordates. 8. All chromodorids are nudibranchs. Some aeolids are nudibranchs. So some chromodorids are aeolids. 9. All professional hockey players can skate. Some three year olds can skate. So some three year olds are professional hockey players. 63

Reasoning and Critical Thinking 10. All lawyers are over eighteen months years old. Some small children are over eighteen months old. So some lawyers are small children. If you are like most people (and most people are), you will have reacted differ­ ently to these arguments, perhaps thinking that some are valid, finding some of the others obviously invalid, and not being immediately sure about some of them. The subjects discussed in the various arguments will likely have in­ fluenced your judgments, as well as affecting the speed with which they were made. However, we are now in a position to see that the reasoning is exactly the same in every one of these arguments, since the Venn diagram for each of them is identical, apart from the labels attached to the circles. The moral of the story that, in order to evaluate reasoning, we need to focus on the form of inference, not the subject matter. But this is very hard to do for most people, and a lot of careful, methodical practice is usually needed to get good at it.

64

C HAPTER 9

O VERVIEW

OF

U NITS 1 -8

U N IT 1: I N T R O D U C T IO N -

Truth

-

Belief Knowledge Reasonable Belief

U N IT 2: A R G U M E N T S , PART 1 -

The nature or arguments Arguments and non-arguments Indicator words Explanations The elements of arguments: * * *

-

Premises Conclusions Inferences

Simple and Complex arguments; intermediate and final conclusions

U N IT 3: A R G U M E N T S , PART 2 -

Arguments in standard form The principle of charity Principles or forms of inference Unstated premises and conclusions Diagramming arguments

U N IT 4: E VALUATING -

I N T R O D U C T IO N

Two questions: * *

-

A R G U M E N T S , AN

Are the premises good (true, reasonable)? Is the reasoning (the inferences) good?

Strength of claims Scope of claims Universal generalizations and counterexamples Inconsistency Evaluating reasoning: focussing on principles or forms 65

Reasoning and Critical Thinking U N IT 5: V A L ID IT Y -

S O U N D N E SS 1

Deductive and non-deductive arguments Validity: a necessary connection between premises and conclusion The sense of possibility Implication and equivalence Validity and soundness

U N IT 6: V A L ID IT Y -

AND

AND

S O U N D N E SS 2

Truth preservation and formal validity: definitions Showing invalidity with parallel instances Proving validity using general arguments Proofs and refutations: *

General arguments: for proving universal claims and disprov­ ing particular claims * Examples/counterexamples: For proving particular claims or refuting universal ones U N IT 7: V ENN D IA G R A M S 1 -

Categorical syllogisms: what are they? A, E, I, O propositions: the propositions occurring in categorical syl­ logisms: * * * *

A : All P are Q (universal affirmative) E: No P is Q ((universal negative) I: Some P is Q (particular affirmative) O: Some P is not Q (particular negative)

- Venn diagrams: basics * * * -

Universe of discourse Extensions of concepts xs and shading

Representing A, E, I, O and similar propositions with Venn dia­ grams.

U N IT 8: V ENN D IA G R A M S 2 -

Constructing Verm diagrams for syllogisms The basic set-up: 3 overlapping circles Universal premises should be represented before particular ones Looking to see if the conclusion is represented in the premise dia­ gram - Morals for the evaluation of reasoning

66

C HAPTER 10

EXERCISES

FOR

R EVIEW

Solutions to these exercises may be found at the end o f these notes. I. Do the following passages contain arguments? If so, identify the premises, final conclusions, and any intermediate conclusions, along with any indicator words. If not, briefly explain why not.

1. Over the past five centuries, the number of independent political entities in Europe has decreased steadily. During this period, wars became less frequent, but the wars that did occur were more intense and caused far more damage. 2. If Joe had gone shopping, there would be some food in the fridge, but there isn't. So he must not have. 3. Enrolments are up in all Ontario Universities this year, but enrolments have not increased as much in some university departments as they have in others. The number of students living in residence is also up compared to last year. 4. One of those three guys must have cleaned up the mess. But it wasn't Harry, because he never helps out. And it wasn't Dick, because he was busy with other things. So it must have been Tom. 5. Current events prove that there is a real need for international law. De­ spite the fact that the United States has the most powerful armed forces in the world today, they got themselves into a big mess in Iraq. Given this, it's obvious that military power alone can't solve all the world's prob­ lems. 6. When we look at the devastation caused by AIDS, malaria, and other diseases in third-world countries, and consider how little money Cana­ dians contribute to help improve the situation, we can easily come to the conclusion that Canadians only care about themselves. But this would be a mistake, for more than half the households in Canada contributed money to help the victims of the Tsunami that hit Indonesia and Thai­ land. If Canadians were completely selfish, they wouldn't have donated all that money. 7. Whether or not the United States was right not to invade Syria, the fact remains that there is now a big mess in that country, and without the sup­ port of the United Nations, things will never get better. Canada should 67

Reasoning and Critical Thinking therefore agree to become involved in a UN mission, under whatever terms are offered. 8. He was unable to continue his studies because he was injured in a car accident, and had to be admitted to the hospital. 9. The colours we see may well exist only in our minds, not in the real world. To see that this is so, consider the following: What we see seems to depend on our eyes and our brain. The only connection between our eyes and our brain is a system of nerves. But nerves only transmit electronic and chemical signals. Colours don't seem to be either of these things. 10. The European Union recently became much larger when it expanded to include a number of former east-bloc countries. It is now one of the biggest political and economic entities in the world, rivalling the size and power of the United States and surpassing Russia. Due to rapid expansion in its industrial sector, China is also becoming an economic powerhouse. Yet many countries remain extremely poor. 11. Tamara couldn't have been in Montreal on Tuesday, because she spent the whole day in the library here in Ottawa. We know that because two independent witnesses said they saw her there. 12. Despite the fact that he received fewer votes than Al Gore in the 2000 election, George W. Bush became President of the US. The Republicans also gained control of both houses of Congress. II. Supply unstated premises and/or conclusions as appropriate.

1. Fred must have made this mess. I know George didn't. 2. If we were meant to fly, we'd be bom with wings. 3. I told you if you stayed out too late you'd get in trouble. 4. The new operating system from Microsoft is sure to be full of bugs. All complex programs are. 5. Of course Sam deserved to fail the course. Everybody knows that's the penalty for plagiarism. III. Rewrite the following arguments in standard form, supplying unstated premises and/or conclusions as necessary, clearly identifying intermediate and final conclusions and showing which premises support them.

1. Gladys and Fred's marriage is probably not going to last much longer. Both of them have had affairs, they are always fighting over how to raise the kids, and money is a real problem for them too. 2. I'm sure Joe came on the bus. Look, either he took the bus or he took a cab. But he couldn't have taken a taxi, since he didn't have enough money to pay for one. 68

Venn Diagrams (2)

3. Featherstone is clearly the best candidate for the job. She works well under pressure; her handling of the crisis in Myanmar is proof of that. She is decisive, as shown by the way she dealt with the Jones case. And in any case, there is no other candidate available. 4. Brand X is clearly the best one to buy. It's not too expensive. And it's by far the most reliable—the tests by the consumers' association show that. 5. Sam is a great baseball player, so he can play in the major leagues. But major league players make lots of money, so it's pretty clear that Sam can be rich if he wants to. 6. Dogs inevitably get fleas, and when they get fleas, they bring them into the house. So if you have a dog, you're sure to have fleas in the house. But the fleas that bite dogs will also bite you given the chance. Thus if you're not willing to put up with a few flea bites, you shouldn't get a dog. 7. Once you have killed someone, there's no way to bring him back to life. So if the state mistakenly executes someone, there's no way to fix the mistake. Yet the justice system is known to make mistakes even in trials for murder and other serious crimes, as is proved by the cases of Donald Marshall and Guy Paul Morin. Thus if we reinstate the death penalty, it is entirely possible that we will commit irreversible injustice by killing innocent people. For this reason, if for no other, we should not do so. 8. Only an idiot would have played with matches next to a gas pump, and Sekeras is no idiot. But the police insist that that's what caused the ex­ plosion. They must be lying. 9. Archy didn't leave the house that day. If he had, the police would have found his footprints in the snow. And if he never left home, there's no way he could have been in town to rob the convenience store. So he didn't do it. 10. The US isn't likely to send a manned mission to Mars within the next 15 years, since manned space travel is incredibly expensive and American voters will demand that the money be spent on other things. IV. Can you refute the following claims by counterexample? If so, do so. If not, explain why not. (Answers may be found at the end of these notes.)

1. No birds can swim. 2. Only mammals are warm blooded. 3. No mammals can fly. 4. Some professional hockey players are fully-qualified brain surgeons. 5. Nuclear power is a completely safe form of electricity generation. 6. Not all diseases can be treated by antibiotics. 69

Reasoning and Critical Thinking 7. Some planets in our solar system have several moons. 8. Only one planet in our solar system has rings. 9. No US President has ever been impeached. 10. An alarming number of countries possess atomic weapons. V. Are the following sets of claims consistent? 1. Able and Baker only ever go to the pub together, and the same holds for Baker and Charlie. Yet only one of the three went to the pub. 2. Able and Baker only ever go to the pub together, and the same holds for Baker and Charlie. Only two of the three went to the pub last night, however. 3. Three reliable witnesses claim Smedley was in Toronto on the night of the 25th. The crime was committed in Toronto on the night of the 25th. Smedley says he was in Montreal all day. Smedley committed the crime. 4. Peters will get into law school provided that she performs well on the LSAT and gets good letters of reference. She will get good letters of refer­ ence, and she will perform well on the LSAT. Still, she won't get into law school. 5. Three reliable witnesses claim Smedley was in Toronto on the night of the 25th. The crime was committed in Toronto on the night of the 25th. Smedley says he was in Montreal all day. Smedley didn't commit the crime. 6. Global temperatures are either on the increase or else they are decreasing. If global temperatures increase, there will be more forest fires. If global temperatures decrease, there will be increased consumption of fossil fuels for heating. Neither the consumption of fossil fuels nor the number of forest fires will increase. 7. The deal will only be a success if Acme Corp, backs down on the dispute resolution clause and Zenith Inc. accepts the proposed financial terms. Acme will back down on the dispute resolution clause, but Zenith won't accept the financial terms. Nonetheless, the deal will be a success. 8. Peters will get into law school provided that she performs well on the LSAT and gets good letters of reference. She will get good letters of ref­ erence, but she won't perform well on the LSAT. Still, she won't get into law school. 9. The Liberals will win the next election, provided that no new scandal arises and the Bloc's popularity decreases. The Bloc's popularity won't decrease even if a new scandal does not arise. Nevertheless, the Liberals will win the next election. 10. The next sentence is false. The previous sentence is true. 70

Venn Diagrams (2) VI. The following arguments are fairly interpreted as using unreliable princi­ ples or forms of inference. Prove that this is so in each case by (a) identifying the principle or form and (b) giving an example of another, obviously bad, ar­ gument where the same principle of reasoning is used.

1. The vast majority of heroin addicts used marijuana first. So it is obvious that marijuana use leads to heroin addiction. 2. If you smoke, you might get lung cancer. Betty doesn't smoke. So she won't get lung cancer. 3. No one has conclusively proven that climate change is caused by human activity. So it's obvious that climate change is not caused by humans. 4. If the NDP won a majority in the last election, they'd be in government. They didn't win a majority. So they are not in government. 5. All even numbers are divisible by two. So every number that's divisible by two is even. VII. Classify the following arguments as valid or invalid. If you say that an

argument is invalid, describe a possible situation in which the premises would be true and the conclusion false. 1. If today is the first day of Spring, then this month is March. Today is not the first day of Spring. So this month is not March. 2. If today is the first day of Spring, then this month is March. This month is March. So today is the first day of Spring. 3. The human race won't survive unless we do something about global warming. We will do something about global warming, however. So the human race will survive. 4. My plants will survive only if I water them. I will water them. So they will survive. 5. If Bill Gates was a successful businessman, he'd be rich. And he is rich. So he must be a successful businessman. 6. Some women play hockey. Some women play golf. So some women play both hockey and golf. 7. If the Conservatives won a majority in the last federal election, they would be in government. And they are in government. So the Conservatives must have won a majority in the last federal election. 8. Not all European countries are members of NATO. So not all members of NATO are European countries.

71

Reasoning and Critical Thinking V I I I . Consider the following pairs of statements. Are they equivalent? If not, does either one imply the other?

1.

(a) Sam and Dave can sing. (b) Sam can sing.

2.

(a) Sam and Dave can sing. (b) Sam or Dave can sing.

3.

(a) Some birds cannot fly. (b) Not all birds can fly.

4.

(a) If Harrison doesn't resign, the Liberals won't win a majority in the next election. (b) The Liberals will win a majority in the next election only if Harrison resigns.

5.

(a) The Conservatives won't win a majority in the next election unless Grabowski is replaced. (b) The Liberals will lose the next election if Grabowski is replaced.

6.

(a) The plants will die unless they're watered. (b) If the plants are watered, they won't die.

7.

(a) The plants will die unless they're watered. (b) If the plants are dead, they must not have been watered.

8.

(a) The plants will die unless they're watered. (b) If the plants are watered, then they'll live.

9.

(a) All vertebrates are chordates. (b) Only chordates are vertebrates.

10.

(a) Whenever I do logic, my brain hurts. (b) Whenever my brain hurts, I do logic.

Classify the following arguments as: (a) Sound; (b) Valid, but not sound; (c) Neither valid nor sound. (Note: the answers to some of the questions will vary depending upon when you do them). IX .

1. Today is Wednesday, and Wednesday is the day after Tuesday. So yester­ day must have been Tuesday. 2.

Not all frogs are green. Some green things are plants. So some frogs are not plants.

3.

Ottawa is more populous than Beijing and Beijing is more populous than Mumbai. So Ottawa is more populous than Mumbai. 72

Venn Diagrams (2)

4. Lester Pearson is the Prime Minister and Lester Pearson is a Conserva­ tive. Therefore, the Prime Minister is a Conservative. 5. Today is not the first day of Summer. Summer begins in June. So today is not in June. 6. Jupiter has more moons than Mars and Mars has more moons than the earth. So the earth has fewer moons than Jupiter. 7. Not all mammals are large and not all mammals are brown. So some small mammals are brown. 8. Whales can't fly and whales are mammals, so not all mammals can fly. 9. Some dogs are not brown. So some brown things aren't dogs. 10. All chordates are vertebrates. So all vertebrates are chordates. 11. There are more days in January than there are in February, and more days in April than there are in January. So there are more days in April than there are in February. 12. Whales are mammals and whales are large. So mammals are large. X. The following arguments are fairly interpreted as being formally invalid. Prove that this is so in each case by (a) identifying the form of the argument and (b) giving an example of another argument of the same form that has true premise(s) and a false conclusion. 1. 2 is not an odd number. So neither 2 nor 4 is an odd number. 2. 3 and 5 are not both even numbers. So 3 is not an even number. 3. All even numbers are divisible by 2. So all numbers that are divisible by 2 are even. 4. Not all children swim. So not all swimmers are children. 5. Some dogs are not pets. So some pets are not dogs. 6. If it doesn't rain, we'll have a picnic. But it will rain. So we won't have a picnic. 7. All dogs are mammals. All dogs are warm-blooded. So all mammals are warm-blooded. 8. My plants would be dead if I had forgotten to water them. They are dead. So I must have forgotten to water them. 9. Not all women play golf. Not all golfers swim. So not all women swim. 10. Some doctors are not surgeons. Some surgeons don't drive Porsches. So some doctors don't drive Porsches.

73

Reasoning and Critical Thinking

XL Give examples of the following, if possible (if it is not possible to provide an example, explain why not): 1. A valid argument with all true premises and a true conclusion. 2. A valid argument with all true premises and a false conclusion. 3. A valid argument with at least one false premise and a true conclusion. 4. A valid argument with at least one false premise and a false conclusion. 5. A sound argument. 6. An argument which is sound but not valid. XII. True or false? Prove that your answer is correct. 1. Some valid arguments have true premises. 2. Some sound arguments have true premises. 3. Some sound arguments have false conclusions. 4. Any argument with all true premises and a false conclusion is invalid. 5. Not all valid arguments are sound. 6. No invalid arguments are sound. XIII. Represent the following statements with Venn diagrams: 1. All people are mortal. (P, M) 2. Some dogs are brown. (D, B) 3. Not all lawyers are rich. (L, R) 4. Some people are happy, and some are not. (P, H) 5. Some non-members will attend. (M, A) 6. There are no carrots, but there are some potatoes. (C, P) 7. There are neither potatoes nor carrots. (C, P) 8. We only have potatoes and carrots (though of course, nothing is both a potato and a carrot). (C, P) 9. No carrots are potatoes. (C, P) 10. Some non-carrots are not potatoes either (C, P).

74

Venn Diagrams (2)

Given the dictionary below, state what is expressed in the following dia­ grams in plain English. Assume that the universe of discourse is restricted to people living in a town called Mudville. X IV .

D IC T IO N A R Y :

L= Lawyer, R= Rich, D= Doctor, H= Happy.

3.

XV.

10.

Construct Venn diagrams to test the following argument forms for validity.

1. All tigers are mammals. All striped animals are tigers. So all striped animals are mammals. (Assume U= the set of all animals) (T, M, S) 2. No tigers are mammals. All striped animals are tigers. So no striped animals are mammals. 3. All tigers are mammals. Some striped animals are tigers. So some striped animals are mammals. 4. All lawyers are educated. Some rich people are educated. So some rich people are lawyers. (U= the set of all people) (L, E, R) 5. No lawyers are educated. Some rich people are educated. So some lawyers are not rich. 75

Reasoning and Critical Thinking 6. No lawyers are educated. All educated people are rich. So no rich people are lawyers. 7. All doctors are educated. No one who is educated is a clown. So no doctor is a clown. (C, E, D) 8. All doctors are educated. Some doctors are clowns. So some clowns are educated. 9. All doctors are educated. Some clowns are uneducated. So not all doctors are clowns. 10. Some students work full time. Everyone who works full time is busy. So some students are busy. (S, W, B) 11. All students work full time. Some people who are busy do not work full time. So some people who are busy are not students. 12. No students work full time. Some people who are busy do not work full time. So some people who are busy are students.

76

C HAPTER 11

S OLUTIONS

TO

EXERCISES

PART I 1. Not an argument: various statements are made, but no conclusions are drawn. 2. An argument. Premises: If Joe had gone shopping, there would be food in the fridge. There is no food in the fridge. Conclusion: Joe did not go shopping. Indicator word: So. 3. Not an argument: various statements are made, but no conclusions are drawn. 4. An argument. Premises: Either Tom, or Dick , or Harry cleaned up the mess. Harry never helps out. Dick was busy with other things. Interme­ diate Conclusions: Harry didn't clean up the mess; Dick didn't clean up the mess. Final conclusion: Tom cleaned up the mess. Indicator words: because (twice), so. 5. An argument. Premise: Despite the fact that the United States has the most powerful armed forces in the world today, they got themselves into a big mess in Iraq. Intermediate conclusion: Military power alone can't solve all the world's problems. Final conclusion: There is a real need for international law. Indicator words: Prove, Given this, it's obvious that. 6. An argument. Premises: More than half the households in Canada con­ tributed money to help the victims of the Tsunami that hit Indonesia and Thailand. If Canadians were completely selfish, they wouldn't have do­ nated all that money. Final conclusion: It would be a mistake to think that Canadians care only about themselves. Indictor word: For 7. An argument: Premises: There is a big mess in Syria. Without the sup­ port of the UN, things will never get better in Syria. Conclusion: Canada should agree to become involved in a UN mission to Syria, under what­ ever terms are offered. Indicator words: therefore 8. An explanation, not an argument. 9. An argument. Premises: What we see seems to depend on our eyes and our brain. The only connection between our eyes and our brain is a system of nerves, nerves only transmit electronic and chemical signals. Colours don't seem to be either electronic or chemical signals. Conclu­ sion: The colours we see may well exist only in our minds, not in the real world, indicator words: To see that this is so, consider the following 77

Reasoning and Critical Thinking 10.

Not an argument: various statements are made, but no conclusions are drawn.

11.

An argument. Premises: Two independent witnesses say they saw Tamara at the library in Ottawa on Tuesday. Intermediate Conclusion: Tamara spent the whole day at the library in Ottawa on Tuesday. Final conclu­ sion: Tamara was not in Montreal on Tuesday. Indicator words: because, we know that because

12.

Not an argument: various statements are made, but no conclusions are drawn.

PART II 1. MP: Either Fred or George made this mess. 2.

MP: We were not bom with wings. MC: We were not meant to fly.

3.

MP: You stayed out too late. MC: You're in trouble.

4.

MP: The new OS is a complex program.

5.

MP: Sam plagiarized.

PART III 1. 1. 2. 3. 4.

Gladys Gladys Gladys Gladys

and and and and

Fred have both had affairs (Pr.) Fred are always fighting about how to rise their kids (Pr.) Fred have money problems (Pr.) Fred's marriage probably won't last much longer (FC, 1,2, 3)

78

Venn Diagrams (2) 2. 1. 2. 3. 4.

Joe either took the bus or he took a cab. (Pr.) Joe didn't have enough money to pay for a cab. (Pr.) Joe didn't take a cab. (IC, 2) Joe came on the bus. (FC, 1,3) Pi

Pi

FC4

1. 2. 3. 4. 5. 6.

Featherstone handled the crisis in Myanmar well under pressure. (Pr.) Featherstone works well under pressure. (IC, 1) Featherstone acted decisively in dealing with the Jones case. (Pr.) Featherstone is decisive. (IC, 3) No other candidate is available. (Pr.) Featherstone is the best candidate for the job (FC, 3,4,5) Pi

IC 2

P3

P5

FC6

79

IC 4

Reasoning and Critical Thinking

4. 1. Tests performed by the consumers' association indicate that Brand X is the most reliable one. (Pr.) 2. Brand X is the most reliable. (IC, 1) 3. Brand X is not too expensive. (Pr.) 4. Brand X is the best one to buy. (FC, 2, 3) Pi

FC±

5. 1. 2. 3. 4.

Sam is a great baseball player. (Pr.) Sam can play in the major leagues. (IC, 1) Major league baseball players make lots of money. (Pr.) Sam can be rich if he wants to. (FC, 2, 3) Pi

IC-2

FCi

80

6. 1. Dogs inevitably get fleas (Pr.) 2. When dogs get fleas, they bring them into the house. (Pr.) 3. If you have a dog, you're sure to have fleas in your house. (IC, 1, 2) 4. The fleas that bite dogs will also bite you given the chance. (Pr.) 5. If you're not willing to put up with a few flea bites, you shouldn't get a dog. (FC, 3,4) Pi

P?

IC 3

Pi

FC 5 7. 1. Once you kill someone, there's no way to bring him back to life. (Pr.) 2. if the state mistakenly executes someone, there's no way to fix the mistake. (IC, 1) 3. In the cases of J.P Morin and DonaldMarshall, the justice system made serious mistakes. (Pr.) 4. The justice system is known to make serious mistakes. (IC, 3) 5. If we reinstate the death penalty, it is entirely possible that we will commit irreversible injustice by killing innocent people. (IC, 2,4) 6. We should not reinstate the death penalty. (FC, 5) Pi

P3

IC 2

IC i

IC 5

FC 6

Reasoning and Critical Thinking

8. 1. 2. 3. 4. 5.

Only an idiot would have played with matches next to a gas pump. (Pr.) Sekeras is no idiot. (Pr.) Sekeras didn't play with matches next to a gas pump. (IC, 1, 2, unstated) The Police say that Sekeras did play with matches next to a gas pump. (Pr.) The Police must be lying. (FC, 3,4) Pi

Pi

IC 3

Pi

FC5

9. 1. If Archy had left the house that day, the police would have found his footprints in the snow. (Pr.) 2. The police didn't find his footprints. (Pr., unstated) 3. Archy didn't leave the house that day. (IC, 1,2) 4. If Archy didn't leave home that day, he couldn't have been in town to rob the convenience store. (Pr.) 5. Archy wasn't in town to rob the convenience store. (IC, 3,4, unstated) 6. Archy didn't rob the convenience store. (FC, 5) Pi

P2

IC 3

P4

IC 5

FC6

82

Venn Diagrams (2)

10. 1. Manned space travel is incredibly expensive. (Pr.) 2. American voters will demand that money be spent on things other than manned space travel. (Pr.) 3. The US isn't likely to send a manned mission to Mars within the next 15 years. (FC, 1, 2) Pi

P2

FC3

PART I V 1. Refutable: ducks, geese, etc. 2. Refutable: birds are warm-blooded, but not mammals. 3. Refutable: Bats 4. Not refutable by counterexample, because it is not a universal statement. 5. Refutable: Chernobyl, Three Mile Island, etc. 6. Not refutable by counterexample, because it is not a universal statement. Besides, it's true. 7. Not refutable by counterexample, because it is not a universal statement. Besides, it's true. 8. Not refutable by counterexample, because it is not a universal statement. 9. Refutable: Johnson, Clinton, Trump. 10. Not refutable by counterexample, because it is not a universal statement. Besides, it's true. PART V 1. Inconsistent

6. Inconsistent

2. Inconsistent

7. Inconsistent

3. Consistent

8. Consistent

4. Inconsistent

9. Consistent

5. Consistent

10. Inconsistent

83

Reasoning and Critical Thinking

PART V I 1.

(a) In the vast majority of cases, A happened before B did. So A leads to B . (b) The vast majority of bank robbers drank milk before turning to bank robbery. So milk drinking leads to bank robbery.

2.

(a) If A happens, B might happen. A won't happen. So B won't happen, (b) If Joe drinks poison, he could die. Joe won't drink poison. So Joe won't die.

3.

(a) No one has conclusively proven that A is true. So A is false, (b) No one has conclusively proven that there are no pink elephants dancing on the dark side of the moon. So there are pink elephants etc.

4.

If P then Q. Not P. So not Q. If Julius Caesar committed suicide, he'd be dead. Caesar did not commit suicide. So he's not dead.

5.

(a) All P are Q. So all Q are P. (b) All dogs are mammals. So all mammals are dogs.

PART V I I 1.

Invalid: suppose it is the 3rd of March.

2.

Invalid. Again, suppose it is the 3rd of March.

3.

Invalid: suppose we solve global warming but the Earth is hit by a giant asteroid.

4.

Invalid: suppose they get eaten by a rabbit.

5.

Invalid: Suppose Bill Gates was terrible at business, but won the lottery.

6.

Invalid: suppose that there were women golfers and women hockey play­ ers, but none of the hockey players played golf and vice versa.

7.

Invalid: Suppose they won a minority.

8.

Invalid: suppose all the non-European members left the alliance.

PART V I I I 1.

(a) implies (b), but not vice versa.

2.

(a) implies (b), but not vice versa.

3.

Equivalent

4.

Equivalent

5.

Neither implies the other. 84

Venn Diagrams (2)

6.

Neither implies the other.

7.

Neither implies the other.

8.

Neither implies the other.

9.

Equivalent

10.

Neither implies the other.

PA R T IX 1.

Sound (a) if answered on Wednesday; otherwise, (b) valid but not sound.

2.

Neither valid nor sound (c).

3.

Valid but not sound (b).

4.

Valid but not sound (b).

5.

Neither valid nor sound (c).

6.

Sound (c).

7.

Neither valid nor sound (c).

8.

Sound (a)

9.

Neither valid nor sound (c).

10.

Neither valid nor sound (c).

11.

Valid but not sound (b).

12.

Neither valid nor sound (c).

PA R T X 1. (a) Not P. So neither P nor Q. (b) I never played in the NHL. So neither I nor Wayne Gretzky ever played in the NHL. 2.

(a) Not both P and Q. So not P. (b) Wayne Gretzky and I didn't both play the in the NHL. So Wayne Gretzky didn't play in the NHL.

3.

(a) All P are Q. So all Q are P . (b) All cars are vehicles. So all vehicles are cars.

4.

(a) Not all P are Q. So not all Q are P. (b) not all mammals are bats. So not all bats are mammals.

5.

(a) Some P are not Q. So some Q are not P. (b) Some amphibians are not frogs. So some frogs are not amphibians. 85

Reasoning and Critical Thinking 6. (a) If not P then Q. P. So not Q. (b) If Napoleon did not live past 1800, he'd be dead now. He did live past 1800. So he's not dead now. 7. (a) All P are Q. All P are B . So all Q are B . (b) All birds are warm­ blooded animals. All birds have feathers. So all warm-blooded animals have feathers. 8. (a) P if Q. P so Q. (b) Julius Caesar would be dead if he'd been killed in battle. Caesar is dead. So he was killed in battle. 9. (a) Not all P are Q, Not all Q are B . So not all P are B . (b) Not all cats are pets. Not all pets are mammals. So not all cats are mammals. 10. (a) Some P are not Q. Some Q are not R. So some P are not R. (b) Some dogs are not pets. Some pets are not mammals. So some dogs are not mammals. PART XI 1. The Sun is bigger than the Earth, and the Earth is bigger than the Moon. So the Sun is bigger than the Moon. 2. This is not possible: if the premises of an argument are true, and the con­ clusion is false, then it is possible for that to happen. But if an argument is valid, it's not possible for the premises to be true and the conclusion false, by definition. 3. The Sun is bigger than the Moon. The Moon is bigger than the Earth. So the Sun is bigger than the Earth. 4. Ottawa is in Nova Scotia. Nova Scotia is in France. So Ottawa is in France. 5. Ottawa is in Ontario. Ontario is in Canada. So Ottawa is in Canada. 6. This is not possible: by definition, a sound argument must have all true premises and be valid. PART XII 1. True. Proof by example: Ottawa is in Ontario. Ontario is in Canada. So Ottawa is in Canada. 2. True. Proof by example: 1+1=2, 2=5-3. So 1+1 =5-3. 3. False. Proof by general argument: By definition, a sound argument is valid and all its premises are true. By definition, if an argument is valid it is impossible for its premises to be true and its conclusion false. Since the premises are true in this case, the conclusion cannot be false, and must itself be true. 86

Venn Diagrams (2)

4. True. Proof by general argument. If the premises of an argument are true and its conclusion is false, then it is possible for this to occur. By definition, however, an argument if valid only if it is not possible for this to occur. So o such argument is valid. 5. True. Proof by example: Ottawa is in Ontario Ontario is in China. So Ottawa is in China. 6. True. Proof by general argument: For an argument to be sound, it must be valid and have true premises. Any argument that is not valid fails to satisfy the first condition, and thus cannot be sound. PART XIII

3.

10.

4.

PART XIV 1.

Some doctors are not happy.

2.

Some rich people aren't lawyers, and some lawyers aren't rich.

3.

Some doctors are not lawyers. 87

Reasoning and Critical Thinking

4. All doctors are lawyers. 5. All lawyers are happy 6. All doctors are happy, and everyone who is happy is a doctor. (All and only doctors are happy.) 7. No one lives in Mudville. 8. There are no lawyers. 9. No lawyers are doctors. 10. Some people are rich but unhappy, and some people are happy but not rich, but no one is both rich and happy. PA R T XV

1. Valid

2. Valid

3. Valid

4. Invalid

88

Venn Diagrams (2)

5. Invalid

6. Invalid

7. Valid

8. Valid

9. Invalid

10. Valid

11. Valid

12: Invalid

89