Learning Objectives
After reading this chapter, you should be able to:
1. Define key terms and concepts in inductive logic, including strength and cogency.
2. Differentiate between strong inductive arguments and weak inductive arguments.
3. Identify general methods for strengthening inductive arguments.
4. Identify statistical syllogisms and describe how they can be strong or weak.
5. Evaluate the strength of inductive generalizations.
5Inductive Reasoning
Iakov Kalinin/iStock/Thinkstock
6. Differentiate between causal and correlational relationships and describe various
types of causes.
7. Use Mill’s methods to evaluate causal arguments.
8. Recognize arguments from authority and evaluate their quality.
9. Identify key features of arguments from analogy and use them to evaluate the strength
of such arguments.
When talking about logic, people often think about formal deductive reasoning. However, most of the
arguments we encounter in life are not deductive at all. They do not intend to establish the truth of the
conclusion beyond any possible doubt; they simply try to provide good evidence for the truth of their
conclusions. Arguments that intend to reason in this way are called inductive arguments. Inductive
arguments are not any worse than deductive ones. Often the best evidence available is not final or
conclusive but can still be very good.
For example, to infer that the sun will rise tomorrow because it has every day in the past is inductive
reasoning. The inference, however, is very strongly supported. Not all inductive arguments are as strong
as that one. This chapter will explore different types of inductive arguments and some principles we can
use to determine whether they are strong or weak. The chapter will also discuss some specific methods
that we can use to try to make good inferences about causation. The goal of this chapter is to enable you
to identify inductive arguments, evaluate their strength, and create strong inductive arguments about
important issues.
Age fotostock/SuperStock
Weather forecasters use inductive
reasoning when giving their
predictions. They have tools at their
disposal that provide support for their
arguments, but some arguments are
weaker than others.
5.1 Basic Concepts in Inductive Reasoning
Inductive is a technical term in logic: It has a precise definition, and that definition may be different from
the definition used in other fields or in everyday conversation. An inductive argument is one in which
the premises provide support for the conclusions but fall short of establishing complete certainty. If you
stop to think about arguments you have encountered recently, you will probably find that most of them
are inductive. We are seldom in a position to prove something absolutely, even when we have very good
reasons for believing it.
Take, for example, the following argument:
The odds of a given lottery ticket being the winning ticket are extremely low.
You just bought a lottery ticket.
Therefore, your lottery ticket is probably not the winning ticket.
If the odds of each ticket winning are 1 in millions, then this argument gives very good evidence for the
truth of its conclusion. However, the argument is not deductively valid. Even if its premises are true, its
conclusion is still not absolutely certain. This means that there is still a remote possibility that you
bought the winning ticket.
Chapter 3 discussed how an argument is valid if our premises guarantee the truth of the conclusion. In
the case of the lottery, even our best evidence cannot be used to make a valid argument for the
conclusion. The given reasons do not guarantee that you will not win; they just make it very likely that
you will not win.
This argument, however, helps us establish the likelihood of its conclusion. If it were not for this type of
reasoning, we might spend all our money on lottery tickets. We would also not be able to know whether
we should do such things as drive our car because we would not be able to reason about the likelihood
of getting into a crash on the way to the store. Therefore, this and other types of inductive reasoning are
essential in daily life. Consequently, it is important that we learn how to evaluate their strength.
Inductive Strength
Some inductive arguments can be better or worse than
others, depending on how well their premises increase the
likelihood of the truth of their conclusion. Some arguments
make their conclusions only a little more likely; other
arguments make their conclusions a lot more likely.
Arguments that greatly increase the likelihood of their
conclusions are called strong arguments; those that do not
substantially increase the likelihood are called weak
arguments.
Here is an example of an argument that could be considered
very strong:
A random fan from the crowd is going to race (in a
100 meter dash) against Usain Bolt.
Usain Bolt is the fastest sprinter of all time.
Oksana
Kostyushko/iStock/Thinkstock
Context plays an
important role in
inductive arguments.
What makes an
argument strong in one
context might not be
strong enough in
another. Would you be
more likely to play the
lottery if your chances
of winning were
supported at 99%?
Therefore, the fan is going to lose.
It is certainly possible that the fan could win—say, for example, if Usain Bolt breaks an ankle—but it
seems highly unlikely. This next argument, however, could be considered weak:
I just scratched off two lottery tickets and won $2 each time.
Therefore, I will win $2 on the next ticket, too.
The previous lottery tickets would have no bearing on the likelihood of winning on the next one. Now
this next argument’s strength might be somewhere in between:
The Bears have beaten the Lions the last four times they have played.
The Bears have a much better record than the Lions this season.
Therefore, the Bears will beat the Lions again tomorrow.
This sounds like good evidence, but upsets happen all the time in sports, so its strength is only moderate.
Considering the Context
It is important to realize that inductive strength and weakness are relative
terms. As such, they are like the terms tall and short. A person who is short
in one context may be tall in another. At 6’0”, professional basketball player
Allen Iverson was considered short in the National Basketball Association.
But outside of basketball, someone of his height might be considered tall.
Similarly, an argument that is strong in one context may be considered
weak in another. You would probably be reasonably happy if you could
reliably predict sports (or lottery) results at an accuracy rate of 70%, but
researchers in the social sciences typically aim for certainty upward of 90%.
In highenergy physics, the goal is a result that is supported at the level of 5
sigma—a probability of more than 99.99997%!
The same is true when it comes to legal arguments. A case tried in a civil
court needs to be shown to be true with a preponderance of evidence,
which is much less stringent than in a criminal case, in which the defendant
must be proved guilty beyond reasonable doubt. Therefore, whether the
argument is strong or weak is a matter of context.
Moreover, some subjects have the sort of evidence that allows for extremely
strong arguments, whereas others do not. A psychologist trying to predict
human behavior is unlikely to have the same strength of argument as an
astronomer trying to predict the path of a comet. These are important
things to keep in mind when it comes to evaluating inductive strength.
Strengthening Inductive Arguments
Regardless of the subject matter of an argument, we
generally want to create the strongest arguments we can. In
general, there are two ways of strengthening inductive
arguments. We can either claim more in the premises or
claim less in the conclusion.
Fuse/Thinkstock
The strength of an inductive argument
can change when new premises are
added. When evaluating or presenting
an inductive argument, gather as
many details as possible to have a
more complete understanding of the
strength of the argument.
Claiming more in the premises is straightforward in theory,
though it can be difficult in practice. The idea is simply to
increase the amount of evidence for the conclusion. Suppose
you are trying to convince a friend that she will enjoy a
particular movie. You have shown her that she has liked
other movies by the same director and that the movie is of
the general kind that she likes. How could you strengthen
your argument? You might show her that her favorite actors
are cast in the lead roles, or you might appeal to the reviews
of critics with which she often agrees. By adding these
additional pieces of evidence, you have increased the
strength of your argument that your friend will enjoy the
movie.
However, if your friend looks at all the evidence and still is
not sure, you might take the approach of weakening your conclusion. You might say something like,
“Please go with me; you may not actually like the movie, but at least you can be pretty sure you won’t
hate it.” The very same evidence you presented earlier—about the director, the genre, the actors, and so
on—actually makes a stronger argument for your new, less ambitious claim: that your friend won’t hate
the movie.
It might help to have another example of how each of the two approaches can help strengthen an
inductive argument. Take the following argument:
Every crow I have ever seen has been black.
Therefore, all crows are black.
This seems to provide decent evidence, provided that you have seen a lot of crows. Here is one way to
make the argument stronger:
Studies by ornithologists have examined thousands of crows in every continent in which they
live, and they have all been black.
Therefore, all crows are black.
This argument is much stronger because there is much more evidence for the truth of the conclusion
within the premise. Another way to strengthen the argument—if you do not have access to lots of
ornithological studies—would simply be to weaken the stated conclusion:
Every crow I have ever seen has been black.
Therefore, most crows are probably black.
This argument makes a weaker claim in the conclusion, but the argument is actually much stronger than
the original because the premises make this (weaker) conclusion much more likely to be true than the
original (stronger) conclusion.
By the same token, an inductive argument can also be made weaker either by subtracting evidence from
the premises or by making a stronger claim in the conclusion. (For another way to weaken or strengthen
inductive arguments, see A Closer Look: Using Premises to Affect Inductive Strength.)
A Closer Look: Using Premises to Affect Inductive Strength
Suppose we have a valid deductive argument. That means that, if its premises are all true, then its
conclusion must be true as well. Suppose we add a new premise. Is there any way that the
argument could become invalid? The answer is no, because if the premises of the new argument
are all true, then so are all the premises of the old argument. Therefore, the conclusion still must
be true.
This is a principle with a fancy name; it is called monotonicity: Adding a new premise can never
make a deductive argument go from valid to invalid. However, this principle does not hold for
inductive strength: It is possible to weaken an inductive argument by adding new premises.
The following argument, for example, might be strong:
99% of birds can fly.
Jonah is a bird.
Therefore, Jonah can fly.
This argument may be strong as it is, but what happens if we add a new premise, “Jonah is an
ostrich”? The addition of this new premise just made the argument’s strength plummet. We now
have a fairly weak argument! To use our new big word, this means that inductive reasoning is
nonmonotonic. The addition of new premises can either enhance or diminish an argument’s
inductive strength.
An interesting “game” is to see if you can continue to add premises that continue to flip the
inductive argument’s degree of strength back and forth. For example, we could make the
argument strong again by adding “Jonah is living in the museum of amazing flying ostriches.”
Then we could weaken it again with “Jonah is now retired.” It could be strengthened again with
“Jonah is still sometimes seen flying to the roof of the museum,” but it could be weakened again
with “He was seen flying by the neighbor child who has been known to lie.” The game
demonstrates the sensitivity of inductive arguments to new information.
Thus, when using inductive reasoning, we should always be open to learning more details that
could further serve to strengthen or weaken the case for the truth of the conclusion. Inductive
strength is a neverending process of gathering and evaluating new and relevant information. For
scientists and logicians, that is partly what makes induction so exciting!
Inductive Cogency
Notice that, like deductive validity, inductive strength has to do with the strength of the connection
between the premises and the conclusion, not with the truth of the premises. Therefore, an inductive
argument can be strong even with false premises. Here is an example of an inductively strong argument:
Every lizard ever discovered is purple.
Therefore, most lizards are probably purple.
Of course, as with deductive reasoning, for an argument to give good evidence for the truth of the
conclusion, we also want the premises to actually be true. An inductive argument is called cogent if it is
strong and all of its premises are true. Whereas inductive strength is the counterpart of deductive
validity, cogency is the inductive counterpart of deductive soundness.
5.2 Statistical Arguments: Statistical Syllogisms
The remainder of this chapter will go over some examples of the different types of inductive arguments:
statistical arguments, causal arguments, arguments from authority, and arguments from analogy. You
will likely find that you have already encountered many of these various types in your daily life.
Statistical arguments, for example, should be quite familiar. From politics, to sports, to science and
health, many of the arguments we encounter are based on statistics, drawing conclusions from
percentages and other data.
In early 2013 American actress Angelina Jolie elected to have a preventive double mastectomy. This
surgery is painful and costly, and the removal of both breasts is deeply disturbing for many women. We
might have expected Jolie to avoid the surgery until it was absolutely necessary. Instead, she had the
surgery before there was any evidence of the cancer that normally prompts a mastectomy. Why did she
do this?
Jolie explained some of her reasoning in an opinion piece in the New York Times.
I carry a “faulty” gene, BRCA1, which sharply increases my risk of developing breast cancer and
ovarian cancer.
My doctors estimated that I had an 87 percent risk of breast cancer and a 50 percent risk of
ovarian cancer, although the risk is different in the case of each woman. (Jolie, 2013, para. 2–3)
As you can see, Jolie’s decision was based on probabilities and statistics. If these types of reasoning can
have such profound effects in our lives, it is essential that we have a good grasp on how they work and
how they might fail. In this section, we will be looking at the basic structure of some simple statistical
arguments and some of the things to pay attention to as we use these arguments in our lives.
One of the main types of statistical arguments we will discuss is the statistical syllogism. Let us start
with a basic example. If you are not a cat fancier, you may not know that almost all calico cats are
female—to be more precise, about 99.97% of calico cats are female (Becker, 2013). Suppose you are
introduced to a calico cat named Puzzle. If you had to guess, would you say that Puzzle is female or male?
How confident are you in your guess?
Since you do not have any other information except that 99.97% of calico cats are female and Puzzle is a
calico cat, it should seem far more likely to you that Puzzle is female. This is a statistical syllogism: You
are using a general statistic about calico cats to make an argument for a specific case. In its simplest
form, the argument would look like this:
99.97% of calico cats are female.
Puzzle is a calico cat.
Therefore, Puzzle is female.
Clearly, this argument is not deductively valid, but inductively it seems quite strong. Given that male
calico cats are extremely rare, you can be reasonably confident that Puzzle is female. In this case we can
actually put a number to how confident you can be: 99.97% confident.
Of course, you might be mistaken. After all, male calico cats do exist; this is what makes the argument
inductive rather than deductive. However, statistical syllogisms like this one can establish a high degree
of certainty about the truth of the conclusion.
Form
If we consider the calico cat example, we can see that the general form for a statistical syllogism looks
like this:
X% of S are P.
i is an S.
Therefore, i is (probably) a P.
There are also statistical syllogisms that conclude that the individual i does not have the property P.
Take the following example:
Only 1% of college males are on the football team.
Mike is a college male.
Therefore, Mike is probably not on the football team.
This type of statistical syllogism has the following form:
X% of S are P.
i is an S.
Therefore, i is (probably) not a P.
In this case, for the argument to be strong, we want X to be a low percentage.
Note that statistical syllogisms are similar to two kinds of categorical syllogisms presented in Chapter 3
(see Table 5.1). We see from the table that statistical syllogisms become valid categorical syllogisms
when the percentage, X, becomes 100% or 0%.
Table 5.1: Statistical syllogism versus categorical syllogism
Statistical syllogism Similar valid categorical syllogism
Example
99.97% of calico cats are female.
Puzzle is calico.
Therefore, Puzzle is female.
All calico cats are female.
Puzzle is calico.
Therefore, Puzzle is female.
Form
X% of S are P.
i is an S.
Therefore, i is (probably) P.
All M are P.
S is M.
Therefore, S is P.
Example
1% of college males are on the football team.
Mike is a college male.
Therefore, Mike is not on the football team.
No college males are on the football team.
Mike is a college male.
Therefore, Mike is not on the football team.
Form
X% of S are P.
i is an S.
Therefore, i is P.
X% of S are P.
i is an S.
Therefore, i is not P.
When identifying a statistical syllogism, it is important to keep the specific form in mind, since there are
other kinds of statistical arguments that are not statistical syllogisms. Consider the following example:
85% of community college students are younger than 40.
John is teaching a community college course.
Therefore, about 85% of the students in John’s class are under 40.
This argument is not a statistical syllogism because it does not fit the form. If we make i “John” then the
conclusion states that John, the teacher, is probably under 40, but that is not the conclusion of the
original argument. If we make i “the students in John’s class,” then we get the conclusion that it is 85%
likely that the students in John’s class are under 40. Does this mean that all of them or that some of them
are? Either way, it does not seem to be the same as the original conclusion, since that conclusion has to
do with the percentage of students under 40 in his class. Though this argument has the same “feel” as a
statistical syllogism, it is not one because it does not have the same form as a statistical syllogism.
Weak Statistical Syllogisms
There are at least two ways in which a statistical syllogism might not be strong. One way is if the
percentage is not high enough (or low enough in the second type). If an argument simply includes the
premise that most of S are P, that means only that more than half of S are P. A probability of only 51%
does not make for a strong inductive argument.
Another way that statistical syllogisms can be weak is if the individual in question is more (or less) likely
to have the relevant characteristic P than the average S. For example, take the reasoning:
99% of birds do not talk.
My pet parrot is a bird.
Therefore, my pet parrot cannot talk.
The premises of this argument may well be true, and the percentage is high, but the argument may be
weak. Do you see why? The reason is that a pet parrot has a much higher likelihood of being able to talk
than the average bird. We have to be very careful when coming to final conclusions about inductive
reasoning until we consider all of the relevant information.
5.3 Statistical Arguments: Inductive Generalizations
In the example about Puzzle, the calico cat, the first premise said that 99.97% of calico cats are female.
How did someone come up with that figure? Clearly, she or he did not go out and look at every calico cat.
Instead, he or she likely looked at a bunch of calicos, figured out what percentage of those cats were
female, and then reasoned that the percentage of females would have been the same if they had looked
at all calico cats. In this sort of reasoning, the group of calico cats that were actually examined is called
the sample, and all the calico cats taken as a group are called the population. An inductive
generalization is an argument in which we reason from data about a sample population to a claim
about a large population that includes the sample. Its general form looks like this:
X% of observed Fs are Gs.
Therefore, X% of all Fs are Gs.
In the case of the calico cats, the argument looks like this:
99.97% of calico cats in the sample were female.
Therefore, 99.97% of all calico cats are female.
Whether the argument is strong or weak depends crucially on whether the sample population is
representative of the whole population. We say that a sample is representative of a population when the
sample and the population both have the same distribution of the trait we are interested in—when the
sample “looks like” the population for our purposes. In the case of the cats, the strength of the argument
depends on whether our sample group of calico cats had about the same proportion of females as the
entire population of all calico cats.
There is a lot of math and research design—which you might learn about if you take a course in applied
statistics or in quantitative research design—that goes into determining the likelihood that a sample is
representative. However, even with the best math and design, all we can infer is that a sample is
extremely likely to be representative; we can never be absolutely certain it is without checking the
entire population. However, if we are careful enough, our arguments can still be very strong, even if they
do not produce absolute certainty. This section will examine how researchers try to ensure the sample
population is representative of the whole population and how researchers assess how confident they
can be in their results.
Representativeness
The main way that researchers try to ensure that the sample population is representative of the whole
population is to make sure that the sample population is random and sufficiently large. Researchers also
consider a measure called the margin of error to determine how similar the sample population is to the
whole population.
Randomness
Suppose you want to know how many marshmallow treats
are in a box of your favorite breakfast cereal. You do not
have time to count the whole box, so you pour out one cup.
You can count the number of marshmallows in your cup and
then reason that the box should have the same proportion
5xinc/iStock/Thinkstock
To ensure a sample is representative,
participants should be randomly
selected from the larger population.
Careful consideration is required to
ensure selections truly represent the
larger population.
One must be careful when making inductive generalizations
based on statistical data. Consider the examples in this video.
Raw numbers can sound more alarming than percentages.
Likewise, rate statistics can be misleading.
Making Inferences From Statistics
of marshmallows as the cup. You found 15 marshmallows in
the cup, and the box holds eight cups of cereal, so you figure
that there should be about 120 marshmallows in the box.
Your argument looks something like this:
A onecup sample of cereal contains 15
marshmallows.
The box holds eight cups of cereal.
Therefore, the box contains 120 marshmallows.
What entitles you to claim that the sample is
representative? Is there any way that the sample may not
represent the percentage of marshmallows in the whole
box? One potential problem is that marshmallows tend to be
lighter than the cereal pieces. As a result, they tend to rise to the top of the box as the cereal pieces settle
toward the bottom of the box over time. If you just scoop out a cup of cereal from the top, then, your
sample may not be representative of the whole box and may have too many marshmallows.
One way to solve this problem might be to shake the box. Vigorously shaking the box would probably
distribute the marshmallows fairly evenly. After a good shake, a particular piece of marshmallow or
cereal might equally end up anywhere in the box, so the ones that make it into your sample will be
largely random. In this case the argument may be fairly strong.
In a random sample, every member of the population has an equal chance of being included.
Understanding how randomness works to ensure representativeness is a bit tricky, but another example
should help clear it up.
Almost all students at my high school have laptops.
Therefore, almost all high school students in the United States have laptops.
This reasoning might seem pretty strong, especially if you go to a large high school. However, is there a
way that the sample population (the students at the high school) may not be truly random? Perhaps if
the high school is in a relatively wealthy area, then the students will be more likely to have laptops than
random American high schoolers. If the sample population is not truly random but has a greater or
lesser tendency to have the relevant characteristic than a random member of the whole population, this
is known as a biased sample. Biased samples will be discussed further in Chapter 7, but note that they
often help reinforce people’s biased viewpoints (see Everyday Logic: Why You Might Be Wrong).
The principle of randomness applies
to other types of statistical arguments
as well. Consider the argument about
John’s community college class. The
argument, again, goes as follows:
85% of community college
students are younger than
40.
John is teaching a
community college course.
Making Inferences From Statistics
From Title: Evidence in Argument: Critical Thinking
(https://fod.infobase.com/PortalPlaylists.aspx?wID=100753&xtid=49816)
Critical Thinking Questions
1. The characteristics of the sample is an important
consideration when drawing inferences from
statistics. Before reading on, what qualities do you
think an ideal sample possesses?
2. How can one ensure that one is making proper
inferences from evidence?
3. What is the danger of expressing things using rates?
What example is given that demonstrates this
danger?
Therefore, about 85% of the
students in John’s class are
under 40.
Since 85% of community college
students are younger than 40, we
would expect a sufficiently large
random sample of community college
students to have about the same
percentage. There are several ways,
however, that John’s class may not be
a random sample. Before going on to
the next paragraph, stop and see how
many ways you can think of on your
own.
So how is John’s class not a random
sample? Notice first that the
argument references a course at a
single community college. The
average student age likely varies
from college to college, depending on
the average age of the nearby
population. Even within this one
community college, John’s class is not
random. What time is John’s class?
Night classes tend to attract a higher
percentage of older students than
daytime classes. Some subjects also
attract different age groups. Finally,
we should think about John himself.
His age and reputation may affect the kind of students who enroll in his classes.
In all these ways, and maybe others, John’s class is not a random sample: There is not an equal chance
that every community college student might be included. As a result, we do not really have good reason
to think that John’s class will be representative of the general population of community college students.
So we have little reason to expect it to be representative of the larger population. As a result, we cannot
use his class to reliably predict what the population will look like, nor can we use the population to
reliably predict what John’s class will look like.
Everyday Logic: Why You Might Be Wrong
People are often very confident about their views, even when it comes
to very controversial issues that may have just as many people on the
other side. There are probably several reasons for this, but one of
them is due to the use of biased sampling. Consider whether you think
Jakubzak/iStock/Thinkstock
Confirmation bias, or
the tendency to seek
out support for our
beliefs, can be seen in
the friends we choose,
books we read, and
news sources we
select.
your views about the world are shared by many people or by only a
few. It is not uncommon for people to think that their views are more
widespread than they actually are. Why is that?
Think about how you form your opinion about how much of the nation
or world agrees with your view. You probably spend time talking with
your friends about these views and notice how many of your friends
agree or disagree with you. You may watch television shows or read
news articles that agree or disagree with you. If most of the sources
you interact with agree with your view, you might conclude that most
people agree with you.
However, this would be a mistake. Most of us tend to interact more with people and information
sources with which we agree, rather than those with which we disagree. Our circle of friends
tends to be concentrated near us both geographically and ideologically. We share similar
concerns, interests, and views; that is part of what makes us friends. As with choosing friends, we
also tend to select information sources that confirm our beliefs. This is a wellknown
psychological tendency known as confirmation bias (this will be discussed further in Chapter 8).
We seem to reason as follows:
A large percentage of my friends and news sources agree with my view.
Therefore, a large percentage of all people and sources agree with my view.
We have seen that this reasoning is based on a biased sample. If you take your friends and
information sources as a sample, they are not likely to be representative of the larger population
of the nation or world. This is because rather than being a random sample, they have been
selected, in part, because they hold views similar to yours. A good critical thinker takes sampling
bias into account when thinking about controversial issues.
Sample Size
Even a perfectly random sample may not be representative, due to bad luck. If you flip a coin 10 times,
for example, there is a decent chance that it will come up heads 8 of the 10 times. However, the more
times you flip the coin, the more likely it is that the percentage of heads will approach 50%.
The smaller the sample, the more likely it is to be nonrepresentative. This variable is known as the
sample size. Suppose a teacher wants to know the average height of students in his school. He randomly
picks one student and measures her height. You should see that this is not a big enough sample. By
measuring only one student, there is a decent chance that the teacher may have randomly picked
someone extremely tall or extremely short. Generalizing on an overly small sample would be making a
hasty generalization, an error in reasoning that will be discussed in greater detail in Chapter 7. If the
teacher chooses a sample of two students, it is less likely that they will both be tall or both be short. The
more students the teacher chooses for his sample, the less likely it is that the average height of the
sample will be much different than the average height of all students. Assuming that the selection
process is unbiased, therefore, the larger the sample population is, the more likely it is that the sample
will be representative of the whole population (see A Closer Look: How Large Must a Sample Be?).
A Closer Look: How Large Must a Sample Be?
In general, the larger a sample is, the more likely it is to be representative of the population from
which it is drawn. However, even relatively small samples can lead to powerful conclusions if
they have been carefully drawn to be random and to be representative of the population. As of
this writing, the population of the United States is in the neighborhood of 317 million, yet Gallup,
one of the most respected polling organizations in the country, often publishes results based on a
sample of fewer than 3,000 people. Indeed, its typical sample size is around 1,000 (Gallup, 2010).
That is a sample size of less than 1 in every 300,000 people!
Gallup can do this because it goes to great lengths to make sure that its samples are randomly
drawn in a way that matches the makeup of the country’s population. If you want to know about
people’s political views, you have to be very careful because these views can vary based on a
person’s locale, income, race or ethnicity, gender, age, religion, and a host of other factors.
There is no single, simple rule for how large a sample should be. When samples are small or
incautiously collected, you should be suspicious of the claims made on their basis. Professional
research will generally provide clear descriptions of the samples used and a justification of why
they are adequate to support their conclusions. That is not a guarantee that the results are
correct, but they are bound to be much more reliable than conclusions reached on the basis of
small and poorly collected samples.
For example, sometimes politicians tour a state with the stated aim of finding out what the people
think. However, given that people who attend political rallies are usually those with similar
opinions as the speaker, it is unlikely that the set of people sampled will be both large enough and
random enough to provide a solid basis for a reliable conclusion. If politicians really want to find
out what people think, there are better ways of doing so.
Margin of Error
It is always possible that a sample will be wildly different than the population. But equally important is
the fact that it is quite likely that any sample will be slightly different than the population. Statisticians
know how to calculate just how big this difference is likely to be. You will see this reported in some
studies or polls as the margin of error. The margin of error can be used to determine the range of
values that are likely for the population.
For example, suppose that a poll finds that 52% of a sample prefers Ms. Frazier in an election. When you
read about the result of this poll, you will probably read that 52% of people prefer Ms. Frazier with a
margin of error of ±3% (plus or minus 3%). This means that although the real number probably is not
52%, it is very likely to be somewhere between 49% (3% lower than 52%) and 55% (3% higher than
52%). Since the real percentage may be as low as 49%, Ms. Frazier should not start picking out curtains
for her office just yet: She may actually be losing!
Confidence Level
We want large, random samples because we want to be confident that our sample is representative of
the population. The more confident we are that are sample is representative, the more confident we can
be in conclusions we draw from it. Nonetheless, even a small, poorly drawn sample can yield informative
results if we are cautious about our reasoning.
If you notice that many of your friends and acquaintances are out of work, you may conclude that
unemployment levels are up. Clearly, you have some evidence for your conclusion, but is it enough? The
answer to this question depends on how strong you take your argument to be. Remember that inductive
arguments vary from extremely weak to extremely strong. The strength of an argument is essentially the
level of confidence we should have in the conclusion based on the reasons presented. Consider the
following ways you might state your confidence that unemployment levels were up, based on noting
unemployment among your friends and acquaintances.
a. “I’m certain that unemployment is up.”
b. “I’m reasonably sure that unemployment is up.”
c. “It’s more likely than not that unemployment is up.”
d. “Unemployment might be up.”
Clearly, A is too strong. Your acquaintances just are not likely to represent the population enough for
you to be certain that unemployment is up. On the other hand, D is weak enough that it really does not
need much evidence to support it. B and C will depend on how wide and varied your circle of
acquaintances is and on how much unemployment you see among them. If you know a lot of people and
your acquaintances are quite varied in terms of profession, income, age, race, gender, and so on, then
you can have more confidence in your conclusion than if you had only a small circle of acquaintances and
they tended to all be like each other in these ways. B also depends on just what you mean by “reasonably
sure.” Does that mean 60% sure? 75%? 85%?
Most reputable studies will include a “confidence level” that indicates how confident one can be that
their conclusions are supported by the reasons they give. The degree of confidence can vary quite a bit,
so it is worth paying attention to. In most social sciences, researchers aim to reach a 95% or 99%
confidence level. A confidence level of 95% means that if we did the same study 100 times, then in 95 of
those tests the results would fall within the margin of error. As noted earlier, the field of physics
requires a confidence level of about 99.99997%, much higher than is typically required or attained in
the social sciences. On the other end, sometimes a confidence level of just over 50% is enough if you are
only interested in knowing whether something is more likely than not.
Applying This Knowledge
Now that we have learned something about statistical arguments, what can we say about Angelina Jolie’s
argument, presented at the beginning of the prior section? First, notice that it has the form of a statistical
syllogism. We can put it this way, written as if from her perspective:
87% of women with certain genetic and other factors develop breast cancer.
I am a woman with those genetic and other factors.
Therefore, I have an 87% risk of getting breast cancer.
We can see that the argument fits the form correctly. While not deductive, the argument is inductively
strong. Unless we have reason to believe that she is more or less likely than the average person with
those factors to develop breast cancer, if these premises are true then they give strong evidence for the
truth of the conclusion. However, what about the first premise? Should we believe it?
In evaluating the first premise, we need to consider the evidence for it. Were the samples of women
studied sufficiently random and large that we can be confident they were representative of the
population of all women? With what level of confidence are the results established? If the samples were
small or not randomized, then we may have less confidence in them. Jolie’s doctors said that Jolie had an
87% chance of developing breast cancer, but there’s a big difference between being 60% confident that
she has this level of risk and being 99% certain that she does. To know how confident we should be, we
would need to look at the background studies that establish that 87% of women with those factors
develop breast cancer. Anyone making such an important decision would be well advised to look at
these issues in the research before acting.
Practice Problems 5.1
Which of the following attributes might negatively influence the data drawn from the
following samples? Click here
(https://ne.edgecastcdn.net/0004BA/constellation/PDFs/PHI103_2e/Answers_PracticeProblems5.1.pdf)
to check your answers.
1. A teacher surveys the gifted students in the district about the curriculum that should be
adopted at the high school.
a. sample size
b. representativeness of the sample
c. a and b
d. There is no negative influence in this case.
2. A researcher for Apple analyzes a large group of tribal people in the Amazon to
determine which new apps she should create in 2014.
a. sample size
b. representativeness of the sample
c. a and b
d. There is no negative influence in this case.
3. A researcher on a college campus interviews 10 students after a yoga class about their
drug use habits and determines that 80% of the student population probably smokes
marijuana.
a. sample size
b. representativeness of the sample
c. a and b
d. There is no negative influence in this case.
iStock/Thinkstock
Sufficient conditions
are present in
classroom grading
systems. If you need a
total of 850 points to
receive an A, the
sufficient condition to
receive an A is earning
850 points.
5.4 Causal Relationships: The Meaning of Cause
It is difficult to say exactly what we mean when we say that one thing causes another. Think about
turning on the lights in your room. What is the cause of the lights turning on? Is it the flipping of the
switch? The electricity in the wires? The fact that the bulb is not broken? Your initial desire for the lights
to be on? There are many things we could identify as a plausible cause of the lights turning on. However,
for practical purposes, we generally look for the set of conditions without which the event in question
would not have occurred and with which it will occur. In other words, logicians aim to be more specific
about causal relationships by discussing them in terms of sufficient and necessary conditions. Recall that
we used these terms in Chapter 4 when discussing propositional logic. Here we will discuss how these
terms can help us understand causal relationships.
Sufficient Conditions
According to British philosopher David Hume, the notion of cause is based
on nothing more than a “constant conjunction” that holds between
events—the two events always occur together (Morris & Brown, 2014). We
notice that events of kind A are always followed by events of kind B, and we
say “A causes B.” Thus, to claim a causal relationship between events of type
A and B might be to say: Whenever A occurs, B will occur.
Logicians have a fancy phrase for this relationship: We say that A is a
sufficient condition for B. A factor is a sufficient condition for the
occurrence of an event if whenever the factor occurs, the event also occurs:
Whenever A occurs, B occurs as well. Or in other words:
If A occurs, then B occurs.
For example, having a billion dollars is a sufficient condition for being rich;
being hospitalized is a sufficient condition for being excused from jury duty;
having a ticket is a sufficient condition for being able to be admitted to the
concert.
Often several factors are jointly required to create sufficient conditions. For
example, each state has a set of jointly sufficient conditions for being able to
vote, including being over 18, being registered to vote, and not having been
convicted of a felony, among other possible qualifications.
Here is an example of how to think about sufficient conditions when thinking about reallife causation.
We know room lights do not go on just because you flip the switch. The points of the switch must come
into contact with a power source, electricity must be present, a working lightbulb has to be properly
secured in the socket, the socket has to be properly connected, and so forth. If any one of the conditions
is not satisfied, the light will not come on. Strictly speaking, then, the whole set of conditions constitutes
the sufficient condition for the event.
We often choose one factor from a set of factors and call it the cause of an event. The one we call the
cause is the one with which we are most concerned for some reason or other; often it is the one that
represents a change from the normal state of things. A working car is the normal state of affairs; a hole in
Stockbyte/Thinkstock
Although water is a necessary
condition for life, it is not a sufficient
condition for life because humans also
need oxygen and food.
the radiator tube is the change to that state of affairs that results in the overheated engine. Similarly, the
electricity and lightbulb are part of the normal state of things; what changed most recently to make the
light turn on was the flipping of the switch.
Necessary Conditions
A factor is a necessary condition for an event if the event would not occur in the absence of the factor.
Without the necessary condition, the effect will not occur. A is a necessary condition for B if the
following statement is always true:
If A is not present, then neither is B.
This statement happens to be equivalent to the statement that if B is present, then A is present. Thus, a
handy way to understand the difference between necessary and sufficient conditions is as follows:
“A is sufficient for B” means that if A occurs, then B occurs.
“A is necessary for B” means that if B occurs, then A occurs.
Let us take a look at a real example. Poliomyelitis, or polio,
is a disease caused by a specific virus. In only a small
minority of those with poliovirus does the virus infect the
central nervous system and lead to the terrible condition
known as paralytic polio. In the large majority of cases,
however, the virus goes undetected and does not result in
paralysis. Thus, infection with poliovirus is not a sufficient
condition for getting paralytic polio. However, because one
must have the virus to have that condition, being infected
with poliovirus is a necessary condition for getting paralytic
polio (Mayo Clinic, 2014).
On the other hand, being squashed by a steamroller is a
sufficient condition for death, but it is not a necessary
condition. Whenever someone has been squashed by a
steamroller, that person is quite dead. However, it is not the
case that anyone who is dead has been run over by a
steamroller.
If our purpose in looking for causes is to be able to produce an effect, it is reasonable to look for
sufficient conditions for that effect. If we can manipulate circumstances so that the sufficient condition is
present, the effect will also be present. If we are looking for causes in order to prevent an effect, it is
reasonable to look for necessary conditions for that effect. If we prevent a necessary condition from
materializing, we can prevent the effect.
The eradication of yellow fever is a striking example. Research showed that being bitten by a certain
type of mosquito was a necessary condition for contracting yellow fever (though it was not a sufficient
condition, for some people who were bitten by these mosquitoes did not contract yellow fever).
Consequently, a campaign to destroy that particular species of mosquito through the widespread use of
insecticides virtually eliminated yellow fever in many parts of the world (World Health Organization,
2014).
Necessary and Sufficient Conditions
The most restrictive interpretation of a causal relationship consists of construing “cause” as a condition
both necessary and sufficient for the occurrence of an event. If factor A is necessary and sufficient for the
occurrence of event B, then whenever A occurs, B occurs, and whenever A does not occur, B does not
occur. In other words:
If A, then B, and if notA, then notB.
For example, to produce diamonds, certain very specific conditions must exist. Diamonds are produced if
and only if carbon is subjected to immense pressure and heat for a certain period of time. Diamonds do
not occur through any other process. If all of the conditions exist, then diamonds will result; diamonds
exist only when all of those conditions have been met. Therefore, carbon subjected to the right
combination of pressure, heat, and time constitutes both a necessary and sufficient condition for
diamond production.
This construction of cause is so restrictive that very few actual relationships in ordinary experience can
satisfy it. However, some scientists think that this is the kind of invariant relationship that scientific laws
must express. For instance, according to Newton’s law of gravitation, objects attract each other with a
force proportional to the inverse of the square of their distance. Therefore, if we know the force of
attraction between two bodies, we can calculate the distance between them (assuming we know their
masses). Conversely, if we know the distance between them, we can calculate the force of attraction.
Thus, having a certain degree of attraction between two bodies constitutes both a necessary and
sufficient condition for the distance between them. It happens frequently in math and science that the
values assigned to one factor determine the values assigned to another, and this relationship can be
understood in terms of necessary and sufficient conditions.
Other Types of Causes
The terms necessary condition and sufficient condition give us concrete and technical ways to describe
types of causes. However, in everyday life, the factor we mention as the cause of an event is rarely one
we consider sufficient or even necessary. We frequently select one factor from a set and say it is the
cause of the event. Our aims and interests, as well as our knowledge, affect that choice. Thus, practical,
moral, or legal considerations may influence our selection. There are three principal considerations that
may lead us to choose a single factor as “the cause,” although this is not an exhaustive listing.
Trigger cause. The trigger cause, or the factor that initiates an event, is often designated the cause of the
event. Usually, this is the factor that occurs last and completes a causal chain—the set of sufficient
conditions—producing the effect. Flipping the switch triggers the lights. All the other factors may be
present and as such constitute the standing conditions that allow the event to be triggered. The trigger
factor is sometimes referred to as the proximate cause since it is the factor nearest the final event (or
effect).
Unusual factor. Let us suppose that someone turns on a light and an explosion follows. Turning on the
light caused an explosion because the room was full of methane gas. Now being in a room is fairly
Hagen/Cartoonstock
Variables, such as buffalo and White
men, can be correlated in two
ways—directly and inversely. Which
type of correlation is being discussed
in this cartoon?
normal, turning on lights is fairly normal, having oxygen in a room is fairly normal, and having an
unsealed light switch is fairly normal. The only condition outside the norm is the presence of a large
quantity of explosive gas. Therefore, the presence of methane is referred to as the cause of the explosion.
What is unusual, what is outside the norm, is the cause. If we are concerned with fixing moral or legal
responsibility for an effect, we are likely to focus on the person who left the gas on, not the person who
turned on the lights.
Controllable factor. Sometimes we call attention to a controllable factor instrumental in producing the
event and point out that since the factor could have been controlled, so could the event. Thus, although
smoking is neither a sufficient nor a necessary condition for lung cancer, it is a controllable factor.
Therefore, over and above uncontrollable factors like heredity and chance, we are likely to single out
smoking as the cause. Similarly, drunk driving is neither a sufficient nor a necessary condition for getting
into a car accident, but it is a controllable factor, so we are likely to point to it as a cause.
Correlational Relationships
In both the case of smoking and drunk driving, neither were necessary nor sufficient conditions for the
subsequent event in question (lung cancer and car accidents). Instead, we would say that both are highly
correlated with the respective events. Two things can be said to be correlated, or in correlation, when
they occur together frequently. In other words, A is correlated with B, so B is more likely to occur if A
occurs, and vice versa. For example, having gray hair is correlated with age. The older someone is, the
more likely he or she is to have gray hair, and vice versa. Of course, not all people with gray hair are old,
and not all old people have gray hair, so age is neither a necessary nor a sufficient condition for gray
hair. However, the two are highly correlated because they have a strong tendency to go together.
Two things that vary in the same direction are said to be
directly correlated or to vary directly; the higher one’s age,
the more gray hair. Things that are correlated may also vary
in opposite directions; these are said to vary inversely. For
example, there is an inverse correlation between the size of
a car and its fuel economy. In general, the bigger a car is, the
lower its fuel economy is. If you want a car that gets high
miles per gallon, you should focus on cars that are smaller.
There are other factors to consider too, of course. A small
sports car may get lower fuel economy than a larger car
with less power. Correlation does not mean that the
relationship is perfect, only that variables tend to vary in a
certain way.
You may have heard the phrase “correlation does not imply
causation,” or something similar. Just because two things
happen together, it does not necessarily follow that one
causes the other. For example, there is a wellknown
correlation between shoe size and reading ability in
elementary children. Children with larger feet have a strong
tendency to read better than children with smaller feet. Of
course, no one supposes that a child’s shoe size has a direct
effect on his or her reading ability, or vice versa. Instead,
both of these things are related to a child’s age. Older children tend to have bigger feet than younger
children; they also tend to read better. Sometimes the connection between correlated things is simple, as
in the case of shoe size and reading, and sometimes it is more complicated.
Whenever you read that two things have been shown to be linked, you should pay attention to the
possibility that the correlation is spurious or possibly has another explanation. Consider, for example, a
study showing a strong correlation between the amount of fat in a country’s diet and the amount of
certain types of cancer in that country (such as K. K. Carroll’s 1975 study, as cited in Paulos, 1997). Such
a correlation may lead you to think that eating fat causes cancer, but this could potentially be a mistake.
Instead, we should consider whether there might be some other connection between the two.
It turns out that countries with high fat consumption also have high sugar consumption—perhaps sugar
is the culprit. Also, countries with high fat and sugar consumption tend to be wealthier; fat and sugar are
expensive compared to grain. Perhaps the correlation is the result of some other aspect of a wealthier
lifestyle, such as lower rates of physical exercise. (Note that wealth is a particularly common
confounding factor, or a factor that correlates with the dependent and independent variables being
studied, as it bestows a wide range of advantages and difficulties on those who have it.) Perhaps it is a
combination of factors, and perhaps it is the fat after all; however, we cannot simply conclude with
certainty from a correlation that one causes the other, not without further research.
Sometimes correlation between two things is simply random. If you search through enough data, you
may find two factors that are strongly correlated but that have nothing at all to do with each other. For
example, consider Figure 5.1. At first glance, you might think the two factors must be closely connected.
But then you notice that one of them is the divorce rate in Maine and the other is the per capita
consumption of margarine in the United States. Could it be that by eating less margarine you could help
save the marriages of people in Maine?
Figure 5.1: Are these two factors correlated?
Although it may seem like two factors are correlated, we sometimes have to look harder
to understand the relationship.
Source: www.tylervigen.com (http://www.tylervigen.com) .
On the other hand, although correlation does not imply causation, it does point to it. That is, when we
see a strong correlation, there is at least some reason to suspect a causal connection of some sort
between the two correlates. It may be that one of the correlates causes the other, a third thing causes
them both, there is some more complicated causal relation between them, or there is no connection at
all.
However, the possibility that the correlation is merely accidental becomes increasingly unlikely if the
sample size is large and the correlation is strong. In such cases we may have to be very thoughtful in
seeking and testing possible explanations of the correlation. The next section discusses ways that we
might find and narrow down potential factors involved in a causal relationship.
5.5 Causal Arguments: Mill’s Methods
Reasoning about causes is extremely important. If we can correctly identify what causes a particular
effect, then we have a much better chance of controlling or preventing the effect. Consider the search for
a cure for a disease. If we do not understand what causes a particular disease, then our chances of being
able to cure it are small. If we can identify the cause of the disease, we can be much more precise in
searching for a way to prevent the disease. On the other hand, if we think we know the cause when we
do not, then we are likely to look in the wrong direction for a cure.
A causal argument—an argument about causes and effects—is almost always an inductive argument.
This is because, although we can gather evidence about these relationships, we are almost never in a
position to prove them absolutely.
The following four experimental methods were formally stated in the 19th century by John Stuart Mill in
his book A System of Logic and so are often referred to as Mill’s methods. Mill’s methods express the
most basic underlying logic of many current methods for investigating causality. They provide a great
introduction to some of the basic concepts involved—but know that modern methods are much more
rigorous.
Used with caution, Mill’s methods can provide a guide for exploring causal connections, especially when
one is looking at specific cases against the background of established theory. It is important to
remember that although they can be useful, Mill’s methods are only the beginning of the study of
causation. By themselves, they are probably most useful as methods for identifying potential subjects for
further study using more robust methods that are beyond the scope of this book.
Method of Agreement
In 1976 an unknown illness affected numerous people in Philadelphia. Although it took some time to
fully identify the cause of the disease, a bacterium now called Legionella pneumophila, the first step in
the investigation was to find common features of those who became ill. Researchers were quick to note
that sufferers had all attended an American Legion convention at the BellevueStratford Hotel. As you
can guess, the focus of the investigation quickly narrowed to conditions at the hotel. Of course, the
convention and the hotel were not the actual cause of getting sick, but neither was it mere coincidence
that all of the ill had attended the convention. By finding the common elements shared by those who
became ill, investigators were able to quickly narrow their search for the cause. Ultimately, the
bacterium was located in a fountain in the hotel.
The method of agreement involves comparing situations in which the same kind of event occurs. If the
presence of a certain factor is the only respect in which the situations are the same (that is, they agree),
then this factor may be related to the cause of the event. We can represent this with something like
Table 5.2. The table indicates whether each of four factors was present in a specific case (A, B, or C) and,
in the last column, whether the effect manifested itself (in the earlier case of what is now known as
Legionnaires’ disease, the effect we would be interested in is whether infection occurred).
Table 5.2: Example of method of agreement
Case Factor 1 Factor 2 Factor 3 Factor 4 Effect
A No Yes Yes No Yes
Case Factor 1 Factor 2 Factor 3 Factor 4 Effect
B No No Yes Yes Yes
C Yes Yes Yes Yes Yes
The three cases all resulted in the same effect but differed in which factors were present—with the
exception of Factor 3, which was present in all three cases. We may then suspect that Factor 3 may be
causally related to the effect. Our notion of cause here is that of sufficient condition. The common factor
is sufficient to account for the effect.
In general, the method of agreement works best when we have a large group of cases that is as varied as
possible. A large group is much more likely to vary across many different factors than a small group.
Unfortunately, the world almost never presents us with two situations wholly unlike except for one
factor. We may have three or more situations that are greatly similar. For example, all of the afflicted in
the 1976 outbreak were members of the American Legion, all were adults, all were men, all lived in
Pennsylvania. Here is where we have to use common sense and what we already know. It is unlikely that
merely being a member of an organization is the cause of a disease. We expect diseases to be caused by
environmental factors: bacteria, viruses, contaminants, and so on. As a result, we can focus our search on
those similarities that seem most likely to be relevant to the cause. Of course, we may be wrong; that is a
hallmark of inductive reasoning generally, but by being as careful and as reasonable as we can, we can
often make great progress.
Method of Difference
The method of difference involves comparing a situation in which an event occurs with similar
situations in which it does not. If the presence of a certain factor is the only difference between the two
kinds of situations, it is likely to be causally related to the effect.
Suppose your mother comes to visit you and makes your favorite cake. Unfortunately, it just does not
turn out. You know she made it in the same way she always does. What could the problem be? Start by
looking at differences between how she made the cake at your house and how she makes it at hers.
Ultimately, the only difference you can find is that your mom lives in Tampa and you live in Denver.
Since that is the only difference, that difference is likely to be causally related to the effect. In fact,
Denver is both much higher and much drier than Tampa. Both of these factors make a difference in
baking cakes.
Let us suppose we are interested in two cases, A and B, in which A has the effect we are interested in
(the cake not turning out right) and B does not. This is outlined in Table 5.3. If we can find only one
factor that is different between the two cases—in this case, Factor 1—then that factor is likely to be
causally related to the effect. This does not tell us whether the factor directly causes the effect, but it
does suggest a causal link. Further investigation might reveal just exactly what the connection is.
Table 5.3: Example of method of difference
Case Factor 1 Factor 2 Factor 3 Factor 4 Effect
A Yes No No Yes Yes
B No No No Yes No
In this example, Factor 1 is the one factor that is different between the two cases. Perhaps the presence
of Factor 1 is related to why Case A had the effect but Case B did not. Here we are seeing Factor 1 as a
necessary condition for the effect.
The method of difference is employed frequently in clinical trials of experimental drugs. Researchers
carefully choose or construct two situations that resemble each other in as many respects as possible. If
a drug is employed in one but not the other, then they can ascribe to the drug any change in one
situation not matched by a change in the other. Note that the two sets must be as similar as possible,
since variation could introduce other possible causal links. The group in which change is expected is
often referred to as the experimental group, and the group in which change is not expected is often
referred to as the control group.
The method of difference may seem obvious and its results reliable. Yet even in a relatively simple
experimental setup like this one, we may easily find grounds for doubting that the causal claim has been
adequately established.
One important factor is that the two cases, A and B, have to be as similar as possible in all other respects
for the method of difference to be used effectively. If your 8yearold son made the cake without
supervision, there are likely to be a whole host of differences that could explain the failure. The same
principle applies to scientific studies. One thing that can subtly skew experimental results is
experimental bias. For example, if the experimenters know which people are receiving the experimental
drug, they might unintentionally treat them differently.
To prevent such possibilities, socalled blind experiments are often used. Those conducting the
experiment are kept in ignorance about which subjects are in the control group and which are in the
experimental group so that they do not even unintentionally treat the subjects differently.
Experimenters therefore, do not know whether they are injecting distilled water or the actual drug. In
this way the possibility of a systematic error is minimized.
We also have to keep in mind that our inquiry is guided by background beliefs that may be incorrect. No
two cases will ever be completely the same except for a single factor. Your mother made the cake on a
different day than she did at home, she used a different spoon, different people were present in the
house, and so on. We naturally focus on similarities and differences that we expect to be relevant.
However, we should always realize that reality may disagree with our expectations.
Causal inquiry is usually not a matter of conducting a single experiment. Often we cannot even control
for all relevant factors at the same time, and once an experiment is concluded, doubts about other
factors may arise. A series of experiments in which different factors are kept constant while others are
varied one by one is always preferable.
Joint Method of Agreement and Difference
The joint method of agreement and difference is, as the name suggests, a combination of the methods
of agreement and difference. It is the most powerful of Mill’s methods. The basic idea is to have two
groups of cases: One group shows the effect, and the other does not. The method of agreement is used
within each group, by seeing what they have in common, and the method of difference is used between
the two groups, by looking for the differences between the two. Table 5.4 shows how such a chart would
look, if we were comparing three different cases (1, 2, and 3) among two groups (A and B).
Table 5.4: Example of joint method of agreement and difference
Case/group Factor 1 Factor 2 Factor 3 Factor 4 Effect
1/A Yes No No Yes Yes
2/A No No Yes Yes Yes
3/A No Yes No Yes Yes
1/B No Yes Yes No No
2/B Yes Yes No No No
3/B Yes No Yes No No
As you can see, within each group the cases agree only on Factor 4 and the effect. But when you compare
the two groups, the only consistent differences between them are in Factor 4 and the effect. This result
suggests the possibility that Factor 4 may be causally related to the effect in question. In this method, we
are using the notion of a necessary and sufficient condition. The effect happens whenever Factor 4 is
present and never when it is absent.
The joint method is the basis for modern randomized controlled experiments. Suppose you want to see
if a new medicine is effective. You begin by recruiting a large group of volunteers. You then randomly
assign them to either receive the medicine or a placebo. The random assignment ensures that each
group is as varied as possible and that you are not unknowingly deciding whether to give someone the
medicine based on some common factor. If it turns out that everyone who gets the medicine improves
and everyone who gets the placebo stays the same or gets worse, then you can infer that the medicine is
probably effective.
In fact, advanced statistics allow us to make inferences from such studies even when there is not perfect
agreement on the presence or absence of the effect. So, in reading studies, you may note that the
discussion talks about the percentage of each group that shows or does not show the effect. Yet we may
still make good inferences about causation by using the method of concomitant variation.
Method of Concomitant Variation
The method of concomitant variation is simply the method of looking for correlation between two
things. As we noted in our discussion of correlation, this cannot be used to conclude conclusively that
one thing causes the other, but it is suggestive that there is perhaps some causal connection between the
two. Stronger evidence can be found by further scientific study.
You may have noticed that, in discussing causes, we are trying to explain a phenomenon. We observe
something that is interesting or important to us, and we seek to know why it happened. Therefore, the
study of Mill’s methods, as well as correlation and concomitant variation, can be seen as part of a
broader type of reasoning known as inference to the best explanation, the effort to find the best or most
accurate explanation of our observations. Because this type of reasoning is sometimes classified as a
separate type of reasoning (sometimes called abductive reasoning), it will be covered in Chapter 6.
In summary, Mill’s methods provide a framework for exploring causal relationships. It is important to
remember that although they can be useful, they are only the beginning of this important field. By
themselves, they are probably most useful as methods for identifying potential subjects for further study
using more robust methods that are beyond the scope of this book.
Practice Problems 5.2
Identify which of Mill’s methods discussed in the chapter relates to the following
examples. Click here
(https://ne.edgecastcdn.net/0004BA/constellation/PDFs/PHI103_2e/Answers_PracticeProblems5.2.pdf)
to check your answers.
1. After going to dinner, all the members of a family came down with vomiting. They all had
different entrées but shared a salad as an appetizer. The mother of the family determines
that it must have been the salad that caused the sickness.
a. method of agreement
b. method of difference
c. joint method
d. method of concomitant variation
2. A couple goes to dinner and shares an appetizer, entrée, and dessert. Only one of the two
gets sick. She drank a glass of wine, and her husband drank a beer. She believes that the
wine was the cause of her sickness.
a. method of agreement
b. method of difference
c. joint method
d. method of concomitant variation
3. In a specific city, the number of people going to emergency rooms for asthma attacks
increases as the level of pollution increases in the summer. When the winter comes and
pollution goes down, the number of people with asthma attacks decreases.
a. method of agreement
b. method of difference
c. joint method
d. method of concomitant variation
4. In the past 15 years there has never been a safety accident in the warehouse. Each day
for the past 15 years Lorena has been conducting the morning safety inspections.
However, today Lorena missed work, and there was an accident.
a. method of agreement
b. method of difference
c. joint method
d. method of concomitant variation
5. Since we have hired Earl, productivity in the office has decreased by 20%.
a. method of agreement
b. method of difference
c. joint method
d. method of concomitant variation
6. In the past, lead was put into many paints. It was found that the number of infant
fatalities increased in relation to the amount of exposure these infants had to leadbased
paints that were used on their cribs.
a. method of agreement
b. method of difference
c. joint method
d. method of concomitant variation
7. It appears that the likelihood of catching the Zombie virus increases the more one is
around people who have already been turned into zombies.
a. method of agreement
b. method of difference
c. joint method
d. method of concomitant variation
8. In order to determine how a disease was spread in humans, researchers placed two
groups of people into two rooms. Both rooms were exactly alike. However, in one room
they placed someone who was infected with the disease. The researchers found that
those who were in the room with the infected person got sick, whereas those who were
not with an infected person remained well.
a. method of agreement
b. method of difference
c. joint method
d. method of concomitant variation
9. In a certain IQ test, students in a specific group performed at a much higher level than
those of the other groups. After analyzing the group, the researchers found that the high
performing students all smoked marijuana before the exam.
a. method of agreement
b. method of difference
c. joint method
d. method of concomitant variation
badahos/iStock/Thinkstock
The ability to think critically about an
authority’s argument will allow you to
determine reliable sources from
unreliable ones, which can be quite
helpful when writing research papers,
reading news articles, or taking advice
from someone.
5.6 Arguments From Authority
An argument from authority, also known as an appeal to authority, is an inductive argument in which
one infers that a claim is true because someone said so. The general reasoning looks like this:
Person A said that X is true.
Person A is an authority on the subject.
Therefore, X is true.
Whether this type of reasoning is strong depends on the issue discussed and the authority cited. If it is
the kind of issue that can be settled by an argument from authority and if the person is actually an
authority on the subject, then it can actually be a strong inductive argument.
Some people think that arguments from authority in general are fallacious. However, that is not
generally the case. To see why, try to imagine life without any appeals to authority. You could not believe
anyone’s statements, no matter how credible. You could not believe books; you could not believe
published journals, and so on. How would you do in college if you did not listen to your textbooks,
teachers, or any other sources of information?
Even in science class, you would have to do every
experiment on your own because you could not believe
published reports. In math, you could not trust the book or
teacher, so you would have to prove every theorem by
yourself. History class would be a complete waste of time
because, unless you had a time machine, there would be no
way to verify any claims about what happened in the past
without appeal to historical records, newspapers, journals,
and so forth. You would also have a hard time following
medical advice, so you might end up with serious health
problems. Finally, why would you go to school or work if
you could not trust the claim that you were going to get a
degree or a paycheck after all of your efforts?
Therefore, in order to learn from others and to succeed in
life, it is essential that we listen to appropriate authorities.
However, since many sources are unreliable, misleading, or
even downright deceptive, it is essential that we learn to
distinguish reliable sources of authority from unreliable
ones. Chapter 7 will discuss how to distinguish between
legitimate and fallacious appeals to authority.
Here are some examples of legitimate arguments from authority:
“The theory of relativity is true. I know because my physics professor and my physics textbook
teach that it is true.”
“Pine trees are not deciduous; it says so right here in this tree book.”
“The Giants won the pennant! I read it on ESPN.com.”
“Mike hates radishes. He told me so yesterday.”
All of these inferences seem pretty strong. For examples of arguments to authority that are not as strong,
or even downright fallacious, visit Chapter 7.
5.7 Arguments From Analogy
An argument from analogy is an inductive argument that draws conclusions based on the use of
analogy. An analogy is a comparison of two items. For example, many object to deficit spending (when
the country spends more money than it takes in) based on the reasoning that debt is bad for household
budgets. The person’s argument depends on an analogy that compares the national budget to a
household budget. The two items being compared may be referred to as analogs (or analogues,
depending on where you live) but are referred to technically as cases. Of the two analogs, one should be
well known, with a body of knowledge behind it, and so is referred to as the familiar case; the second
analog, about which much less is known, is called the unfamiliar case.
The basic structure of an argument from analogy is as follows:
B is similar to A.
A has feature F.
Therefore, B probably also has feature F.
Here, A is the familiar case and B is the unfamiliar case. We made an inference about thing B based on its
similarity to the more familiar A.
Analogical reasoning proceeds from this premise: Since the analogs are similar either in many ways or in
some very important ways, they are likely to be similar in other ways as well. If there are many
similarities, or if the similarities are significant, then the analogy can be strong. If the analogs are
different in many ways, or if the differences are important, then it is a weak analogy. Conclusions arrived
at through strong analogies are fairly reliable; conclusions reached through weak analogies are less
reliable and often fallacious (the fallacy is called false analogy). Therefore, when confronted with an
analogy (“A is like B”), the first question to be asked is this: Are the two analogs very similar in ways that
are relevant to the current discussion, or are they different in relevant ways?
Analogies occur in both arguments and explanations. As we saw in Chapter 2, arguments and
explanations are not the same thing. The key difference is whether the analogy is being used to give
evidence that a certain claim is true—an argument—or to give a better understanding of how or why a
claim is true—an explanation. In explanations, the analogy aims to provide deeper understanding of the
issue. In arguments, the analogy aims to provide reasons for believing a conclusion. The next section
provides some tips for evaluating the strength of such arguments.
Evaluating Arguments From Analogy
Again, the strength of the argument depends on just how much A is like B, and the degree to which the
similarities between A and B are relevant to F. Let us consider an example. Suppose that you are in the
market for a new car, and your primary concern is that the car be reliable. You have the opportunity to
buy a Nissan. One of your friends owns a Nissan. Since you want to buy a reliable car, you ask a friend
how reliable her car is. In this case you are depending on an analogy between your friend’s car and the
car you are looking to buy. Suppose your friend says that her car is reliable. You can now make the
following argument:
The car I’m looking at is like my friend’s car.
My friend’s car is reliable.
Therefore, the car I’m looking at will be reliable.
How strong is this argument? That depends on how similar the two cases are. If the only thing the cars
have in common is the brand, then the argument is fairly weak. On the other hand, if the cars are the
same model and year, with all the same options and a similar driving history, then the argument is
stronger. We can list the similarities in a chart (see Table 5.5). Initially, the analogy is based only on the
make of the car. We will call the car you are looking at A and your friend’s car B.
Table 5.5: Comparing cars by make
Car Make Reliable?
B Nissan Yes
A Nissan ?
The make of a car is relevant to its reliability, but the argument is weak because that is the only
similarity we know about. To strengthen the argument, we can note further relevant similarities. For
example, if you find out that your friend’s car is the same model and year, then the argument is
strengthened (see Table 5.6).
Table 5.6: Comparing cars by make, model, and year
Car Make Model Year Reliable?
B Nissan Sentra 2000 Yes
A Nissan Sentra 2000 ?
The more relevant similarities there are between the two cars, the stronger the argument. However, the
word relevant is critical here. Finding out that the two cars have the same engine and similar driving
histories is relevant and will strengthen the argument. Finding out that both cars are the same color and
have license plates beginning with the same letter will not strengthen the argument. Thus, arguments
from analogy typically require that we already have some idea of which features are relevant to the
feature we are interested in. If you really had no idea at all what made some cars reliable and others not
reliable, then you would have no way to evaluate the strength of an argument from analogy about
reliability.
Another way we can strengthen an argument from analogy is by increasing the number of analogs. If you
have two more friends who also own a car of the same make, model, and year, and if those cars are
reliable, then you can be more confident that your new car will be reliable. Table 5.7 shows what the
chart would look like. The more analogs you have that match the car you are looking at, the more
confidence you can have that the car you’re looking at will be reliable.
Table 5.7: Comparing multiple analogs
Car Make Model Year Reliable?
B Nissan Sentra 2000 Yes
C Nissan Sentra 2000 Yes
D Nissan Sentra 2000 Yes
A Nissan Sentra 2000 ?
In general, then, analogical arguments are stronger when they have more analogous cases with more
relevant similarities. They are weaker when there are significant differences between the familiar cases
and the unfamiliar case. If you discover a significant difference between the car you are looking at and
the analogs, that reduces the strength of the argument. If, for example, you find that all your friends’ cars
have manual transmission, whereas the one you are looking at has an automatic transmission, this
counts against the strength of the analogy and hence against the strength of the argument.
Another way that an argument from analogy can be weakened is if there are cases that are similar but do
not have the feature in question. Suppose you find a fourth friend who has the same model and year of
car but whose car has been unreliable. As a result, you should have less confidence that the car you are
looking at is reliable.
Here are a couple more examples, with questions about how to gauge the strength.
“Except for size, chickens and turkeys are very similar birds. Therefore, if a food is good for
chickens, it is probably good for turkeys.”
Relevant questions include how similar chickens and turkeys are, whether there are significant
differences, and whether the difference in size is enough to allow turkeys to eat things that would be too
big for chickens.
“Seattle’s climate is similar, in many ways to the United Kingdom’s. Therefore, this plant is
likely to grow well in Seattle, because it grows well in the United Kingdom.”
Just how similar is the climate between the two places? Is the total about of rain about the same? How
about the total amount of sun? Are the low and high temperatures comparable? Are there soil
differences that would matter?
“I am sure that my favorite team will win the bowl game next week; they have won every game
so far this season.”
This example might seem strong at first, but it hides a very relevant difference: In a bowl game, college
football teams are usually matched up with an opponent of approximately equal strength. It is therefore
likely that the team being played will be much better than the other teams played so far this season. This
difference weakens the analogy in a relevant way, so the argument is much weaker than it may at first
appear. It is essential when studying the strength of analogical arguments to be thorough in our search
for relevant similarities and differences.
Analogies in Moral Reasoning
Analogical reasoning is often used in moral reasoning and moral arguments. Examples of analogical
reasoning are found in ethical or legal debates over contentious or controversial issues such as abortion,
gun control, and medical practices of all sorts (including vaccinations and transplants). Legal arguments
are often based on finding precedents—analogous cases that have already been decided. Recent
arguments presented in the debate over gun control have drawn conclusions based on analogies that
compare the United States with other countries, including Switzerland and Japan. Whether these and
similar arguments are strong enough to establish their conclusions depends on just how similar the
cases are and the degree and number of dissimilarities and contrary cases. Being aware of similar cases
Jupiterimages/BananaStock/Thinkstock
Retailers such as bookstores
commonly use arguments from
analogy when they suggest purchases
that have already occurred or that are occurring in other areas can vastly improve one’s wisdom about
how best to address the topic at hand.
The importance of analogies in moral reasoning is sometimes captured in the principle of equal
treatment—that if two things are analogous in all morally relevant respects, then what is right (or
wrong) to do in one case will be right (or wrong) to do in the other case as well. For example, if it is right
for a teacher to fail a student for missing the final exam, then another student who does the same thing
should also be failed. Whether the teacher happens to like one student more than the other should not
make a difference, because that is not a morally relevant difference when it comes to grading.
The reasoning could look as follows:
Things that are similar in all morally relevant respects should be treated the same.
Student A was failed for missing the final exam.
Student B also missed the final exam.
Therefore, student B should be failed as well.
It follows from the principle of equal treatment that if two things should be treated differently, then
there must be a morally relevant difference between them to justify this different treatment. An example
of the application of this principle might be in the interrogation of prisoners of war. If one country wants
to subject prisoners of war to certain kinds of harsh treatments but objects to its own prisoners being
treated the same way by other countries, then there need to be relevant differences between the
situations that justify the different treatment. Otherwise, the country is open to the charge of moral
inconsistency.
This principle, or something like it, comes up in many other types of moral debates, such as about
abortion and animal ethics. Animal rights advocates, for example, say that if we object to people harming
cats and dogs, then we are morally inconsistent to accept to the same treatment of cows, pigs, and
chickens. One then has to address the question of whether there are differences in the beings or in their
use for food that justify the differences in moral consideration we give to each.
Other Uses of Analogies
Analogies are the basis for parables, allegories, and forms of
writing that try to give a moral. The phrase “The moral of
the story is . . .” may be featured at the end of such stories, or
the author may simply imply that there is a lesson to be
learned from the story. Aesop’s Fables are one wellknown
example of analogy used in writing. Consider the fable of the
ant and the grasshopper, which compares the hardworking,
industrious ant with the footloose and fancyfree
grasshopper. The ant gathers and stores food all summer to
prepare for winter; the grasshopper fiddles around and
plays all summer, giving no thought for tomorrow. When
winter comes, the ant lives warm and comfortable while the
grasshopper starves, freezes, and dies. The fable argues that
we should be like the ant if we want to survive harsh times.
The ant and grasshopper are analogs for industrious people
and lazy people. How strong is the argument? Clearly, ants
based on their similarity to other
items.
and grasshoppers are quite different from people. Are the
differences relevant to the conclusion? What are the
relevant similarities? These are the questions that must be
addressed to get an idea of whether the argument is strong or weak.
Practice Problems 5.3
Determine whether the following arguments are inductive or deductive. Click here
(https://ne.edgecastcdn.net/0004BA/constellation/PDFs/PHI103_2e/Answers_PracticeProblems5.3.pdf)
to check your answers.
1. All voters are residents of California. But some residents of California are Republican.
Therefore, some voters are Republican.
a. deductive
b. inductive
2. All doctors are people who are committed to enhancing the health of their patients. No
people who purposely harm others can consider themselves to be doctors. Therefore,
some people who harm others do not enhance the health of their patients.
a. deductive
b. inductive
3. Guns are necessary. Guns protect people. They give people confidence that they can
defend themselves. Guns also ensure that the government will not be able to take over its
citizenry.
a. deductive
b. inductive
4. Every time I turn on the radio, all I hear is vulgar language about sex, violence, and drugs.
Whether it’s rock and roll or rap, it’s all the same. The trend toward vulgarity has to
change. If it doesn’t, younger children will begin speaking in these ways and this will
spoil their innocence.
a. deductive
b. inductive
5. Letting your kids play around on the Internet all day is like dropping them off in
downtown Chicago to spend the day by themselves. They will find something that gets
them into trouble.
a. deductive
b. inductive
6. Many people today claim that men and women are basically the same. Although I believe
that men and women are equally capable of completing the same tasks physically as well
as mentally, to say that they are intrinsically the same detracts from the differences
between men and women that are displayed every day in their social interactions, the
way they use their resources, and the way in which they find themselves in the world.
a. deductive
b. inductive
7. Too many intravenous drug users continue to risk their lives by sharing dirty needles.
This situation could be changed if we were to supply drug addicts with a way to get clean
needles. This would lower the rate of AIDS in this highrisk population as well as allow
for the opportunity to educate and attempt to aid those who are addicted to heroin and
other intravenous drugs.
a. deductive
b. inductive
8. I know that Stephen has a lot of money. His parents drive a Mercedes. His dogs wear
cashmere sweaters, and he paid cash for his Hummer.
a. deductive
b. inductive
9. Dogs are better than cats, since they always listen to what their masters say. They also
are more fun and energetic.
a. deductive
b. inductive
10. All dogs are warmblooded. All warmblooded creatures are mammals. Hence, all dogs
are mammals.
a. deductive
b. inductive
11. Chances are that I will not be able to get in to see Slipknot since it is an over 21 show, and
Jeffrey, James, and Sloan were all carded when they tried to get in to the club.
a. deductive
b. inductive
12. This is not the best of all possible worlds, because the best of all possible worlds would
not contain suffering, and this world contains much suffering.
a. deductive
b. inductive
13. Some apples are not bananas. Some bananas are things that are yellow. Therefore, some
things that are yellow are not apples.
a. deductive
b. inductive
14. Since all philosophers are seekers of truth, it follows that no evil human is a seeker after
truth, since no philosophers are evil humans.
a. deductive
b. inductive
15. All squares are triangles, and all triangles are rectangles. Therefore, all squares are
rectangles.
a. deductive
b. inductive
16. Deciduous trees are trees that shed their leaves. Maple trees are deciduous trees.
Therefore, maple trees will shed their leaves at some point during the growing season.
a. deductive
b. inductive
17. Joe must make a lot of money teaching philosophy, since most philosophy professors are
rich.
a. deductive
b. inductive
18. Since all mammals are coldblooded, and all coldblooded creatures are aquatic, all
mammals must be aquatic.
a. deductive
b. inductive
19. I felt fine until I missed lunch. I must be feeling tired because I don’t have anything in my
stomach.
a. deductive
b. inductive
20. If you drive too fast, you will get into an accident. If you get into an accident, your
insurance premiums will increase. Therefore, if you drive too fast, your insurance
premiums will increase.
a. deductive
b. inductive
21. The economy continues to descend into chaos. The stock market still moves down after it
makes progress forward, and unemployment still hovers around 10%. It is going to be a
while before things get better in the United States.
a. deductive
b. inductive
22. Football is the best sport. The athletes are amazing, and it is extremely complex.
a. deductive
b. inductive
23. We should go to see Avatar tonight. I hear that it has amazing special effects.
a. deductive
b. inductive
24. Pigs are smarter than dogs. It’s easier to train them.
a. deductive
b. inductive
25. Seventy percent of the students at this university come from upperclass families. The
school budget has taken a hit since the economic downturn. We need funding for the
three new buildings on campus. I think it’s time for us to start a phone campaign to raise
funds so that we don’t plunge into bankruptcy.
a. deductive
b. inductive
26. Justin was working at IBM. The last person we got from IBM was a horrible worker. I
don’t think that it’s a good idea for us to go with Justin for this job.
a. deductive
b. inductive
27. If she wanted me to buy her a drink, she would’ve looked over at me. But she never
looked over at me. So that means that she doesn’t want me to buy her a drink.
a. deductive
b. inductive
28. Almost all the people I know who are translators have their translator’s license from the
ATA. Carla is a translator. Therefore, she must have a license from the ATA.
a. deductive
b. inductive
29. The economy will not recover anytime soon. Big businesses are struggling to keep their
profits high. This is due to the fact that consumers no longer have enough money to
purchase things that are luxuries. Most of them buy only those things that they need and
don’t have much left over. Those same businesses have been firing employees left and
right. If America’s largest businesses are losing employees, then there won’t be any jobs
for the people who are already unemployed. That means that these people will not have
money to pump back into the system, and the circle will continue to descend into
recession.
a. deductive
b. inductive
Determine which of the following forms of inductive reasoning are taking place.
30. The purpose of ancient towers that were discovered in Italy are unknown. However,
similar towers were discovered in Albania, and historical accounts in that country
indicate that the towers were used to store grain. Therefore, the towers in Italy were
probably used for the same purpose.
a. argument from analogy
b. statistical syllogism
c. inductive generalization
d. causal argument
31. After the current presidential administration passes a bill that increases the amount of
time people can be on unemployment, the unemployment rate in the country increases.
Economists studying the bill claim that there is a direct relation between the bill and the
unemployment rate.
a. argument from analogy
b. statistical syllogism
c. inductive generalization
d. causal argument
32. When studying a group of electricians, it was found that 60% of them did not have
knowledge of the new safety laws governing working on power lines. Therefore, 60% of
the electricians in the United States probably do not have knowledge of the new laws.
a. argument from analogy
b. statistical syllogism
c. inductive generalization
d. causal argument
33. In the state of California, studies found that violent criminals who were released on
parole had a 68% chance of committing another violent crime. Therefore, a majority of
violent criminals in society are likely to commit more violent crimes if they are released
from prison.
a. argument from analogy
b. statistical syllogism
c. inductive generalization
d. causal argument
34. Psilocybin mushrooms cause hallucinations in humans who ingest them. A new species of
mushroom shares similar visual characteristics to many forms of psilocybin mushrooms.
Therefore, it is likely that this form of mushroom has compounds that have neurological
effects.
a. argument from analogy
b. statistical syllogism
c. inductive generalization
d. causal argument
35. A recent survey at work indicates that 60% of the employees believe that they do not
make enough money for the work that they do. It is likely that a majority of the people
that work for this company are unhappy in their jobs.
a. argument from analogy
b. statistical syllogism
c. inductive generalization
d. causal argument
36. A family is committed to buying Hondas because every Honda they have owned has had
few problems and been very reliable. They believe that all Hondas must be reliable.
a. argument from analogy
b. statistical syllogism
c. inductive generalization
d. causal argument
Summary and Resources
Chapter Summary
The key feature of inductive arguments is that the support they provide for a conclusion is always less
than perfect. Even if all the premises of an inductive argument are true, there is at least some possibility
that the conclusion may be false. Of course, when an inductive argument is very strong, the evidence for
the conclusion may still be overwhelming. Even our best scientific theories are supported by inductive
arguments.
This chapter has looked at four broad types of inductive arguments: statistical arguments, causal
arguments, arguments from authority, and arguments from analogy. We have seen that each type can be
quite strong, very weak, or anywhere in between. The key to success in evaluating their strength is to be
able to (a) identify the type of argument being used, (b) know the criteria by which to evaluate its
strength, and (c) notice the strengths and weaknesses of the specific argument in question within the
context that it is given. If we can perform all of these tasks well, then we should be good evaluators of
inductive reasoning.
Critical Thinking Questions
1. What are some ways that you can now protect yourself from making hasty generalizations
through inductive reasoning?
2. Can you think of an example that relates to each one of Mill’s methods of determining causation?
What are they, and how did you determine that it fit with Mill’s methods?
3. Think of a time where you reasoned improperly about correlation and causation. Have you seen
anyone in the news or in your place of employment fall into improper analysis of causation?
What did they do, and what errors did they make?
4. Learning how to evaluate arguments is a great way to empower the mind. What are three forms
of empowerment that result when people understand how to identify and evaluate arguments?
5. Why do you believe that superstitions are so prevalent in many societies? What forms of
illogical reasoning lead to belief in superstitions? Are there any superstitions that you believe
are true? What evidence do you have that supports your claims?
6. Think of an example of a strong inductive argument, then think of a premise that you can add
that significantly weakens the argument. Now think of a new premise that you can add that
strengthens it again. Now find one that makes it weaker, and so on. Repeat this process several
times to notice how the strength of inductive arguments can change with new premises.
Web Resources
http://austhink.com/critical/pages/stats_prob.html
(http://austhink.com/critical/pages/stats_prob.html)
This website offers a number of resources and essays designed to help you learn more about statistics
and probability.
http://www.nss.gov.au/nss/home.nsf/pages/Sample+size+calculator
(http://www.nss.gov.au/nss/home.nsf/pages/Sample+size+calculator)
The Australian government hosts a sample size calculator that allows users to approximate how large a
sample they need.
http://www.gutenberg.org/ebooks/27942 (http://www.gutenberg.org/ebooks/27942)
Read John Stuart Mill’s A System of Logic, which is where Mill first introduces his methods for identifying
causality.
Key Terms
appeal to authority
See argument from authority.
argument from analogy
Reasoning in which we draw a conclusion about something based on characteristics of other similar
things.
argument from authority
An argument in which we infer that something is true because someone (a purported authority) said
that it was true.
causal argument
An argument about causes and effects.
cogent
An inductive argument that is strong and has all true premises.
confidence level
In an inductive generalization, the likelihood that a random sample from a population will have
results that fall within the estimated margin of error.
correlation
An association between two factors that occur together frequently or that vary in relation to each
other.
inductive arguments
Arguments in which the premises increase the likelihood of the conclusion being true but do not
guarantee that it is.
inductive generalization
An argument in which one draws a conclusion about a whole population based on results from a
sample population.
joint method of agreement and difference
A way of selecting causal candidates by looking for a factor that is present in all cases in which the
effect occurs and absent in all cases in which it does not.
margin of error
A range of values above and below the estimated value in which it is predicted that the actual result
will fall.
method of agreement
A way of selecting causal candidates by looking for a factor that is present in all cases in which the
effect occurs.
method of concomitant variation
A way of selecting causal candidates by looking for a factor that is highly correlated with the effect in
question.
method of difference
A way of selecting causal candidates by looking for a factor that is present when effect occurs and
absent when it does not.
necessary condition
A condition for an event without which the event will not occur; A is a necessary condition of B if A
occurs whenever B does.
population
In an inductive generalization, the whole group about which the generalization is made; it is the group
discussed in the conclusion.
proximate cause
See trigger cause.
random sample
A group selected from within the whole population using a selection method such that every member
of the population has an equal chance of being included.
sample
A smaller group selected from among the population.
sample size
The number of individuals within the sample.
statistical arguments
Arguments involving statistics, either in the premises or in the conclusion.
statistical syllogism
An argument of the form X% of S are P; i is an S; Therefore, i is (probably) a P.
strong arguments
Inductive arguments in which the premises greatly increase the likelihood that the conclusion is true.
sufficient condition
A condition for an event that guarantees that the event will occur; A is a sufficient condition of B if B
occurs whenever A does.
trigger cause
The factor that completes the cause chain resulting in the effect. Also known as proximate cause.
weak arguments
Inductive arguments in which the premises only minimally increase the likelihood that the conclusion
is true.
Choose a Study Mode
Place this order or similar order and get an amazing discount. USE Discount code “GET20” for 20% discount
