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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.

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.

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

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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.

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.

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

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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%?

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 high-energy 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

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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.

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.

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 never-ending 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.

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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

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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.

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 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 one-cup 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).

Making Inferences From Statistics

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.

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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?

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.

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

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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.

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 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 well-known 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.

1. “I’m certain that unemployment is up.”

2. “I’m reasonably sure that unemployment is up.”

3. “It’s more likely than not that unemployment is up.”

4. “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 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.

1. sample size

2. representativeness of the sample

3. a and b

4. 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.

1. sample size

2. representativeness of the sample

3. a and b

4. 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.

1. sample size

2. representativeness of the sample

3. a and b

4. There is no negative influence in this case.

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

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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.

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 real-life 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 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.

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Although water is a necessary condition for life, it is not a sufficient condition for life because humans also need oxygen and food.

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 not-A, then not-B.

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 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.

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?

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 well-known 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 .

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.

5.6 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?

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.

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.

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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 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

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Retailers such as bookstores commonly use arguments from analogy when they suggest purchases based on their similarity to other items.

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 well-known 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 fancy-free 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 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.

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.