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Reasoning and critical thinking book
Reasoning and Critical Thinking (University of Ottawa)
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Reasoning and Critical Thinking
(Course notes for PHI 1101, Version 2.1)
P. Rusnock
Copyright c 2020 Paul Rusnock
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C ONTENTS
Introduction
1 Arguments
1.1 Introduction . . . . . . . . . . . . .
1.2 Explanations and arguments . . . .
1.3 Simple and complex arguments . .
1.4 Arguments in standard form . . . .
1.5 Real life is messy . . . . . . . . . . .
1.6 The principle of charity . . . . . . .
1.7 Principles or forms of inference . .
1.8 Unstated premises and conclusions
1.9 Diagramming arguments . . . . . .
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2 Evaluating Arguments
2.1 Introduction . . . . . . . . . . . . . . . .
2.2 Evaluating premises . . . . . . . . . . . .
2.3 Evaluating reasoning . . . . . . . . . . .
2.4 Deductive and non-deductive reasoning
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3 Validity and Soundness
3.1 Validity . . . . . . . . . . . . . . . . . .
3.2 Truth preservation and formal validity
3.3 Validity, truth, and soundness . . . . .
3.4 Implication and equivalence . . . . . .
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4 Basic Propositional logic
4.1 Introduction . . . . . . . . . . . . . . . . . . . .
4.2 Some common logical concepts; symbolization
4.3 Some common valid argument forms . . . . . .
4.4 Some common invalid argument forms . . . . .
4.5 Chains of valid inferences, or proofs . . . . . . .
4.6 Proofs by reduction to absurdity . . . . . . . . .
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5 Basic Syllogistic and Venn Diagrams
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
5.2 Syllogisms . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Venn diagrams: the basics . . . . . . . . . . . . . . . .
5.4 Two terms; categorical propositional forms . . . . . .
5.5 Some simple inferences justified with Venn diagrams
5.6 Representing syllogistic forms with Venn diagrams .
5.7 Symbolization: quantifiers . . . . . . . . . . . . . . .
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Reasoning and Critical Thinking
6 Non-deductive arguments
7 Inductive and Causal Reasoning
7.1 Introduction . . . . . . . . .
7.2 Inductive generalizations . .
7.3 Causes and effects . . . . . .
7.4 Applications . . . . . . . . .
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8 Language and argumentation
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Virtues and vices of language use . . . . . . . . . . . . . . . . . .
8.3 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9 A Bucketful of Fallacies
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10 Solutions to Even-Numbered Exercises
Solutions for Chapter 1 . . . . . . . . . .
Solutions for Chapter 2 . . . . . . . . . .
Solutions for Chapter 3 . . . . . . . . . .
Solutions for Chapter 4 . . . . . . . . . .
Solutions for Chapter 5 . . . . . . . . . .
Solutions for Chapter 6 . . . . . . . . . .
Solutions for Chapter 7 . . . . . . . . . .
Solutions for Chapter 8 . . . . . . . . . .
Solutions for Chapter 9 . . . . . . . . . .
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I NTRODUCTION
Most mammals are able to move about and fend for themselves at least to some
extent within hours of being born. 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, before 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, walking, 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 practice and training can make us better at these things. Singers, trumpet players,
and swimmers have much to gain from learning how to breathe in the way
1 Hamlet,
Act 4, Scene 4.
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Introduction
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 performance is the result of practice, methodical observation, and adjustment.
So too, it can be argued, with thinking. By slowing down and paying attention 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.
This book, intended for use in an introductory course in critical thinking
or informal logic, presents a variety of techniques that can be used in our own
reasoning and in evaluating the reasoning 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 another throughout your studies, whatever their
particular focus may be, and also in life beyond the walls of the academy. The
presentation is also introductory 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.
Each chapter contains a number of exercises, with solutions to the evennumbered problems given at the back of the book. These are a very important
part of the book, as skill in reasoning and thinking critically can only be acquired through attentive practice. Again, they are only meant to provide a start
and to help point you in the right direction. Our hope is that you will continue
to improve these skills by applying and refining them as you progress through
your studies.
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Here is a brief overview of the contents of the book:
Chapter One deals with the identification and analysis of arguments, how
to tell when an argument is or is not present and, when there is an argument,
how to get a clear view of its elements and how they fit together. Chapter
Two discusses in general terms how to tell whether or not a given argument
is a good one. Here, we separate that question into two parts, one concerned
with the acceptability of the premises, the other with the quality of the reasoning. With respect to the second question, we distinguish deductive and
non-deductive reasoning.
The next three chapters consider deductive reasoning in greater detail. Chapter Three presents a general discussion of the important concepts of validity,
formal validity, and soundness. The two following chapters introduce techniques for dealing with important classes of deductive argument forms, Chapter Four treating propositional logic, and Chapter Five categorical syllogisms.
Chapter Six presents a general discussion of non-deductive arguments, while
Chapter Seven deals with several important kinds of them (inductive and causal
arguments).
Chapter Eight deals with issues involving language use and argumentation
Chapter Nine, finally, presents a variety of common fallacies, bad arguments
which many people nevertheless find convincing.
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C HAPTER 1
A RGUMENTS
1.1
I NTRODUCTION
In this chapter, we will look at the 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 conclusion, and an inference linking premises and conclusion. Here is an example:
Socrates was the victim of injustice. No one, however, can act unjustly 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 hearing 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 argument 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, alongside 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.
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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 expressions 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 cannot be justified.
[T]heir Lordships have come to the conclusion that the word ‘persons’ in sec. 24 includes members both of the male and female sex
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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 intended 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.
1.2
E XPLANATIONS
AND ARGUMENTS
One particularly tricky case is that of explanations. In explanations, as in arguments, 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 convince 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 disqualified 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.
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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:
a2 − b2 = (a − b)(a + b)
a-b
b
a
3
b
1
2
a-b
a
For when we remove the square 3, whose area is b2 , from the larger square
whose area is a2 , 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:
a2 − b2 = (a − b) · a + (a − b) · b = (a − b)(a + b)
1.3
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 arguments, 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.
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Arguments
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 sciences 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 thousands of years. With the help of computers, the boundaries have been pushed
still farther.
1.4
A RGUMENTS
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.
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Reasoning and Critical Thinking
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.
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 unemployment 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 expenditures 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 support 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:
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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)
1.5
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 policeman 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 preventing that from happening. They’re not perfect, and will sometimes 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 problem would be solved once and for all by a simple change: give all
policemen tasers.
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As is often the case, there are many words here, but not a whole lot of argument. 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 rhetorical 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 summarized as follows:
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)
1.6
T HE
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 interpretation 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 under discussion, we may do well to choose an interpretation that makes for the
strongest possible argument compatible with what was actually said (and allowing 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 of 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.
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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:
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.
1.7
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:
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The hamster X had babies.
So X is female.
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:
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 unreliable 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.
1.8
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 immediately 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.
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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.
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 explicitly stated is perfectly acceptable, but an unstated premise that is essential
to the argument is not.
1.9
D IAGRAMMING
ARGUMENTS
It can also be a useful exercise to draw a diagram of an argument’s structure. Especially 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 conclusions.
Let’s begin with the simplest case, where there is only a single premise supporting the conclusion.
[P1] 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:
P1
C
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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:
P1: Smith has a violent temper.
P2: Smith is frequently intoxicated.
C: You shouldn’t hire Smith as a babysitter.
In this example, each premise provides support to the conclusion, regardless
of whether we consider the other.
Consider, by contrast, the following argument:
P1: 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:
P1: Smith has a violent temper.
P2: Smith is frequently intoxicated.
C: You shouldn’t hire Smith as
a babysitter.
P1
P1: If Archie doesn’t go, Betty won’t
go either.
P2: Archie’s not going.
C: Betty won’t go.
P2
P1
C
P2
C
The same options are available when there are three or more premises, as
in the following examples:
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P1: Tom, Dick or Harry stole the
bread.
P2: Tom didn’t do it.
P3: Dick didn’t do it.
C: Harry did it.
P1: This apartment is filthy.
P2: It’s infested with fleas.
P3: The landlord is a crook.
C: You shouldn’t rent it.
P1
P2
P3
P1
C
P2
P3
C
It is in the case of complex arguments that diagrams are the most helpful.
For example:
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.
P1 : The Honda is cheap.
P2 : The Honda has low
mileage.
P3 : The Honda is in good repair.
IC4 : You should buy the
Honda (from 1, 2, 3).
P5 : You can only afford one car.
F C6 : You shouldn’t buy
the Chevy. (from 4, 5)
P1
P2
P3
IC4
P5
F C6
For a particular messy final example, let’s diagram the argument we looked
at above (p. 8):
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)
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P1
P2
P4
IC3
P2
IC5
P6
IC7
F C8
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
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 conclusion IC3 , as only in conjunction with P2 does P1 support IC3 . For similar
reasons, we would lose support for IC7 . But if IC3 turned out to be unsupported, we would also lose support for IC5 , and thus for the final conclusion.
All of this is apparent at a glance from the diagram.
By contrast, even if premise P1 in the previous example turned out to be
unacceptable, it is clear from the diagram that the intermediate and final conclusions would still enjoy some support.
E XERCISES :
(Answers to even numbered exercises may be found at the back of the book.)
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. Sam didn’t steal the necklace, so Joe must have done it, because only
those two had the opportunity.
3. If Joe had gone shopping, there would be some food in the fridge, but
there isn’t. So he must not have.
4. Since Finland last won the world junior hockey championship, Canada
has won it five times.
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5. 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.
6. You should take the apartment: it isn’t too expensive, it’s relatively clean,
and it’s close to where you work.
7. 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.
8. The more people own guns, the safer everyone is. Some people try to
deny this, but they are just plain wrong. It is completely mistaken to
think that fewer guns makes people safer.
9. Current events prove that there is a real need for international law. 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. It’s obvious
that military power alone can’t solve all the world’s problems.
10. Most Canada geese fly south for the winter. They seem to be prompted
to do this by a combination of shorter day-length and decreasing temperatures.
11. When we look at the devastation caused by AIDS, malaria, and other
diseases in third-world countries, and consider how little money Canadians 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 Thailand. If Canadians were completely selfish, they wouldn’t have donated
all that money.
12. Going to university is a privilege that students should have to pay for.
Most of them can easily afford to pay more if they stop spending so much
on things like cell phones, beer and cigarettes. So it’s obvious that students should be paying higher tuition. Besides, there’s no other way to
raise the money the university needs.
13. 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 support of the United Nations, things will never get better. Canada should
therefore agree to become involved in a UN mission, under whatever
terms are offered.
14. University tuition fees are high enough as it is. Society as a whole benefits
from the presence of educated people, and high tuition prevents many
qualified people from obtaining an education.
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15. He was unable to continue his studies because he was injured in a car
accident, and had to be admitted to the hospital.
16. Tuition fees vary considerably across Canada, and even more across North
America as a whole. They are lowest in Québec, highest in a number of
private universities and colleges in the US. In some countries overseas,
there are no tuition fees at all.
17. 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.
18. The reason tuition fees are lower in Québec than elsewhere in Canada is
because the students are better organized politically.
19. 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.
20. According to the doctor, it’s either measles or chicken pox. But it must be
chicken pox, because with measles you get a high fever, and his temperature is normal.
21. 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.
22. Different people make different judgements about ethical questions: that
is a simple fact of experience. What one person, or one culture sees as
totally wrong may be seen by another as a good thing. This shows us
that, unlike trees and mountains, good and bad are not out there in the
world for us to perceive. If further proof is needed, just ask yourself this
question: which of the five senses are we supposed to use to perceive
good and bad?
23. 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.
24. During the twentieth century, mankind developed technologies like nuclear bombs and biological/chemical weapons that have the potential for
killing all human beings. These technologies were accessible to surprisingly large numbers of people, a situation made even more frightening
with the collapse of the USSR. Reflecting on this, we can see that it is
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something of a miracle that the human race managed to survive up till
now.
25. Clarity should be avoided at all cost by authors who wish to be thought
profound. The most commonplace thoughts, when expressed in a tortured and deliberately obscure way, take on an air of mystery, just as the
most humble outhouse, surrounded in mist, becomes a castle in the common imagination. When we reflect, further, that it is far easier to adopt
a turgid style than to develop truly deep thoughts, the wisdom of this
advice is plain.
26. I really don’t see how anyone could deny that abortion is wrong. It’s as
plain as day.
II. Supply unstated premises and/or conclusions as appropriate.
1. Fred must have made this mess. I know George didn’t.
2. She can’t be a supermodel. She’s too short.
3. If we were meant to fly, we’d be born with wings.
4. When tax revenues increase, the government inevitably feels pressure to
waste money. Since the only way to avoid this is to cut taxes, it’s obvious
what needs to be done.
5. I told you if you stayed out too late you’d get in trouble.
6. If he had nothing to hide, his answers wouldn’t be so evasive.
7. The new operating system from Microsoft is sure to be full of bugs. All
complex programs are.
8. Only Russian or Bulgarian spies could have done something so sophisticated, and I know for a fact that it wasn’t the Russians.
9. Of course Sam deserved to fail the course. Everybody knows that’s the
penalty for plagiarism.
10. Housing prices have risen steadily over the past decade, even though
incomes have remained flat. We know when that happens, you’ve got a
housing bubble. Draw your own conclusion.
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. Then draw a diagram which displays the argument’s structure.
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.
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2. The television set is on fire, so I’m sure it’s broken. That means you’ll
have to buy a new one.
3. 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.
4. Buddy has sent me hundreds of e-mails, he’s written dozens of letters to
me, and he keeps phoning. He must want to get in touch with me.
5. 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.
6. 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.
7. 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 consumer association show that.
8. Only the Liberals and the Conservatives have a real chance at forming a
government after the election, so you’ll have to vote for one of those two
parties, because you want your local MP to be a government member.
9. 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.
10. The fire was either caused by arson or by a mechanical problem with the
boiler. Now it couldn’t have been a mechanical problem, because that
would have set off the alarm. So it must have been arson. The police and
fire departments should start an investigation.
11. 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.
12. If the cook didn’t kill Weatherby, then the butler must have. But we know
that the cook didn’t do it, because if he had, he couldn’t have been at the
fish shop. So the butler did it.
13. 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.
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14. There’s no way that the federal government could assume full responsibility for health care, for it would be impossible to do that without the
unanimous approval of the provinces. And you know that won’t happen
because Quebec and Alberta are sure to oppose any such change.
15. 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 explosion. They must be lying.
16. Mehitabel committed the murder. We know she was at the scene of the
crime, because her fingerprints were found there. Also, she surely had a
motive, as she was having an affair with the victim’s husband.
17. Archy didn’t leave the house that day. If he had, the police would have
found his footprints in the snow. But they didn’t find any. 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.
18. The current scandal will cause problems for the party, since it will make
the party look dishonest, force some high level members to resign, and
increase voters’ cynicism about politics.
19. 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.
20. If the current problems with the US mortgage market spill over into the
main economy—and they will—the US dollar is going to continue to decline in value relative to the Canadian dollar. But that means that Canadian retail sales will suffer, because when the Canadian dollar is worth
more, many Canadians go cross border shopping.
IV. Many newspaper columnists present extended arguments in their contributions. Find a few such articles and provide concise summaries of the arguments
they contain.
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C HAPTER 2
E VALUATING A RGUMENTS
2.1
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 different 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 conclusion. 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 questions 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 Broadway 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.:
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• 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 examples 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.
2.2
E VALUATING
PREMISES
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
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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 of
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 according 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 rejecting 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
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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 certain 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 proportion 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.
Universal generalizations and counterexamples A universal generalization is
a claim made about all things of a certain kind. For example:
• All mammals are warm-blooded.
• All politicians are crooks.
• Every even number is equal to the difference of two odd numbers.
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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 generalizations, 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 impossible. 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 instance:
There has never been a female Canadian PM.
—can be refuted by the counterexample of Kim Campbell.
1 This
is interesting advice. Why?
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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.
E XERCISES
Can you refute the following claims by counterexample? If so, do so. If not,
explain why not.
1. No birds can swim.
2. All birds can fly.
3. Only mammals are warm blooded.
4. All mammals are warm blooded.
5. No mammals can fly.
6. All mammals can fly.
7. Some professional hockey players are fully-qualified brain surgeons.
8. Many politicians have law degrees.
9. Nuclear power is a completely safe form of electricity generation.
10. All of Canada’s Prime Ministers have come from Quebec.
11. Not all diseases can be treated by antibiotics.
12. Several members of the Canadian parliament don’t hold university degrees.
13. Some planets in our solar system have several moons.
14. No planet in our solar system has no moons.
15. Only one planet in our solar system has rings.
16. Every country that fought in one of the World Wars in the twentieth century emerged stronger after the conflict.
17. No US President has ever been impeached.
18. The Pope is Catholic.
19. An alarming number of countries possess atomic weapons.
20. No universal generalization is false.
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Sources of Information Often, we accept claims because we heard or read
them somewhere, for example, from a friend or teacher, on the internet, in a
newspaper, on the radio or television, etc. This can be a reasonable practice,
but it has its limits. Here are some of the things that are worth thinking about
in this connection:
• Is the source reliable? Has it ever steered you or others wrong before?
• Is the source competent and in a position to know the relevant facts?
• Is there any reason to expect the source to be biased? Is there a conflict of
interest, for example?
• Do other independent and reliable sources agree? How many?
• Do any reliable sources disagree? How many?
Part of what one learns from experience is whom to trust. It is a sad fact of
public life today that people who have repeatedly been caught lying or otherwise spreading false information often continue to enjoy the privilege of having their opinions spread far and wide. While it can sometimes be entertaining
to see what such people have been saying lately, you would generally be well
advised to avoid listening to them too much, unless you have strong evidence
that they have changed their ways. In any case, you should not unreservedly
trust what they say.
Unreliability is not always a matter of bad intentions, however. Some people pass on false information while firmly believing that they are spreading
the truth. This is a common human failing, studied under the names of false
confidence or simply overconfidence; no one is entirely immune.
Incompetence and ignorance are also common features of advocates. Interestingly, they are frequently joined with false confidence, since those who are incompetent are often unable to detect their own incompetence (a sort of anosognosia). People who speak forcefully about topics when they lack the knowledge required to judge the matter at hand are not hard to find. A short glance
at the comments sections of the on-line editions of most newspapers should be
enough to satisfy you on that point, especially since one usually finds equally
vehement tirades on different sides of a given issue—they can’t all be right,
though sometimes they perform the neat trick of all being wrong.
Bias and outright fraud are other sources of unreliability. Some people deliberately misrepresent things for their own personal reasons. Others are paid by
others to promote a certain point of view, and may either lie or distort the facts
in order to do so. The history of the Tobacco industry’s fight against the claim
that smoking causes cancer provides many excellent examples of commonlyused techniques. But bias is also quite often unconscious, and may not be
motivated by self-interest at all. The sources of information someone habitually relies upon, selectivity, prejudice, etc., can all produce significant biases.
When relying on sources, then, we do well to ask ourselves whether someone
might be biased concerning the matter at hand, whether consciously or unconsciously. Does the source, for example, have a conflict of interest? If so,
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what assurance do we have that it has not impaired his or her judgment in this
instance? And so on.
It is important to remember that the mere number of voices speaking in
favour of a given claim is by itself no guarantee of the claim’s truth. For it
can and does happen that many people are either mistaken, or else have been
organized to spread a particular opinion, regardless of whether it is true. In
politics and public relations, for example, the use of talking points ensures that
many opinions will be repeated thousands of times, first by those consciously
using the talking points, then by others who simply repeat what they have
heard from the former. In such cases, the number of voices speaking in favour
of a certain proposition is merely the result of artificial amplification, and adds
nothing to the trustworthiness of the claim. This is quite different from what
occurs when a large number of independent witnesses speak in favour of some
proposition (e.g., report seeing a meteor). They could all be mistaken, it is true;
but it is far less likely than when a thousand voices are mere parrots, simply
repeating what one person has said.
The multiplicative potential of modern communications technologies has
made it possible for errors to be spread with astonishing rapidity. This is why,
as the old saying goes:
A lie can travel halfway around the world while the truth is putting
on its shoes.
Experts and Authorities Some sources we rely upon play a special role, in
that we do not simply trust them on a single question, but rather on an entire
range of questions having to do with specific areas of knowledge. For example,
we consult a doctor about medical problems, a mechanic about car problems,
and so on. Much of what you learned in school, moreover, came from textbooks
written by people who are supposed to be authorities on their subjects, people
whose job it is to know certain things.
But the trust we place in authorities can be misplaced. We shouldn’t consult
an expert on, say, playing hockey, concerning which coffee to buy. Nor should
we trust so-called “experts” when there is no field of expertise, e.g., fortune
tellers, astrologers, phrenologists, etc. Then there are incompetents and frauds
of various kinds, as well as experts who are not to be trusted on account of bias,
whether innocent or corrupt, conscious or unconscious. Fortunately, there are
still many reliable, sensible, honest authorities out there. But separating them
from the others can take some work.
Reliance on authorities in public debate often suffers from the error of selectivity (or cherry-picking). Rather than approaching authorities with an open
mind to see what they have to say, one seeks precisely those authorities who
support one’s own position and ignores the others. And since there is money
to be made and attention to be won, there is no shortage of ready-made authorities willing to speak on any side of pretty much any question. This is hardly a
rational way to get closer to the truth.
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Here are a few rules of thumb for assessing the claims of experts:
1. There should be a genuine field of expertise. No one can be an expert in
the fields of phrenology or orgone therapy, for example.
2. The authority should be an expert in the relevant field. Someone who is a
reliable source for matters in his or her field may well be a crackpot when
speaking on other subjects.
3. Get a second, third, . . . opinion. For there are many cases where experts
in a given field disagree.
4. You may still decide to take the side of one party of experts even when
there is no consensus on a particular topic. If you do this, however, you
should at least try to explain how the other party may have come to hold
what you consider to be a mistaken view.
Common knowledge In most contexts, we can count on a certain number of
claims being accepted as common knowledge, i.e., truths that everyone someone
is addressing can reasonably be expected to know. I won’t need to spend any
time convincing someone who lives in Montreal, for example, that it will most
likely snow there next winter, or that both French and English are spoken in
Canada.
Common knowledge varies from one group to another. When a number of
doctors get together, for example, they are usually able to count on a sizeable
store of common knowledge having to do with their profession. When they
step outside this group, however, many of these things will not be commonly
known, and would have to be argued for or explained. Similar remarks apply
to other groups, e.g., a group of teenagers, of auto mechanics, people who ride
horses, dog owners, etc. Depending on the context in which one presents an
argument, then, one may or may not be able to count on certain claims being
accepted on this basis.
Of course, many claims that are accepted by all the members of some group
are anything but truths, so we will do well to distinguish between commonly
held beliefs and common knowledge. And while it is true that one wouldn’t
need to argue for a false claim when it is accepted by all those to whom a given
argument is addressed, this would hardly be the right thing to do if we knew
that it was false. At most, we might formulate hypothetical arguments based
on the false claim, like this:
Even if what you say were true (which I do not admit), such and
such would also be true, etc.
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 following claims, for example, are inconsistent, since there is no way both of them
could be true:
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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 immediately 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
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.
E XERCISES
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. Peters won’t get into law school unless she performs well on the LSAT
and gets good letters of reference. She will get good letters of reference, and she will perform well on the LSAT. Still, she won’t get into
law school.
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3. 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.
4. If the value of the dollar rises, our exports will suffer, but if the value of
the dollar goes down, our imports will suffer. If either our exports or our
imports suffer, the economy will be harmed. The economy will not be
harmed.
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 committed the crime.
6. At least one of Able, Baker, and Charlie was in the pub last night. Able
never goes to the pub without taking either Baker or Charlie with him.
Baker only goes to the pub if Charlie comes along. Only two of the three
went to the pub last night.
7. 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 reference, and she will perform well on the LSAT. Still, she won’t get into law
school.
8. If the Americans prevail in Iraq, Iran will lose power in the region. If the
Americans withdraw, Iran won’t lose power in the region, but the Americans will. The Americans won’t lose power in the region, but neither will
Iran.
9. 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.
10. Able, Baker and Charlie never all go to the pub together. If Charlie goes,
he always takes Able with him. Baker would never go to the pub without
Charlie. And Able never goes to the pub without one of the other two.
11. 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.
12. 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 lied when he said that.
Smedley didn’t commit the crime.
13. 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.
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14. The government will either raise or lower taxes. If the government raises
taxes, people will have less money in their pockets, and so will be worse
off. However if the government lowers taxes, public services will have to
be cut, and the people will also be worse off.
15. 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 reference, but she won’t perform well on the LSAT. Still, she won’t get into
law school.
16. The deal won’t be a success unless either Acme backs down on the dispute resolution clause or Zenith accepts the financial terms. Zenith won’t
accept the financial terms, though Acme will back down on the dispute
resolution clause. All the same, the deal will not be a success.
17. 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.
18. John is a tall man. He is also a professional basketball player, though not
a tall one.
19. The next sentence is false. The previous sentence is true.
20. Among three friends, Able, Baker and Charlie, the tallest is also the oldest, while the shortest is not the youngest. Charlie is neither the tallest
nor the shortest. Able is both younger and taller than Baker.
2.3
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 inferences 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 principles. 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.
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• 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.
• Hitler, Stalin and Pol Pot started as dictators and finished as mass murderers. 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 presidents 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.
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As we see from the examples given above, principles of inference are sometimes 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 the following:
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.
E XERCISES
The following arguments are fairly interpreted as using unreliable principles 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, argument
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. As the number of people who smoke increased, the number of cases of
lung cancer also increased. So it’s obvious that smoking causes lung cancer.
3. If you smoke, you might get lung cancer. Betty doesn’t smoke. So she
won’t get lung cancer.
4. Some women play hockey. Some women play golf. So some women play
both hockey and golf.
5. 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.
6. Smoking can cause lung cancer. Betty died from lung cancer. So she must
have been a smoker.
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7. 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.
8. You think agricultural co-ops are a good idea? That’s funny—so did
Stalin.
9. All even numbers are divisible by two. So every number that’s divisible
by two is even.
10. Not all mammals live in the ocean. So some animals that live in the ocean
aren’t mammals.
2.4
D EDUCTIVE
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 guarantee 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 corner 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 the next three chapters, we will take a closer look at deductive reasoning,
first discussing the topic in general terms, and then developing techniques for
dealing with some important kinds of deductive arguments. Afterwards, we
will do the same with non-deductive reasoning.
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C HAPTER 3
D EDUCTIVE A RGUMENTS : VALIDITY AND
S OUNDNESS
3.1
VALIDITY
We said above that if an argument is valid, then there is no way its conclusion could be false if all its premises were true. The following argument, for
example, is like this:
Sue is older than Farah, and Farah is twenty-one. So Sue is not a
teenager.
With an invalid argument, by contrast, it is possible for the premises to be true
and the conclusion false, as in the following example:
If Mickey is a major-league baseball player, he makes a lot of money.
Mickey makes a lot of money. So he must be a major-league baseball
player.
For, clearly, Mickey might make his pile some other way.
Considerations such as these motivate a common conception of validity in
terms of possibility and necessity:
Definition: An argument is said to be valid if and only if it
would be impossible for its conclusion to be false if all of its
premises were true.
Or, equivalently, in a formulation that goes back to Aristotle:
An argument is said to be valid if and only if its conclusion would
have to be true if all its premises were true.
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:
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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. We can thus confidently declare the argument to be valid.
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
be true, we recognize, if Joe has $5.01 or any larger amount of money. Unlike
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, as well as in the following exercises:
E XERCISES
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 not March. So today is not the first day of Spring.
3. 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.
4. If today is the first day of Spring, then this month is March. Today is the
first day of Spring. So this month is March.
5. 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.
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6. My plants would be dead if I hadn’t watered them. They are dead. So I
must not have watered them.
7. My plants will survive only if I water them. I will water them. So they
will survive.
8. My plants will live only if I water them. I won’t water them, however. So
they won’t live.
9. If Bill Gates was a successful businessman, he’d be rich. And he is rich.
So he must be a successful businessman.
10. If Bill Gates wasn’t a successful businessman, he wouldn’t be rich. But
he is rich. So he must be a successful businessman.
11. Some women play hockey. Some women play golf. So some women play
both hockey and golf.
12. Fred and George are twins. So they have the same birthday.
13. 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.
14. The number of murders in Chicago didn’t increase last year compared to
the year before. So it decreased.
15. Not all European countries are members of NATO. So not all members of
NATO are European countries.
16. Some Americans are Buddhists. So some Buddhists are Americans.
17. Not all Americans are Buddhists. So not all Buddhists are Americans.
18. Most members of the Lions Club are members of the Rotary Club. Most
members of the Rotary Club are members of the Loyal Order of Moose.
So most members of the Lions Club are members of the Loyal Order of
Moose.
19. 6 is an odd number. So 1+1=2.
20. Flipper is a dolphin. So Flipper is a mammal.
3.2
T RUTH
PRESERVATION AND FORMAL VALIDITY
As mentioned previously, the reasoning in individual arguments usually embodies 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 argument, 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.
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Let us take the following argument as an example:
All lawyers are educated. Some rich people are educated. So some
lawyers are rich.
Noting that certain words are repeated, we can discern the following pattern:
All A are B.
Some C are B.
Some A are C.
On the reasonable assumption that this is the form of inference used in the argument, we can then ask ourselves whether it is a trustworthy one, by considering various other arguments that result when different terms are substituted
for ‘A’,‘B’, and ‘C’. For instance:
(T) All whales are mammals.
(T) Some carnivores are mammals.
(T) Some whales are carnivores.
(F) All whales are reptiles.
(T) Some snakes are reptiles.
(F) Some whales are snakes.
(F) All birds fly.
(T) Some mammals fly.
(F) Some birds are mammals.
(F) All cats fly.
(F) Some dogs fly.
(F) Some cats are dogs.
(T) All whales live in the ocean.
(T) Some fish live in the ocean.
(F) Some whales are fish.
(F) All people are dogs.
(F) Some lawyers are dogs.
(T) Some people are lawyers.
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.
This approach is quite different from the one we discussed in the previous section. Instead of thinking about what our premises say, and considering
whether those very claims would be true or false in various possible circumstances, 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 recognized as independent questions, namely: (1) are the premises and conclusion
true or false? and (2) is the reasoning good? Consider again the argument we
started with:
All lawyers are educated. Some rich people are educated. So some
lawyers are rich.
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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.
Let us now introduce some terminology. Beginning with a particular argument, for example:
All mice are rodents. Not all mammals are rodents. So some mammals 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 R, we obtain:
All P are Q. Not all R are Q. So some R are not P .
If we now stipulate that P , Q and R 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 R. 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
aren’t 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 distinction, depending on whether the basis of the truth-preservation is or is not necessary. By way of example, consider the following argument forms:
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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 contrast, we see that the connection between premises and conclusion is more robust, 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 formally 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:
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.
E XERCISES
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 neither 3 nor 5 is an even number.
3. 3 and 5 are not both even numbers. So 3 is not an even number.
4. Some women don’t swim. So some swimmers aren’t women.
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5. Not all children swim. So not all swimmers are children.
6. All triangles are three-sided figures. So all three-sided figures are triangles.
7. If it doesn’t rain, we’ll have a picnic. But it will rain. So we won’t have a
picnic.
8. If it rains, we won’t have a picnic. It won’t rain. So we will have a picnic.
9. Not all dogs bark. Some animals that don’t bark have tails. So some dogs
have tails.
10. No dogs bark. Some animals that don’t bark have tails. So some dogs
have tails.
11. Smith’s roses will bloom only if they are fertilized. They will be fertilized.
So they will bloom.
12. Cleveland would only have won the series if James had performed brilliantly. They didn’t win the series. So James didn’t perform brilliantly.
13. All dogs are mammals. All dogs are warm-blooded. So all mammals are
warm-blooded.
14. Trump won’t be indicted unless Manafort cooperates with Mueller. Manafort will cooperate with Mueller. So Trump will be indicted.
15. My plants would be dead if I had forgotten to water them. They are dead.
So I must have forgotten to water them.
16. Some women play golf. Some golfers play tennis. So some women play
tennis.
17. Not all women play golf. Not all golfers swim. So not all women swim.
18. Some men don’t play basketball. Some basketball players play golf. So
some golfers are not men.
19. Some doctors are not surgeons. Some surgeons don’t drive Porsches. So
some doctors don’t drive Porsches.
20. No professional basketball players are jockeys. No jockeys are sumo
wrestlers. So no professional basketball players are sumo wrestlers.
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3.3
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 conclusion 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.
E XERCISES
I. 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. An invalid argument with all true premises and a true conclusion.
3. A valid argument with all true premises and a false conclusion.
4. An invalid argument with all true premises and a false conclusion.
5. A valid argument with at least one false premise and a true conclusion.
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6. An invalid argument with at least one false premise and a true conclusion.
7. A valid argument with at least one false premise and a false conclusion.
8. An invalid argument with at least one false premise and a false conclusion.
9. A sound argument.
10. An argument which is valid but not sound.
11. An argument which is sound but not valid.
12. An argument which is neither valid nor sound.
II. The next problems ask you to determine whether certain statements are true
or false, and to justify your answer.
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’.
Note too 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. 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.
Similarly, we recall that a single counterexample is enough to prove that a
universal claim is false. To prove a universal claim true, however, requires a
general argument that covers all the cases.
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. The following argument is valid (that is, the reasoning is fine) 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. By definition, a sound argument is a valid argument
with true premises. It follows that every sound argument is valid,
and hence that none are invalid.
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Problems: True or false? Explain.
1. Some valid arguments have true premises.
2. Some sound arguments have false premises.
3. Some sound arguments have true premises.
4. Some valid arguments have false conclusions.
5. Some sound arguments have false conclusions.
6. A valid argument with a false conclusion must have at least one false
premise.
7. Any argument with all true premises and a false conclusion is invalid.
8. Any argument with all true premises and a true conclusion is valid.
9. Not all valid arguments are sound.
10. Not all sound arguments are valid.
11. No invalid arguments are sound.
12. No sound arguments are invalid.
III. 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).
1. Today is Wednesday, and Wednesday is the day after Tuesday. So yesterday must have been Tuesday.
2. Some men like sports. All men like beer. So some men like both sports
and beer.
3. Not all frogs are green. Some green things are plants. So some frogs are
not plants.
4. All frogs are green. No plants are green. So no frogs are plants.
5. Ottawa is more populous than Beijing and Beijing is more populous than
Mumbai. So Ottawa is more populous than Mumbai.
6. The average temperature in Ottawa is colder than the average temperature in Tunis. Therefore, it is never warmer in Ottawa than in Tunis.
7. Lester Pearson is the Prime Minister and Lester Pearson is a Conservative. Therefore, the Prime Minister is a Conservative.
8. Today is the first day of Summer. Summer begins in June. So today is in
June.
9. Today is not the first day of Summer. Summer begins in June. So today
is not in June.
10. Today is not in June. Summer begins in June. So today is not the first day
of Summer.
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11. Jupiter has more moons than Mars and Mars has more moons than the
earth. So the earth has fewer moons than Jupiter.
12. You can’t be President of the US unless you are older than 35. Ronald
Reagan is President of the US. So he’s not 25 years old.
13. Not all mammals are large and not all mammals are brown. So some
small mammals are brown.
14. Whales can’t fly and whales are mammals, so no mammals can fly.
15. Whales can’t fly and whales are mammals, so not all mammals can fly.
16. Some dogs are brown. So some brown things are dogs.
17. Some dogs are not brown. So some brown things aren’t dogs.
18. No dogs are beagles. So no beagles are dogs.
19. All chordates are vertebrates. So all vertebrates are chordates.
20. All surgeons are doctors. Some women are doctors. So some women are
surgeons.
21. 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.
22. Some politicians are crooks. All crooks are dishonest. So not all politicians are honest.
23. There are more than 12 students in this class. So at least two of the students in this class were born in the same month.
24. Jorge Mario Bergoglio is the Pope and Jorge Mario Bergoglio is Argentinian. So the Pope is Argentinian.
25. Whales are mammals and whales are large. So mammals are large.
3.4
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.
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E XERCISES
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) Dave and Sam can sing.
3.
(a) Sam and Dave can sing.
(b) Sam or Dave can sing.
4.
(a) Either Sam or Dave can sing.
(b) If Sam can’t sing, then Dave can.
5.
(a) Some birds cannot fly.
(b) Not all birds can fly.
6.
(a) If the Conservatives win a majority in the next election, Harrison
will resign.
(b) If the Conservatives do not win a majority in the next election, Harrison will not resign.
7.
(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.
8.
(a) Neither the Conservatives nor the Liberals will win a majority in the
next election.
(b) The Conservatives and the Liberals won’t both win a majority in the
next election.
9.
(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.
10.
(a) The Conservatives will win a majority in the next election if Grabowski
is replaced.
(b) The Liberals will lose the next election if Grabowski is replaced.
11.
(a) The plants will die unless they’re watered.
(b) If the plants are watered, they won’t die.
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12.
(a) The plants will die unless they’re watered.
(b) The plants won’t live if they are not watered.
13.
(a) The plants will die unless they’re watered.
(b) If the plants are dead, they must not have been watered.
14.
(a) The plants will die unless they’re watered.
(b) The only way the plants will live is if they are watered.
15.
(a) The plants will die unless they’re watered.
(b) If the plants are watered, then they’ll live.
16.
(a) Joe knows that Sam is lying.
(b) Sam is lying.
17.
(a) Joe believes that Sam is lying.
(b) Sam is lying.
18.
(a) All chordates are vertebrates.
(b) All vertebrates are chordates.
19.
(a) All vertebrates are chordates.
(b) Only chordates are vertebrates.
20.
(a) Some small mammals can fly.
(b) Some flying mammals are small.
21.
(a) Whenever I do logic, my brain hurts.
(b) Whenever my brain hurts, I do logic.
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C HAPTER 4
B ASIC P ROPOSITIONAL LOGIC
4.1
I NTRODUCTION
In this chapter, we will study a special kind of argument forms, those in which
the variable components are complete sentences or propositions.
Here are some examples of such forms, some valid, some invalid, along
with an instance of each:
Not P .
Not both P and Q.
I don’t have beans.
I don’t have both beans and cornbread.
Not both P and Q.
Not P .
I don’t have both beans and cornbread.
I don’t have beans.
Either P or Q.
Not P .
Q.
Either Jasmine went or Kate did.
Jasmine didn’t go.
Kate went.
If P then Q.
Not Q.
Not P .
If Kate goes, Jasmine will too.
Jasmine won’t go.
Kate won’t go.
If P then Q.
Not P .
Not Q.
If Kate goes, Jasmine will too.
Kate won’t go.
Jasmine won’t go.
We’ll also take this opportunity to introduce symbolic logic. The use of special symbols here has two important advantages: by providing symbols for
frequently occurring logical concepts, it allows us to abbreviate, giving us a
clearer view of the forms of propositions and arguments; and, by taking us one
step farther from our original arguments, the symbolic representations help us
to concentrate on the forms of arguments, reducing the risk of being distracted
by their subject matter when we assess their reasoning.
4.2
S OME
COMMON LOGICAL CONCEPTS ; SYMBOLIZATION
In this section, we present a handful of concepts which are central to the study
of propositional logic, along with a set of symbols commonly used to designate
them.
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Negation The concept of negation is one of the simplest logical concepts. As
we use it here, negation is a propositional operator, one that converts true
propositions into false ones, and false ones into true ones. Thus if a proposition is true, e.g.,
February comes after January
—its negation
February does not come after January.
is false. While if a proposition is false, e.g.,
Great white sharks are vegetarians.
—its negation:
Great white sharks are not vegetarians.
is true.
Logicians mostly use the symbol ‘¬’ to represent negation: Thus ‘¬A’ symbolizes ‘not A’, or ‘It is not the case that A’.1
Negation is expressed in English in a variety of different ways: sometimes
with the help of the word ‘not’ or a contracted form of the same word (‘can’t’),
other times with the help of prefixes, including ‘in-’, ‘im’, ‘un-’ ‘a-’, ‘dis-’. Thus
for example, if we replace ‘Sam can walk’ with S, then each of the following
sentences can be represented as ¬S:
Sam is not able to walk.
Sam is unable to walk.
Sam isn’t able to walk.
Sam is incapable of walking.
Because negation simply changes the truth-value of a given proposition and
there are only two truth-values (true and false), applying negation twice in
a row simply ends you right back where you started. Thus ‘not not A’ will
always have the same truth-value as ‘A’ itself, as will ‘not not not not A’, ‘not
not not not not not A’, and so on. Thus, for example, if ‘Sam is not unable to
walk’ is true, so is ‘Sam is able to walk’; and if the former is false, so is the latter.
Note, however, that this equivalence only applies in cases where two negations occur in a row. In other situations, two negations may not “cancel out”.
The Lottery corporation, for example, would be in deep trouble if the two negations in the following sentence cancelled each other out:
Joe didn’t win the lottery and Sam didn’t either.
With the above notation, we can write the above result as follows:
For any proposition A, A and ¬¬A always have the same truthvalue (as do ¬¬¬¬A, ¬¬¬¬¬¬A, etc.) .
1 Other
symbols you may see are ‘∼ A’, ‘A’ or ‘−A’.
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Conjunction The concept of conjunction is most often expressed in English by
the word ‘and’. But the same meaning can be and often is conveyed by words
such as ‘but’, ‘however’, ‘nevertheless’, ‘although’, and several others. We will
use the symbol ‘& ’ to represent conjunction.2 Propositions of the form A&B
are also called conjunctions, and the propositions A, B occurring in them are
called conjuncts.
Now that we have two logical concepts, we can also express certain others
in terms of them. A noteworthy case here is the logical operator expressed in
English by ‘not both . . . and . . . ’, as used, for example, in the sentence:
Joe and Sam didn’t both get a raise.
With negation and conjunction, we can express this as follows:
It is not the case that Joe got a raise and Sam got a raise.
Or, in symbols:
¬(J&S)
Note that we introduce parentheses here to indicate what part of the sentence
is covered by the negation.
E XERCISES
I. How would the following expressions read in English? Are there any simpler
expressions that are equivalent?
1. ¬J&S
5. ¬J&¬S
2. ¬S&¬¬J
6. ¬(¬J&S)
3. S&¬S
7. ¬(J&¬S)
4. J&¬S
8. ¬(¬J&¬S)
II. Symbolize the following (where J, S are as above)
1. Joe didn’t get a raise, and neither did Sam.
2. Sam got a raise, but Joe didn’t.
3. Joe got a raise but Sam didn’t.
4. Joe failed to get a raise, but Sam got one.
5. Joe failed to get a raise, but Sam didn’t.
6. Both Joe and Sam failed to get a raise.
7. Neither Joe nor Sam got a raise.
8. Neither Joe nor Sam failed to get a raise.
2 Other
common symbols are ‘∧’ and ‘·’.
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Disjunction is usually expressed in English by the word ‘or’. Logicians use
the symbol ‘∨’. A proposition of the form ‘A ∨ B’ is called a disjunction, and the
propositions A, B occurring in it are called disjuncts.
As logicians use this word, ‘A or B’ should be understood to mean the same
as ‘At least one of A, B is true’; thus, in saying ‘A or B’, we do not exclude the
possibility that both A and B are true. (Sometimes the expression ‘and/or’ is
used to represent this sense). This sense of ‘or’ is sometimes called inclusive, as
opposed to the exclusive sense, according to which ‘A or B’ comes out false if
both A and B are true. We won’t introduce a special symbol for the exclusive
‘or’. If we want to express it, we can do so in terms of inclusive disjunction,
negation, and conjunction, as follows:
(A orexcl B) = (A orincl B) but not both (A and B)
In symbols:
(A ∨ B)&¬(A&B)
We can also express ‘neither . . . nor’ in terms of disjunction and negation as
follows:
¬(A ∨ B)
Recall from the exercises above that ‘neither A nor B’ can also be expressed
in terms of negation and conjunction as follows:
¬A&¬B
Thus ‘¬(A ∨ B)’ and ‘¬A&¬B’ say the same thing in the sense that whenever
one of them is true the other is, and when one of them is false the other is
as well. (We say that two propositions of these forms have the same truthconditions.) We have already encountered the same phenomenon—recall ‘A’
and ‘¬¬A’. Such formulas are said to be equivalent (see above, p. 47).
Conditionals are most often expressed in English by the words ‘if . . . then’.
We will symbolize ‘if . . . then’ with the arrow ‘→’.3 Other common forms involving conditionals are ‘A only if B’, ‘A unless B’, and ‘A provided that B’.
In a conditional statement A → B, the first part, A, is called the antecedent,
and the second part, B, the consequent.
Symbolizing ‘if . . . then . . . ’ statements is relatively straightforward, though
it should be noted that English allows one to reverse the clauses in such statements, so that:
1. He will have a fit if you come home late tonight.
2. If you come home late tonight, he will have a fit.
are equivalent, and should both be symbolized as L → F .
To deal with ‘only if’, it is perhaps easiest to paraphrase first in English.
Thus, for example:
3 Another
commonly used symbol is the horseshoe: ‘⊃’.
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He will buy the motorcycle only if you lower the price.
Is clearly equivalent to:
If you do not lower the price, he will not buy the motorcycle.
Which we can symbolize as:
We can also symbolize as follows:
¬L → ¬B
B→L
as the two formulas are equivalent.
We can use the same trick of paraphrasing first in English to deal with the
connective ‘. . . unless . . . ’. Consider, for example the statement:
He will stay home unless you invite him.
Clearly, this is equivalent to:
If you do not invite him, he will stay home.
Which we can symbolize as:
¬I → S
Converse Given a conditional
A→B
its converse is the conditional
B→A
When a conditional is true, its converse may or may not be. Here are a pair
of examples to prove the point:
Conditional
(T) If Joe is older than Kate, then
then Kate is younger than Joe.
(T) If you’ve been in Moosonee
you’ve been in Ontario.
Converse
(T) If Kate is younger than Joe,
then Joe is older than Kate.
(F) If you’ve been in Ontario, then
you’ve been in Moosonee.
Biconditional When both a conditional A → B and its converse B → A are
true, we may assert a biconditional:
(A → B)&(B → A)
Often, biconditionals are abbreviated with the help of the double-arrow ‘↔’:
A↔B
Recall that ‘B → A’ (i.e., ‘A ← B’) can be read as ‘A if B’ and ‘A → B’ as ‘A
only if B’; this is why biconditionals are often stated in the following form:
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A if and only if B.
Or, using ‘iff’ to abbreviate ‘if and only if’
A iff B.
Definitions are often stated as biconditionals. A little reflection should make
the reason for this obvious. A standard definition is just a statement indicating
that two expressions are synonymous. For example:
A number is said to be even if and only if it is divisible by two.
This tells us that wherever we apply either one of the two expressions ‘even’,
‘divisible by two’ to a number, we may also apply the other—hence the biconditional.
Contrapositive The contrapositive of a conditional A → B is the conditional:
¬B → ¬A
While a conditional and its converse may differ in truth-value, this is never
the case with a conditional and its contrapositive. For whenever a conditional
is true, its contrapositive is too, and conversely. That is, a conditional is always
equivalent to its contrapositive. Here are a pair of examples:
Conditional
(T) If Joe is older than Kate, then
Kate is younger than Joe.
(F) If you’ve been in Ontario, then
you’ve been in Moosonee.
Contrapositive
(T) If Kate is not younger than Joe,
then Joe is not older than Kate.
(F) If you haven’t been in Moosonee,
then you haven’t been in Ontario.
Necessary and Sufficient Conditions This seems a good place to discuss the
distinction between necessary and sufficient conditions. A is said to be a sufficient condition for B if the presence of A guarantees the presence of B. Since
there are only 12 months in a year, for example, the presence of 13 people in a
room is a sufficient condition for at least two of them to have been born in the
same month.
A is called a necessary condition for B, by contrast, if B cannot be present
unless A is. In order to have a wedding, for example, at least two people must
be present. The presence of at least two people is thus a necessary condition
for a wedding.
The examples we have given so far indicate that there are sufficient conditions that are not necessary (we could have two people born in the same month
in a room even if fewer than 13 were present), and also necessary conditions
that are not sufficient (a wedding does not take place every time at least two
people are present).
This being said, there are conditions that are both necessary and sufficient.
A number’s being divisible by two, for instance, is both a necessary and a
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sufficient condition for its being even, and receiving more votes than any other
candidate is a necessary and sufficient condition for winning an election in the
first-past-the-post system.
The difference between necessary and sufficient conditions is nicely captured by the direction of the associated conditionals. If A is a sufficient condition for B, we can say that if A occurs B will too, i.e.:
A→B
While if A is a necessary condition for B, we can say that if A doesn’t occur, B
won’t either, i.e.:
¬A → ¬B
As we have seen, this is equivalent to its contrapositive:
B→A
Thus the distinction between necessary and sufficient conditions corresponds
to the difference between ‘if’ and ‘only if’ among conditional statements, and
hence to a simple difference in the way the arrow points. Finally, a condition
that is both necessary and sufficient will be expressed by means of a biconditional, e.g.:
A↔B
For example:
A candidate wins the election if and only if he or she receives more
votes than any other candidate.
E XERCISES
I. Symbolize the following, using the dictionary provided:
D ICTIONARY: S: Sam will go/goes/went to the store. D: Dave will go/goes/went
to the store. M: Sam will be/is/was out of milk. T: Dave will be/is/was tired.
C: The store will be/is/was closed.
1. Sam and Dave both went to the store.
2. Sam went to the store but Dave didn’t.
3. Sam and Dave didn’t both go to the store.
4. Only one of them went to the store.
5. Neither of them went to the store.
6. Sam was out of milk, but neither he nor Dave went to the store.
7. Sam and Dave went to the store, but it was closed.
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8. If Sam goes to the store, Dave will too.
9. One of them will go to the store if Sam runs out of milk.
10. Sam will go to the store if it’s not closed.
11. Sam only goes to the store if Dave is tired.
12. Sam won’t go to the store unless he’s out of milk.
13. Dave won’t go to the store if he’s tired.
14. If Sam’s not out of milk, he won’t go the store, but Dave will.
15. Dave goes to the store if Sam is out of milk, unless he’s tired.
16. Neither Sam nor Dave goes to the store if it’s closed.
17. Sam and Dave will both go to the store only if Sam is out of milk and the
store’s not closed.
18. Dave won’t go to the store with Sam unless Sam is out of milk.
19. If the store is open, and he’s not tired, Dave will go, provided that Sam
goes too.
20. Only one of them will go to the store if Sam runs out of milk, and it won’t
be Dave if he’s tired.
II. Using the dictionary given above, render in plain English:
1. S&¬C
2. (¬M &(S&D))&C
3. ¬(S&D)
4. ¬(S ∨ D)
5. S → D
6. ¬S → D
7. ¬S → ¬D
8. C → (¬S&¬D)
9. (S&¬D) ∨ (D&¬S)
10. (¬M &T ) → ¬(S ∨ D)
11. M → ((¬T &¬S) → D)
12. ¬(S&¬D)
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13. ((C ∨ ¬M ) ∨ T ) → ¬D
14. M → ((¬S&¬T ) → D)
15. (T &¬M )&(S&D)
16. T ∨ D
17. ¬(C ∨ T ) → (¬S → (M → D))
18. S → C
19. ¬((S ∨ D) ∨ M )
20. ¬M
III. In the following questions, state whether A is a necessary and/or a sufficient condition for B. Provide a brief explanation of your findings.
1. A: breaking eggs; B: making an omelette
2. A: a number n is divisible by 4; B: the number n is even
3. A: the number n is divisible by 3; B: the number n is divisible by 9
4. A: a bill is passed by the House of Commons; B: the bill becomes law
5. A: Sally has a neighbour; B: Sally is someone’s neighbour
6. A: a geometrical plane figure has two right angles; B: the figure is not a
triangle
7. A: Martha has a son; B: Martha is a parent
8. A: Martha is a parent; B: Martha has a son
9. A: Smith received an absolute majority of votes in the election; B: Smith
won the election
10. A: Fred is heavier than George; B: Fred weighs 90 Kg. and George weighs
88 Kg.
11. A: Fred is taller than someone; B: Someone is shorter than Fred
12. A: The number n is divisible by 3; B: The number n is even
13. A: mammals exist; B: warm-blooded animals exist
14. A: Plants exist; B: mammals exist
15. A: Smith didn’t win the race; B: Neither Smith nor Jones won the race.
16. A: bears exist; B: mammals exist
17. A: Smith didn’t win a prize; B Smith and Jones did not both win prizes.
18. A: Jones was not convicted of any crime; B: Jones has done nothing
wrong.
19. A: Jones is a smoker; B: Jones has or will have lung cancer.
20. A: Smith and Jones did not both win prizes; B: Neither Smith nor Jones
won a prize.
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4.3
S OME
COMMON VALID ARGUMENT FORMS
There are infinitely many valid argument forms (or patterns of inference), and
infinitely many invalid ones, so there is no point in trying to list them all (in
advanced logic courses, we develop general methods for testing many argument forms for validity). Nevertheless, some argument forms (both valid and
invalid) occur so frequently that they have been given special names. We will
list a few of these here.
(DN) Adding or dropping double negation Since a proposition A and its
double negation ¬¬A are always equivalent, it is always safe to infer one from
the other, for if one is true, so is the other. We can write this rule of inference as
follows:
A
¬¬A
Where the double line indicates that the inference may go in either direction. This rule justifies inferences such as the following:
Sam did not fail in his attempt to get a raise.
Therefore, Sam got a raise.
Note that the rule applies not only to simple propositions but to complex
ones as well. For instance, if (P → Q) is true, then so is ¬¬(P → Q), and
conversely. We use the script letter A to indicate that we may apply the rule to
any proposition or formula (and not just a simple one).
4.3.1
I NFERENCE
FORMS INVOLVING CONJUNCTION
Conjunction Our first rule tells us that if two propositions A and B are both
true, then so is the more complex proposition A&B:
A
B
A&B
The order of the premises does not matter: one might equally well conclude
B&A.
Simplification Similar remarks apply to the rule(s) called simplification:
A&B
B
A&B
A
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4.3.2
F ORMS
INVOLVING DISJUNCTION
Weakening is the pair of rules:
A
A∨B
B
A∨B
This rule is so-called because the conclusion one reaches is, generally speaking,
logically weaker than the premise. For if we know, for example, that A is true,
we also know that at least one of the propositions A, B is true, but the converse
doesn’t hold.
Disjunctive syllogism
A∨B
¬A
B
A∨B
¬B
A
This rule is the basis of arguments which proceed by the elimination of
alternatives, for example:
You’ll either pay me now or pay me later.
You won’t pay me now.
So you’ll pay me later.
Similar rules might be formulated for cases in which there are more than two
possible outcomes, for example: if it’s either A or B or C, but it isn’t A and it
isn’t B, then it must be C. In the memorable phrase of Sherlock Holmes:
When you have eliminated all which is impossible, then whatever
remains, however improbable, must be the truth.
Recall that negation (¬) simply changes true statements into false ones, and
false statements into true ones. With respect to truth and falsity, then, applying
negation twice just gets you back where you started. With this in mind, we
will sometimes tacitly add or drop double negations. Thus, for example, we
will count the inferences on the left below as “sort of” instances of Disjunctive
syllogism:
“Sort of” DS
Actual DS
¬P ∨ Q, P ∴ Q
¬P ∨ Q, ¬¬P ∴ Q
P ∨ ¬Q, Q ∴ P
P ∨ ¬Q, ¬¬Q ∴ P
¬P ∨ ¬Q, P ∴ ¬Q
¬P ∨ ¬Q, ¬¬P ∴ ¬Q.
¬P ∨ ¬Q, Q ∴ ¬P
¬P ∨ ¬Q, ¬¬Q ∴ ¬P
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4.3.3
F ORMS
INVOLVING CONDITIONALS
Our first two rules can be justified based on the observation that if a conditional
A → B is true, we do not have both A true and B false. Hence if the conditional
is true and one of the latter two things occurs, the other does not.
Modus ponens deals with the case where A is true, which rules out the falsity
of B:
A→B
A
B
It is usually applied so automatically that people are not even aware of having
used it.
Modus tollens deals with the case where B is false, which rules out the truth
of A:
A→B
¬B
¬A
This rule is used in the following inference:
If you had locked the door, it would be closed.
The door isn’t closed.
So you didn’t lock it.
With modus tollens, for the reasons discussed above, we will sometimes
tacitly add or drop double negations. Thus, for example, we will count the
following as “sort of” instances of modus tollens:
“Sort of” MT
Actual MT
¬P → Q, ¬Q ∴ P
¬P → Q, ¬Q ∴ ¬¬P
P → ¬Q, Q ∴ ¬P
P → ¬Q, ¬¬Q ∴ ¬P
¬P → ¬Q, Q ∴ P
¬P → ¬Q, ¬¬Q ∴ ¬¬P
Hypothetical syllogism is the rule which justifies chaining conditionals in
certain circumstances:
A→B
B→C
A→C
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4.4
S OME
COMMON INVALID ARGUMENT FORMS
Common mistakes in reasoning rooted in the use of invalid argument forms
(in the belief that they are in fact valid) are called formal fallacies. We mention
two of them here, both having to do with conditionals.
Affirming the consequent
A→B
B
A
Perhaps because of their similarity to modus ponens arguments, many people
are fooled by arguments of this form. The following example should suffice to
show that the form is invalid:
If this cat just had kittens, then it is a female.
It is a female.
Therefore it just had kittens.
Denying the Antecedent: again, a common fallacy, perhaps due to the similarity between this argument form and the valid modus tollens.
A→B
¬A
¬B
A modification of the above example will prove the invalidity of this form:
If this cat just had kittens, then it is a female.
It didn’t just have kittens.
Therefore it’s not a female.
E XERCISES
I. Symbolize the following arguments using the dictionary provided, and identify the form of inference used (either modus ponens, modus tollens, affirming
the consequent, or denying the antecedent—including the “sort of” cases), stating at the same time whether or not the argument forms are valid.
D ICTIONARY: P = We’ll have a picnic. R = It will rain. D = The plants die. W =
The plants are watered. J = Joe gets off work by ten. T = Joe gets his essay done
on time. E = Joe gets an extension.
1. If it rains, we won’t have a picnic. But it will rain. So we won’t have a
picnic.
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2. If it rains, we won’t have a picnic. We won’t have a picnic. So it will rain.
3. If it doesn’t rain, we’ll have a picnic. We will have a picnic. So it won’t
rain.
4. If it doesn’t rain, we’ll have a picnic. But it will rain. So we won’t have a
picnic.
5. We’ll only have a picnic if it doesn’t rain. We won’t have a picnic. So it’s
going to rain.
6. We’ll only have a picnic if it doesn’t rain. It’s going to rain. So we won’t
have a picnic.
7. We’ll only have a picnic if it doesn’t rain. It’s not going to rain. So we will
have a picnic.
8. We’ll have a picnic unless it rains. It isn’t going to rain. So we’ll have a
picnic.
9. The plants will die unless they are watered. But they will be watered. So
they won’t die.
10. The plants will die unless they’re watered. But they won’t be watered.
So they’re going to die.
11. The plants would be dead if they hadn’t been watered. The plants aren’t
dead. So they must have been watered.
12. The plants would be dead if they hadn’t been watered. The plants are
dead. So they must not have been watered.
13. Joe will finish his English essay on time only if he gets off work by 10. He
will get off work by 10. So he’ll finish his English essay on time.
14. Joe would only have finished his essay on time if he had gotten off work
by 10. He didn’t finish his essay on time. So he must not have gotten off
work by 10.
15. Joe would only have finished his essay on time if he had gotten off work
by 10. He did finish his essay on time. So he must have gotten off work
by 10.
16. Joe would only have finished his essay on time if he had gotten off work
by 10. He didn’t get off work by 10. So he must not have finished his
essay on time.
II. Symbolize the following arguments as best you can given the dictionary
provided. Identify the symbolized forms, and say whether or not they are
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valid. In some cases, the symbolized forms are invalid even though the original
arguments are valid. Identify these cases and explain what is going on.
D ICTIONARY: O = I’ve told you once. T = I’ve told you a thousand times.
1. If I’ve told you once, I’ve told you a thousand times. But I’ve told you
once. So I’ve told you a thousand times.
2. If I’ve told you once, I’ve told you a thousand times. But I’ve told you a
thousand times. So I’ve told you once.
3. If I’ve told you once, I’ve told you a thousand times. But I haven’t told
you once. So I haven’t told you a thousand times.
4. If I’ve told you once, I’ve told you a thousand times. But I haven’t told
you a thousand times. So I haven’t told you once.
4.5
C HAINS
OF VALID INFERENCES , OR PROOFS
A formally valid inference necessarily preserves truth: if we begin with one
or more propositions A, B, C, . . . , and validly infer a different proposition, M,
from them, then that proposition must also be true provided that all of A, B, C, . . .
are. If we then validly infer yet another proposition N from the larger set of
propositions A, B, C, . . . , M, we know that it must be true if all of A, B, C, . . . , M
are. But since M has to be true whenever all of A, B, C, . . . are, we also know
that N must be true whenever A, B, C, . . . are. Consequently, whatever conclusion we reach from a set of premises by a sequence of valid inferences will
follow from (be implied by) this set of premises, even if other premises, deduced along the way, are used in some of these inferences. This shows that
chains of valid inferences (or, as they are sometimes called, proofs) themselves
constitute valid inferences.
Let us give a simple example. We will show that the conclusion C&D follows from the premises A&B, A → C, B → D by deducing it in a series of
steps using some of the valid forms mentioned above. We begin by listing our
premises, drawing a line under the last one, and drawing a line along the left
side:
1.
2.
3.
A&B
A→C
B→D
Premise
Pr.
Pr.
We now note that A follows from A&B by the rule we called simplification.
We make this inference the next step in our proof, using the notation ‘Simp., 1’
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to indicate that we inferred A from line 1 using this rule:
1.
2.
3.
4.
A&B
A→C
B→D
A
Premise
Pr.
Pr.
Simp., 1
Next, noting that B also follows from line 1, we add a new line:
1.
2.
3.
4.
5.
A&B
A→C
B→D
A
B
Premise
Pr.
Pr.
Simp., 1
Simp., 1
We now use modus ponens on lines 2 (A → C) and 4 (A) to infer the conclusion
C, and similarly infer D from B → D (line 3) and B (line 5):
1.
2.
3.
4.
5.
6.
7.
A&B
A→C
B→D
A
B
C
D
Premise
Pr.
Pr.
Simp., 1
Simp., 1
m.p., 2,4
m.p., 3, 5
Finally, we apply the rule called conjunction to lines 6 and 7 to obtain the conclusion C&D:
1.
2.
3.
4.
5.
6.
7.
8.
A&B
A→C
B→D
A
B
C
D
C&D
Premise
Pr.
Pr.
Simp., 1
Simp., 1
m.p., 2,4
m.p., 3, 5
conj., 6, 7
What we have here is a complete proof, showing how the conclusion C&D follows from the premises (lines 1–3) via a series of valid inferences.
This is a formal proof: we derive a sequence of formulas from given premise
formulas using valid forms of inference. But it also provides a proof pattern
that shows the formal validity of countless arguments, e.g.:
The government will pass tax cuts yet unemployment will continue
to increase. If the tax cuts are passed, however, revenues will decrease. On the other hand, if unemployment continues to increase,
expenditures will also increase. So revenues will decrease and expenditures will increase.
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Some Valid Forms of inference:
N EGATIONS
DN
DN
¬¬A
A
A
¬¬A
C ONJUNCTIONS
Conj.
Simp.
Simp.
A
B
A&B
A&B
A
A&B
B
D ISJUNCTIONS
Weak.
Weak.
DS
DS
A
A∨B
B
A∨B
A∨B
¬A
B
A∨B
¬B
A
C ONDITIONALS
MP
MT
HS
A→B
A
B
A→B
¬B
¬A
A→B
B→C
A→C
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E XERCISES
I. Each of the following questions contains a list of premises and a conclusion.
Show that the conclusion follows from the premises by constructing an appropriate chain of inferences, or proof, as illustrated above.
1. Premises: A&B, B → C; Conclusion: A&C
2. Premises: A → (B&C), ¬(B&C), A ∨ D; Conclusion: D
3. Premises: A → B, A ∨ C, D, D → ¬B; Conclusion: C
4. Premises: A → ¬B, A&D, ¬B → C, (D ∨ E) → F ; Conclusion: C&F
5. Premises: A → B, A ∨ C, ¬B; Conclusion: C
6. Premises: A → B, A ∨ C, ¬C; Conclusion: B
7. Premises: A, ¬¬A → ¬¬B; Conclusion: B
8. Premises: A&B, A → C, B → D, ¬(C&D) ∨ E; Conclusion: E
9. Premises: A → (B ∨ C), A&¬B; Conclusion: C
10. Premises: (A ∨ B) → (C ∨ D), A&¬D; Conclusion: C
11. Premises: A → B, B → C, ¬C; Conclusion: ¬A
12. Premises: A&(B&C), A → D, (B∨E) → F ; Conclusion: (C∨G)&(D&F )
13. Premises: A → B, B → C, D → E, E → F, ¬C&¬F ; Conclusion:
¬A&¬D
14. Premises: U ∨ W, P → R, Q → S, P &Q, (R&S) → (T &¬U ); Conclusion: W &T
15. Premises: ¬A → ¬B, B; Conclusion: A
16. Premises: (A ∨ B) ∨ C, C → ¬D, ¬A&D; Conclusion: B
17. Premises: A → C, B → D, (C&D) → (E&¬F ), E → (F ∨ G), A&B;
Conclusion: G
18. Premises: A&¬B, (C ∨ A) → (B ∨ D), C → ¬D; Conclusion: ¬C
19. Premises: (A&B) → (C ∨ D), ¬A → E, F → ¬D, (B&¬E)&F ; Conclusion: C
20. Premises: A, A → C, B → D, ¬C ∨ ¬D; Conclusion: ¬B
II. Symbolize the following arguments, then show that they are valid by constructing proofs of their conclusions from their premises.
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1. If Al didn’t go to Toronto, then either Bob or Carol did. But Al didn’t go
and neither did Carol. So Bob went to Toronto.
2. If Peters went to the lecture, then Quine didn’t. Either Quine went, or
Russell didn’t. If Sellars went, then Russell did. But Peters did go. So
Sellars didn’t.
3. If Peters went to the lecture, then Russell did too. But if Quine went,
Sellars did too. Either Peters or Quine went, but Sellars didn’t. So Russell
went.
4. Either Peters or Quine didn’t go to the lecture. If Sellars went, then Quine
did too. Now Peters did go to the lecture. So Sellars didn’t.
5. Peters went to the lecture along with Quine, but Russell didn’t go. If
Sellars had gone, Russell would have gone too. If Quine had gone but
Sellars didn’t, then either Tarski or Veblen would have gone. But Veblen
didn’t go. So Tarski must have.
6. If Peters had gone to the lecture, then Quine would have as well. Either
Peters or Russell went. If Sellars went, then Quine didn’t. And Sellars
did go. So Russell went.
7. If Peters had gone to the lecture, Quine would have gone too. Now either
Quine didn’t go, or else Russell did. But we know that Russell didn’t go.
So Peters didn’t go.
8. If Peters goes, the Quine will too. And if Quine goes, Russell will be sure
to tag along. But Russell isn’t going. So neither are Peters and Quine.
9. If Peters went to the lecture, then if Quine didn’t go, Russell and Sellars
did. Now Peters went, but not both Russell and Sellars did. So Quine
went too.
10. Carnap only goes to the pub if Wittgenstein doesn’t. But Wittgenstein
went to the pub, and so did Neurath. And Neurath never goes to the
pub unless Hahn does too. But whenever Hahn and Neurath both go
to the pub, either Carnap or Schlick goes too. So Schlick was there, but
Carnap wasn’t.
4.6
P ROOFS
BY REDUCTION TO ABSURDITY
Our final topic in this chapter is a special kind of argument called argument by
reduction to absurdity (in Latin, reductio ad absurdum), indirect arguments or sometimes also apagogic arguments. In such arguments, we add to the premises the
opposite (i.e., the negation) of the proposition we wish to prove. We then proceed to deduce, by valid inference, a proposition which cannot be true (generally speaking, this will be a contradiction, though sometimes arguments which
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arrive at an obviously false conclusion are also called arguments by reduction
to absurdity). We then reason as follows: if all our premises (including the assumption) had been true, then (since our reasoning was valid) the conclusion
would also have to be true. But since this conclusion isn’t true not all of our
premises can be true (i.e., they are inconsistent). It follows that if all the original premises were true, the additional assumption would have to be false. But
then it follows that if all the original premises had been true, the opposite (i.e.,
the negation) of the additional assumption would also have to be true. That is,
the opposite of our assumption follows from the original premises.
Reductio arguments are very common in mathematics. Here is an example:
Theorem: If a square number n2 is even, then so is its root n.
Proof. Suppose that n2 is even, but that n is not even (this is the opposite of
what we want to prove). Then n is odd, and so bigger by one than some even
number; we can therefore say that n = 2k + 1 for some number k. Then, doing
the algebra:
n2 = (2k + 1)2 = 4k 2 + 4k + 1 = 2(2k 2 + 2k) + 1
That is, n2 is one bigger than an even number (namely, 2[2k 2 + 2k]), and hence
is odd. But n2 is also even, by assumption. Hence the number n2 must be both
even and odd (contradiction). This is impossible. So our assumption that n is
not even must be false. We conclude that n is even.
Here is another example, due to Galileo. Galileo Galilei (1564-1642) is famous for, among other things, having discovered the law of free fall, in particular for the somewhat counterintuitive discovery that bodies of different
weights fall at the same speed. His opponents, following Aristotle, thought
otherwise: they maintained that heavier bodies fall faster than lighter ones, all
other things being equal. Here is a thought experiment Galileo used to refute
them.
Suppose that a lighter body is attached to a heavier body by a rope. If the
lighter body falls more slowly than the heavier one, it will act as a drag on the
heavier one, and the two together will fall more slowly than the heavier body
alone. But if we gradually shorten the rope until the two bodies are touching,
we get a single, heavier body, which should (according to Aristotle) fall more
quickly. So if heavier bodies fall more rapidly than lighter ones, the two bodies
would have to fall both more and less rapidly than the heavy body alone. This
is impossible. So the light and the heavy body must fall at the same speed.
One final example is drawn from philosophy. In the dialogue entitled Euthyphro, Socrates discusses the nature of piety with a fellow named Euthyphro.
Euthyphro suggests that pious acts can be defined as those which are pleasing
to the Gods. At first, Socrates interprets this to mean that an act is pious if it is
pleasing to one or more of the Gods, and impious if it is displeasing to one or
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more of them. He then notes that, according to what people say, the Gods have
many disagreements: what pleases Zeus may not please Poseidon, and so on.
But if so, on Euthyphro’s definition, the same actions will be both pious and
impious, which is absurd. Hence the proposed definition must be incorrect.
We can add a rule to accommodate reductio proofs to the simple proof system we used above. According to this rule, we may add an assumption to our
premises. Once we have derived a contradiction, we may conclude the opposite of our assumption. Here is a simple example, showing that ¬A follows
from ¬(A ∨ B).
1.
2.
3.
4.
¬(A ∨ B)
A
A∨B
¬A
Premise
assumption for Reductio
Weakening, 2
RAA, contradiction on lines 1, 3
E XERCISES
Prove the following by reduction to absurdity:
1. Premise: A; Conclusion: ¬¬A
2. Premise: A&¬B; Conclusion: ¬(A → B)
3. Premise: ¬A; Conclusion: ¬(A&B)
4. Premise: A&B; Conclusion: ¬(A → ¬B)
5. Premises: ¬(A&B), A; Conclusion: ¬B
6. Premises: ¬A&¬B; Conclusion: ¬(A ∨ B)
7. Premises: ¬A&B; Conclusion: ¬(A ∨ ¬B)
8. Premises: A&B; Conclusion: ¬(¬A ∨ ¬B)
9. Premises: A → B, A ∧ ¬C; Conclusion: ¬(B → C)
10. Premises: A → B, C → D, ¬B&¬D; Conclusion: ¬(A ∨ C)
11. Premise: ¬A&¬B; Conclusion: ¬(A&B)
12. Premise: ¬(A ∨ B); Conclusion: ¬(A&B)
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C HAPTER 5
B ASIC S YLLOGISTIC AND V ENN D IAGRAMS
5.1
I NTRODUCTION
This chapter deals mainly with the class of argument forms called categorical
syllogisms. It shows 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 variety 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; some of them are studied in courses devoted to formal or symbolic
logic.
5.2
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, syllogisms were the primary topic of most logicians through the middle ages and
even well into the modern 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 modern usage. It is too narrow because it is restricted to valid arguments, while modern usage recognizes
1 Prior
analytics I, 1 (24b 18f).
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both valid and invalid syllogisms. It is also too broad, because not all valid
arguments are of the form now called syllogistic. Modus tollens, for example:
P → Q, ¬Q ∴ ¬P
is not considered a syllogism, though it is a valid form, and seems to fit Aristotle’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 ourselves even further, by ignoring the modal syllogisms and restricting our attention 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 modern logicians understand claims of the form ‘All P are Q’ also
takes some getting used to. The odd case is when there are no P s. 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 P s.
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):
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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).
5.3
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 of discourse 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 different 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:
U
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:
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A
U
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.
Something is A.
A
x
U
While an ‘x’ outside the circle indicates that at least one thing is not A:
Something is not A.
x
A
U
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.
A
x
x
U
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Finally, if we simply wish to indicate that there is something, without specifying whether or not it is an A, we may put an ‘x’ on the circumference of the
A-circle, like this:
There is something.
x
A
U
Shading is used to indicate that a region is empty. If, for example, we shade
in the entire interior of the A-circle, this represents a situation where nothing
whatsoever is an A:
Nothing is A.
A
U
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:
Everything is A.
A
U
We would encounter this situation, for example, if the universe of discourse
were again restricted to people, and the term A was ‘mortal’.
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It is also entirely possible that the entire universe of discourse is empty:
There is nothing
A
U
This would be the case, for example, if the universe of discourse were restricted 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 inferences. We shall see a few examples of such disagreements later on.
5.4
T WO
TERMS ; CATEGORICAL PROPOSITIONAL FORMS
When two terms are involved, we draw two overlapping circles, like this:
Two terms; the basic set-up
A
B
U
These circles divide the universe of discourse into four separate regions:
Regions
1
2
A
3
4
B
U
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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 A-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.
Some A is not B
1
x
2
A
3
4
B
U
Similarly,
Some B is not A
1
2
A
3
4
x
B
U
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
1
2
A
3
x
4
B
U
An x outside of both circles indicates that there is something that is neither
A nor B:
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Something is neither A nor B
1
3
2
A
4
B
x
U
Finally, multiple xs will express combinations of such propositions, e.g.,
Something is both A and B and something is neither A nor B
1
3
x
2
A
4
B
x
U
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 A-circle, but on the circumference of the B-circle, like this:
Something is A
A
x
B
U
Similar possibilities exist for shading. If, to begin with, we were to shade
in the part of the A-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
1
2
A
3
4
B
U
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Similarly:
All B are A
1
2
A
3
4
B
U
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:
No A is B
1
2
A
3
4
B
U
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 B-circle, or both. More familiarly:
Everything is either A or B.
1
2
A
3
4
B
U
Here are a few other possibilities, including combinations of shading and
xs:
All and only A are B.
1
2
A
3
Nothing is either A or B.
4
B
1
U
2
A
3
4
B
U
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Nothing is A but something is B
1
2
A
3
Nothing is B and everything is A
4
x
B
1
U
2
A
3
4
B
U
E XERCISES
I. Represent the following statements with Venn diagrams:
1. All people are mortal. (P, M)
2. No dog is a fish. (D, F)
3. Some dogs are brown. (D, B)
4. Some dogs are not brown. (D, B)
5. Not all lawyers are rich. (L, R)
6. Only women can get pregnant. (W, P)
7. Some people are happy, and some are not. (P, H)
8. All and only even numbers are divisible by 2. (E, D)
9. Some non-members will attend. (M, A)
10. No people are unhappy. (P, H)
11. There are no carrots, but there are some potatoes. (C, P)
12. There are both potatoes and carrots. (C, P)
13. There are neither potatoes nor carrots. (C, P)
14. The only thing we have is carrots. (C)
15. We only have potatoes and carrots (though of course, nothing is both a
potato and a carrot). (C, P)
16. All is lost, but something has been found. (L, F)
17. There is something, but we don’t know if it’s a carrot. (C)
18. There is something, and it’s definitely not a carrot. (C)
19. There is something, and it is a carrot. (C)
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20. There is something, but we’re not sure if it’s a potato or a carrot. Maybe
it’s neither. (C, P)
21. No carrots are potatoes. (C, P)
22. All potatoes are non-carrots. (P, C)
23. Some non-carrots are not potatoes either (C, P).
24. There is something. (U)
25. Nothing whatsoever exists. (U)
II. Given the dictionary below, state what is expressed in the following diagrams in plain English. Assume that the universe of discourse is restricted to
people living in a town called Mudville.
D ICTIONARY: L= Lawyer, R= Rich, D= Doctor, H= Happy.
6.
1.
x
x
D
H
H
R
U
U
7.
R x L
U
D
L
8.
U
R
H
9.
H
L
H
L
U
5.
10.
x
D
H
14.
U
U
L
R
U
4.
R
L
13.
U
x
H
D
U
3.
x
L
H
12.
U
x
R
D
U
2.
U
11.
R
15.
H
U
H
L
U
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16.
x
L
18.
H
R
U
19.
x
R
H
U
x
H
U
20.
17.
x
L
D
H
U
L
U
5.5
S OME
SIMPLE INFERENCES JUSTIFIED WITH
DIAGRAMS
V ENN
A side-by-side comparison of some of the above diagrams reveals logical connections between certain propositional forms. Consider, to begin with, the following two diagrams:
Some A are not B
1
x
2
A
3
All A are B
4
B
1
U
2
A
3
4
B
U
The left diagram indicates that there is something in region 2, while the one
on the right indicates that region 2 is empty. There is no way both can hold
simultaneously. Thus if ‘Some A are not B’ is true, ‘All A are B’ must be false,
and if the latter is true, the former must be false. These forms (A, O) are what
logicians call contradictories: whenever one of them is true, the other is false.
One consequence of this is that the statement forms:
• Some A is not B.
• Not all A are B.
are equivalent.
This relationship is the basis of the method of proof by counterexample, whereby
we show the falsity of a universal proposition (‘All P are Q’) by giving a socalled counterexample, an object that is P but not Q. For example, we can prove
the falsity of the universal claim
All birds can fly.
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by pointing to a single bird that can’t, e.g., an ostrich or a kakapo.
The same holds for the E- and I- forms, as the following pair of diagrams
shows.
No A is B
Some A is B
1
2
A
3
x
1
4
B
2
A
3
4
B
U
U
Thus we can use the method of counterexamples to disprove universal negative claims as well. For example, to show the falsity of the proposition:
No bird can swim.
it is enough to point to one bird that can, e.g., a penguin.
Older logicians maintained that the following forms of inference (from the
universal to the particular) were also valid:
No P is Q.
∴ Some P is not Q.
All P are Q.
∴ Some P is Q.
Modern logicians deny this, something that is reflected in Venn diagrams. For
when we depict a proposition of the form ‘All P are Q’, we simply shade in
region 2:
All P are Q
1
2
P
3
4
Q
U
We do not enter an ‘x’ in region 3, as would be required to represent the proposition ‘Some P is Q’.2 Thus the particular does not follow from the universal
according to our system.
Sometimes, the consideration of a single diagram reveals valid inferences.
Consider again the diagram for ‘Some A is B’:
2 The difference of opinion is sometimes described by saying that the ancients took universal
propositions to have existential import, while modern logicians do not. We can adapt Venn diagrams
to ancient practice by interpreting their universal propositions as follows: the ancient ‘All P are Q’
= the modern ‘All P are Q and there are Ps.’
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Some A is B
1
2
A
3
x
4
B
U
Notice that this diagram is symmetrical: it will also serve as a representation of ‘Some B is A’. Thus these two forms are equivalent; whenever one
statement of this form is true, the other is as well, and we are justified in inferring either one from the other.
Similarly in the case of ‘No A is B’:
No A is B
1
2
A
3
4
B
U
Again, note the symmetry: the same diagram also represents ‘No B is A.’
Thus these two forms are equivalent; whenever one statement of this form is
true, the other is as well, and we may again infer either one of the pair from
the other.3
By contrast, a side-by side comparison of the following diagrams shows
that in cases where the diagrams are asymmetrical, there is no equivalence:
Some A is not B
1
x
2
A
3
Some B is not A
4
B
1
U
2
A
3
4
x
B
U
3 It is interesting that in both of these above cases English has two different sentences corresponding to one Venn diagram. If we were to think of the system of Venn diagrams as a kind of
picture language, we could say that in these cases at least, the picture-language more accurately
reflects the states of affairs it represents. The reason for this difference is that English is a linear
language, where one word has to come after the other.
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All A are B
1
2
A
3
All B are A
4
B
1
U
2
A
3
4
B
U
E XERCISES
Construct pairs of Venn diagrams to determine which of the following inferences are valid.
1. Some politicians aren’t crooked. So not all politicians are crooked.
2. Some politicians are crooked. So it’s not true to say that none are.
3. Some people who aren’t politicians are crooked. So not all politicians are
crooked.
4. Some people who aren’t politicians are crooked. So not everyone who is
crooked is a politician.
5. Some politicians are crooked. So some aren’t.
6. All politicians are crooked. So some are.
7. Some politicians are crooked. So some crooked people are politicians.
8. All politicians are crooked. So all crooked people are politicians.
9. Some birds can’t swim. So some animals that can swim aren’t birds.
10. Some birds can’t swim. So some animals that can’t swim are birds.
11. No marsupials can fly. So some marsupials can’t fly.
12. No marsupials can fly. So nothing that can fly is a marsupial.
13. All birds lay eggs. So every animal that lays eggs is a bird.
14. Some birds lay eggs. So some animals that lay eggs are birds.
15. Some birds lay eggs. So all birds lay eggs.
16. Some elephants can’t fly. So no elephants can fly.
17. Some dogs aren’t brown. So some brown things aren’t dogs.
18. Some brown things aren’t dogs. So not all dogs are brown.
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19. Not all animals that lay eggs are birds. So some birds don’t lay eggs.
20. Not all animals that lay eggs are birds. So some animals that aren’t birds
lay eggs.
21. No politicians are perfectly frank. So anyone who is perfectly frank is not
a politician.
5.6
R EPRESENTING
SYLLOGISTIC FORMS WITH
DIAGRAMS
V ENN
Since the categorical syllogisms we are interested in involve three terms, we
will use three overlapping circles to depict them:
Three circles
B
A
C
U
These circles divide the universe of discourse into 8 distinct regions:
Three circles: regions
8
B
5
6
7
A
1
C
2
3
4
U
The objects contained within each of these regions may be characterized as
follows:
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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
In this case, there are hundreds of ways to shade regions and place xs. Instead of taking the time to consider some of these these separately, we will
proceed directly to diagramming syllogisms. In a nutshell, the procedure is as
follows: 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 B-circle (regions 2 and 3):
Barbara: first premise
8
B
5
A
1
2
6
3
7
4
C
U
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
8
B
5
A
1
2
6
3
7
4
C
U
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When we superimpose these two diagrams, we obtain this one:
Barbara: both premises
8
B
5
A
1
2
6
3
7
4
C
U
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 diagram. That conclusion, recall, looks like this:
Barbara: conclusion
8
B
5
A
1
2
6
3
7
4
C
U
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: both premises
Barbara: conclusion
8
B
8
B
5
A
1
2
6
3
5
7
4
C
A
1
2
U
U
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3
7
4
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By way of contrast, consider the related argument form EEE:
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
8
B
5
A
1
2
6
3
7
4
C
U
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
8
B
5
A
1
2
6
3
7
4
C
U
When these two are superimposed, we obtain this one:
EEE: both premises
8
B
5
A
1
2
6
3
7
4
C
U
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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
8
B
5
A
1
2
6
3
7
4
C
U
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: both premises
EEE: conclusion
8
B
8
B
5
A
1
2
6
3
5
7
4
C
A
1
2
6
3
7
4
C
U
U
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.
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We begin by representing the universal premise ‘All R are Q’:
Baroco: universal premise
8
Q
5
P
1
2
6
3
7
4
R
U
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
8
Q
5
P
x 2
1
6
3
7
4
R
U
But now the conclusion, ‘Some P is not R’ is clearly represented in our diagram: 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.’:
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First premise: All P are Q
8
Q
5
P
1
2
6
7
3
4
R
U
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? 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
8
Q
5
P
1
2
6
x
3
7
4
R
U
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.
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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 subregions 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 depicted.
(a) If so, the form is valid.
(b) Otherwise, invalid.
E XERCISES
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. All tigers are mammals. All tigers are striped. So all mammals are striped.
3. No tigers are mammals. All striped animals are tigers. So no striped
animals are mammals.
4. No tigers are mammals. All tigers are striped. So no striped animals are
mammals.
5. All tigers are mammals. Some striped animals are tigers. So some striped
animals are mammals.
6. Some tigers are mammals. Some mammals are striped. So some tigers
are striped.
7. All lawyers are educated. Some rich people are educated. So some rich
people are lawyers. (U= the set of all people) (L, E, R)
8. No educated people are lawyers. Some rich people are educated. So
some rich people are not lawyers.
9. No lawyers are educated. Some rich people are educated. So some lawyers
are not rich.
10. No lawyers are educated. All rich people are educated. So no rich people
are lawyers.
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11. No lawyers are educated. All educated people are rich. So no rich people
are lawyers.
12. No lawyers are uneducated. All educated people are rich. So all lawyers
are rich.
13. All doctors are educated. No one who is educated is a clown. So no
doctor is a clown. (C, E, D)
14. All doctors are educated. No clown is educated. So no clown is a doctor.
15. All doctors are educated. Some doctors are clowns. So some clowns are
educated.
16. All doctors are educated. No clowns are educated. So no clowns are
doctors.
17. All doctors are educated. Some clowns are uneducated. So not all doctors
are clowns.
18. No doctors are uneducated. No clowns are educated. So no doctors are
clowns.
19. Some students work full time. Everyone who works full time is busy. So
some students are busy. (S, W, B)
20. All students are busy. Everyone who works full time is busy. So all students work full time.
21. All students work full time. Some people who are busy do not work full
time. So some people who are busy are not students.
22. Some students work full time. Some students are busy. So some people
who work full time are busy.
23. No students work full time. Some people who are busy do not work full
time. So some people who are busy are students.
24. All radicals are extremists. All fascists are extremists. So all fascists are
radicals. (R, E, F)
25. All fascists are extremists. Some radicals are not extremists. So not all
radicals are fascists.
5.7
S YMBOLIZATION :
QUANTIFIERS
Finally, it seems worthwhile to briefly discuss how modern logicians symbolize
A-, E-, I, and O-propositions and related forms. The starting point is the pair
of logical operators that are most often expressed in English by the words ‘all’
and ‘some’, as for instance in the sentences:
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All hockey players can skate.
Some hockey players have a full set of teeth.
The operator ‘all’ is called the universal quantifier, and is usually symbolized
with an upside-down A, ‘∀’; while ‘some’ is called the existential quantifier, and
symbolized with a backwards E: ‘∃’. In symbolic logic, quantifiers are always
used in conjunction with variables of quantification. Thus we never see a quantifier all by itself. Rather, we see them connected to variables such as x, y or z.
To illustrate: logicians will understand a sentence like:
Everything is beautiful.
to mean:
For every x, x is beautiful.
They write this as follows:
∀xBx
Similarly,
∃xBx
would symbolize:
There exists an x such that x is beautiful.
Or, in plain English:
Something is beautiful.
Two points should be recalled about the way logicians use the word ‘some’
(cf. p. 72, above)
1. When they say Some A is B, this should not be taken to imply that some
A is not B. Thus ‘Some A is B’ still counts as true when all A are B.
2. Logicians say that ‘Some A is B’ is true even in cases where there is only
one such A. Thus, in order to avoid any possible misunderstanding, you
should read ‘∃xBx’ as follows: ‘There exists at least one x such that Bx’.
Note that our notation gives us two places to insert a negation symbol in
formulas such as ‘∀xBx’ or ‘∃xBx’. This allows us to express the following
(where Bx = x is beautiful, as above):
∀xBx
¬∀xBx
∀x¬Bx
¬∀x¬Bx
∃xBx
¬∃xBx
∃x¬Bx
¬∃x¬Bx
Everything is beautiful.
Not everything is beautiful.
Everything is not beautiful, or
Nothing is beautiful.
Not everything is not beautiful, or
Something is beautiful.
Something is beautiful.
Nothing is beautiful.
Something is not beautiful.
Nothing is not beautiful, or
Everything is beautiful.
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We note in passing the equivalence of the following pairs of formulas:
Everything is B.
Not everything is B.
Everything is non-B.
Not everything is non-B.
∀xBx
¬∀xBx
∀x¬Bx
¬∀x¬Bx
¬∃x¬Bx
∃x¬Bx
¬∃xBx
∃xBx
There isn’t anything that isn’t B.
Something is not B.
Nothing is B.
Something is B.
The Categorical Forms of Traditional Logic
Let us now work out symbolizations for the categorical forms of traditional
syllogistic:
• (A) Universal affirmative: All A are B.
• (E) Universal negative: No A is B.
• (I) Particular affirmative: Some A is B.
• (O) Particular negative: Some A is not B.
Let us begin with the form ‘All A are B.’ Here, ‘A’ and ‘B’ mark places for
predicates, things that may be said (either truly or falsely) of certain objects, for
example, ‘is a dog’ and ‘is a mammal’, as in ‘Fido is a dog.’ It is natural in many
cases to think of predicates as standing for certain properties, e.g., the property
of being a dog or a mammal, etc. Given this, we can give the truth conditions
for a claim of the form ‘All A are B’ in either of the following two ways:
• The predicate ‘B’ applies (is true of) every object to which the predicate
‘A’ applies. More simply, everything that can truly be said to be A can
also truly be said to be B.
• Every object that has the property designated by ‘A’ also has the property
designated by ‘B’.
Now we already know how to deal with the word ‘everything’ as it occurs
here—namely, by introducing a quantifier, ‘∀x’. The remainder of the truth
conditions is nicely captured by the conditional: if x has the property A then it
also has the property B. Thus we arrive at the following symbolization:4
∀x(Ax → Bx)
To say that no A is B is the same as saying that all As are non-B, which leads
us to the following symbolization:
∀x(Ax → ¬Bx)
4 We note in passing that this symbolization does not exactly coincide with the way older logicians understood sentences of this form, since it turns out to be true even when there are no As.
The ancient logicians denied this. For them, the above proposition would count as false if its subject (i.e. A) was an empty term. To adequately formalize the ancient understanding of ‘All A are
B’, we would have to render it as ∃xAx ∧ ∀x(Ax → Bx).
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At the same time, to say that no A are B also says that there is nothing that is
both A and B, which gives rise to an alternative symbolization:
¬∃x(Ax&Bx)
Finally, the forms called (I) and (O) are easily dealt with. To say that Some A
are B amounts to saying that there is something that has both of the properties
A and B, or, in symbols:
∃x(Ax&Bx)
While to say that some A are not B is just to say that there is something that
has the property A but lacks the property B, i.e.:
∃x(Ax&¬Bx)
Thus we have the following:
Propositional Form
English
Symbolization
Universal Affirmative (A)
All A are B
Universal negative (E)
No A is B
∀x(Ax → Bx)
Particular affirmative (I)
Some A is B.
Particular negative (O)
Some A is not B.
∀x(Ax → ¬Bx)
∃x(Ax&Bx)
∃x(Ax&¬Bx)
Because ‘All A are B’ can be symbolized as
∀x(Ax → Bx)
and ‘All B are A’ as
∀x(Bx → Ax)
simply by reversing the direction of the conditional, i.e. by replacing ‘Ax →
Bx’ with its converse ‘Bx → Ax’, a statement of the form:
All B are A
is often called the converse of the statement
All A are B
The converse ‘All B are A’ may also be stated in English as ‘Only A are B’.
This is why sentences of the form ‘All and only A are B’ are sometimes used to
express ‘All A are B and all B are A.’
Similarly, the statement ‘All non-B are non-A’
In symbols: ∀x(¬Bx → ¬Ax)
may be called the contrapositive of ‘All A are B,’ i.e., ∀x(Ax → Bx).
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E XERCISES
I. Symbolize, using the dictionary provided.
Dictionary: V x = x is a vertebrate; Ix = x is an insect; M x = x is a mammal;
Ax = x is an animal; U x = x is a unicorn; W x = x is white.
1. All mammals are vertebrates.
11. There are no unicorns.
2. Some vertebrates are not mammals.
12. All unicorns are mammals.
13. Some insects are not unicorns.
3. All insects are invertebrates.
14. Some unicorns are mammals.
4. There are animals.
15. There isn’t anything that isn’t an
animal.
5. Some animals are insects.
6. Everything is an animal.
16. Not everything is not an insect.
7. Not everything is a mammal.
17. All unicorns are white.
8. There is something that is not an
insect
18. There are no white insects.
9. No mammals are insects.
19. There are white mammals.
20. Some mammals are not white.
10. No vertebrates are insects.
II. For each of the following statements, state (a) the converse and (b) the contrapositive.
1. All mammals are warm blooded.
2. If Joe wins the lottery, he’ll buy a Ferrari.
3. If an insect lays eggs, it’s a female.
4. All prime numbers larger than 2 are odd.
5. Whenever I catch a cold, my lungs get congested.
6. All bats are flying mammals.
7. Every differentiable function is continuous.
8. All flowering plants produce seeds.
9. All of Shakespeare’s comedies are written in blank verse.
10. If Bernhard wrote it, it’s sure to be depressing.
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C HAPTER 6
N ON - DEDUCTIVE ARGUMENTS
Just as there are many kinds of deductive arguments, so too is there a great
variety of non-deductive arguments, and in both cases a variety of methods
are used to assess them. In the next chapter, we will look at some special kinds
of non-deductive arguments in greater detail. For the moment, however, the
discussion will be limited to some general remarks.
We say that in a good argument, the premises support the conclusion. This
kind of talk suggests a useful analogy. Think of constructing an argument as
being something like ordinary construction, say, of a house. If I want to build
walls that will hold up a roof, I need to make sure that:
• The wood I use is solid;
• It is pointed in the right direction (i.e., so that it bears weight); and
• There is enough of it to hold up the roof.
Similarly, in an argument, we can ask:
• Are the premises acceptable?
• Do they provide any support at all for the conclusion? (Are they relevant?)
• If so, do they provide enough support? (Are they adequate?)
In light of this analogy, an unacceptable premise is like a rotten two-by-four.
Even if we put it in the right place, it wouldn’t help to hold anything up. An
irrelevant premise is like a piece of wood that’s not connected to the structure—
it’s not doing anything to support the conclusion, even if it is solid. An inadequate set of premises, finally, is like a structure that isn’t strong enough to hold
all the weight it is supposed to.
Normally, the questions are asked and answered in the sequence indicated
above. For if the premises of an argument are unacceptable, we can already
dismiss the argument as a bad one; and if the premises of an argument are
completely irrelevant to the conclusion, then they provide no support at all for
it, and this can never be enough for any conclusion worth arguing for.
There is an important difference in the way the three questions must be
asked. The question of acceptability is always asked about each premise separately, and this is also often the case for the question of relevance. The question
of adequacy, by contrast, concerns the entire set of premises that are deemed
both acceptable and relevant: we want to know whether it is rational to accept
the conclusion based upon all the evidence that has been presented and found
worthy, not just some piece of it.
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Failures in all three aspects are quite common. Many arguments start from unacceptable or irrelevant premises, and conclusions are often not well-supported.
Some forms of failure, called fallacies, are so widespread that they have been
given special names. A number of these are described in Chapter 9.
Assessing relevance can often be tricky. Many people mistakenly judge
some claims to be relevant to others, but we also often fail to see relevance
where it does exist, since the connections between things are often anything
but obvious. Who knew, for example, that the introduction of flushable baby
wipes to the market would lead to the emergence of fatbergs? So a policy of
caution, and a willingness to revise your opinions when new evidence comes
in, seem to be called for.
When asking about adequacy, the best place to begin is with a careful consideration of the conclusion. We should ask: How strong is it? What is its
scope? A weaker conclusion obviously requires less support than a strong
one. The question is one of how well the evidence provided by the premises
matches the strength with which the conclusion is stated. You will often find
that people go too far in drawing conclusions, putting forward claims that
aren’t adequately supported by the evidence they provided. In such cases,
we will do well to look for any weaker conclusions that are supported by the
given premises. More rarely, people draw conclusions that are weaker than the
evidence warrants. Here, too, we do well to consider different conclusions, in
this case stronger ones.
For example, suppose we are told that three people living close to a chemical plant have been stricken with cancer. Now consider the following possible
conclusions:
• The chemical plant is causing cancer at an alarming rate in the local population.
• The chemical plant is causing cancer in certain individuals in the vicinity.
• The chemical plant is quite likely the source of the local cancer epidemic.
• The chemical plant is probably the cause of some local cases of cancer.
• The chemical plant may be the cause of some local cases of cancer.
It should be clear that only the weakest of these is adequately supported by the
premise.
Often, when someone argues for a given conclusion with the aim of persuasion (of others or even of him- or herself), the reasons that speak in favour
of the conclusion are prominently displayed, while those that speak against it
are somehow minimized or even left unstated. For this reason, it is also often
important when assessing adequacy to ask whether reasons counting against
the conclusion have been given proper consideration.
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E XERCISES
I. Are the following claims acceptable without supporting arguments? If so, to
what extent and on what basis? If not, why not?
1. The roof won’t collapse while I sleep.
2. 1+1=2
3. Penguins can’t fly.
4. Mammals are the only warm-blooded animals.
5. No parrots can swim.
6. All parrots can fly.
7. All swans are white.
8. Aliens have visited Earth in flying saucers.
9. The Earth is at the centre of the universe and does not move.
10. The Earth revolves around the sun, and is in constant motion.
11. No sound is a colour.
12. Red is a colour.
13. Space is not a discursive concept of things as such; rather, it is a pure
intuition.
14. Elvis lives.
15. Taxation is theft.
16. eπi = −1
17. If everyone carried a gun, we would all be much safer.
18. If no one carried a gun, we’d all be much safer.
19. I have a pretty good chance of winning the lottery.
20. Someone has a good chance of winning the lottery.
21. There’s a good chance that someone will win the lottery.
22. Some people who smoke a pack of cigarettes a day don’t get cancer.
23. I won’t get cancer if I smoke only one pack of cigarettes a day.
24. Global warming is a myth invented by environmentalists.
25. Global warming is a serious problem.
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26. Abortion is wrong in all circumstances.
27. Abortion is permissible under some circumstances.
28. Abortion is permissible under all circumstances.
29. The Prime Minister is elected by all Canadians.
30. Commonly used vaccines cause autism.
II. In each of the following questions, suppose that claim (a) is used as a premise
in support of claim (b). Assess the relevance of (a) to (b) in each case, and
explain your verdicts. In some cases, you may find that further information
would be required to give a definitive judgment. If that is the case, explain
what more we would need to know.
1.
(a) Owls eat meat.
(b) No birds are vegetarians.
2.
(a) James Garfield was shot and killed while President of the USA.
(b) Not all elected politicians complete their terms of office.
3.
(a) Kim Campbell became leader of the federal Progressive Conservative party only after the party had become deeply unpopular.
(b) Women only get an opportunity to lead in politics when they don’t
stand a chance to win.
4.
(a) Bruce lives in Australia.
(b) Bruce is a philosopher.
5.
(a) The president of the Tobacco growers’ association has stated that
smoking does not cause lung cancer.
(b) Smoking does cause lung cancer.
6.
(a) The number of cases of brain cancer per capita in the USA increased
almost immediately after Nutrasweet began to be used there in diet
drinks.
(b) Nutrasweet causes brain cancer.
7.
(a) Some people who believe in ghosts are insane.
(b) Ghosts don’t exist.
8.
(a) Smith was convicted of murder.
(b) Smith committed murder.
9.
(a) Smith committed murder.
(b) Smith either was, or will be, charged with murder.
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10.
(a) Vancouver may play in the Stanley Cup finals this year.
(b) The Vancouver Police Department should be prepared to pay more
overtime than usual this year.
11.
(a) Some wild mushrooms are deadly poisonous.
(b) Everything that is natural is good for you.
12.
(a) Many people who favour tax cuts would themselves benefit if tax
cuts were approved.
(b) Tax cuts would be bad for the economy.
13.
(a) Hitler was strongly opposed to vivisection.
(b) Vivisection is an acceptable practice.
14.
(a) Jones’ family would suffer greatly if he went to jail for theft.
(b) Jones did not steal anything.
15.
(a) The crime rate is declining.
(b) The government’s policies on crime are working.
16.
(a) The economy is performing well.
(b) The government deserves to be re-elected.
17.
(a) Enderby has not been charged with any crime.
(b) Enderby has done nothing wrong.
18.
(a) What O’Brien did was against the law.
(b) What O’Brien did was wrong.
19.
(a) Joyce knows that Sam is in Montreal.
(b) Sam is in Montreal.
20.
(a) Carruthers believes that the President has committed crimes.
(b) The President has committed crimes.
III. Put the following arguments into standard form, and them assess them by
answering the following questions. Are the premises acceptable? Are they relevant to the conclusion? Do they provide adequate support for the conclusion?
1. Sometimes police officers have to deal with people who are violent and/or
mentally disturbed, and all too often these people are carrying knives or
other weapons. In such circumstances, it is hardly surprising that the police do not want to get too close, and may find it necessary to use their
guns, often with tragic results. But it doesn’t have to be this way. There
is a better alternative. Tasers also allow police to deal with potentially
violent, armed people at a distance but, unlike guns, they are never fatal.
Yet police departments have been reluctant to issue tasers to all officers
on patrol. Why? It’s obvious that this is the right thing to do.
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2. It is a well-known fact that fewer women than men pursue careers in
highly technical fields like engineering and computer science. Talking to
my nieces, who are in their teens, I have discovered the main reason for
this. Although they are both good at math and science, it’s obvious that
the lifestyle of a scientific career just doesn’t appeal to them. And who
can blame them? Who wants to spend most of their life in a lab with no
windows, where people barely speak to one another, and everyone wears
a white coat?
3. There is a broad scientific consensus that global warming is a major problem that is already costing the world economy billions of dollars annually, and is very likely to cause even more serious problems, political as
well as economic, in the future. It is clear that controlling the emission of
greenhouse gases would, over the long term, help to reduce these costs
significantly. Yet government regulation is both inefficient and ineffective
in dealing with such problems, so it is clear that a market-based solution
is the best way to go. And here there is really only one viable option:
namely, a cap and trade system. So it’s obvious that we should support
cap and trade.
4. North Korea, as a communist country, has a centrally planned economy,
while the USA has a free-market economy. The results speak for themselves: North Korea is one of the poorest countries in the world, the USA
one of the richest. It is clear that free-market economies perform better
than centrally planned ones.
5. The past two centuries have shown us one example after another of religious and ethnic conflict: Israelis and Palestinians, Serbs and Croats,
the Catholics and Protestants of Northern Ireland, the Chinese and the
Tibetans, and so on. At the same time, we cannot point to a single case
of different ethnic or religious groups living together without conflict. It
just isn’t possible for different ethnic or religious groups to live together
in peace and harmony.
6. It is well known that mistakes are sometimes made in our judicial system—
innocent people do get convicted from time to time. We know from experience, too, that such mistakes may not be corrected right away—it may
take years for the truth to come to light. Indeed, we cannot be certain that
some mistakes will never be discovered and corrected. These reflections
should make us think twice about supporting capital punishment. For if
killing an innocent person is such a serious crime that we think it merits
death as a punishment, do we really want the state to kill people who
may for all we know be wrongly convicted?
7. Taxation is just a bad idea. For one thing, it costs a huge amount of
money just to collect the taxes: did you know that at least fifty percent
of the amount actually collected by the federal government simply goes
to pay the employees of the Canada Revenue Agency? In the second
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place, the government wastes money all the time. Individual citizens are
never foolish with their own money: they always know better what to do
with it than the government does. This is obvious as soon as we think
about what the government does with tax money. Can anyone point to
a single example where money is more wisely spent by the government
than by citizens?
8. Going to university is a privilege that students should have to pay for.
Most of them can easily afford to pay more if they stop spending so much
on things like cell phones, beer and cigarettes. So it’s obvious that students should be paying higher tuition. Besides, there’s no other way to
raise the extra money the university needs.
9. We all know smokers are far more likely to develop lung cancer than nonsmokers, but it seems that the emphasis on discouraging young people
from smoking has been misplaced. According to Statistics Canada, Canadians between the ages of 20 and 40 are 40 times more likely to die in a
car accident than from lung cancer, while for Canadians aged 60–80 years,
the pattern is reversed: Canadians between these ages are 25 times more
likely to die from lung cancer than in car accidents. It follows that young
people have far more to fear from driving than from smoking. Instead
of discouraging smoking among the young, we should encourage safe
driving; it is the older people who need to stop smoking, not the young.
10. Government-run gambling—lotteries, casinos, and the like—are a significant source of revenue for Canadian provinces, and helps to pay for
important government services. It is true that gambling can be addictive, and that it causes serious problems for some people and especially
for their families. We should not forget, however, that if the government
were not providing opportunities to gamble, others would do so, and
that there would be problem gamblers no matter what. In addition, only
if gambling is government-run can we be sure that adequate measures
are in place to deal with problem gambling. So it is best for governments
to continue in the gambling business.
11. A number of public health professionals have supported the opening of
more safe injection sites, where intravenous drug users can obtain clean
needles, and inject themselves under medical supervision. This is a bad
idea. For one thing, by making drug use easier, safe injection sites lead
to an increase of drug use, which makes the problem worse. In addition,
these sites have to go somewhere, but wherever they are put, they are a
huge problem for the people who live nearby, who have to deal with drug
use in their neighbourhood. On the whole, then, there is every reason to
oppose the proposal.
12. In the first-past-the-post system, the candidate who receives more votes
than any of the others is elected. In Canada, where there are more than
two competitive parties, this means that a candidate can get elected while
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receiving fewer than half the votes cast. This happens a lot. Indeed, majority governments have been elected with fewer than 40 percent of the
votes cast. The result is that very often the majority of voters do not see
their choices reflected in the composition of government. The system of
proportional representation does not have this problem: using this system, the composition of the legislature mirrors the percentage of votes
cast for each party—if your party got twenty percent of the votes, for instance, it gets twenty percent of the seats. Voters get what they voted for,
which is much fairer. In light of this, it’s clear that we should replace the
outdated first-past-the-post system with proportional representation.
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C HAPTER 7
I NDUCTIVE AND C AUSAL R EASONING
7.1
I NTRODUCTION
In valid deductive arguments, the truth of the premises guarantees the truth
of the conclusion. But many arguments are not like this, among them most of
those in which we draw conclusions from experience. This chapter looks at
three kinds of such non-deductive arguments, namely:
• Inductive generalizations: arguments in which our premises tell us about
a number of things of a certain kind and we draw a conclusion about a
larger number of things of that kind.
• Causal arguments, in which we draw a conclusion stating that one thing
or kind of thing causes another.
• Applications: Arguments in which a general claim is used to support a
conclusion about a particular thing or things of a certain kind.
Here are some examples of arguments of these kinds:
• Inductive generalization: A recent poll of residents of Toronto indicates that
only 30% of those surveyed believe that the Mayor should resign. So the
majority of Torontonians do not think that the Mayor should resign.
• Causal argument: In communities where Fluoride is added to the municipal water supply, the rate of dental cavities is lower than in communities where Fluoride is not added. Also, in communities where Fluoride
is added to the water supply, we find a higher rate of cavities in families
who drink bottled water, which does not contain added Fluoride. Finally,
studies have found a strong positive correlation between the presence of
Fluoride in saliva and the rate at which tooth enamel remineralizes, and
the latter has been associated with a lower incidence of cavities. So there
is some reason to believe that Fluoride prevents cavities.
• Application: On average, Canadian women live longer than Canadian
men. Alice and Ralph, a Canadian woman and and a Canadian man,
respectively, are the same age. So Ralph will likely die before Alice does.
In the remainder of this chapter, we’ll take a closer look at each of these kinds
of argument.
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7.2
I NDUCTIVE
GENERALIZATIONS
Many inductive generalizations belong to the following form:
x% of things of kind A observed so far have had property P.
So approximately x% of all things of kind A have property P.
Here, based upon what we find in a number of observed cases, we draw a conclusion covering all cases, whether we have observed them or not. When, for
example, we conclude that all (i.e., 100% of) crows are black because all the
ones we have seen so far have been black, we make an inference of this sort.
The observed cases are usually called the sample, while the collection of
things spoken of in the conclusion is called the population. In an inductive generalization, then, we draw a conclusion about a population based on information concerning a sample.
A little reflection should suffice to convince you that we constantly rely on
inductive generalizations. Most of our knowledge about the general features
of nature, for example, is derived by means of such arguments. We know that
fire burns, that snow is cold, that leaves turn colour and fall in autumn, that
day follows night, that people grow old and eventually die, that blackflies bite,
and so on, because we and a great many others have had repeated experiences
of fire burning, etc., and have drawn general conclusions from them.
By their very nature, inductive generalizations are almost always fallible,
since the part of the population we have not seen might upset the expectations
we have formed, however reasonably, from looking at a sample.1 Even if all the
crows we have seen so far were black, the next one might happen to be white,
or green, etc. Even so, inductive generalizations are not all equally likely to
lead us astray: there are better and worse ones. And the following factors play
an important role in determining how reliable an inductive inference will be.
• Quality of data
• Representativeness of sample
• Sample size
7.2.1
Q UALITY OF
DATA
No matter how good your reasoning is, if the data you begin with are unreliable, you should not be surprised if the conclusions you draw are too. As they
say in computer science:
Garbage in, garbage out.
1 This will not be the case if the sample includes every object in the population, in which case we
may speak of complete induction. The claim that no twentieth-century US President was born in
South Dakota, for example, can be verified in this way.
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In many cases, the source of problems with data is quite obvious—beginning
students in high school chemistry, for example, should not be expected to make
measurements as accurately as professional chemists. And even the most accurate measurements can be recorded or transcribed incorrectly, etc. Equipment
can malfunction, as can the computers or software used to record and analyze
data. When people report their own opinions, there are even more sources of
error. In surveys, for example, the wording of a question can significantly affect responses, as can the order in which questions are asked, the number of
choices presented, and so on.
Bias and fraud are also common causes of unreliable data. In many cases,
this is obvious. In some cases, only the results favouring a certain conclusion
are recognized as valid. The others may either be suppressed or discounted.
Consider, for example, some of the most common methods used to commit
electoral fraud: ballots for the “wrong” candidates may be destroyed, or declared “spoiled”. If necessary, extra marks may be added to make sure that
they are spoiled. And extra ballots may be added to the box to round out the
total appropriately. Given this, it is hardly surprising to find an overwhelming majority of the votes deemed valid supporting the dictator in charge of
the election. Similarly, students will sometimes fudge their data to obtain the
conclusion they think their teacher wants, throwing out some measurements,
altering or even inventing others, etc. A number of prominent cases show that
more than a few professional scientists are also prepared to commit fraud.
Could someone with a financial or other interest deliberately falsify data?
You bet. This is why you always do well to carefully consider the source of the
data used in inductive arguments.
As noted above in Chapter 3, bias can also be unintentional and even unconscious. A measuring instrument that is not calibrated correctly, for example, will predictably produce skewed results, and it is clear that this can happen
quite easily without us noticing it. Furthermore, the expectations of those performing measurements or making observations can also influence their perceptions. In the early days of the microscope, for instance, people imagined they
saw all sorts of marvellous things when they looked through the lens.2 Not to
speak of UFOs.
7.2.2
R EPRESENTATIVENESS
OF THE SAMPLE
When a sampling method is random, every individual in the population has an
equal chance of being selected. Most sampling methods used in actual life do
not measure up to this ideal. As a result, there is an even greater risk that what
we find in the sample will not accurately reflect the real situation. We call such
samples unrepresentative, biased, or skewed.
Suppose, for example, that a professor of education wants to do research
on student engagement, and has developed a questionnaire for this purpose.
It might seem that nothing could be easier than to find a suitable sample. One
2 See, e.g., http://www.wellcomecollection.org/full-image.aspx?page=999&image=a-monster-soup.
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way would be to ask other professors to distribute the questionnaire in their
classes. Another would be to send it by e-mail to all the students enrolled at the
university. Yet both of these methods would be flawed, because the possibility
cannot be discounted that less engaged students would be less likely to be
present in class and less likely to respond to such a survey, thus skewing the
results.
Many opinion polls are conducted by telephone. This method can produce a biased sample for a number of reasons. First, some people do not have
phones at all, and hence would never be sampled. Others who do have phones
may not even answer, or refuse to respond to the poll if they do. It is hardly
obvious that those who do answer will be representative of the general population in their responses to the questions asked in a given poll.
Still worse are self-selected samples. For example, a web-site may provide
readers with the opportunity to participate in a poll. Only those who choose
to do so are sampled, and we have every reason to fear that they won’t be representative of the general population (not no mention the possibility of some
people taking the poll repeatedly). In self-selected samples of this sort, it only
costs time to participate. In others, money may also be required. Needless
to say, those willing to pay to participate in an opinion poll make up a very
special subset of the general population.
Though, as noted above, most sampling methods fall short of the ideal of
perfect randomness, some are more flawed than others, and more likely to
result in weak inductive arguments. So when evaluating an inductive generalization, you should think carefully about possible sources of bias in sampling,
and adjust your estimate of the strength of the argument accordingly. Similarly,
when thinking about drawing a conclusion yourself, you should take care not
to claim more than is justified given the sampling method that produced the
data.
7.2.3
S AMPLE SIZE
The size of the sample, finally, is always an important factor to consider when
assessing the strength of an inductive generalization. All other things being
equal, a larger sample makes for a stronger argument. This being said, the improvement is not directly proportional to the increase of the sample size. After
a while, the benefit of increasing the sample size by a constant amount becomes
progressively smaller. And at a certain point, it can become prohibitively expensive to attain a significant increase in accuracy.
When reputable firms report the results of public opinion polls, they will
indicate the number of people surveyed as well as an estimate of the survey’s
reliability. These usually take the following form:
The results are deemed to be accurate to within ±x% nineteen times
out of twenty.
In a typical poll of Canadian public opinion, for example, the sample size is
somewhere between 1000 and 2000, and the results claimed to be accurate to
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within 3 or 4 percent nineteen times out of twenty.
You should not hurry past this claim. What it says is that there is a 95%
chance that the results of the poll accurately reflect the true situation to within
x%. Since unlikely things do sometimes occur, you should not entirely discount the possibility that this particular poll does not attain that degree of accuracy. For this reason, it is always best to look at several polls on the same
subject, and even at aggregate polls, where the results of different polls are combined.
The other thing that is important to note is that even if the poll were accurate to within x%, the true value could still depart significantly from the
reported one. Put otherwise, what the poll provides is a range of values, rather
than one precise value. Suppose, for example, we read a poll of voting preference in Ontario that tells us that the Progressive Conservatives (PCs) have
35% support and the New Democrats (NDP) 32%, where the poll is claimed
to be accurate within ±4% nineteen times out of twenty. Even if we make the
most likely assumption that the actual numbers are within ±4% of the sample numbers, the poll just tells us that support for the PCs is between 31 and
39%, while support for the NDP is between 28 and 36%. In particular, we do
not have enough information to say that the PCs enjoy more support than the
NDP. All the same, if you pay attention you will find quite a few news reports
where just that sort of mistaken inference is made.
7.2.4
C OMMON FLAWS
OF INDUCTIVE GENERALIZATIONS
Some mistakes made in inductive generalizations are so common that they
have been given special names. We consider a few of these here.
To begin with, we have a deeply ingrained tendency to think that there
must be a pattern to events, a tendency that can lead us to find order when the
data indicate no such thing (the predictable world bias). This inclination is no
doubt the ultimate source of many of the errors we make in inductive generalizations. It should be resisted. For even if nothing is completely random, in the
vast majority of cases we will never be in a position to know whatever order
and structure there is.
One prominent type of failure in this respect is called the Texas sharpshooter
fallacy, in memory of the mythical gunman who shot repeatedly at the side
of a barn and then painted a bull’s-eye where a number of bullet holes were
clustered purely by chance. In real life, one encounters this fallacy when people
tinker with raw data until they find a way to make them look significant, and
then stop.
For example, suppose we obtain information concerning the home addresses of people who have developed cancer in a given city over the last ten years.
We then check to see if there is any significant difference in the rates of cancer between those who live within a certain distance from high voltage power
lines and those who live farther away. By varying the distance, we might well
find one for which there is a significant difference. At this point, many people
would be ready to conclude that this difference proves that it’s dangerous to
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live close to high-voltage wires. They shouldn’t, any more than we should consider the Texas gunman an excellent shot. For the distance, like the location of
the bull’s-eye, was chosen precisely because it yielded a significant result. But
the clumping up of cancer cases, like that of bullet holes, can and does occur
purely by chance, and the existence of some distance that produce a significant
difference is to be expected. This is not to mention even more serious problems involved in lumping all the different forms of cancer together, and other
problems besides.
When people favour a certain conclusion, they may be more likely to notice
individual cases that favour that conclusion and to overlook those that count
against it. This effect is known as the confirmation bias. When this is done deliberately, and data are selected and presented precisely because they support
a given conclusion, we encounter the fallacy of cherry picking.
One source of unconsciously biased sampling is the conspicuousness or
salience of certain events or things. We have a natural tendency to remember dramatic or shocking events while forgetting the less exciting ones. For
instance, plane crashes, though relatively uncommon, are quite spectacular.
Automobile crashes, by contrast, tend to receive less attention, despite the fact
that they are far more common and often just as deadly. Perhaps this is why
some people who are terrified of getting on an airplane don’t think twice before driving a car, perhaps even while talking on a cell phone, texting, etc. For
similar reasons, people may easily be convinced that crime rates are increasing
even when they aren’t because the crimes that are committed are prominently
discussed in the news, while the absence of crime is rarely remarked upon.
Another very common source of bias is a preference for the data that are
easiest to obtain (the availability bias). A researcher, for example, may prefer
to look at trees from his truck rather than to take a long walk through woods
teeming with biting flies (practising roadside forestry). When they can, many
people will rely entirely on data they can find with a simple internet search.
The vast majority of people rely on a small number of sources for news. It
should be obvious from the start that these are not ideal sampling methods.
With respect to the size of the sample, two flaws are usually singled out.
When a conclusion is drawn based on a sample that is too small to support it,
we speak of a hasty generalization. There is also the fallacy of apriorism, where
the sample size is the smallest possible, namely, zero. This is, unfortunately, the
sample size favoured by many people on certain issues, nicely summed up in
the famous quote:
My mind’s made up. Don’t confuse me with facts.
Finally, since inductive generalizations are by their nature fallible, it is important not to regard their conclusions as definitive. If new data come forward
which cast doubt on a conclusion we drew previously, we should be prepared
to revise it. Indeed, in most cases we should continue to seek new data. This
can lead to surprises.3
3 For
an interesting discussion of this and some related topics, see J. Lehrer, “The truth wears
off,” New Yorker, December 13, 2010.
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7.3
C AUSES
AND EFFECTS
Much of what we know, or want to know, or claim to know, concerns causes
and effects. We know, for example, that a hammer blow will drive in a nail
or hurt our thumb, that the Moon’s gravity is responsible for the tides, that
smoking can cause lung cancer, and so on. We would like to know what causes
certain diseases, whether there are any general causes of poverty, societal problems of various sorts, and so on.
Knowledge of causes is valued because it brings understanding and, in
some cases at least, the possibility of changing things for the better. But such
knowledge is in many cases hard to come by: it is not a simple matter to produce a convincing proof that one thing causes another, and easy enough to
make mistakes in this business. But we are an optimistic species on the whole,
and often expect that it will be fairly easy to know the things we want to know,
in particular, what makes things tick, their causes and effects. Not surprisingly,
we are more often than not mistaken in such judgments, especially with respect
to the confidence with which we make them.
We saw above how difficult it can be to produce a truly convincing inductive generalization. Making a solid case for causal claims is even harder, for in
their case we do not merely need to show (by inductive generalization) that A
and B are connected, but also that this connection is not merely accidental. For
in any concrete case, there will be many things that go along with both A and
B. To single out A as the cause of B, we have to convince ourselves that even
if those other things hadn’t been present, A would still have brought about B.
7.3.1
C ORRELATION
AND CAUSATION
We speak of a positive correlation between events, features, or objects of kinds
A and B when the presence of B is more likely given the presence of A. For
instance, one study reported that approximately 17% of men who smoke regularly, but only about 1.5% of those who do not, develop lung cancer. The
chances that a randomly selected smoker will develop lung cancer are thus
considerably (more than ten times) higher than those that a randomly selected
non-smoker will. Thus we have a positive correlation in men between smoking and lung cancer. We speak of a negative correlation, by contrast, when the
presence of A makes the absence of B more likely. The hotter it is outside, for
example, the fewer people go skiing. Temperature is thus negatively correlated with the number of skiers. In a great many cases, of course, we have no
evidence of any significant correlation whatsoever.
If A is a cause of B, then we should expect that the presence of A will make
the presence of B more likely. Thus in order to establish causation, we must
first check to see whether there is any correlation. But while the failure to find
any positive correlation makes a good case against causation, the presence of
a positive correlation is not enough by itself to prove causation. The size of
a person’s vocabulary, for example, is positively correlated with his shoe size,
but this hardly proves that learning new words causes your feet to grow.
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In the most straightforward cases, we may find that there is a positive correlation between A and B:
• Because A is a cause of B;
• Because B is a cause of A;
• Because some third thing causes both A and B; or
• By pure chance.
The first two options show that even when a correlation does exist on account
of a causal relationship, it is possible to get things backwards. Finding a correlation between insomnia and depression might lead us to conclude that depression causes insomnia (i.e., when people are depressed, this prevents them
from sleeping), when in fact the causation might just as easily run the other
way.
The third possibility is easily illustrated by a familiar example. My neighbour’s thermometer goes up and down in parallel to mine. There is thus an
almost perfect correlation, but obviously my thermometer going up or down
doesn’t cause his to do the same. Rather, a common cause, namely, the temperature, is responsible for the correlation.
The last possibility, finally, is too frequently neglected. Many correlations
do occur by pure chance. For instance: in the 1990s, the cities of Kitchener
and Waterloo in Ontario had populations of roughly the same age composition. Waterloo had fluoridated water, while Kitchener did not. About twice as
many people live in Kitchener as in Waterloo, so we should expect the number
of babies born each year in Waterloo to be about one-half the number of those
born in Kitchener. Yet, year in and year out, fewer than one-tenth as many babies were born in Waterloo. There was thus a very strong negative correlation
between fluoridation and the number of births in these two cities. Yet it existed
purely by chance.
By itself, then, mere correlation is not enough to establish causation. What
is? Two things are generally thought to be highly desirable in this connection.
First, we must make every reasonable effort to rule out other possible causes
of a given effect. In some cases, this can be done by running a series of controlled
experiments. In others, by the collection of additional data or reanalysis of existing data.
Second, unless we think that the causal relation in question is fundamental, we should try to discover more basic causes that give rise to the observed
relation. In the case of smoking and lung cancer, for example, it was found
that some of the chemicals present in cigarette smoke had been observed to
react with DNA, and cause genetic changes which had in turn been associated
with cancer. The discovery of this mechanism strengthens the case for causation
by showing how more generally operative and better confirmed causes might
have combined to give rise to this particular effect.
Of course, neither of these procedures will ever result in complete certainty:
such causal arguments, like all non-deductive arguments, are fallible. On the
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other hand, there are plenty of cases (smoking and lung cancer being just one)
where the evidence in favour of causal links is strong enough to justify acting
on the assumption that they do exist. So we should be careful to distinguish
cases where there is insufficient proof of causation from those where there is
substantial, but not absolute, proof. In both cases, we might say that proof is
lacking, or that the science is not settled. But, given that absolute proof is never
to be had, such a claim will usually only be noteworthy if the former is meant.
7.3.2
C ONTROLLED EXPERIMENTS
AND BLINDING
The history of science bears witness to many of the subtleties involved in trying
to verify that causal links exist. A first, important step was the development
of the concept of a controlled experiment. In these experiments, a sample, or
population, is divided into two groups, one of which is treated in a certain
way (e.g., given a certain medication), while the other (the control group) is
simply left alone. Afterwards, we compare the condition of the two groups, to
see if there is any significant correlation between treatment and outcome. For
example, a group of people suffering from arthritis might be divided into two,
and half given an experimental drug. If it turns out that many of the people
receiving the drug report feeling less pain, we might want to conclude that the
drug works. But if a similar proportion of the control group also report similar
improvement, we would not be tempted to draw that conclusion. The control
group thus acts as a check on our natural enthusiasm to leap to conclusions.
So far, so good. Unfortunately, however, the data we obtain in this way
might not be good enough, on account of the so-called placebo effect. It turns
out that people who believe they are receiving treatment often report feeling
better, even when they receive no treatment at all. So even if there was a significant difference between the treatment and control groups, we might still have
grounds to doubt that the people in the treatment group are better because of
the medicine.
The response to this problem was the blind or masked controlled experiment,
where the participants in the study do not know whether they belong to the
treatment or to the control group. In the case of drug studies, for example,
the control group might be given a pill that looks and tastes just like the pill
given to the treatment group, but one that contains only inert ingredients. Since
neither group knows whether or not it has received treatment, it is reasoned,
the placebo effect can be discounted.
Alas, even blind controlled trials have been found to be unreliable in some
cases, because those who conduct the experiments still know who is really receiving treatment and who is in the control group, and they can affect the data
in a variety of subtle ways, for example, by giving cues to the participants, or
showing bias (whether conscious or unconscious) in the recording and interpretation of data.
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These problems are addressed in a double-blind experiment, where neither
the experimenters nor the participants know who belongs to which group. A
further refinement is a so-called triple-blind experiment, where the data obtained in a double-blind experiment are analyzed by someone who does not
know which group is which (they are simply called, say, group A and group
B).
7.3.3
C OMMON MISTAKES
IN CAUSAL REASONING
As noted above, we human beings are quick to assume we know the causes of
things, even when the evidence we possess is completely insufficient to justify
our conclusions. Quite commonly, for example, people latch onto a personal
experience they and perhaps several other people have had, and then leap immediately to the conclusion that there is a causal connection. Parents who observe that their child’s earache disappeared soon after they gave her antibiotics
may conclude that the antibiotic caused the infection to go away. Others, noting that symptoms of autism appeared soon after their child was vaccinated,
have concluded that the vaccines caused the autism. When large windmills are
installed, people living nearby may conclude that some health problems they
encounter soon afterwards were somehow caused by them. And so on. To
draw such conclusions is to commit the fallacy called post hoc, ergo propter hoc
(this happened after that, so that caused this).
While understandable, these conclusions are in no way justified. For personal experiences of this sort are not even sufficient to establish a correlation,
still less causation. Often enough, the testimony people rely on is cherrypicked, or interpreted so as to commit the Texas Sharpshooter fallacy. This
being said, the personal experiences should simply not be discounted either.
Such anecdotal evidence is sometimes the first indication we obtain of causal
connections. And just because it fails to prove that there is a causal connection,
we should not conclude that we have proof that there is no causal connection.
The absence of proof is not a proof of absence.4 This, in turn, is not to say
that every case where anecdotal evidence is presented warrants the expenditure of scarce resources on studies aimed at discovering whether or not there
is a connection.5
Even when there is a genuine correlation between two things, it may, as
noted above, exist for a number of reasons. We can err by assuming that causation must exist when the correlation is there by pure chance; or by assuming
that causation runs one way when in fact it runs the other way (confusing cause
and effect); by overlooking the possibility that a third thing is responsible for
both A and B (common cause).
We may also go wrong by neglecting the possibility of mutual causation or
causal loops. If we just think about hammers and nails and the like, it’s hard to
see how causation could be mutual, A causing B and B also causing A. But
4 See
“Appeal to ignorance”, below, p. 144.
instructive recent case is the “liberation therapy” for Multiple Sclerosis championed by Dr.
Paolo Zamboni.
5 An
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in many situations where we speak of cause and effect, this is exactly what
occurs. Take, for example, the relation between overeating and not getting
enough sleep. Overeating can produce indigestion, etc., which can certainly
interfere with our sleep. On the other hand, it appears that lack of sleep can
also cause people to eat more. As a consequence, the assumption that causation
can only be one-way is just as mistaken as the assumption that correlation must
always be due to causal connections.
It’s Simple (only it isn’t) Finally, this seems a good place to give a general
warning about drawing conclusions about causes and effects in complex situations or systems (e.g., ecosystems, human societies, economic systems). It is
this: it’s hard even for highly intelligent, careful people to make accurate judgments about how such systems work. In fact, people who have studied such
systems the most are often the least confident in making such claims. All the
same, there is rarely a shortage of people willing to say loud and clear that it’s
all very simple (and hence that there is a simple solution).
Is there a problem with crime, for example? Well, it’s all very simple: we
don’t punish criminals severely enough. Persistently high unemployment and
depressed wages? The greed of the wealthy explains it all. Why did the teenage
pregnancy rate go up? Salacious music videos. Why did Detroit go bankrupt?
Unions. No, corrupt city government. No, greedy corporations. Etc., etc. As
H. L. Mencken wrote:
For every complex problem there is an answer that is clear, simple,
and wrong.
7.4
A PPLICATIONS
We turn, finally, arguments in which generalizations are applied to particular
cases. In the simplest cases, these arguments take the form of statistical syllogisms:
x% of P s are Q.
A is a P .
So there is an x% chance that A is Q.
Statistical syllogisms are in fact valid arguments. This is not to say that all
are equally good, however. Much depends upon what we know about A in
addition to its being a P .
Suppose, for example, that Bruno is a Canadian man who happens to be a
heavy smoker. Recent data indicate that about 7.5% of Canadians will develop
lung cancer during their lifetime. With this in mind, I might conclude that
Bruno has a 7.5% chance of developing lung cancer. While this is correct as
far as it goes (namely, insofar as we merely consider Bruno as a Canadian), it
doesn’t go nearly far enough. For the data also show that men are more likely
than women to develop lung cancer (approximately 9% vs. 6%). Considering
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Bruno as a Canadian man, then, we get a different estimate: he has a 9% chance.
If we also add that he is a smoker, the data will provide yet another estimate: he
has a 17% chance. If we add that he is a heavy smoker, things look still worse
for him. Clearly, Bruno would be deluding himself if he stopped at the first
estimate.
Note that additional information can change the estimate in both directions.
Considered merely as a Canadian, for example, Susan has about a 7.5% chance
of developing lung cancer. If we consider her as a Canadian woman, the estimate goes down to roughly 6%. But if we add that she is a smoker, it goes up
again, to roughly 12%.
Generally speaking, the more precisely we determine the subject of our applications relative to the available data, the stronger the argument. Thus thinking of Bruno as a Canadian male, heavy smoker will give us a stronger argument than thinking of him as a Canadian male smoker, which in turn would
give us a better argument than considering him merely as a Canadian male,
etc.
If resources were never an issue, we would like to use all available data
and base our applications upon the most precise determinations of their subjects relative to this data. In this way, all the information available to us would
be used. As nice as this would be, we are often in no position to realize this
ideal. Consider a doctor treating a patient. The medical literature is vast, containing all sorts of generalizations based on controlled experiments. But to use
much of this information, the doctor would have to perform certain diagnostic
tests, which may use scarce resources, cost a lot of money, make demands on
the patient’s time and health, etc. Because running all the tests that might be
relevant is impractical, not to say inconceivable, the doctor will rarely if ever
be in a position to use all the relevant information that is available. In this,
as in many things medical, choices impose themselves, and a sort of triage is
necessary. We have to weigh the odds of various possible courses of action,
and choose the one that, among those that are feasible, seems most likely to
produce a successful result. Many real world applications involve complex
trade-offs of this sort.
This much can be said, however. If, in making an application, we neglect to
consider reliable data that might have been obtained with relatively little effort,
our argument will be weaker through our own fault. That this is the case for
most of us much of the time can hardly be disputed.
E XERCISES
I. Comment on the strengths and weaknesses of the sampling methods described below.
1. Conducting public opinion polls by means of automated telephone calls.
2. Measuring public opinion by means of a voluntary internet poll posted
on the website of the Ottawa Sun.
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3. Conducting public opinion polls by means of a bulk-mailed survey: (a)
with a prepaid return envelope (b) without one.
4. Looking at several social media sites to determine how the provincial
government is perceived by the voting public.
5. Doing a word-frequency search on the archives of three Canadian newspapers in order to come up with a list of the most important events in the
world in 2013.
6. Comparing the time devoted to various topics on the television newscasts of the national networks in Canada to determine the most discussed
Canadian news stories of 2013.
7. Asking for reports from local bird-watchers in order to estimate populations of various species present in Ottawa during a certain month.
8. Estimating the incidence of Attention Deficit Hyperactivity Disorder in
children worldwide by looking at data from developed countries such as
the USA, Canada, or Japan.
9. Putting out a light trap on a summer night to collect insects in order to
determine which species are present in a given area and to estimate the
size of their populations.
10. Randomly surveying 1500 45–50 year old Ontario residents by means of
personal interviews in order to determine how many people between
these ages in Ontario (a) have saved money for retirement; (b) have cheated
on their taxes.
II. Discuss the strengths and weaknesses of the following arguments, which
involve inductive and/or causal reasoning.
1. Everywhere in the world that needle exchanges have been established,
the rate of heroin use is high. Isn’t it obvious that making things easier
for addicts just encourages drug use, and so makes the problem ten times
worse? We should abolish needle exchanges and throw addicts in jail.
2. Police reports indicate that the number of robberies in Ottawa increased
sharply in February 2011. We may conclude that the number of criminals
engaged in such robberies also increased sharply at that time.
3. Over her four years of study in the Faculty of Engineering at the University of Ottawa, Yasmina has had 35 male instructors and only 5 female
instructors. She concludes that a solid majority of instructors at the University of Ottawa are male.
4. Data collected by Statistics Canada consistently show that, on average,
people with university degrees have significantly higher incomes than
people without such degrees. The difference is large enough, moreover,
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to more than make up for the earning potential lost during the years of
study. In the long run, university graduates simply earn more than the
rest of the population, and also pay more taxes. It is clear that university education creates economic benefits both for individuals and for the
country as a whole.
5. Since university education creates these economic benefits, it seems clear
that it is in the interest of society to send more people to university. That
way, a higher proportion of people can enjoy above-average incomes.
6. A survey conducted by a Canadian polling firm indicated that while
55% of Americans surveyed said they did not object to terrorism suspects being tortured, an even greater proportion (57%) said that they did
not object to the use of enhanced interrogation techniques on terrorism suspects. The same poll indicates that while torture of terrorism suspects
was said to be always justified by 13% of those polled, enhanced interrogation techniques were deemed always justified by 26%. Since ‘torture’
and ‘enhanced interrogation techniques’ are just two names for the same
thing, this shows beyond a doubt that euphemisms do make a difference
to public opinion. (The survey asked the questions to a randomly selected group of more than 1000 Americans; the results are claimed to be
accurate to within plus or minus 3.5% nineteen times out of twenty).6
7. As the deadline for RRSP contributions approaches, many financial institutions advertise their investment products. Prominently featured are
the performance figures for the previous year. This is obviously important information. For someone making a contribution, it makes sense to
go with the fund that earned the most money last year, right?
8. Data collected by the National Institutes of Health in the USA indicate
that, on average, poor people have significantly more health problems
than people who are not poor. It should be obvious from this that the
stress of living in poverty makes people unhealthy.
9. A recent study found that people who had sex four or more times a week
had, on average, significantly higher salaries than those who had sex less
frequently. The Globe and Mail reported on the study under the headline:
“Want a bigger paycheque? Have more sex, new study suggests.” Briefly
discuss the appropriateness of this headline.
10. The average life expectancy in the world today is a little short of 67 years.
So Fred, who has lived his whole life in Vancouver, should not expect to
live past 70.
11. Students at the Fairleigh Normal University were surveyed to see if there
was interest in raising fees in order to provide greater access to recreational and sports facilities on weekends. The survey was carried out
6 Angus Reid, as reported by UPI, 24 February, 2010:
http://www.upi.com/Top News/US/2010/02/24/.
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last Saturday: questioners were posted at various places on campus, and
attempted to question every twentieth student they encountered. In all,
over 300 students were questioned. The results showed that 70% of those
surveyed were in favour. It seems, then, that a substantial majority of
students support the proposed changes.
12. I have used the library every Friday night for the past three years while
carrying out my research, and I have consistently found that most of the
people working there are part-time workers, who don’t know enough to
help researchers. What is more, there seem to be very few people working in the library, and when I have tried to complain to the management,
I have had a hard time finding someone. The library is obviously understaffed and poorly run.
13. Cardiovascular disease is the world’s biggest health problem—it kills
more Canadians than any other cause.
14. The 1960s the trend towards married women working outside the home
began to emerge. Within a few years we began to see a significant increase in the divorce rate, which has now reached alarming proportions.
Obviously, if we value the family as an institution we should try to prevent married women from working outside the home.
15. Many people were appalled by the use of nuclear weapons against Japanese cities in World War II, and also by the massive bombing raids launched
against civilian populations in Germany and Japan. As horrible as these
acts of war might have been, however, they were completely justified.
Before, both Japan and Germany were dominated by militaristic and imperialistic cultures, and had embarked upon wars of increasing scale and
viciousness. After the massive bombing raids, however, they became two
of the most pacifistic countries on earth. These campaigns of terror bombing cured these two nations of militarism.
16. It has been documented that during the sixties and seventies, the incidence of sexually transmitted diseases and pregnancy among teenagers
increased in direct proportion to the number of students who had sex education programs in the schools. Obviously, the sex education programs
served to increase the number of teenagers having sex.
17. An extensive study in North America and Europe has shown that the
incidence of heart attacks and high blood pressure is lower in men who
drink a moderate amount of red wine on a regular basis. Clearly, there
must be something in the red wine that improves cardiovascular health
in men.
18. In spring, the days get progressively longer, the temperature gets progressively warmer, and all sorts of dormant life—insects, trees, seeds, hibernating mammals—becomes active again. The increase in temperature
must be the trigger for all this activity.
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19. Your mother always told you to bundle up when the weather is cold, and
it turns out she was right—statistics prove that colds occur much more
frequently when the temperature is lower, so it is obviously exposure to
cold temperatures that causes us to catch colds.
20. Malaria is still the world’s most deadly disease: it kills more people every
year than any other disease, including AIDS. It is therefore the greatest
risk to your life: you are more likely to die of malaria than any other
disease.
21. Studies have shown that the number of dental cavities is higher on average for the children of families that drink bottled water compared to
those who drink municipal tap water. This is alarming news for the companies that sell bottled water, since it means that something in their product must be causing cavities.
22. School X consistently has the highest test scores in the city. They must
have a much better teaching staff there than in the other schools.
23. In recent tests administered by the OECD, Canadian schoolchildren did
worse in mathematics than they had in previous years. The decline was
worst in Manitoba and Alberta, which had recently adopted an “inquirybased” approach to teaching mathematics. It’s pretty obvious from this
that the “new, improved” method of teaching was no improvement at all,
and only made things worse.
III. Questions for Discussion
1. Often, when a crime has been committed and a suspect arrested, the police organize a so-called line-up, where the suspect is asked to stand next
to several other people of similar appearance, and the victim asked to
identify the perpetrator. Do the policemen who organize such line-ups
know who the suspect is? Should they?
2. In 2011, the federal government eliminated the mandatory long-form
census in Canada, replacing it by the National Household Survey. Citizens were legally obliged to complete and return the long-form census,
while the National Household Survey is voluntary. Discuss the potential
problems in comparing census data from years before and after 2011, and
possible implications for public policy.
3. Statistics Canada reports the population of Ontario in 2013 as 13,538.0 (in
thousands). They appear to have rounded off the number to the nearest
hundred. Is this a reasonable practice? Why not give an exact figure?
4. Often, when an election result is close, one of the candidates asks for a
recount. Often enough, the recount produces a different result than the
original count. Given this, is it appropriate for election authorities to
report exact vote counts?
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5. If doctors have doubts about the effectiveness of a surgical procedure,
they may think it appropriate to organize a controlled trial. In order fully
to account for the possibility of placebo effects, however, it would seem
to be necessary to perform “placebo surgery”, where participants in the
study are prepared for surgery, and even cut open and stitched up afterwards, without the procedure in question being performed. Discuss
the ethical justifiability of placebo surgery. Could such a trial be doubleblind? Explain.
6. Readers of scientific journals may not be interested in reports of studies
that did not find any significant correlation between the factors considered. For this reason, editors of scientific journals might be less inclined
to publish such results. Supposing this to be the case, how might it affect
the pursuit of truth?
7. Polls conducted before a 2013 Federal by-election in Manitoba produced
wildly varying results. Though the Conservatives ended up winning the
seat by a close margin, one poll showed the Liberals with double-digit
lead in the final days of the campaign, and another, based on a small
sample, indicated that the Liberals had 59% support among likely voters.
Given the result, many concluded that there must have been flaws in the
methods used by the polling firms. Was this reasonable?
8. In many cases that we care about, inductive arguments provide us with
information about the past when we want to know about the future.
When thinking about career options, for example, learning that people
with computer science degrees found good jobs in the past will not help
us at all if things have changed in the meantime and such jobs are not
going to be so plentiful three or four years down the road. Given that circumstances may change and hence that past performance is no guarantee
of future results, why should we look at past data at all?
9. Inductive reasoning, broadly speaking, may be said to rely on the principle that, provided that there are no significant changes in the meantime,
what we will observe in the future will resemble what we have observed
in the past. We expect the sun to rise tomorrow morning, for example,
because it has always done so in the past, and we are not aware of any
change that might prevent it happening tomorrow. What is the basis of
our confidence in this principle?
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C HAPTER 8
L ANGUAGE AND ARGUMENTATION
8.1
I NTRODUCTION
In this chapter, we take a quick look at some features of language use that have
a significant bearing on the evaluation of arguments. The chapter is divided
into two main parts, the first dealing with aspects of language use that make
for better (or worse) arguments, and the second with definitions.
8.2
V IRTUES
AND VICES OF LANGUAGE USE
Language is a tool that can be used for a variety of purposes. Depending upon
what these purposes are, certain features may be considered either a good or
a bad thing. When it comes to reasoning and argumentation, where our main
interest is in discovering the truth, the following generally count as virtues:
• Clarity
• Orderliness
• Precision
• Consistent usage
• Avoidance of prejudicial language
Clarity and Obscurity To begin with, a clear presentation, all other things
being equal, is almost always preferable to an unclear (or obscure) one, since it
is difficult to evaluate an argument when we don’t know what the argument
is.
We say that a passage is clear if it is easy to understand, and readers are
not left in any doubt about what is meant. What counts as clear, however,
can vary from one person and context to the next. Arguments presented to
specialists in a given field, for example, may appear highly obscure to outsiders
who do not know the terminology, haven’t mastered the relevant concepts, and
so on. For those with the appropriate knowledge, however, these arguments
may be perfectly clear. So common courtesy dictates that we should choose the
level of our presentation with its intended audience in mind, explaining where
necessary to ensure that our words are clearly understood.
Obscurity is often unintentional: people may not want to be hard to understand, but simply lack the skills required to express their thoughts clearly.
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In some cases, however, people are deliberately obscure, intentionally writing or
speaking in a way that makes it difficult for others to understand. This can
take the form of using convoluted grammatical constructions, or unfamiliar
words, acronyms, jargon, etc. Sometimes, people take refuge in obscurity because they want others to think that they are deep thinkers, highly educated,
or in on some important secrets. In other cases, it is done in order to avoid
difficult questions or to deceive people. Though there may be good reasons for
deliberate obscurity in rare situations, in most cases it is best avoided.
Orderliness The kind of clarity we just considered is the sort that is found
in the fine details of arguments—in the choice of words, the construction of
sentences, and so on. But clarity may also be found on a larger scale, when
the way an argument is presented leaves no doubt about what its structure is.
Especially in longer arguments, such clarity is an obvious virtue. Arguments
whose structure is not clear, by contrast, pose the same sorts of problems as
those afflicted by other kinds of obscurity: how can we evaluate an argument
when we’re not even sure what it is?
Precision and Vagueness Sometimes, the things we talk about can’t really be
described in precise terms. It might happen, for example, that we find a piece
of music beautiful, but can’t say exactly why. Or we might say that someone
was kind of funny-looking and just leave it at that.
If not much is at stake, or if we are trying to communicate something that is
a little fuzzy, such a lack of precision, or vagueness, doesn’t pose a serious problem. In other cases, however, we need to know with a fair degree of precision,
and vagueness is unacceptable. If, for example, we want to know if we can
catch a bus that runs regularly, it may be enough to be told that one is is coming soon. By contrast, if we want to catch a bus that only runs once a day, we
will probably want to know the exact time it is scheduled to leave. Similarly,
the vague indication that a truck is large may suffice when we want to know if
we can fit a small amount of furniture in it, while a more precise one is needed
in case we want to know whether it will fit under a low overpass. And so on.
People are sometimes deliberately vague, since it helps them avoid commitments that might prove inconvenient. Politicians say things like this, for
example:
We are taking steps to ensure that our children receive the best education possible.
where neither the steps that are to be taken nor the measure of what counts as
better education are specified, so that any number of things might be counted
as partially fulfilling the promise.
And every parent has had this conversation:
When will you clean your room (or take out the recycling or get a
job, a haircut, married, etc.)?
Soon.
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Astrologers, palm-readers, crystal-ball gazers, psychics, and forecasters also
often resort to vagueness, thereby increasing the odds that their predictions
will be counted as true. For instance:
• Aquarius: The problems in your life are serious, but they will not defeat
you. Takes some small steps today to deal with just one of them—which
one doesn’t matter—and you will find yourself on the path to dealing
with the others too.
• Your lifeline is very long. It shows that deep down you are a warm and
caring person. But the Mars regions are also quite rough. I sense there
may be some difficulties somewhere in your relationships.
• Next year will be a turbulent one: we can expect many disasters, both
natural and human, and violent conflicts in various parts of the world.
You can hardly go wrong with these, which tells you how little information is
actually contained in them.
How much precision is enough? Heed the following, admittedly somewhat
vague, advice of Aristotle:
. . . it is the mark of an educated man to look for precision in each
class of things just so far as the nature of the subject admits; it is evidently foolish to accept probable reasoning from a mathematician
and to demand from a rhetorician scientific proofs.1
Consistent Use of Terms and Ambiguity Many words in English have more
than one meaning, and many of them have different shades of meaning. And
it is quite easy to shift from one meaning to another while speaking or writing.
By itself, this needn’t cause a problem, since in many cases the context makes
it sufficiently clear which meaning was intended. Often enough, though, we
can’t be sure which of several possible meanings an author or speaker had in
mind. And this is a problem, because in that case, we have more than one
argument to consider, and can’t always be sure which was intended.
An expression is said to be ambiguous if it has more than one meaning, or
can be interpreted in more than one way. Consider, for example, the sentence:
Mary had a little lamb.
Here, two quite different interpretations are possible, nicely distinguished by
asking whether the following question is appropriate:
With mint sauce?
Ambiguity can give rise to difficulties both in the evaluation and in the production of arguments. If a sentence can be interpreted in several different ways,
to begin with, we may not be sure what claim the arguer actually intended to
1 Nicomachean
Ethics, Book 1, Ch. 3, tr. W. D. Ross.
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make, and hence not know whether or not we should accept it. The problem
is compounded if an ambiguous expression occurs repeatedly in an argument,
for then we may have a considerable number of possible interpretations to sort
through before evaluating what was said.
Ambiguities also complicate the evaluation of the reasoning employed in
an argument. For, as noted above, we often evaluate reasoning by focussing
on patterns or forms of inference, as embodied in the verbal expression of an
argument. But if the same expression is used with different meanings in the
course of an argument, we can easily be fooled into thinking that the reasoning
follows a valid pattern when in fact it does not. Consider, for example, the
following argument:
An average woman in Canada has two children.
Jane is an average Canadian woman.
So Jane has two children.
Superficially, we seem to have an argument here of the valid form:
A Y is a Z.
X is a Y
So X is a Z.
Clearly, though, the original argument is anything but valid.
The problem here is ambiguity, namely, that the expression ‘an average
Canadian woman’ is used with different meanings in the two premises, so that
it isn’t appropriate to represent the argument as belonging to the above form.
Arguments of this kind are said to commit the fallacy of equivocation.2
One especially important kind of ambiguity is very common in discussions
of economic policy. As you know, currencies are subject to both inflation and
(much more rarely) deflation. When comparing amounts from different years,
therefore, we can either report the so-called nominal amounts (the amounts actually paid at the time), or instead report all figures in so-called constant dollars
(where the Consumer Price Index is used to adjust for inflation/deflation). It is
entirely possible, for example, for both of the following statements to be true:
The average family income increased in Canada from 1990 to 1995.
The average family income decreased in Canada from 1990 to 1995.
Where the former claim refers to nominal amounts, the latter to inflation-adjusted
figures.
Ideally, when there is a risk of such misunderstandings, people should
clearly define the terms they use, and consistently use the defined expressions
with the meanings they set out in their definitions. Someone writing about
economic policy, for example, can state at the beginning of her article that all
amounts will be reported in constant dollars as determined by the Consumer
Price Index, and then stick with that decision throughout.
2 For
further discussion, see p. 149, below.
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Use and Mention A special sort of ambiguity is the basis of the distinction
between using and mentioning linguistic expressions. We can illustrate this distinction with a simple example. When I say:
Toronto is a large city.
I am using the word ‘Toronto’, namely, in order to talk about the city. By contrast, when I say:
The last letter in Toronto is a vowel.
I am talking, not about the city, but rather about the word. In cases like this,
we will say that the word is mentioned. Philosophers generally use quotation
marks or some other device to indicate when a word is being mentioned rather
than used, e.g.,
The last letter of ‘Toronto’ is a vowel.
or
The last letter of Toronto is a vowel.
Because use and mention are not always indicated in this way, ambiguities
can arise. Here is one case: Einstein, according to Bennet Cerf (Try and Stop me,
p. 166),
was fascinated by American slang. He listened carefully three times
to the story of the employer who told his secretary, “There are two
words I must ask you never to use in my presence. One of them is
lousy, the other is swell.” “That’s all right by me,” said the secretary.
“What are the two words?” When he finally comprehended, he
threw back his head and roared with laughter.
Unbiased versus Prejudicial Language; emotionally charged language and
euphemism In addition to neutral, descriptive information, many expressions also convey attitudes towards the things described. Consider an ordinary member of the legal profession, for example. She might be described in
fairly neutral terms as a lawyer, or more positively as learned counsel, or quite
negatively as someone’s mouthpiece or even as an ambulance chaser. These expressions might well be taken to refer to the same group of professionals, but
the coloration added by the value-judgments is quite different.
Excessive use of emotionally charged language in arguments can be a problem, since it gets in the way of a reasonable consideration of the evidence provided to support conclusions. When this happens, we say that the language
used pre-judges the issue, and accordingly call the language prejudicial. Sometimes, indeed, it seems that the prejudicial language not only colours the argument, but is all there is to it. Here is an example:
There has been some discussion lately about raising support payments to single-parent families. Now this is the sort of soft-headed,
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warm and fuzzy thinking we’ve come to expect from lazy left-wingers,
who can’t be bothered to work themselves, but who are quite happy
to take money from hard-working people and hand it over to other
loafers like themselves. Anyone with two brain cells to rub together
can see that welfare is a massive fraud.
It is also possible to go wrong in removing evaluative or emotional content
from language. When, for example, the term ‘border rectification’ is used to
describe the forcible dispossession and deportation of millions of people, the
language fails to do justice to the reality; so, too, when the deaths of hundreds
of civilians in a bombing raid is referred to as ‘collateral damage’ or torture as
an ‘enhanced interrogation technique’. In such cases, we speak of euphemisms.
Interestingly, some expressions start out as euphemisms but over time lose
their ability to fulfil that role. In the earlier part of the twentieth century, for
example, the term ‘depression’ was used as a euphemism for what had widely
(and more alarmingly) been called financial panics. After the great depression
of the 1930s, however, ‘depression’ no longer had the same soothing effect.
The terms ‘mentally retarded’, and ‘juvenile delinquent’ similarly, began life as
euphemisms but gradually lost that character.
With arguments aimed at truth rather than at mere persuasion, unbiased
language is generally best. For language that is heavy with emotional or evaluative content often distracts us from the matter at hand, while euphemisms
and similar devices, when successful, prevent us from seeing clearly what is
being discussed.
8.3
D EFINITION
Clarity and precision, as just noted, are generally good things in arguments.
Definitions are in some cases a good way to attain them.
It is customary to distinguish several kinds of definitions. Lexical (or dictionary) definitions are simply reports on common usage: they aim to tell us
how words are in fact used by most people who speak the language. Stipulative definitions, by contrast, are used by authors who wish to assign a particular meaning to a given expression, when common usage either attaches no
meaning at all to it or else a different one. Such definitions are quite common
in science, where new terms are frequently introduced by authors, and also
in law, where many terms that are in common use have special meanings attached to them. In 1982, for example, the term ‘prion’ was given the following
stipulative definition:
Because the dominant characteristics of the scrapie agent resemble those of a protein, an acronym is introduced to emphasize this
feature. In place of such terms as ‘unconventional virus’ or ‘unusual slow virus-like agent’, the term ‘prion’ (pronounced pree-on)
is suggested.3
3 Prusiner,
“Novel proteinaceous infectious particles cause scrapie,” Science, 9 April, 1982, pp.
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The word ‘charm’, to take another example, has been given a special meaning
in physics, where it is used to name a kind of quark.
Both ‘prion’ and ‘charm’ were already part of the English language. Still,
the scientific definitions can’t really be criticized as giving rise to confusion,
since the meanings they stipulate are so clearly different from the already existing ones.
Now anyone is at liberty to ask that a new and different meaning be attached to a word, regardless of whether the word is newly coined or already
in use. We can also introduce new words if we like. But these freedoms are
easily abused. For the sake of mutual comprehension, we should stick with
established meanings (as given in dictionaries, etc.) unless very strong reasons
speak against this. The discovery of a new, previously unnamed but important
concept is one case where it makes sense to introduce a new term, or assign a
new meaning to an existing term. Being too lazy to find out whether a suitable
word already exists, just as obviously, is not a compelling reason to do so.
Persuasive definitions, finally, are stipulative definitions masquerading as
lexical ones. They often contain words such as ‘genuine’ or ‘true’, and are the
starting point of many a poor argument. Here is one example:
A: That’s not music, that’s just noise.
B: Well, it was played by the Toronto Symphony Orchestra, during
a concert, in a concert hall.
A: That may well be, but that doesn’t mean it’s music. Real music
has a melody and a beat. That was just banging and crashing, not
music.
Or again:
A: All Canadians play hockey.
B: What about Fred? He lives in Moose Jaw, but he doesn’t play
hockey.
A: That just goes to show he’s not a true Canadian. Genuine Canadians are people who play hockey and buy coffee and donuts at
Tim’s.
8.3.1
G OALS
OF DEFINITION
In most cases, definitions attempt to convey the meanings of words by using
other words. Obviously, this won’t succeed unless the meanings of the words
used to define are already known to the people for whom the definition is
intended, at least to some extent. Roughly speaking, then, we can say that in a
definition we use words whose meaning is already known to try to convey the
meanings of other words.
In the most familiar case, a definition of a term A provides another expression B that is claimed to have the same meaning as A. We might define the
term ‘square’, for example, by saying that a square is a rectangle with four
136–144.
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equal sides. The expressions A (‘square’) and B (‘rectangle with four equal
sides’) are thus taken to be synonymous. In many instances, like the one just
mentioned, the defining expression for a single word consists of several words.
It is possible, however, to offer a single word, as in the following example:
‘Pusillanimous’ means the same as ‘cowardly’.
Extensional adequacy Often, the words we seek to define apply to various
objects. An adjective such as ‘red’, for example, applies to objects like apples,
cars, shirts, and so on, while a common noun like ‘politician’ applies to Barack
Obama, Stephen Harper, and others. We call the set of all objects to which
an expression applies its extension. The extension of the expression ‘Maritime
province’, for example, is the set:
{P EI, N B, N S}
while the extension of the term ‘even number’ is the infinite set:
{0, 2, 4, 6, . . .}.
When we seek to define expressions that have extensions, an important
question is whether the expression we seek to define (the definiendum) and the
proposed definition (the definiens) have the same extension. For if two expressions do not apply to exactly the same objects, how could they have the same
meaning? Thus if in a definition B is claimed to give the meaning of A, we
should always ask whether the following statements are true:
• All As are also Bs.
• All Bs are also As.
Put otherwise, being a B should be both a necessary and a sufficient condition
for being an A. And to test these claims, as we know, we look for counterexamples.
If the definiendum applies to objects that the definiens does not, i.e., if there
are As that are not Bs, we say that the definition is too narrow. If, for example,
we define a mammal as a warm-blooded, fur-bearing, land animal, our definition does not apply to some animals (e.g., whales, dolphins) that the word to
be defined applies to. If, by contrast, there are Bs that are not As, we say the
the proposed definition is too wide. The definition of snow as a form of frozen
water is of this sort.
It is possible for a definition to fail on both counts. The definition of a fish
as an animal that lives in the ocean is both too narrow (since there are fish that
do not live in the ocean) but also too wide (since there are animals other than
fish that live in the ocean.
An extensionally correct definition, finally, will avoid all of these flaws: Every
A will be a B, and every B an A.
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We can use diagrams to represent the extensions of expressions, and thus to
picture the extensional success of definitions. In the system of Venn diagrams
(Chapter 5), we recall, the extension of an expression is represented by a circle,
like this:
A
The idea here is that all the objects to which the expression ‘A’ applies are
inside the circle.
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.
Something is A.
A
x
While an ‘x’ outside the circle indicates that at least one thing is not A:
Something is not A.
x
A
xs both inside and outside will then indicate that something is, and something
else is not, A:
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Something is is A and something is not A.
A
x
x
Shading, finally, is used to indicate that a region is empty. If, for example,
we shade in the entire interior of the A-circle, this represents a situation where
nothing whatsoever is an A:
Nothing is A.
A
With this system, we can depict various flaws of definitions, as well as their
extensional adequacy, as follows:
T OO NARROW,
x
BUT NOT TOO WIDE
A
B
E XAMPLE: A bird (A) is a feathered animal that flies (B). x can be, e.g., an
ostrich.
T OO WIDE ,
BUT NOT TOO NARROW
A
B
x
E XAMPLE: A chair (A) is a piece of furniture (B). x might be, e.g., a table.
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T OO NARROW
AND TOO WIDE
x1
x2
A
B
E XAMPLE: A train (A) is a vehicle used to transport people (B). x1 might be,
e.g., a freight train and x2 a bus.
E XTENSIONALLY CORRECT
A
DEFINITION
B
U
E XAMPLE: An even number (A) is one that is divisible by 2 (B).
Further Features of Successful Definitions Extensional adequacy is a necessary condition for a good definition, but it is not sufficient. If we define
an expression using terms that our audience doesn’t know, for example, we
won’t succeed in communicating its meaning, even if it is extensionally correct. The definition of endocarditis as inflammation of the endocardium, for example, will not help many people understand what is meant by the word. By
itself, then, this definition might be criticized as obscure. Of course, we might
follow up the above definition with a definition of ‘endocardium’ and, if necessary, of ‘inflammation’. Provided that these definitions are stated in familiar,
clear terms, we could no longer complain of obscurity. Similar remarks apply
in cases where more than two rounds of definitions are required.
In some cases, an expression is defined in terms of itself, either immediately,
or via a chain of definitions. We speak in such cases of circular definitions. If we
are simply told, for example, electrical charge is an attribute of physical objects
that makes them electrically charged, we will be none the wiser for this. And
if we are told that that an oncologist is someone who practices oncology, while
oncology is the specialization practiced by oncologists, it should be obvious
that we are no closer to our goal of understanding.
Similarly, if an algebraic number is defined as one that isn’t transcendental,
and a transcendental number, in turn, as one that is non-algebraic, we ultimately define ‘algebraic’ in terms of itself. No one with just these definitions
to work with would be in a position to know what ‘algebraic’ is supposed to
mean. The case is the same when a true proposition is defined as one that isn’t
false, and a false proposition as one that isn’t true.
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A good definition, then, won’t be circular. Nor will it be redundant, in the
sense that it contains unnecessary parts. The definition of a square as a rectangle
with four equal sides and four equal angles is an example, since the part ‘and
four equal angles’ can be left out without altering the extension (since every
rectangle already has four equal angles).
Extensionally correct definitions are sometimes criticized because they fail
to indicate the essence of the thing defined, to tell us what it really is. Compare,
for example, the definition of water as the sort of liquid I drank first thing this
morning with the definition: Water is H2 O. While both of these definitions are
extensionally correct, the second might well be preferred because it tells us
what water is, while the first does not (beyond the mere indication that it is a
liquid).
8.3.2
S PECIAL
KINDS OF DEFINITION
This part briefly describes some interesting special kinds of definitions.
Recursive definitions In the simplest cases, recursive definitions have three
parts: a base case, a recursion clause, and an exclusion clause. Here are a couple of
examples:
(1) Joe’s children are Joe’s descendants. (Base case)
(2) Any child of one of Joe’s descendants is one of Joe’s descendants.
(Recursion clause)
(3) No one else is a descendent of Joe. (Exclusion clause)
(1) 1 is an odd number. (Base case)
(2) If n is an odd number, then so is n + 2. (Recursion clause)
(3) Nothing else is an odd number. (Exclusion clause)
Recursive definitions (especially the recursion clause) can easily appear circular. But, if properly formulated, they are not. The key is the base case, which
provides us with a starting point to build on. In the case of ‘odd number’,
for example, we are told that 1 is odd. Given this, the recursion clause can be
applied to obtain:
If 1 is odd, then so is 3.
From which, along with the base case, we may conclude that 3 is odd. We
can then repeat this indefinitely, showing that 5, then 7, 9, etc., are odd. The
exclusion clause, finally, rules out all numbers that are not arrived at in this
way.
Between them, then, the three clauses precisely determine an extension for
the term ‘odd number’, and since it is the right extension, this is a perfectly
good definition.
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Contextual definitions are another philosophically important kind. Here, we
do not simply provide a synonymous expression for the one we wish to define,
but instead show how for certain sentences containing the expression there
are equivalent sentences, composed entirely of already known terms. Since this
only provides us with a means for dealing with the expression in certain cases
or contexts, we speak of a contextual definition.
Consider, for example, the concept of the direction of a line.4 The statement:
Line AB goes in the same direction as line CD.
is equivalent to the statement:
Line AB is parallel to line CD.
So whenever we encounter a statement of the first form, we can replace it by
one of the second form. If we already know what ‘parallel’ means, we can see
that this allows us to define the term ‘direction’ (i.e., clearly indicate its meaning) in a sense, but only insofar as it occurs in such contexts. So contextual definitions are usually only partial definitions.
E XERCISES
I. In the following passages, identify any cases of: i) euphemism, ii) emotionally charged language, iii) ambiguity, iv) vagueness. If you identify a euphemism, say in plain language what is meant by the expression. If you identify ambiguities, indicate the possible meanings of the words or expressions in
question. In case of vagueness, explain how the indicated words or phrases are
vague by indicating what information would be required to make them more
precise. If a passage contains none of the above flaws, simply say so.
1. He was socially promoted to grade 6.
2. In a leap year, the month of February has 29 days.
3. At the party, Bob told James that he had made a serious mistake.
4. Prince Edward Island is Canada’s smallest province.
5. The government has provided enough funding to meet the most important needs of the community.
6. The government is taking steps to ensure that the rate of child poverty
will fall to an acceptable level.
7. A recent survey shows that more young women are smoking now than a
few years ago.
8. Buy the new Chevy Impala: it has all the best features at a price you can
live with.
4 Cf.
Frege’s Foundations of Arithmetic, §64 ff.
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9. The Auditor noted some irregularities in the Ministry’s accounts.
10. The fears that we will soon run out of oil are not well-founded. There is
plenty of oil in the ground, enough to keep us going for years.
11. Mary was upset to learn that Sally told Frances that she would not get
the job.
12. Men are more interested in hockey than women.
13. Soon to retire, she can look forward to enjoying her golden years.
14. This house has everything. You should buy it.
15. John told Sam that he had sold his car to Joanne.
16. We regret to inform you that, due to market pressures, our company is
forced to downsize, and as a consequence you will be free as of next week
to pursue new opportunities.
17. There are many confirmed reports every year of Unidentified Flying Objects (UFOs). But UFOs are spaceships. It is clear that we have evidence
of alien life.
18. The company’s auditor did say that there were irregularities in the accounts, but that none of them was very serious.
19. The General admitted that a few of his soldiers may have been guilty of
some inappropriate conduct during the war.
20. People say that Creationism shouldn’t be taught in the public schools,
because it is just a theory, not proven fact. But Evolution is also a theory,
yet it gets taught in the schools. So there’s no reason why one should be
taught and the other not.
21. When confronted with evidence during the divorce trial, Mr. A admitted
that he had, on occasion, not fully lived up to the spirit of his marriage
vows.
22. The minister was, admittedly, somewhat economical with the truth.
23. The government has provided a substantial amount of funding; the critics who complain about it are a bunch of lazy whiners.
24. For years, liberals have been waging an all-out war against gun rights for
law-abiding citizens. The main argument used by these anti-Americans
is that fewer guns equals a safer society. Is there a poor, pathetic leftist
out there who actually believes that criminals buy guns in stores?
25. Environmentalists have been saying for years that there have to be fundamental changes in the way we live. They are absolutely right—anyone
can see that without such changes we are headed for disaster.
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26. Aries (Thursday, Nov. 1): Big changes are in store for you. Live life to the
fullest. Now is not the time to be wary.
27. By having the amenities of prison life pointed out to him, he was encouraged to cooperate with the police.
28. The Minister has taken all necessary steps to ensure that no serious problems will arise because of Canada’s immigration policy. The people who
continue to criticize her are just narrow-minded bigots.
II. Supply ordinary language corresponding to the following euphemisms:
1. The man was subjected to extraordinary rendition, and transferred to a
black site, where enhanced interrogation techniques were used.
2. After the invasion of Poland in 1939, German troops carried out many
pacification operations behind the lines.
3. Information operations are a crucial part of every war.
4. We believe that regime change is the best option for Iraq.
5. The insurgents were neutralized.
6. The air force carried out a surgical strike.
7. There were several casualties.
8. He was unfortunately the victim of friendly fire.
9. The soldiers employed area denial munitions to impede the progress of
the enemy.
10. Ethnic cleansing was carried out in various parts of the former Yugoslavia.
11. The brave freedom fighters continue their struggle.
12. The enemy troops have failed utterly to advance beyond the city of X.
13. We should expect a period of negative growth in the short to medium
term.
14. The firm is downsizing.
15. Several of the posts at the firm were declared redundant.
16. His position was terminated.
17. The government is delaying a planned reduction in the rate of tax growth.
18. This house is a real handyman’s special.
19. For rent: Cozy, 1 bedroom apartment with a funky, retro look; in a lively
neighbourhood, close to everything.
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20. He went to the rest room.
21. Francis passed away last night.
22. He has some motivation issues.
23. The dog was put to sleep.
24. For sale: 2008 Chevy. Just needs a little TLC. Perfect for the mechanically
inclined.
III. Evaluate the following definitions. Are they too wide? too narrow? both?
circular? obscure? redundant? Explain.
1. A book is a printed document.
2. An equilateral triangle is a triangle with three equal sides and three equal
angles.
3. A square is a polygon with four equal sides.
4. A musician is a man who plays a musical instrument.
5. A violin is a musical instrument with strings.
6. Hockey is a game played on ice in Canada.
7. A statement is true if it is factual.
8. A banker is someone who works in a bank.
9. An even number is a number that is divisible by two.
10. A mammal is a warm-blooded animal.
11. A table is a piece of furniture with four legs.
12. A valid argument is one with true premises and a true conclusion.
13. An invalid argument is one with true premises and a false conclusion.
14. A sound argument is one with true premises and a true conclusion.
15. A professor is a man who teaches at a University.
16. Knowledge is true belief.
17. Torture is any act by which severe pain or suffering, whether physical
or mental, is intentionally inflicted on a purpose for such purposes as
obtaining from him or a third person information or a confession.
18. (in the previous definition) Severe pain is pain caused by serious physical injury, such as organ failure, impairment of bodily function, or even
death, and mental pain and/or suffering is severe only if it lasts for months
or even years.
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19. A teacher is someone who instructs others.
20. A fish is an animal with fins which lives in the ocean.
21. A dog is a small, four-legged domestic animal.
22. February is the shortest month of the year.
23. New Brunswick, Nova Scotia, and Prince Edward Island are the Maritime
Provinces of Canada.
24. French is the language spoken by the vast majority of people living in
France.
25. A cause is something that produces an effect.
26. Murder is intentionally killing a person.
27. An action is wrong if and only if it is against the law.
28. A paper clip is a metal device used to hold several pieces of paper together.
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C HAPTER 9
A B UCKETFUL OF FALLACIES
A fallacy is a pattern of reasoning that should not convince anyone but which
many people nevertheless find convincing. Some are so common that they
have been given names. This chapter will present a selection of them.
A fallacy is sometimes called a non-sequitur, Latin for “it [i.e., the conclusion] doesn’t follow.” Many fallacies work by distracting our attention from
the matter at hand—by stirring up our emotions, getting us to confuse similar
but nonetheless different things, and so on. For this reasons, people sometimes
speak of certain fallacies as red herrings, devices used to throw us off the scent
just as a tasty smoked fish can draw a beagle’s attention away from a fox that
somebody would like her to chase.
Against the person (ad hominem) In the ad hominem fallacy, the person putting
forth an argument, rather than the argument or claim itself, is attacked. Here
are a couple of examples:
Jones says that there’s too much violence in hockey. That’s garbage—
he’s just a wimp who’s afraid of getting hurt.
The Senate committee said that marijuana should be legalized. But
they’re just a bunch of old pot-heads left over from the sixties. There’s
no way it should be legal.
In the vast majority of cases, ad hominems are completely irrelevant. Notice
in the first example, for instance, that no reason whatsoever is given bearing on
the issue of violence in hockey. Instead, we are simply told something negative
about one person who thinks there is too much. That this is irrelevant easily
seen by performing a simple mental variation: suppose the very same claim
to be put forward by someone else who cannot be accused of cowardice in the
face of hockey violence. In such a case, the attack on the person fails, even
though the claim remains the same.
When is an attack on the person relevant? Here is one such case. Suppose
that a witness claims to have seen John Doe break in to a house across the street.
In this case, the personal characteristics of the witness are relevant, since the
credibility of that witness’s testimony is what is at issue when we ask whether
the premise is acceptable. We might ask, for example: is the witness prone to
exaggeration, a habitual liar, does she have poor eyesight, a beef against Doe,
etc.? If so, mentioning personal characteristics of someone making a claim
might well be relevant, for example:
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Smith says that John Doe robbed her neighbour’s house—she saw
him do it. But her eyesight is extremely poor and it was very dark
at the time. There’s no way she could have seen the robber clearly.
So you shouldn’t just take her word for it.
Circumstantial ad hominem In a circumstantial ad hominem argument, we attack the proponent of a given claim based upon his or her personal circumstances. For example:
Fairweather says that corporate taxes should be lower. She would
say that—after all, she’s President of a large corporation. Corporate
taxes are low enough already.
Notice that no reason whatsoever has been given to support the conclusion
that taxes should not be lowered. We may indeed have reason to suspect that
Fairweather’s opinion may be biased, but at most this would permit us to conclude that we shouldn’t accept her word for it, not the stronger claim that what
she says is false.
Tu quoque (you too!) The tu quoque ad hominem seeks to deflect attention from
the misdeeds of a given person or group by pointing out that others (most often
the accusers or a group they belong to) are guilty of the same transgressions.
Very common in long-term relationships and politics. When the Canadian government supported sanctions against South Africa on account of the Apartheid
system, for example, the South African foreign minister Pik Botha said:
Canada should mind its own business, and take a look at the way
it treats its own indigenous people before passing judgement.1
A somewhat more recent example involves the Canadian Senate:
Liberal: Duffy and Wallin, not to mention Brazeau, fudged their
expenses bigtime. Conservative senators sure play fast and loose
with taxpayer dollars.
Conservative: You’re a fine one to talk. What about the Liberal
senators, like Mac Harb or even Raymond Lavigne, who went to
jail for fraud? They’re hardly choirboys when it comes to spending.
For obvious reasons, the tu quoque is extremely effective in debate. All the
same, it is irrelevant, and the appropriate response is that condemning a flaw in
one person, group, or institution by no means prevents us from condemning it
in another. Spectators are by no means required to takes sides in such a debate,
for they have a better option. In the words of Shakespeare:
A plague on both your houses.
1 From the CBC digital archives:
http://www.cbc.ca/archives/categories/politics/internationalpolitics/canada-and-the-fight-against-apartheid/canada-announces-sanctions-against-southafrica.html.
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Guilt by association Another variant of the ad hominem, the guilt by association fallacy attempts to discredit an argument or claim by linking it to some
undesirable group or other. For example:
Richard Dawkins is an unapologetic atheist. That’s interesting—so
were Stalin and the other Soviet communists who murdered millions upon millions of people. He’s a smart man; he should know
by now what atheism is all about.
Reductio ad Hitlerum Perhaps the most used variety of guilt by association,2
the reductio ad Hitlerum attempts to convince us that something is a bad idea
because Hitler either was or would have been in favour of it.
You’re a vegetarian, eh? So was Hitler. For my part, I’ll continue to
eat meat.
Councillor Smith has been complaining that the O-train is always
late. Perhaps he wants them to run on time like they did in the
Third Reich. Me, I’m happy to wait a few minutes and live in a free
country.
Appeal to force (Ad baculum) The appeal to force is a favourite of tyrants
and also of many would-be tyrants, along with certain middle-managers. It
tells us that we should accept something or else. . . .
The Central Committee has ruled that Comrade Lysenko’s theory
of genetics is correct. Those who oppose the theory are therefore
enemies of the state who will be dealt with accordingly. Now what
was that you were saying about Darwin?
You can write that in your column if you like. But if you do, I’ll
fire you, and make sure you never work for any other newspaper
in this country. Perhaps you’d like to reconsider your opinion?
There is no doubt that the appeal to force often gives us good reason to
pretend to believe something. But that, of course, is not the same thing as a
good reason to believe it.
Appeal to pity (Ad misericordiam) Just because we feel sorry for someone
does not mean it would be reasonable to believe what he says. But such appeals
to pity often convince people. Here are a couple of examples:
She’s just lost her job and has been evicted from her apartment.
How dare you say she’s not telling the truth?
2 According to a well-known conjecture (called Godwin’s law), the probability that someone in
an internet thread will mention Hitler approaches one as the length tends to infinity.
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You may think that this young man deserves a long prison term
for murdering his parents, but you should reconsider, and give him
five years or less. His has been a life of poverty and deprivation for
many years.3
Appeal to popular opinion/bandwagon appeal (Ad populum) Many people
try hard to fit in, and the ad populum fallacy exploits this by asking us to accept
a claim because lots of other people do. But popularity is by itself no guarantee
of truth, and the problem is only compounded when the claim that everyone
believes something turns out to be bogus.
Examples:
Obviously, everyone wants taxes lowered and no cuts to public
spending, so obviously it’s the right thing to do. Don’t you agree?
Everybody’s using Brand X. Shouldn’t you be too?
Complex question The complex question is another debater’s trick, much
favoured in Parliament. It consists in asking a question which seems to demand a yes or no answer, when either answer will saddle the responder with
an unpleasant admission. Perhaps the best-known example is:
Have you stopped beating your wife? Yes or no?
The answer “yes” then brings the follow-up: “How long did you beat her?”
while the answer “no”, equally predictably, leads to: “When will you stop,
then?”
Most question periods contain at least one question along the following
lines:
When will the Minister stop lying to Canadians and admit that his
department has failed miserably under his leadership?
The standard defence against the complex question is to separate the two
issues, like this:
I reject the premise of your question. I have never lied to Canadians
and . . .
Appeal to ignorance (Ad ignorantiam) Recall that when it comes to the truth
of claims, there are just two options, true or false. In the case of belief, by contrast, there are three: accept, reject, or remain neutral. The fallacy called appeal
to ignorance tries to muddy the waters by confusing the two issues. It points
out that we have no compelling proof that some claim is false and asks us to
therefore accept it as true, i.e.:
We have no proof that claim C is false. Therefore it’s true.4
3 Besides,
4 The
now he’s an orphan.
appeal to ignorance also occurs in the following form:
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But we might well have no compelling proof that the claim is true either, in
which case the appeal to ignorance would counsel us also to accept it as true.
And this can’t be right. The thing to remember here is that:
The lack of disproof is not a proof.
So too:
The lack of proof is not disproof.
We thus have the option of remaining neutral, even of weighing probabilities
and favouring one side without taking a definitive stand.
Here are a couple of examples.
Of course 9/11 was an inside job. Nobody has proved that it wasn’t.
Some environmentalists say that genetically modified foods aren’t
safe. But they have never proved it. It’s pretty obvious that they are
perfectly safe.
Post hoc Short for post hoc, ergo proper hoc, this happened before that, so this
caused that. For example:
My child had an earache, so I went to the doctor, who prescribed
antibiotics. Within days, the earache had cleared up thanks to the
antibiotics.
A quite common move in reasoning, it nonetheless is not to be trusted. Earaches often clear up on their own, with or without antibiotics, so it would be
rash to assume that the drugs were responsible. So too with the following argument:
Since the superpowers obtained massive stocks of nuclear weapons,
there hasn’t been a single major war. Obviously, the threat of nuclear annihilation has made the world a safer place.
The following examples show that post hoc reasoning is anything but reliable.
Baskin-Robbins has been selling ice cream since 1945, and there
hasn’t been a single major war since then. Baskin-Robbins frozen
treats are obviously crucial to world peace.
In 1990, Madonna scored a big hit with “Justify my love.” The next
year, the Soviet Union was dissolved. Coincidence? I think not.
The moral of this tale is: it takes a lot more work than a lazy post hoc argument
to make a case for causation.
We have no proof that claim C is true. Therefore it’s false.
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Hasty generalization We discussed above the pitfalls of making inductive
generalizations from samples that are too small. The fallacy called hasty generalization is a variant of this unsafe procedure, consisting in leaping to a strong,
general conclusion after only a small number of data have come in.
The human species seems to have a strong propensity for hasty generalization: it is quite common for people to assume they know exactly what’s going
on after looking at a small handful of data points. This often leads to painful
correction, or to bitter accusations that reality is at fault for failing to conform
to theory.
Here are a couple of examples:
Last week, I bought stock in Bees’ Knees Corp, and it went up immediately. Then I invested heavily in BMI, which also went up. I
obviously have a real talent for picking stocks. Next week, I’m going to buy even more.
I’ve dated five different men in the last three years, and they were
inconsiderate slobs. All men are pigs.
Gambler’s fallacy Closely related to the hasty generalization, this fallacy is
so-called in honour of the eternally hopeful gamblers who think that because
they have had a string of bad luck, things are sure to turn around on the next
throw of the dice, spin of the wheel, etc. For example:
The Leafs failed to win the Stanley Cup yet again. It’s been decades
since the last time they won. Their luck is due to turn. They’re sure
to win this year.
The federal NDP have never won enough seats to form a government. A string of bad luck like that can’t last forever: next time,
you’ll see, they’ll win a majority.
There are two problems confronting the gambler’s fallacy. First of all, continued negative results may well indicate that the odds favour that outcome after
all. Second, even if the odds of winning really were 50%, the next throw of the
dice, just like every other, would be as likely as not to lose, regardless of what
had happened previously.
Slippery Slope One thing inevitably leads to another. . . except when it doesn’t.
An argument commits the fallacy of slippery slope when it falsely claims that
if A happens, so will B, and then C, D, . . . , and finally Z, where Z is usually an
obviously unacceptable consequence. Since we don’t want Z, it continues, we
should at all costs avoid A. The thing to remember here is that even if the links
in such a chain are fairly strong, the chain itself may be weak. Suppose, for
example that the probability of B happening given that A happens is as high
as 80%, and similarly for the probability of C happening if B happens, and so
on. In this case, the probability of E happening if A happens is less than 50%,
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and the likelihood of F even lower (33%). The deterioration is even more rapid
when the links are not so strong, as is usually the case.
Examples:
If marijuana is made legal, then how are we going to be able to say
that other “soft” drugs should be illegal? But once we have made
enough soft drugs legal, there will be no good reason to ban harder
drugs like cocaine. In the end, you’ll see, heroin will have to be
made legal as well, and before you know it schoolchildren will be
shooting up during recess.
Once a law is passed to ban private ownership of machine guns, it’s
clear what the next step will be. Soon, semi-automatic rifles will be
made illegal, then handguns, and finally rifles. There will no longer
be any obstacle to government tyranny, and freedom will disappear
from this great country.
This is not to say that no slopes are slippery, however. Walk around in
January and February and you will probably encounter one or two. And so too
in life: you would be well advised to avoid heroin, Oxycodone, membership in
organized crime groups, etc., because sometimes one thing does quite reliably
lead to another.
False dichotomy In this fallacy, we are presented with a list of alternatives
(most commonly two) that is supposed to be exhaustive but isn’t. Often, this
is a prelude to further deductions which, though valid, won’t be sound because of the premise falsely restricting the available alternatives. The false dichotomy is especially popular in politics, for example, when we are asked to
choose sides between two parties in conflict, even though various intermediate
positions are possible.
Examples:
You’re either with us or with the terrorists.
It’s obvious that free trade with China is the best thing for our country. What alternative is there? Do you really want us to do no business at all with the Chinese? That would be economic suicide.
Begging the Question (petitio principii) An argument begs the question when
it uses a premise that is either equivalent to or stronger than the conclusion it
is supposed to support. Perhaps the most easily-remembered example is the
following:
A: How do you know he’s telling the truth?
B: He told me he was.
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Often, it is not at all obvious that an argument begs the question, because what
is essentially the same information can be conveyed in so many different ways.
Here is a simple example:
A: I’m sure you’ll make a profit on this transaction.
B: That isn’t at all obvious to me. How do you know?
A: Well, let me tell you. You’ll have to pay some expenses (including my fee), I admit that. But you’ll also get some income, right?
And, see, the income will be a lot more than the expenses.
Begging the question is closely related to the fallacy of arguing in a circle,
exemplified in the following passage:
When asked by a reporter about grade inflation, the university president said: “It’s true that average grades have been getting higher
in recent years. But those grades are justified, because we’re doing
a better job teaching and our students are learning more.” The reporter then asked: “Are you sure they’re learning more? How do
you know?” The president replied, “The fact that they are learning
more is evident from their improved grades.”
Begging the question is in fact the smallest circle in reasoning possible, namely:
A. Therefore A.
Of course an argument would never succeed in convincing anyone worth convincing if it presented things so baldly. Most often, there is a fair amount of
distracting clutter, e.g.:
A, B, C, D, . . . Oh yes, and M, N, O, . . . . Did I mention C? And P?
Therefore B.
Note on terminology: The expression ‘begs the question’ has in recent years been
widely used to mean ‘raises the following question’. Because of this, some
people prefer to use the Latin term ‘petitio principii’ when there is a risk of confusion.
Fallacy of composition This fallacy is committed when we infer that a whole
has (or lacks) a certain property or characteristic because each of its parts does.
It is easy to see that this reasoning is invalid. Consider, for example, the following instances:
This rock is easy to carry, and so is that rock and that one...
So it will be easy to carry all of these rocks if I put them in a bag.
The steering wheel on the bus can’t get me across town.
Nor can the left rear wheel or the exhaust manifold, etc.
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In fact, none of the parts of the bus can get me across town.
Hence the bus, which is made up of these parts, can’t get me across
town.
Here are a couple of examples where the unreliability of the reasoning may
be less obvious:
All the players on the team are very talented. So the team is certain
to be good.
All of the singers in the choir are very good. So the choir is good.
Fallacy of division This fallacy goes in the opposite direction, inferring that
a part of some whole has (or lacks) a characteristic because the whole does.
There are cases where the reasoning is obviously invalid, e.g.:
This bus can take me across town, so the exhaust pipe of the bus
can too.
And, again, cases, where this may not be so obvious.
Jones played on the great Red Wings teams of the last decade, so he
must have been a great hockey player.
The government is very unpopular, and Witherspoon is in the cabinet, so she must be unpopular.
Equivocation In the fallacy of equivocation, the same expression is used in
different senses in the course of an argument, when the validity of the reasoning depends upon it being used in the same sense throughout. The word ‘cat’,
for instance, is sometimes used in a narrower sense to refer to house-cats, but
sometimes used in a broader sense that also covers lions, leopards and panthers, etc. So the following argument can easily appear to be sound:
Tigers are cats.
Cats make nice pets.
So Tigers make nice pets.
For the reasoning appears to be valid and the premises can both be taken in
senses in which they are true. Yet the conclusion is clearly false.
We can see the fallacy of equivocation as bearing on the cogency of the reasoning or on the acceptability of the premises. If we take ‘cat’ in the same sense
in both premises, then the reasoning will certainly be valid, since it follows the
valid pattern:
A is B.
B is C.
So A is C.
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But in this case, one or the other of the premises will be false: the first if we
take ‘cats’ in the narrower sense, then second if we take it in the broader.
On the other hand, if ‘cat’ is taken to mean different things in the two
premises, then while both premises can be true, the reasoning will no longer
be valid, since the form will be:
A is B.
C is D.
So A is D.
Where A marks the spot for ‘cat’ in the broad sense and D for ‘cat’ in the narrow. (For this reason, equivocation is sometimes called the fallacy of four terms).
Either way, the argument should be declared unsound.
Much of our reasoning is carried out symbolically, and the switch from one
meaning attached to a symbol to another can easily go unnoticed. For this
reason equivocation, either conscious or unconscious, is quite common.
Faulty appeal to authority (ad verecundiam) Sometimes, it is quite appropriate to consult an authority when we want to find out about something.
We might, for example, consult a doctor about medical problems, a mechanic
about car problems, and so on. Much of what we learned in school, moreover,
came from textbooks written by people who are supposed to be authorities
on their subjects. But the trust we place in authorities can be misplaced. We
shouldn’t consult an expert on, say, playing hockey, concerning which coffee
to buy. Nor should we trust so-called “experts” when there is no field of expertise, e.g., fortune tellers, astrologers, phrenologists, etc. Then there are incompetents and frauds of various kinds, as well as experts who are not to be
trusted on account of bias, whether innocent or corrupt, conscious or unconscious. Fortunately, there are still plenty of reliable, sensible, honest authorities
out there. But finding them can take some work.
Reliance on authorities in public debate often suffers from the error of selectivity (or cherry-picking). Rather than approaching authorities with an open
mind to see what they have to say, one seeks precisely those authorities who
support one’s own position and ignores the others. And since there is money
to be made and attention to be won, there is no shortage of ready-made authorities willing to speak on any side of pretty much any question. This is hardly a
rational way to get closer to the truth.
Here are a few rules of thumb for assessing the claims of experts:
1. There should be a genuine field of expertise. No one can be an expert in
the fields of phrenology or orgone therapy, for example.
2. The authority should be an expert in the relevant field. Someone who is a
reliable source for matters in his or her field may well be a crackpot when
speaking on other subjects.
3. Get a second, third, . . . opinion. For there are many cases where experts
in a given field disagree.
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4. You may still decide to take the side of one party of experts even when
there is no consensus on a particular topic. If you do this, however, you
should try to explain how the other party may have come to hold what
you consider to be a mistaken view.
Here are a few examples of faulty appeals to authority:
Linus Pauling won the Nobel Prize for Chemistry and then the Nobel Peace prize. He’s obviously a genius of the first order. And he
said that taking massive doses of Vitamin C is the key to a longer
and healthier life. So it’s clear that we should all be taking Vitamin
C every day.
MS is caused by restriction of the arteries leading to the brain. Dr.
Paolo Zamboni, a well-respected Italian surgeon, tells us that this is
so, and we can take his word for it, even if other medical researchers
disagree.
The idea of global warming is a hoax cooked up by a bunch of socialists, peaceniks and other undesirables. Dr. Dagmar Teersand, a
chemist who works for Statoil, has conclusively proved this.
Straw man It is easier and far less dangerous to knock over a man of straw
than a real one. If the lighting is just right, we may convince the gullible that
we have scored a famous victory. So too in arguments: one can gain the appearance of refuting an argument by taking on a similar, but much weaker
one.
The fallacy called straw man occurs in the context of responding to someone
else’s argument; it consists in putting words into someone else’s mouth, words
which are easier to refute than what the person actually said. Does your opponent think that there is too much pollution? Why not accuse him of wanting to
do away with industry and the jobs it brings? Does she think factory farming
unnecessarily cruel? Then accuse her of wanting to make us all vegetarians.
And so on.
Examples:
X thinks we should increase welfare payments. Obviously, he thinks
the role of the state is to take away all sense of personal responsibility.
Y thinks that the price of gasoline should be much higher than it
is. Clearly his goal is to reduce the hard working people of this
country to poverty and misery.
Z says that we should all recycle and use less energy; in short, we
should all be environmentally friendly. But this is impossible—we
can’t all live in mud huts and collect nuts and berries.
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Two wrongs Related to the tu quoque, this fallacy asks us to condone behaviour because someone else has done it, hoping that, against what our mothers told us, two wrongs do make a right after all. For example:
You can hardly blame him for plagiarizing his essay. After all, other
students did the same thing.
So what if Lance Armstrong was doping when he won the Tour de
France all those times? He didn’t do anything wrong—other riders
were doping too.
Argumentum ad nauseam A form of persuasion that has become increasingly
popular in the present age, the argumentum ad nauseam attempts to bludgeon
us into accepting a proposition by repeating it endlessly. Schematically: P, P,
P, P, P, P, P, P, . . . . So P already. One common manifestation of this style of
argumentation is the so-called talking points issued by political organizations.
Another is repeated exposure to advertising.
Mrs. Lincoln fallacy An argument which asks us to disregard things that
should under no circumstances be disregarded. So-called in memory of the
reporter who asked the president’s wife: “Setting aside the fact that your husband was shot, how did you like the play?”
Examples:
After the fall of communism, a number of prominent western economists recommended a “shock treatment” for Poland designed to
bring about a quick transition to a market economy. These recommendations were followed, and were an unqualified success,
as Poland’s growth was the strongest in the former east bloc over
the years 1990–1994. Setting aside the persistently high unemployment and prices for food and housing that were so high most Poles
couldn’t afford to pay them, the economic performance was excellent.
The operation was a complete success. Unfortunately, the patient
died.
Lying Perhaps the most common fallacy of all, it is a particular favourite of
the shameless. Many excellent examples are available for viewing on the internet, among them a succession of athletes earnestly denying ever taking performance enhancing drugs. These repay careful study.
Examples:
“I have never doped. I can say it again, but I’ve said it for seven
years.” (Lance Armstrong)
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“I did not have sexual relations with that woman, Miss Lewinsky.”
(Bill Clinton)
“The fact that Mr. Schreiber may or may not have had any business
dealings was not my principal preoccupation. I had never had any
dealings with him.” (Brian Mulroney)
“Manafort had nothing to do with my campaign.” (Donald Trump)
C OMMON
FALLACIES DISCUSSED IN THIS CHAPTER
• Ad hominem
• Slippery slope
• Tu quoque
• False dichotomy
• Guilt by association
• Begging the question
• Reductio ad Hitlerum
• Fallacy of composition
• Appeal to force
• Fallacy of division
• Appeal to pity
• Equivocation
• Ad populum
• Faulty appeal to authority
• Complex question
• Straw man
• Appeal to ignorance
• Two wrongs
• Post hoc
• Argumentum ad nauseam
• Hasty generalization
• Mrs. Lincoln fallacy
• Gambler’s fallacy
• Lying
E XERCISES :
I. Spot the fallacy: Most but not all of the following passages contain arguments, and many of these commit one or more of the fallacies discussed above.
Identify fallacies where they occur.
1. Should we have Federally subsidized day-care? Kathy Penner thinks we
should. She’s always ready to give away someone else’s money, just like
her tax-and-spend liberal friends. Besides, Federally subsidized day care
is just another government intrusion into private life. One they get the
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Reasoning and Critical Thinking
foot in this door, there’s no limit to what they’ll do next. You can kiss
your freedom goodbye.
2. We can either re-introduce the death penalty for murder, or else turn our
society over to the criminals. No one wants to live in a society dominated
by fear. So it’s clear that re-introducing the death penalty is the right
thing to do.
3. Ever since the Liberals came to power in Ontario, there has been an increase in all forms of immoral behaviour. It’s pretty clear that the Liberal
party’s policies are causing a breakdown of our most cherished values.
4. John Dough recently wrote an article criticizing certain academic policies
at Carleton University. That’s interesting. Did you know that he graduated from the University of Ottawa? It’s obvious that his article is biased
and worthless, and that the policies at Carleton are just fine as they are.
5. Testifying before the US Congress, Professor Diana Findlay, a respected
expert on ecology, stated that even if global warming continues at the current pace, there will not be serious impacts on animal and plant species
or on humans. It is true that other ecologists disagree, but I, for one, think
that Prof. Findlay has got things right. She’s the expert, after all. So it’s
safe to say the threat of global warming has been exaggerated.
6. No one has ever proved conclusively that a second gunman wasn’t involved in the assassination of John F. Kennedy. Clearly, the official US
government report on the Kennedy assassination, which claimed that Lee
Harvey Oswald was the lone gunman, was a cover-up.
7. Bombing Afghanistan was clearly the right thing to do. Some people say
there shouldn’t be any bombing, but apparently they don’t care whether
or not innocent people are murdered by terrorists.
8. Kowalski should do what’s right for the country and resign before the
next election. He isn’t getting any younger, and polls consistently show
that a majority of Canadians want him to step down.
9. Theft is wrong, because stealing is immoral.
10. Some people say they only want to eat natural foods and avoid artificial foods. But everything that exists is part of nature, even chemicals produced in laboratories. So all foods are natural, and those people
shouldn’t worry.
11. We can only deal with the problem of terrorism by negotiation or by war.
But there is no way to negotiate with terrorists. So war is our only option.
12. Why do I think he’s the only man for the job? Well, let me tell you: no
other candidate is suitable.
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13. The government should bring back capital punishment for serious crimes
like murder. After all, polls consistently show that the majority of Canadians favour the death penalty.
14. Laws restricting the genetic engineering of plants should not be passed.
After all, nobody knows of any evidence that genetic engineering poses
a threat to human well-being or the environment.
15. Senator B says that Congress should vote in favour of legalizing soft
drugs. In saying this, he is buying into the views of groups which almost
brought an end to this country—the hippies, the yippies, the weathermen, radicals one and all. It is unbelievable that a man like Senator B.
would join forces with this crowd. All drugs should remain illegal.
16. There has been a lot of talk lately about monitoring the internet to watch
for terrorist communications. This is a potentially dangerous idea, since it
involves a violation of people’s privacy without due cause for suspicion.
17. Dexter Moolah, the president and CEO of Acme Inc., says that now is a
good time to buy stock in his company. He’s certainly in a good position
to know. If he says the company is in great shape, you can be sure it is.
18. Yeah, maybe I had a few beers once or twice before driving. I suppose
you’re all perfect, and never did that?
19. Look, either you sign the contract or you don’t. If you do, we’ll both be
better off. If you don’t sign, I’ll still be better off even if you aren’t. So no
matter what happens, I’ll be better off.
20. Often, when a sports team wins (or loses) a championship, there are riots
in the city where the team is based. So it is a good idea for there to be
extra police on duty in cities whose teams might win on the days when
the championship might be decided.
21. Thousands upon thousands of teenagers love Brittney Spears, so there
must be something about her music that appeals to many young people.
22. Buddy Whasisname tells you that he is a good candidate for public office,
but you should not believe what he says. His record speaks for itself:
when he was Mayor of Erewhon from 1990-1994, he was charged, and
ultimately convicted, of fraud and theft. He is obviously dishonest, and
there’s no way you should vote for him.
23. Of course Saddam Hussein had weapons of mass destruction. Has anyone ever proved that he didn’t? He must have moved them to Syria just
before the War.
24. Professors Smith and Jones, a respected chemist and a highly successful
petroleum engineer, have stated that global warming is a myth. It seems
clear that the people supporting the Kyoto accord don’t have science on
their side.
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25. I’m going to wear this Iron Maiden t-shirt under my uniform every time
we play, and I’ll tell you why. Last week I wore it to both of our games,
and we won. I also scored a hat trick in first game, and got two goals in
the second. It’s pretty obvious that the shirt brings me good luck.
26. John Kerry says that the United States should have sought approval from
the United Nations before starting a pre-emptive war in Iraq. This is a
very dangerous position to take, and really amounts to nothing short of
a policy of automatic surrender when challenged by any enemy. Do we
really want to get on our hands and knees and beg the President of France
to allow us to defend ourselves when we are under attack?
27. If we don’t stop admitting so many refugees into Canada soon, the consequences will be disastrous. Word is already spreading throughout the
world that Canada is an easy country to get into. This will lead to a flood
of refugees. Soon there will be so many that we won’t be able to afford
public services such as medicare. And when they send for their families
and friends, yet another flood will come, and we will all be swamped.
Soon English and French will be minority languages in this country.
28. I’ve used cocaine three or four times, and I had no problems whatsoever.
People really exaggerate the harm caused by drugs. It’s obvious from my
experience that cocaine is perfectly safe.
29. Look, the government can either spend money on health care or on education. Money spent on health care goes mostly to the old, while money
spent on education goes mostly to the young. We should look towards
the future. It doesn’t make sense to throw away money on the past. Education is obviously the way to go.
30. For every one of the past ten years, it has rained during the Remembrance
Day ceremonies on Parliament Hill. That trend can’t continue forever, so
it’s sure to be sunny this time.
31. Being in favour of legal abortion basically boils down to saying it’s OK to
kill anyone who is small enough. Clearly, abortion in any form would be
illegal.
√
32. Thomas Hobbes said that π = 10. He’s a brilliant philosopher, so it
must be true.
33. Environmentalists keep saying that if we don’t cut back on our consumption of fossil fuels, a variety of disasters will result. But which of these
people has ever run a company? Met a payroll? Provided jobs for the
people who need them? Environmentalists enjoy the standard of living
provided by these hard-working people; they should stop criticizing all
the time.
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34. So what if he made a few false claims on his expense account? That’s not
really dishonest—everybody does it.
35. I hear that the students in Québec are demonstrating again for the government to keep tuition fees low. Why don’t they study for once in their
lives, instead of complaining all the time?
36. The US government’s deficit increased sharply in recent years due to two
factors: an increase in military expenditures, especially for the war in
Iraq, and a decline in revenues caused by deep tax cuts.
37. Every day this week when I was on the highway, people were speeding
like crazy and there were no cops to be seen. The police must never stop
anyone for speeding on that road.
38. Ever since television became available to most people, people have been
getting dumber. It’s clear that television is responsible for the increase in
stupidity and ignorance.
39. Some people say we should allow 16 year olds to vote. What a ridiculous
idea! If we allow 16 year olds to vote, what reason could we have for
denying the vote to 14 years olds? But once they get it, even younger
people will demand their rights, and before you know it, kids in diapers
will be deciding who gets to form the new government. No thank you. I
say leave the voting age where it is.
40. Some Ukrainians think they have right to criticize the Russian government for its so-called “interference” in their affairs. They should remember where the natural gas pipeline comes from, and who controls the tap.
41. Jones says that inheritance taxes should be higher. But just think about
what that means. The government takes away, after I die, more and more
of my hard earned money. Once they get the idea in their heads that that
sort of confiscation is okay, they’ll soon move on to others. One day it’ll
be a 100% inheritance tax, the next day, they’ll confiscate your house and
send you off to public housing. In no time at all, private property will
cease to exist.
42. The failure of socialism is obvious to any impartial observer. Just look
at Sweden, the most socialist country in Europe. They have the lowest
standard of living of any European country, and the highest suicide rate.
43. “Mr. Speaker, the government is preparing to read Canadians’ emails and
track their movements through cellphone signals, in both cases, without a warrant. How can we trust the Conservatives with such sweeping
powers when they use Facebook to keep law-abiding Canadians out of a
public meeting? Is this 2012 or 1984? How can we trust them not to use
private information to intimidate law-abiding Canadians gathering, for
example, to protest a pipeline or to protest pension cuts?”
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“Mr. Speaker, I thank the member for the opportunity to tell him that every province unanimously supported moving forward with the legislation, legislation that was introduced first under the Liberal government,
by his party. As technology evolves, many criminal activities, such as the
distribution of child pornography, become much easier. We are proposing measures to bring our laws into the 21st century and to provide the
police with the lawful tools that they need. He can either stand with us
or with the child pornographers.” (Quoted after Hansard, 13 Feb, 2013)
44. Individuals, one and all, eventually grow old, decay, and die, unless they
are carried off by some misfortune when young. From this we see that all
societies and nations, which are made up of individuals, must also decay
and eventually die.
45. Joe has a pain in his foot. His foot is in his shoe. So Joe has a pain in his
shoe.
46. I see that the French government is once again disputing the right of the
Americans to use military force in the Middle East. Is there no end to
their pointless complaints?
47. Last weekend, there were three murders in Toronto alone. I’m telling
you, violent crime is completely out of control in this country.
48. You say I’ve been sleeping on the job. Well, what about you and your
two hour “coffee breaks”?
49. So you think you should report the illegal activities of your boss? I hope
you like being unemployed.
50. I know I did terrible job on the term paper, but couldn’t you give me a
good mark anyway? I need it to keep my scholarship.
51. We’ve seen the Prime Minister scramble for months trying to find a way
out of the sponsorship scandal. Why is it that he is so unwilling to take responsibility for the fraud he and other members of his government committed?
52. For as long as anyone can remember, either the Liberals or the Conservatives have been the governing party of Canada. The NDP is due for a
win, and is sure to form the next government.
53. Botswana is a country that is rich in natural resources. So Ghanzi, as one
of the provinces of Botswana, must also be rich in natural resources.
54. MP1: The government is proposing to close a safe-injection facility in
Vancouver for drug-addicts which has been proved to reduce fatalities,
health-care costs, and crime significantly. What is their justification for
the proposed measure?
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MP2: MP1 wants to have luxurious public institutions where drug-addicts
can shoot up. Why doesn’t she care about the safety of the women and
children who live in those neighbourhoods? Why is she so concerned
about making the addicts comfortable? Why is she so pro-heroin and
anti-family? Perhaps she can explain that, Mr. Speaker.
55. Sure, he’s frequently drunk, has bought and used illegal drugs, associated with known criminals, driven while intoxicated, lied to the public,
interfered with the police, and brought disgrace and dishonour upon the
city, but setting such insignificant issues aside, Frank Smedley is the best
mayor anyone could ever wish for.
56. Some people say that men and women should have equal rights. That’s
nonsense: should men have the right to pregnancy leave, for example?
57. Some Americans think that there should be restrictions on the right of
citizens to own guns. They say that no ordinary citizen should be permitted to possess a machine gun. But if the government makes it illegal
to own a machine gun, it’s just a matter of time before they ban handguns
as well. Soon they will make all guns illegal. And when that happens,
when only the government has guns, the honest citizens will have no
protection against the government. It happened in Nazi Germany, and it
would happen here too. So there should be no restrictions on Americans’
right to bear arms.
58. The Conservatives have been complaining non-stop for more than a decade
about the sponsorship scandal. Have they forgotten about Airbus? What
about Gazebo Tony?
59. Jones must be a crook. After all, he worked for the law firm Dewey,
Cheatham, and Howe, and they were famous for their shady dealings.
60. Charging tolls for the use of roads in the city is just plain wrong. Why
can’t you get that into your thick skull? You might think you have good
reasons for introducing such tolls, but you’re mistaken. There are no
good reasons. As a policy, it’s misguided, ill-conceived, and in fact downright crazy. Can’t you see that? What part of wrong don’t you understand? Nothing could possibly justify charging tolls in the city. Nothing,
I tell you.
61. The vast majority of Ontarians see nothing wrong with talking on cellphones while driving. So the government of Ontario was wrong to pass
a law making it illegal to drive while talking on a cell-phone.
62. During WWII, the Soviet Union fought on the side of the Allies, against
the evil menace that was Nazi Germany. So Stalin was one of the good
guys.
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63. Smith never took a bribe. Sure, some people testified in court that he did,
but they were all losers who were jealous of him.
64. There is no doubt that Acme cleanser is the superior product. After all,
it’s clearly better than other cleansers on the market.
65. The economist’s model for determining the price of Gold is absolutely
brilliant. No wonder he won the Nobel Prize. It’s true that the predictions
produced by the model turned out to be off by a wide margin in almost
all cases. But this is a minor point.
66. The company sold various items that could be used to assemble weapons,
it’s true, but that doesn’t mean they violated the arms embargo. Never
once did they sell a weapons system.
67. You can’t find him guilty; that would ruin all his important plans for the
next ten to fifteen years.
68. We have to decide: do you want jobs and a healthy economy, or a substantial reduction of carbon dioxide emissions? Do you want to put people out of work? Not me. It’s clear that we should fight against attempts
to force reductions in CO2 emissions.
II. Look at the comments section of any newspaper. Collect as many fallacies
as you can. First one to twenty wins.
III. The official record of parliamentary business in Canada, Hansard is another
rich source for fallacies. It is available on-line at:
http://www.parl.gc.ca/housechamberbusiness/
Find the transcript for a recent question period and see how many fallacies
you can find. Watch especially for the complex question, the red herring, the
ad hominem, the tu quoque, and the straw man.
IV. Look at a variety of advertisements, in print, on television, on-line, etc.
Identify fallacies as they occur.
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C HAPTER 10
S OLUTIONS TO E VEN -N UMBERED E XERCISES
S OLUTIONS
FOR
C HAPTER 1
Exercises, pp. 16 ff., Part I:
2. An argument. Premises: 1) Sam didn’t steal the necklace. 2) Only Sam
and Joe had an opportunity to steal the necklace. Conclusion: Joe stole
the necklace. ‘So’ is a conclusion indicator, ‘because’ a premise indicator.
4. Not an argument. Merely reports a fact, without drawing any conclusions.
6. An argument. Premises: 1) The apartment isn’t too expensive. 2) The
apartment is relatively clean. 3) The apartment is close to where you
work. Conclusion: You should take the apartment. There are no indicator
words.
8. Not an argument. No reasons are given to support any of the claims.
10. Not an argument, but an explanation. Reasons are given to explain why
geese fly south, rather than to convince us that they do.
12. An argument. Premises: 1) Going to university is a privilege that students should have to pay for. 2) Most students can easily afford to pay
more if they stop spending so much on things like cell phones, beer and
cigarettes. 3) There’s no other way to raise the money the university
needs. Conclusion: Students should be paying higher tuition. ‘So it’s
obvious that’ is a conclusion indicator.
14. An argument. Premises: 1) Society as a whole benefits from the presence of educated people. 2) High tuition prevents many qualified people
from obtaining an education. Conclusion: University tuition fees are high
enough as it is. There are no indicator words.
16. Not an argument. Facts are reported, but no conclusions are drawn.
18. Not an argument, but an explanation. Reasons are given to show why
tuition fees are lower in Québec, not in an attempt to convince you that
this is so.
20. An argument. Premises 1) According to the doctor, he has either measles
or chicken pox. 2) With measles you get a high fever. 3) His temperature
is normal. Conclusion: He has chicken pox. ‘He must have’ is a conclusion indicator, ‘because’ a premise indicator.
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22. An argument. Premises: 1) Different people make different judgments
about ethical questions. 2) None of the five senses seems capable of perceiving good and bad. Conclusion: 3) Good and bad are not in the world
for us to perceive. The second sentence might be seen either as support
for P1 or as a repetition of P1. ‘This shows us that’ is a conclusion indicator; ‘further proof’ indicates a premise.
24. An argument. Premises: first two sentences. Conclusion: Last sentence.
‘Reflecting on this, we can see that’ is a conclusion indicator.
26. Not an argument. No reasons are given to support the claims.
Part II:
2. MP: Supermodels are tall.
4. MP: Tax revenues are increasing. MC: Taxes should be cut.
6. MP: He is being evasive. MC: He has something to hide.
8. MC: It was the Bulgarians.
10. MC: We are in the midst of a housing bubble.
Part III.
2. 1. The TV is on fire.
2. The TV is broken. (IC, from 1)
4. You’ll have to buy a new TV. (FC, from 2)
P1
IC2
F C3
4. 1. Buddy has sent me hundreds of e-mails.
2. Buddy has written dozens of letters to me.
3. Buddy keeps phoning.
4 Buddy must want to get in touch with me. (FC, from 1, 2, 3)
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Solutions to Even-Numbered Exercises
P1
P2
P3
C4
6. 1. One of Tom, Dick, Harry cleaned up the mess.
2. Harry never helps out.
IC3. Harry didn’t clean up the mess. (P2)
P4. Dick was busy with other things.
IC5. Dick didn’t clean up the mess. (P3)
6. Tom cleaned up the mess. (FC, from 1, 3, 5)
P1
P2
P4
IC3
IC5
F C6
8. 1. Only the Liberals and the Conservatives have a real chance at forming
a government after the election.
2. You want your local MP to be
a government member.
3. You’ll have to vote for either the Liberals or the Conservatives. (FC,
from 1, 2)
P1
P2
F C3
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10. 1. If the fire had been caused by a mechanical problem, the alarm would
have been set off.
2. The alarm was not set off. (MP)
3. The fire was not caused by a mechanical problem. (IC, from 1, 2)
4. The fire was caused by arson
or by a mechanical problem.
5. The fire was caused by arson. (IC, from 3, 4)
6. The police and fire departments should start an investigation of the
fire. (FC, from 5)
P1
M P2
P4
IC3
IC5
F C7
12. 1. If the cook had killed Weatherby, we couldn’t have been at the fish
shop.
2. The cook was at the fish shop. (Missing premise)
3. The cook didn’t kill Weatherby. (IC, from 1, 2)
4. If the cook didn’t kill Weatherby, the butler must have.
5. The butler killed Weatherby. (FC, from 3, 4)
M P2
P1
IC3
M P4
F C5
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Solutions to Even-Numbered Exercises
14. 1. Quebec and Alberta are certain to oppose any proposal to make
health care the sole responsibility of the federal government.
2. The provinces won’t unanimously approve a proposal to make health
care the sole responsibility of the federal government. (IC, from 1)
3. Without unanimous consent of the provinces, the federal
government cannot assume full responsibility for health care.
4. The federal government won’t assume full responsibility for health
care. (FC, from 2, 3)
P1
IC2
P3
F C4
16. 1.
2.
3.
4.
5.
Mehitabel’s fingerprints were found at the scene of the crime.
Mehitabel was at the scene of the crime. (IC, from 1)
Mehitabel was having an affair with the victim’s husband.
Mehitabel had a motive for murder. (IC, from 3)
Mehitabel committed the murder. (FC, from 2, 4)
P1
P3
IC2
IC4
F C5
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18. 1. The current scandal will make the Party look dishonest.
2. It will force some high-level members of the party to resign.
3. It will increase voters’ cynicism about politics.
4. The current scandal will cause problems for the Party (FC, from 1, 2,
3).
P1
P2
P3
F C4
20. 1. If the current problems with the US mortgage market spill over into
the main economy, then the US dollar is going to continue to decline in
value relative to the Canadian dollar.
2. The current problems with the US mortgage
market will spill over into the main economy.
3. The US dollar will continue to decline in value relative to the
Canadian dollar. (Unstated conclusion)
4. When the Canadian dollar is worth more, many Canadians go cross
border shopping.
5. Cross border shopping harms Canadian retail sales. (Missing premise)
6. . Canadian retail sales will suffer. (FC, from 3, 4, 5)
P2
P1
M IC3
P4
M P5
F C6
S OLUTIONS
FOR
C HAPTER 2
Exercises on Counterexamples (p. 27):
2. Refutable: e.g., penguins, ostriches
4. True, so not refutable.
6. Refutable: e.g., bears
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8. Not a universal claim, so not refutable by counterexample.
10. Refutable: e.g., Diefenbaker
12. Not a universal claim, so not refutable by counterexample.
14. Refutable: e.g., Mercury
16. Refutable: e.g., The Ottoman Empire
18. A subtle question. The statement could be treated as equivalent to a universal claim, namely: ‘whoever is Pope at a given time is Catholic.’ Even
so, it would not be refutable by counterexample, because it’s true.
20. Refutable: e.g., ‘Every number is odd.’ Incidentally, since the statement
is itself a false universal generalization, it serves as a counterexample to
itself.
Exercises on Consistency (p. 31 ff):
2. Consistent: the stated conditions were not claimed to guarantee admission.
4. Consistent. Suppose the dollar’s value doesn’t change.
6. Consistent. Suppose, e.g., Able and Charlie went, but Baker stayed home.
8. Consistent. The Americans might neither prevail nor withdraw.
10. Consistent. Provided Baker never goes to the pub, everything said here
could be true.
12. Consistent. Smedley could have lied for all sorts of reasons while not
having any connection to the crime.
14. Consistent. Life is like that sometimes.
16. Consistent. The stated condition was not claimed to guarantee success.
18. Consistent. Someone can be tall compared to the general population
while not tall relative to a subset.
20. Inconsistent. If Charlie is to be neither the tallest nor the shortest, he must
be intermediate in height. For Able to be taller than Baker, then, he would
have to be the tallest, and Baker the shortest. This would make Able the
oldest according to the first claim, which is incompatible with his being
younger than Baker.
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Reasoning and Critical Thinking
Exercises on Principles of Reasoning (p. 35):
2. If the number of cases where X happens increases as the number of cases
where Y does, then Y leads to X. (b) The number of people who drown
while swimming increases as the number of people using air conditioning does. So it’s obvious that air conditioning leads to drowning.
4. (a) Some A are B. Some A are C. So some A are both B and C. (b) Some
men are sumo wrestlers. Some men are olympic-champion sprinters. So
some men who are sumo wrestlers are also champion sprinters.
6. (a) X can cause Y . Y did happen. So X happened. (b) Falling off the top
of the Empire State Building can cause death. Julius Caesar is dead. So
Julius Caesar fell off the top of the Empire State Building.
8. (a) A thoroughly evil person thinks X is true. So X is false. (b) Pol Pot
thought that 2+2=4. So 2 + 2 6= 4.
10. (a) Not all A are B. So some B are not A. (b) Not all mammals are bears.
So some bears are not mammals.
S OLUTIONS
FOR
C HAPTER 3
Exercises on Validity (p. 38 f.):
2. Valid
4. Valid
6. Invalid. Suppose the plants were watered, but died for some other reason, e.g., insect attack or disease.
8. Valid
10. Valid
12. Invalid; they might have been born around midnight, but on different
dates, for example.
14. Invalid; the number might not have changed.
16. Valid
18. Invalid. Suppose, e.g., that of the 10 Lions Club members, 8 are members of the Rotary Club, and that of the 100 Rotary Club members, 90 are
members of the Loyal Order of Moose, which has 300 members.
20. Valid
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Solutions to Even-Numbered Exercises
Exercises on Formal Validity (p. 42 f.):
2. (a) Not both P and Q. So neither P nor Q. (b) Obama and Romney didn’t
both win the 2012 Presidential election. So neither Obama nor Romney
won the election.
4. (a) Some P is not Q. So some Q is not P . (b) Some mammals are not cats.
So some cats are not mammals.
6. (a) All P are Q. So all Q are P . (b) All fish live in water. So everything
that lives in water is a fish.
8. (a) If P then not Q. Not P . So Q. (b) If Napoleon committed suicide, he
would not be alive today. He did not commit suicide. So he’s alive.
10. (a) No P is Q. Some non-Q is R. So some P is R. (b) No whale flies. Some
animals that don’t fly are reptiles. So some whales are reptiles.
12. (a) P only if Q. Not P . So not Q. (b) Romney could only have won the
2012 Presidential election if he had been on the ballot. He didn’t win the
election. So he wasn’t on the ballot.
14. (a) Not P unless Q. Q. So P (b) Napoleon could not have won at Waterloo
unless he fought. He did fight at Waterloo. So he won.
16. (a) Some P is Q. Some Q is R. So some P is R. (b) Some mammals are
vertebrates. Some vertebrates are fish. So some mammals are fish.
18. (a) Some P is not Q. Some Q is R. So some R is not P . (b) Some mammals don’t fly. Some of the things that fly are bats. So some bats are not
mammals.
20. (a) No P is Q. No Q is R. So no P is R. (b) No mammals are reptiles. No
reptiles are cats. So no mammals are cats.
Exercises on Validity and Soundness (p. 44 ff.):
I. Give examples of the following, if possible (if it is not possible to provide an
example, explain why not):
2. An invalid argument with all true premises and a true conclusion.
[Some dogs bark. So some cats have blue eyes.]
4. An invalid argument with all true premises and a false conclusion.
[Ottawa is in Canada. Ottawa is in Ontario. So Canada is in Ottawa.]
6. An invalid argument with at least one false premise and a true conclusion.
[Ottawa is in Germany. Germany lost the second world war. So Ottawa
is in Canada.]
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Reasoning and Critical Thinking
8. An invalid argument with at least one false premise and a false conclusion.
[All dogs can tap dance. So everyone who can tap dance is a dog.]
10. An argument which is valid but not sound.
[All dogs tap dance. All who tap dance can whistle Dixie. So all dogs can
whistle Dixie.]
12. An argument which is neither valid nor sound.
[Some dogs are brown. So some cats eat mice.]
II. True or false? Explain.
2. Some sound arguments have false premises.
[False. By definition, all the premises of a sound argument are true.]
4. Some valid arguments have false conclusions.
[True. For example: Chicago is in Germany. Germany is in Asia. So
Chicago is in Asia.]
6. A valid argument with a false conclusion must have at least one false
premise.
[True. If all the premises of a valid argument were true, the conclusion
would also have to be true. So if the conclusion isn’t true, not all of the
premises can be true; hence at least one of them must be false.]
8. Any argument with all true premises and a true conclusion is valid.
[False. Counterexample: Some dogs are brown. So some cats have stripes.]
10. Not all sound arguments are valid.
[False. A sound argument is valid by definition.]
12. No sound arguments are invalid.
[True. A sound argument is valid by definition. Hence any argument
that is invalid cannot be sound.]
III.
2. Valid but not sound (b).
4. Valid but not sound (b).
6. Neither valid nor sound (c)
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Solutions to Even-Numbered Exercises
8. Sound (a) if answered on the first day of summer; otherwise, valid but
not sound (b).
10. Sound (a) if answered in a month other than June. If answered in June,
valid but not sound (b).
12. Valid but not sound (b).
14. Neither valid nor sound (c).
16. Sound (a).
18. Valid but not sound (b).
20. Neither valid nor sound (c).
22. Sound (a).
24. Sound (a).
Exercises (p. 48ff.): Consider the following pairs of statements. Are they equivalent? If not, does either one imply the other?
2.
(a) Sam and Dave can sing.
(b) Dave and Sam can sing.
[Equivalent]
4.
(a) Either Sam or Dave can sing.
(b) If Sam can’t sing, then Dave can.
[Equivalent]
6.
(a) If the Conservatives win a majority in the next election, Harrison
will resign.
(b) If the Conservatives do not win a majority in the next election, Harrison will not resign.
[Neither implies the other.]
8.
(a) Neither the Conservatives nor the Liberals will win a majority in the
next election.
(b) The Conservatives and the Liberals won’t both win a majority in the
next election.
[a implies b, but not vice versa.]
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Reasoning and Critical Thinking
10.
(a) The Conservatives will win a majority in the next election if Grabowski
is replaced.
(b) The Liberals will lose the next election if Grabowski is replaced.
[a implies b, but not vice versa.]
12.
(a) The plants will die unless they’re watered.
(b) The plants won’t live if they are not watered.
[Equivalent]
14.
(a) The plants will die unless they’re watered.
(b) The only way the plants will live is if they are watered.
[Equivalent]
16.
(a) Joe knows that Sam is lying.
(b) Sam is lying.
[a implies b, but not vice versa.]
18.
(a) All chordates are vertebrates.
(b) All vertebrates are chordates.
[Neither implies the other.]
20.
(a) Some small mammals can fly.
(b) Some flying mammals are small.
[Equivalent]
S OLUTIONS
TO
C HAPTER 4
Exercises (p. 52): Part I. How would the following expressions read in English?
2. ¬S&¬¬J [Sam didn’t get a raise, and Joe was not unsuccessful in getting
a raise. Equivalently but more simply: Sam didn’t get a raise but Joe did.]
4. J&¬S
[Joe got a raise but Sam didn’t.]
6. ¬(¬J&S)
[It is not true to say that Joe didn’t get a raise but Sam did.]
8. ¬(¬J&¬S)
[They didn’t both fail to get a raise.]
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Solutions to Even-Numbered Exercises
II. Symbolize the following (where J, S are as above)
2. Sam got a raise, but Joe didn’t.
[S&¬J]
4. Joe failed to get a raise, but Sam got one.
[¬J&S]
6. Both Joe and Sam failed to get a raise.
[¬J&¬S]
8. Neither Joe nor Sam failed to get a raise.
[¬¬J&¬¬S]
Exercises, p. 56 ff.
Part I.
2. Sam went to the store but Dave didn’t.
[S&¬D]
4. Only one of them went to the store.
[(S&¬D) ∨ (D&¬S)]
6. Sam was out of milk, but neither he nor Dave went to the store.
[M &¬(S ∨ D)]
8. If Sam goes to the store, Dave will too.
[S → D]
10. Sam will go to the store if it’s not closed.
[¬C → S]
12. Sam won’t go to the store unless he’s out of milk.
[¬M → ¬S or S → M ]
14. If Sam’s not out of milk, he won’t go the store, but Dave will.
[¬M → (¬S&D)]
16. Neither Sam nor Dave goes to the store if it’s closed.
[C → ¬(S ∨ D)]
18. Dave won’t go to the store with Sam unless Sam is out of milk.
[¬M → ¬(D&S)]
20. Only one of them will go to the store if Sam runs out of milk, and it won’t
be Dave if he’s tired.
[M → (((S&¬D) ∨ (D&¬S))&(T → ¬D))]
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Reasoning and Critical Thinking
Part II.
2. (¬M &(S&D))&C
[Even though Sam wasn’t out of milk, he and Dave went to the store, but
it was closed.]
4. ¬(S ∨ D)
[Neither of them went to the store.]
6. ¬S → D
[Dave will go if Sam doesn’t.]
8. C → (¬S&¬D)
[If the store is closed, neither of them will go.]
10. (¬M &T ) → ¬(S ∨ D)
[If Sam’s not out of milk and Dave’s tired, neither of them will go to the
store.]
12. ¬(S&¬D)
[Sam won’t go to the store without Dave.]
14. ¬M → ((¬S&T ) → D)
[If Sam is out of milk, then Dave will go to the store, provided that Sam
doesn’t go and Dave isn’t tired.]
16. T ∨ D
[Either Dave is tired or he’ll go to the store.]
18. S → C
[If Sam goes to the store, it will be closed.]
20. ¬M
[Sam’s not out of milk.]
Part III.
2. A is a sufficient but not a necessary condition for B. Any number divisible by 4 (= 2 × 2) is even but there are even numbers that aren’t divisible
by 4, e.g., 6.
4. A is a necessary, but not a sufficient condition for B. To become law, a bill
must be passed by the House but also by the Senate.
6. A is a sufficient condition for B, as a triangle can have at most one right
angle. However A is not necessary for B, as there are figures that are not
triangles which do not have two right angles, e.g., a regular pentagon.
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Solutions to Even-Numbered Exercises
8. A is a necessary, but not a sufficient condition for B; Martha could not
have a son unless she was a parent, but would still be a parent if she only
had daughters.
10. A is necessary but not sufficient for B. If Fred were not heavier than
George, there is no way they could have the stated weights. Yet Fred
could still be heavier if they had different weights (e.g. Fred 92, George
90).
12. A is neither necessary nor sufficient for B. There are numbers divisible
by two but not by three (e.g., 4) as well as numbers divisible by three but
not by two (e.g., 9).
14. Debatable. It might be argued that A is a necessary condition for B, since
mammals (which do not produce their own food) could not exist unless
organisms that do produce their own food existed. On the other hand,
such organisms might not be plants. We can say in any case that A is not
a sufficient condition for B, since at one point there were plants but no
mammals.
16. A is a sufficient but not a necessary condition for B, since once bears exist
there are mammals, while there could still be mammals (e.g., squirrels)
even if there were no bears.
18. A is neither a necessary nor a sufficient condition for B. Jones may have
done something wrong without being caught or prosecuted, or simply
have done something that is wrong without being illegal, so A is not
sufficient for B. Nor is A necessary for B, since Jones might have done
nothing wrong but have been convicted nonetheless (either wrongly, or
else under an unjust law).
20. A is a necessary, but not a sufficient, condition for B. For A not to hold,
both would have to have won prizes, in which case B would be false. On
the other hand, A can be satisfied even though B isn’t—if, for instance,
Jones wins but Smith doesn’t.
Exercises, p. 62: I. Symbolize the following arguments, and identify the form
of inference used, stating at the same time whether or not the argument forms
are valid.
2. If it rains, we won’t have a picnic. We won’t have a picnic. So it will rain.
[R → ¬P, ¬P ∴ R Affirming the consequent; invalid.]
4. If it doesn’t rain, we’ll have a picnic. But it will rain. So we won’t have a
picnic.
[¬R → P, R ∴ ¬P Denying the antecedent, invalid.]
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Reasoning and Critical Thinking
6. We’ll only have a picnic if it doesn’t rain. It’s going to rain. So we won’t
have a picnic.
[P → ¬R, R ∴ ¬P Modus tollens; valid.] or
[R → ¬P, R ∴ ¬P Modus ponens, valid.]
8. We’ll have a picnic unless it rains. It isn’t going to rain. So we’ll have a
picnic.
[¬R → P, ¬R ∴ P Modus ponens, valid.]
10. The plants will die unless they’re watered. But they won’t be watered.
So they’re going to die.
[¬W → D, ¬W ∴ D Modus ponens, valid.]
12. The plants would be dead if they hadn’t been watered. The plants are
dead. So they must not have been watered.
[¬W → D, D ∴ ¬W Affirming the consequent, invalid.]
14. Joe would only have finished his essay on time if he had gotten off work
by 10. He didn’t finish his essay on time. So he must not have gotten off
work by 10.
[¬J → ¬T, ¬T ∴ ¬J Affirming the consequent, invalid.] or
[T → J, ¬T ∴ ¬J Denying the antecedent, invalid.]
16. Joe would only have finished his essay on time if he had gotten off work
by 10. He didn’t get off work by 10. So he must not have finished his
essay on time.
[¬J → ¬T, ¬J ∴ ¬T Modus ponens, valid.] or
[ T → J, ¬J ∴ ¬T Modus tollens, valid.]
II. Symbolize the following arguments as best you can given the dictionary
provided. Identify the symbolized forms, and say whether or not they are
valid. In some cases, the symbolized forms are invalid even though the original
arguments are valid. Identify these cases and explain what is going on.
Dictionary: O = I’ve told you once. T = I’ve told you a thousand times.
2. If I’ve told you once, I’ve told you a thousand times. But I’ve told you a
thousand times. So I’ve told you once.
[O → T, T ∴ O Affirming the consequent, invalid. The original argument is valid, as can be seen by considering the form: I’ve done X a
thousand times. So I’ve done X once.]
4. If I’ve told you once, I’ve told you a thousand times. But I haven’t told
you a thousand times. So I haven’t told you once.
[O → T, ¬T ∴ ¬O Valid, modus tollens.]
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Solutions to Even-Numbered Exercises
Exercises I, p. 67 Each of the following questions contains a list of premises
and a conclusion. Show that the conclusion follows from the premises by constructing an appropriate chain of inferences, or proof.
2)
1.
2.
3.
4.
5.
A → (B&C)
¬(B&C)
Pr.
Pr.
¬A
D
MT, 1, 2
DS, 3,4
A∨D
Pr./ Show D
4)
1.
2.
A → ¬B
A&D
Pr.
Pr.
3.
4.
¬B → C
(D ∨ E) → F
Pr.
Pr./ Show C&F
5.
6.
7.
8.
9.
10.
11.
A→C
HS, 1, 3
A
C
simp., 2
m.p., 5, 6
D
D∨E
simp., 2
weak., 8
m.p., 4, 9
conj., 7, 10
F
C&F
6)
1.
2.
3.
4.
5.
A→B
Pr.
A∨C
¬C
Pr.
Pr./Show B
A
DS, 2, 3
B
MP, 1, 4
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Reasoning and Critical Thinking
8)
1.
A&B
Pr.
2.
3.
A→C
B→D
Pr.
Pr.
A
B
simp., 1
simp., 1
7.
8.
C
D
m.p., 2, 5
m.p., 3, 6
9.
10.
C&D
¬¬(C&D)
conj., 7, 8
DN, 9
4.
5.
6.
11.
¬(C&D) ∨ E
E
Pr./Show E
DS, 4, 10
10)
1.
2.
(A ∨ B) → (C ∨ D) Pr.
A&¬D
Pr./Show C
3.
4.
A
A∨B
simp., 2
weak., 3
¬D
C
simp., 2
DS, 5, 6
5.
6.
7.
C∨D
MP, 1, 4
12)
1.
2.
A&(B&C)
A→D
Pr.
Pr.
(B ∨ E) → F
Pr./ Show (C ∨ G)&(D&F )
6.
7.
B&C
B
simp., 1
simp., 6
8.
9.
B∨E
F
weak., 7
MP, 3,8
10.
11.
C
C∨G
simp., 6
weak., 10
3.
4.
5.
12.
13.
A
D
simp., 1
MP, 2, 4
D&F
conj., 5,9
(C ∨ G)&(D&F ) conj., 11, 12
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Solutions to Even-Numbered Exercises
14)
1.
U ∨W
Pr.
P →R
Q→S
Pr.
Pr.
P
simp., 4
7.
8.
Q
R
simp., 4
MP, 2, 6
9.
10.
S
R&S
MP, 3, 7
conj., 8, 9
11.
12.
T &¬U
T
MP, 5, 10
simp., 11
13.
14.
¬U
W
simp., 11
DS, 1, 12
15.
W &T
conj., 12, 14
2.
3.
4.
5.
6.
P &Q
Pr.
(R&S) → (T &¬U ) Pr./ Show W &T
16)
1.
2.
3.
4.
5.
6.
7.
8.
9.
(A ∨ B) ∨ C
C → ¬D
Pr.
Pr.
D
¬¬D
simp., 3
DN, 4
¬A&D
¬C
A∨B
Pr./Show B
MT, 2, 5
DS, 1, 6
¬A
B
simp., 3
DS, 7, 8
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Reasoning and Critical Thinking
18)
1.
A&¬B
2.
3.
(C ∨ A) → (B ∨ D) Pr.
C → ¬D
Pr./Show ¬C
4.
5.
6.
7.
8.
9.
10.
Pr.
A
simp., 1
C∨A
B∨D
weak., 4
MP, 2, 5
¬B
D
simp., 1
DS, 6, 7
¬¬D
¬C
DN, 8
MT, 3, 9
20)
1.
A
Pr.
2.
A→C
Pr.
C
¬¬C
MP, 1, 2
DN, 5
3.
4.
5.
6.
7.
8.
B→D
¬C ∨ ¬D
Pr.
Pr./Show ¬B
¬D
DS, 4, 6
¬B
MT, 3, 7
II. Symbolize the following arguments, then show that they are valid by constructing proofs of their conclusions from their premises.
2. If Peters went to the lecture, then Quine didn’t. Either Quine went, or
Russell didn’t. If Sellars went, then Russell did. But Peters did go. So
Sellars didn’t.
P → ¬Q, Q ∨ ¬R, S → R, P ∴ ¬S
1.
2.
3.
4.
5.
6.
7.
P → ¬Q
Pr.
Q ∨ ¬R
S→R
Pr.
Pr.
¬R
¬S
DS, 2, 5
MT, 3,6
P
¬Q
Pr.
MP, 1, 4
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Solutions to Even-Numbered Exercises
4. Either Peters or Quine didn’t go to the lecture. If Sellars went, then Quine
did too. Now Peters did go to the lecture. So Sellars didn’t.
¬P ∨ ¬Q, S → Q, P ∴ ¬S
1.
2.
3.
4.
5.
6.
¬P ∨ ¬Q
S→Q
Pr.
Pr.
¬Q
¬S
DS, 1, 4
MT, 2, 5
P
¬¬P
Pr.
DN, 3
6. If Peters had gone to the lecture, then Quine would have as well. Either
Peters or Russell went. If Sellars went, then Quine didn’t. And Sellars
did go. So Russell went.
P → Q, P ∨ R, S → ¬Q, S ∴ R
1.
2.
3.
4.
5.
6.
7.
P →Q
P ∨R
Pr.
Pr
S → ¬Q
S
Pr.
Pr.
¬Q
¬P
MP, 3, 4
MT, 1, 5
R
DS, 2, 6
8. If Peters goes, the Quine will too. And if Quine goes, Russell will be sure
to tag along. But Russell isn’t going. So neither are Peters and Quine.
P → Q, Q → R, ¬R ∴ ¬P &¬Q
1.
2.
3.
4.
5.
6.
P →Q
Q→R
¬R
¬Q
Pr.
Pr.
Pr.
MT, 2, 3
¬P
MT, 1, 4
¬P &¬Q conj., 4,5
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Reasoning and Critical Thinking
10. Carnap only goes to the pub if Wittgenstein doesn’t. But Wittgenstein
went to the pub, and so did Neurath. And Neurath never goes to the
pub unless Hahn does too. But whenever Hahn and Neurath both go
to the pub, either Carnap or Schlick goes too. So Schlick was there, but
Carnap wasn’t.
C → ¬W, W &N, N → H, (H&N ) → (C ∨ S) ∴ S&¬C
1.
2.
3.
C → ¬W
Pr.
W &N
N →H
Pr.
Pr.
6.
7.
H
H&N
MP, 3,5
conj., 5,6
8.
9.
C ∨S
W
MP, 4,7
simp., 2
10.
¬¬W
DN, 9
¬C
S
MT, 1, 10
DS, 8, 11
S&¬C
conj., 11, 12
4.
5.
11.
12.
13.
(H&N ) → (C ∨ S) Pr.
N
simp., 2
Exercises, p. 70 Prove the following by reduction to absurdity:
2. Premise: A&¬B; Conclusion: ¬(A → B)
1.
A&¬B
Pr.
2.
3.
A→B
A
Assumption for RAA
simp., 1
4.
5.
B
¬B
MP, 2, 3
simp., 1
6.
¬(A → B) RAA, contradiction on lines 4,5
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Solutions to Even-Numbered Exercises
4. Premise: A&B; Conclusion: ¬(A → ¬B)
1.
A&B
Pr.
2.
A → ¬B
Assumption for RAA
B
simp., 1
3.
4.
5.
6.
A
¬B
simp., 1
MP, 2, 3
¬(A → ¬B) RAA, contradiction on lines 4,5
6. Premise: ¬A&¬B; Conclusion: ¬(A ∨ B)
1.
2.
3.
4.
5
6.
¬A&¬B
P r.
A∨B
Assumption for RAA
B
¬B
DS, 2, 3
simp., 1
¬A
simp., 1
¬(A ∨ B)
RAA, contradiction on lines 4, 5
8. Premises: A&B; Conclusion: ¬(¬A ∨ ¬B)
1.
A&B
Pr.
2.
3.
¬A ∨ ¬B
A
Assumption for RAA
simp., 1
4.
5.
¬¬A
¬B
DN, 3
DS, 2, 3
6
7.
B
¬(¬A ∨ ¬B)
simp., 1
RAA, contradiction on lines 5, 6
10. Premises: A → B, C → D, ¬B&¬D; Conclusion: ¬(A ∨ C)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
A→B
Pr.
A∨C
Assumption for RAA
C→D
¬B&¬D
Pr.
Pr.
¬B
¬A
simp., 3
MT, 1, 5
C
D
DS, 4, 6
MP, 2, 7
¬D
¬(A ∨ C)
simp., 3
RAA, contradiction on lines 8, 9
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Reasoning and Critical Thinking
12. Premise: ¬(A ∨ B); Conclusion: ¬(A&B)
1.
2.
3.
4.
5
¬(A ∨ B) P r.
A&B
Assumption for RAA
A
A∨B
simp., 2
weak, 3
¬(A&B)
RAA, contradiction on lines 1, 4
S OLUTIONS
FOR
C HAPTER 5
Exercises, p. 80 ff, Part I:
2.
D
18.
10.
F
U
P
H
U
U
4.
x
D
20.
12.
x
B
U
P
C
P
P
C
C
U
U
8.
24.
16.
D
C
22.
14.
U
U
P
U
6.
E
x
x
U
W
x
C
L
x
x
F
U
U
Part II.
2. Some of the lawyers in Mudville are rich.
4. Some of the happy people in Mudville aren’t rich.
6. Some people in Mudville are neither happy nor rich.
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8. None of the rich people in Mudville are happy.
10. Everyone in Mudville is either rich or happy.
12. There are no doctors or lawyers in Mudville.
14. There are no lawyers in Mudville and everyone there is happy.
16. Some of the lawyers in Mudville are unhappy.
18. All the happy people in Mudville are rich.
20. There are no lawyers in Mudville, and someone there is unhappy.
Exercises, p. 85 ff.
2. Valid:
Some P are C
P
x
No P are C.
C
P
U
C
U
4. Valid
Some non-P are C
P
C
Not all C are P
x
P
U
C
x
U
6. Invalid
Some P are C
All P are C
P
P
C
x
C
U
U
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Reasoning and Critical Thinking
8. Invalid
All P are C
P
All C are P.
C
P
U
C
U
10. Valid
Some B are not S.
Some non-S are B.
x
x
B
S
B
U
S
U
12. Valid
No M are F
No F are M
M
M
F
U
F
U
14. Valid
Some B are E
B
x
Some E are B
E
B
U
x
U
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16. Invalid
No E are F
Some E are not F.
x
E
E
F
F
U
U
18. Invalid
Some B are not D.
Not all D are B
x
B
D
B
U
D
x
U
20. Valid
Some non-B are E.
Not all E are B.
x
x
E
E
B
B
U
U
Exercises, p. 93 ff:
M
2. Invalid
4. Invalid
T
T
S
S
U
M
U
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6. Invalid
8. Valid
M
E
x x
T
x
S
R
U
L
U
10. Valid
12. Valid
E
E
R
L
L
U
R
U
14. Valid
16. Valid
E
E
C
D
C
U
D
U
D
18. Valid
20. Invalid
E
B
C
S
U
U
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22. Invalid
24. Invalid
S
E
x x
W
B
F
U
R
U
Exercises, p. 98 ff, Part I
2. ∃x(V x&¬M x)
12. ∀x(U x → M x)
4. ∃xAx
14. ∃x(U x&M x)
6. ∀xAx
16. ¬∀x¬Ix or ∃xIx
8. ∃x¬Ix
18. ¬∃x(Ix&W x)
20. ∃x(M x&¬W x)
10. ∀x(V x → ¬Ix)
Part II:
2.
(a) If Joe buys a Ferrari, he’ll win the lottery.
(b) If Joe doesn’t win the lottery, he won’t buy a Ferrari.
4.
(a) All odd numbers are prime numbers which are greater than 2.
(b) If a number isn’t odd, then it isn’t a prime number greater than 2.
6.
(a) All flying mammals are bats.
(b) If something isn’t a flying mammal, then it isn’t a bat.
8.
(a) Everything that produces seeds is a flowering plant.
(b) Something that doesn’t produce seeds isn’t a flowering plant.
10.
(a) If it’s depressing, then Bernhard wrote it.
(b) If it’s not depressing, Bernhard didn’t write it.
S OLUTIONS
FOR
C HAPTER 6
Exercises 6, Part I:
2. Acceptable as common knowledge
4. Not acceptable because false: any bird is a counterexample.
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6. Not acceptable because false; if you didn’t know that, you should ask for
supporting arguments in any case.
8. Not acceptable; would require strong supporting arguments.
10. Acceptable as common knowledge.
12. Acceptable as common knowledge.
14. Not acceptable since commonly known to be false.
16. If you know a fair amount of mathematics, this might pass as common
knowledge; otherwise, would require supporting arguments.
18. Not acceptable. Would require supporting arguments. What about the
police or the army, for example?
20. Not acceptable since false; the odds of any one person winning are extremely small.
22. Acceptable as common knowledge.
24. Not acceptable, would require strong supporting arguments; the odds
are heavily against this claim.
26. Not acceptable. Since people disagree strongly on this matter, the claim
could not be taken for granted. It would require strong supporting arguments.
28. Not acceptable. Since people disagree strongly on this matter, the claim
could not be taken for granted. It would require strong supporting arguments.
30. Not acceptable. Would require strong supporting arguments; the odds
are heavily against this claim.
Part II.
2. (a) is relevant, and indeed completely adequate to prove (b).
4. Irrelevant; there is no reason to suppose that Australians (even those
named Bruce) are more likely to be philosophers than other people.
6. (a) is relevant to (b), in the sense that it tells us that a causal link cannot
be ruled out. This being said, the relevance is minimal.
8. Relevant, provided that the trial took place in a well-functioning judicial
system. Though some people who are convicted are innocent, a conviction in a fair trial provides a fairly strong reason to believe in the guilt of
the person charged. In a poorly functioning or corrupt judicial system,
however, relevance may be lacking.
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10. Relevant; the crowd control required would cost the Police Department
extra money.
12. Irrelevant: The fact that some people would benefit if a certain policy
were enacted does not mean that the policy would be harmful. Suppose, for example, that everyone would be better off if taxes were lowered. In that case, too, those in favour of tax cuts would also benefit, but
we couldn’t say that the cuts were harmful in that case.
14. Irrelevant: Guilty or innocent, we would expect his family to suffer hardship if he were jailed.
16. Relevance is not obvious: we would want to know whether the good
performance was connected in any way to the government’s policies and
decisions.
18. Relevant in normal circumstances, since laws are usually passed against
actions that are deemed wrong. However, there are cases where illegal
actions are not considered wrong: for example, if the law is unjust, or in
some cases of civil disobedience.
20. Relevance is not obvious. It all depends upon Carruthers. If he is credulous, unreliable, etc., we would tend to find his beliefs irrelevant. If, on
the other hand, he is a careful, thoughtful, and well-informed person, we
would find them relevant.
Part III.
2. Argument in standard form:
1. The arguer’s nieces say that they are not interested in pursuing scientific careers because the lifestyle doesn’t appeal to them.
2. This is understandable, because scientific careers involve spending
most of your life in a lab, barely talking to other people, and everyone
wearing a white coat.
3. The main reason why fewer women than men pursue highly technical
careers is that the lifestyle doesn’t appeal to them. (FC, from 1, 2)
Acceptability: Premise one is hearsay. We have only the arguer’s word
for what his nieces actually told him. Whether this is the whole story,
or whether he has even reported their words accurately, is far from obvious.The second premise is not acceptable, as many jobs in computer
science and engineering do not fit the description. Many engineers work
outdoors, for example, and few computer scientists work in labs or wear
lab coats.
Relevance: If true, the nieces’ testimony would be relevant, as it would
provide some explanation of their reluctance to pursue a career in a highly
technical field.
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Adequacy: The premises, even if acceptable, would provide only minimal support for the conclusion. The secondhand testimony of two people, even if accurately reported, is not nearly enough to support such
sweeping claims.
On the whole, then, this is a very poor argument.
4. Argument in standard form:
1. North Korea is a communist state with a centrally planed economy.
2. North Korea is one of the poorest countries in the world.
3. The USA has a free-market economy.
4. The USA is one of the richest countries in the world.
5. Free-market economies outperform centrally planned ones. (FC, from
1, 2, 3, 4)
Acceptability. All the premises are acceptable as common knowledge;
those who are not aware of the truth of these claims can easily verify
them with a little research.
Relevance: All premises are relevant for the conclusion, as they establish that, in the case of two particular countries, a free-market economy
outperforms a centrally planned one.
Adequacy: The evidence offered (one pair of countries) is too meagre
to support the sweeping generalization made in the conclusion. It is
like arguing that Fred, who is from Alberta, is taller than George from
Saskatchewan, so people in Alberta are taller than people in Saskatchewan.
On the whole, a very weak argument.
6. Argument in standard form:
1. Innocent people get convicted in our judicial system.
2. Such mistakes may take years to discover and correct.
3. We cannot even be certain that all such mistakes will eventually be
discovered and corrected.
4. It is wrong for anyone to kill an innocent person, including employees
of the state.
5. We should think twice before supporting capital punishment. (FC,
from 1, 2, 3, 4)
Acceptability: Premises 1–3 are acceptable as common knowledge, or
can be verified after a little research There have been quite a few welldiscussed cases of wrongful convictions. Premise 4 seems a quite reasonable moral principle, and can also be accepted as common knowledge.
Relevance: The premises are obviously relevant to the conclusion, since
they establish that there is a risk of killing an innocent person if capital
punishment is reinstated, and that that would be wrong.
Adequacy: The premises are more than adequate for the conclusion. The
important thing to notice here is just how weak the conclusion is. It does
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not say that we should not support capital punishment, but rather only
that we should think twice before doing so. The premises fully support
this modest claim.
8. Argument in standard form:
1. Going to university is a privilege that students should have to pay for.
2. Most students can easily afford to pay more if they stop spending so
much on things like cell phones, beer and cigarettes.
3. Tuition increases are the only way to raise the extra money the university needs.
4. Students should pay higher tuition fees. (FC, from 1, 2, 3)
Acceptability: The first half of premise 1 (attending university is a privilege) seems acceptable, but the second half (students should have to pay
for this privilege) is controversial. If a student had to work long hours in
order to pay tuition, for example, his or her studies would inevitably suffer, and that would seem counterproductive. Premise 2 is unacceptable
as it stands. Most students spend no money on cigarettes, for example,
and more than a few spend nothing on beer. A more reasonable claim
might be that students sometimes spend money on things that are not
necessities. P3 is clearly unacceptable, since universities could also obtain more money from the provincial government or private sources. P3
also presupposes that the university does need more money, but no reason
is given to believe that this is so.
Relevance: If the first premise were acceptable, it would certainly be relevant, as it would at least support the claim that students should pay
something for their education. Premise 2, if it were true, would also be
relevant, as it would establish students’ ability to pay more if it were true.
Since premise P3 is unacceptable, we do not need to assess its relevance.
Adequacy: The premises do not provide adequate support for the conclusion. Even supposing premise 1 to be acceptable, it would only show that
university students should pay something for their education. It would
not, even with the help of the dubious premise P2, show that students are
not paying enough or even too much now. It would at most show that
they could pay more, not that they should.
10. Argument in standard form:
1. Government-run gambling generates a lot of revenue for the provinces.
2. These revenues help to pay for important services.
3. Opportunities to gamble would exist even if the government got out
of the business.
4. Even though gambling is addictive and causes serious problems, only
if it is government-run can we be sure that proper measures will be taken
to combat problem gambling.
5. Governments should remain in the gambling business. (FC, from 1–4)
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Acceptability: Premises 1, 2, and 3 are acceptable, either as common
knowledge for a reasonably well-informed citizen, or verifiable with a
little research. The first part of premise 4 (that gambling causes serious
problems) is also acceptable as common knowledge. The second part,
however, is not acceptable without supporting arguments (which are not
provided here)—for instance, couldn’t similar measures be put in place
if gambling were run by the private sector but regulated by the government?
Relevance: Premises 1 and 2 are relevant, since they inform us that government-run gambling has at least some positive impacts. The relevance of
premise 3 is less clear: just because people would engage in an activity
even if the government had nothing to do with it does not give us a reason to accept that the government should be involved with it (consider,
for example, glue sniffing or car surfing). Premise 4 would have been
relevant if acceptable, but, as just noted, it isn’t.
Adequacy: This argument gives reasons in favour of government-run
gambling, and also raises and responds to the important objection that
gambling causes serious social problems. That response, however, is
weak, based on the unacceptable claim contained in premise 4. In addition, a number of other fairly obvious objections that might be raised
have not been addressed: for instance, does government-run gambling
(along with advertising, etc.) create more problem gambling? In sum,
the conclusion is not adequately supported.
12. Argument in standard form:
1. In the first-past-the-post system, candidates and can and frequently do
get elected with fewer than half the votes cast.
2. Majority governments have been elected with less than 40 percent of
the votes cast.
3. Governments are often elected against the wishes of a majority of voters. (IC, from 1, 2)
4. With proportional representation, the choices of voters are accurately
reflected in the composition of the legislature.
5. We should replace the first-past-the-post system with proportional representation. (FC, from 3, 4)
Acceptability: All premises are acceptable as common knowledge for a
reasonably well-informed citizen.
Relevance: The premises are also relevant, since they establish a point
in which proportional representation performs better than first-past-thepost.
Adequacy: The premises do not provide adequate support for the conclusion. Although we are told of one point that favours proportional
representation, no consideration whatsoever is given to points that might
count against it, nor to any points that might count in favour of first-pastthe-post. At most, we have here support for a much weaker conclusion,
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such as: In at least one respect, PR performs better than FPTP, and thus
merits further consideration.
S OLUTIONS
FOR
C HAPTER 7
Exercises, p. 118ff., Part I:
2. Voluntary polls, to begin with, do not provide a random sample, since
only those who decide to answer them are counted. We might expect,
for example, that those with strong opinions on the matter will be overrepresented, as will those who have more time on their hands to bother
with such things. Both are potential sources of bias. A poll from a single
newspaper is also problematic, since different newspapers tend to have
different readerships, so that we would only be obtaining a non-random
subset of a non-random subset of the public. This method is therefore
highly unreliable.
4. To begin with, much would depend upon which sites were consulted. If,
for example, we checked a Facebook page devoted to opposing a provincial government initiative, we shouldn’t expect to obtain a random sample of public opinion. Newspaper comments sections are often cluttered
up with contributions of paid commenters, so they too can be unreliable
reflections of what people actually think. Finally, there are many people
who do not use social media at all, who would not be sampled. Since
social media use is negatively correlated with age, this would be yet another source of bias. On the whole, this would not be a good sampling
method.
6. This would be a fairly good sample for this specific question. It could be
made better by including local stations and other forms of media, however.
8. This sampling method would have advantages as well as drawbacks. On
the plus side, developed countries tend to have more extensive medical
services, so we might expect the number of reported cases of ADHD to
be closer to the actual number than in countries where fewer children are
seen by medical personnel. On the minus side, there might be factors
present in developing countries that favour the development of ADHD.
If so, and the incidence is lower in less developed countries, the sample
would not accurately reflect the global population. Finally, the particular choice of developed countries might make a difference, since rates of
diagnosis could vary considerably from one to another. On the whole,
then, this is a poor method of sampling.
10. On question (a), this sampling method would be fairly good, though
there are other, better ways of getting information concerning this question (e.g., statistics on RRSP contributions, pension plans, etc.). This
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method would be less reliable for question (b) since, in a personal interview, we have reason to fear that people might not tell the truth.
II.
2. There are two concerns here. First, as unlikely as it seems, it might be
that the number of reported robberies increased sharply even though the
actual number did not. A more significant flaw is the assumption that an
increase in robberies indicates a corresponding increase in robbers. For
it could well be that a few robbers were very active, etc. On the whole,
then, this is a weak argument.
4. This argument makes a causal claim based only on the correlation between university study and income, and hence is an extremely weak one.
University admissions are selective: it is the students who did better in
high school or CEGEP who get in. It is entirely possible that those people
would have ended up with higher incomes whether or not they attended
university. Nothing in this argument rules out this possibility.
6. Two sets of results are reported here. The first does not support the conclusion, since, with the margin of error, support for torture likely lies between 51.5% and 58.5%, while support for enhanced interrogation techniques likely lies between 53.5% and 60.5%. Thus there is not enough
information to say that one receives more support than the other. The
second set of results is significant even with the margin of error, and thus
supports the conclusion. This being said, it provides only limited support
for that conclusion, since the poll is only claimed to be accurate nineteen
times out of twenty, and also concerns only one comparison of attitudes
involving a euphemism. The argument would be stronger if other polls
(on the same and on related questions) produced similar results. Finally,
insofar as the expression “makes a difference” is taken to express a causal
claim, it would not be well-supported, given that we only have a correlation, and that in only one of the two cases.
8. A short leap from correlation to causation, and thus a weak argument.
In some cases, at least, the causation would run the opposite way, since
health care expenditures can quickly impoverish people. The claim that
stress is more particularly responsible is another completely unsupported
causal claim. A weak argument.
10. Life expectancy data for Canadians are readily available, so the use of a
worldwide figure makes for a weak argument. It is also quite likely that
other information about Fred (e.g., does he smoke, have a family history
of heart disease, diabetes, etc.) would yield an even better estimate without too much effort.
12. The sample here is biased, since the observations were all made on Friday
nights, when one would expect the regular staff not to be working, and
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fewer employees working overall. The argument is consequently a weak
one. Observations made at various other times during the week would
be required in order to support the conclusion adequately.
14. Again, we have a quick move from correlation to causation—sloppy, post
hoc reasoning. It seems obvious, too, that causation must have run the
opposite way in many cases, as divorced women had to find jobs to feed
their families. A final weakness in the argument is the assumption that
the only way to give the institution of the family its due is to prevent
women from working. Even if women working did put a strain on families, surely other possibilities exist for strengthening the institution?
16. Again, post hoc reasoning, and again overlooking the possibility that causation may have run the other way in at least some cases. Couldn’t
it be that some schools brought in sex ed because officials were concerned about the increase in sexual activity among students and the consequences of ignorance?
18. A very weak argument, that moves in one step from correlation to causation. Helpfully, two factors (increased temperature, increased daylength) are mentioned here, so that we can see quite clearly that at least
one alternative explanation exists, and that nothing has been done to exclude it. Not to mention the possibility that one factor might be responsible in some cases, the other in others, and a combination of the two in still
others. Or that other factors, not mentioned here, might also be involved.
20. Since data on causes of death for Canadians (and many other countries)
are readily available, the reliance on worldwide data makes for a very
weak argument. If you live here and do not travel to certain parts of the
world, malaria is not a significant risk.
22. One final leap from correlation to causation. How do we know the teachers are responsible for the higher test scores? It is entirely possible, for
instance, that the school just happens to have students who would have
performed better on the tests regardless of who had taught them.
S OLUTIONS
TO
C HAPTER 8
Exercises I (p. 136ff.):
2. In a leap year, the month of February has 29 days.
[no flaws]
4. Prince Edward Island is Canada’s smallest province.
[‘smallest’ is ambiguous: smallest in population or in area? Interestingly,
the claim is true either way.]
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6. The government is taking steps to ensure that the rate of child poverty
will fall to an acceptable level.
[‘taking steps’ and ‘acceptable’ are both vague: what steps? and what
level is acceptable?]
8. Buy the new Chevy Impala: it has all the best features at a price you can
live with.
[‘all the best features’ and ‘price you can live with’ are both vague: what
features? how much?]
10. The fears that we will soon run out of oil are not well-founded. There is
plenty of oil in the ground, enough to keep us going for years.
[‘plenty’ and ‘for years’ are both vague: how much? and how many
years?]
12. Men are more interested in hockey than women.
[Ambiguous: are men more interested in hockey than women are or is
their attention drawn to hockey in preference to women?]
14. This house has everything. You should buy it.
[‘Everything’ is vague: what exactly does the house have?]
16. We regret to inform you that, due to market pressures, our company is
forced to downsize, and as a consequence you will be free as of next week
to pursue new opportunities.
[‘downsize’ is a euphemism for ‘firing employees’; ‘free to pursue new
opportunities’ a euphemism for ‘out of work’]
18. The company’s auditor did say that there were irregularities in the accounts, but that none of them was very serious.
[‘irregularities’ is vague, and perhaps also a euphemism for evidence of
fraud, etc.; ‘not very serious’ is also vague: how serious is not very?]
20. People say that Creationism shouldn’t be taught in the public schools,
because it is just a theory, not proven fact. But Evolution is also a theory,
yet it gets taught in the schools. So there’s no reason why one should be
taught and the other not.
[‘theory’ is used ambiguously here, first with the meaning of ‘unsubstantiated speculation’ and then to mean a body of claims that has received
substantial confirmation of various kinds (by experiments, etc.)]
22. The minister was, admittedly, somewhat economical with the truth.
[A euphemism for lying or not telling everything he should have.]
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24. For years, liberals have been waging an all-out war against gun rights for
law-abiding citizens. The main argument used by these anti-Americans
is that fewer guns equals a safer society. Is there a poor, pathetic leftist
out there who actually believes that criminals buy guns in stores?
[There is a fair amount of prejudicial, emotionally charged language here,
e.g., ‘anti-Americans’, ‘poor, pathetic leftist’, even ‘actually’, since it might
be taken to suggest that such belief could only be the product of idiocy.]
26. Aries (Thursday, Nov. 1): Big changes are in store for you. Live life to the
fullest. Now is not the time to be wary.
[‘big changes’, ‘to the fullest’ and ‘wary’ are all vague; what changes, or at
least what kind of changes? How exactly does one live life to the fullest?
Wary of what?]
28. The Minister has taken all necessary steps to ensure that no serious problems will arise because of Canada’s immigration policy. The people who
continue to criticize her are just narrow-minded bigots.
[Vagueness and prejudicial, emotionally-charged language: ‘all necessary steps’—what steps? ‘serious’—what does she mean by this? Finally,
‘narrow-minded bigots’ is obviously prejudicial.]
II. Exercises on euphemisms
2. pacification operations = murder of thousands of civilians, terrorization of
civilian populations
4. regime change = violent overthrow of the government
6. surgical strike = bombing raid
8. friendly fire= killed by his own troops
10. ethnic cleansing = removal of an ethnic or religious group from an area
where it lives by murder, threats, deportation, etc.
12. The enemy troops have failed utterly to advance beyond the city of X.= They
have advanced right up to city X.
14. downsizing= firing employees
16. His position was terminated = he was fired.
18. handyman’s special = in serious need of repair
20. rest room= toilet
22. He has some motivation issues = he’s lazy.
24. needs a little TLC. Perfect for the mechanically inclined= in serious need of
repair
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Reasoning and Critical Thinking
III. Exercises on definitions.
2. An equilateral triangle is a triangle with three equal sides and three equal
angles.
[Extensionally correct, but redundant, since any triangle with three equal
sides already has three equal angles.]
4. A musician is a man who plays a musical instrument.
[Too narrow: overlooks female musicians, and male singers.]
6. Hockey is a game played on ice in Canada.
[Too narrow: Hockey is played in other countries as well. Too wide: e.g.,
ringette, curling.]
8. A banker is someone who works in a bank.
[Too wide, e.g., a janitor might also work in a bank.]
10. A mammal is a warm-blooded animal.
[Too wide, e.g., birds.]
12. A valid argument is one with true premises and a true conclusion.
[Too wide and too narrow. There are valid arguments with false premises
and/or false conclusions, but also arguments with true premises and conclusions that aren’t valid.]
14. A sound argument is one with true premises and a true conclusion.
[Too wide; here is a counterexample: Fish live in water. So mammals are
warm-blooded.]
16. Knowledge is true belief.
[Too wide: we can have true belief without knowledge, e.g., if we guess
right and believe in our guess.]
18. Severe pain is pain caused by serious physical injury, such as organ failure,
impairment of bodily function, or even death, and mental pain and/or
suffering is severe only if it lasts for months or even years.
[Too narrow: there are many cases where pain would be called severe
when none of the stated conditions are met; and the restriction concerning mental pain is wholly arbitrary, as there are obvious cases of severe,
but short-lived mental suffering.]
20. A fish is an animal with fins which lives in the ocean.
[Too wide, e.g., dolphins; also too narrow, e.g., fresh water fish such as
walleye.]
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Solutions to Even-Numbered Exercises
22. February is the shortest month of the year.
[This definition is extensionally correct.]
24. French is the language spoken by the vast majority of people living in
France.
[This definition is fine as it stands. There is no circularity, as we can define
France without reference to the French language.]
26. Murder is intentionally killing a person.
[Too wide: not all cases of intentional killing are murder, e.g., in war, or
cases of self-defence.]
28. A paper clip is a metal device used to hold several pieces of paper together.
[Too wide: e.g., staples. Also too narrow, since not all paper clips are
made of metal.]
S OLUTIONS
FOR
C HAPTER 9
Exercises, p. 153 ff, Part I:
2. False dichotomy
4. Circumstantial ad hominem
6. Appeal to ignorance
8. Ad populum
10. Equivocation (‘natural’)
12. Begs the question
14. Appeal to ignorance
16. No fallacy
18. Tu quoque
20. No fallacy
22. No fallacy; though there is an attack on the person, that person’s character is what is in question.
24. Faulty appeal to authority
26. Straw man
28. Hasty generalization
30. Gambler’s fallacy
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Reasoning and Critical Thinking
32. Faulty appeal to authority
34. Two wrongs/ ad populum
36. An explanation, not an argument
38. Post hoc
40. Appeal to force
42. Lying
44. Composition
46. Complex question: presupposes that the complaints are pointless.
48. Tu quoque
50. Appeal to pity
52. Gambler’s fallacy
54. Straw man + a series of complex questions
56. Equivocation (‘equal’)
58. Tu quoque
60. Argument ad nauseam
62. A kind of innocence by association
64. Begs the question
66. Fallacy of composition
68. False dichotomy
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