Discussion questions

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

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

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

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

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.

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).

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

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.

“I’m certain that unemployment is up.”

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

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

“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.

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 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.

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.

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.

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 Bellevue-Stratford 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

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

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 8-year-old 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, so-called 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,  and 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

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.

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.”

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

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

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

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

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.

6.1 Contrasting Deduction and Induction

Remember that in logic, the difference between induction and deduction lies in the connection between the premises and conclusion. Deductive arguments aim for an absolute connection, one in which it is impossible that the premises could all be true and the conclusion false. Arguments that achieve this aim are called valid. Inductive arguments aim for a probable connection, one in which, if all the premises are true, the conclusion is more likely to be true than it would be otherwise. Arguments that achieve this aim are called strong. (For a discussion on common misconceptions about the meanings of induction and deduction, see A Closer Look: Doesn’t Induction Mean Going From Specific to General?). Recall from Chapter 5 that inductive strength is the counterpart of deductive validity, and cogency is the inductive counterpart of deductive soundness. One of the purposes of this chapter is to properly understand the differences and connections between these two major types of reasoning.

There is another important difference between deductive and inductive reasoning. As discussed in Chapter 5, if you add another premise to an inductive argument, the argument may become either stronger or weaker. For example, suppose you are thinking of buying a new cell phone. After looking at all your options, you decide that one model suits your needs better than the others. New information about the phone may make you either more convinced or less convinced that it is the right one for you—it depends on what the new information is. With deductive reasoning, by contrast, adding premises to a valid argument can never render it invalid. New information may show that a deductive argument is unsound or that one of its premises is not true after all, but it cannot undermine a valid connection between the premises and the conclusion. For example, consider the following argument:

All whales are mammals.

Shamu is a whale.

Therefore, Shamu is a mammal.

This argument is valid, and there is nothing at all we could learn about Shamu that would change this. We might learn that we were mistaken about whales being mammals or about Shamu being a whale, but that would lead us to conclude that the argument is unsound, not invalid. Compare this to an inductive argument about Shamu.

Whales typically live in the ocean.

Shamu is a whale.

Therefore, Shamu lives in the ocean.

Now suppose you learn that Shamu has been trained to do tricks in front of audiences at an amusement park. This seems to make it less likely that Shamu lives in the ocean. The addition of this new information has made this strong inductive argument weaker. It is, however, possible to make it stronger again with the addition of more information. For example, we could learn that Shamu was part of a captive release program.

An interesting exercise for exploring this concept is to see if you can keep adding premises to make an inductive argument stronger, then weaker, then stronger again. For example, see if you can think of a series of premises that make you change your mind back and forth about the quality of the cell phone discussed earlier.

Determining whether an argument is deductive or inductive is an important step both in evaluating arguments that you encounter and in developing your own arguments. If an argument is deductive, there are really only two questions to ask: Is it valid? And, are the premises true? If you determine that the argument is valid, then only the truth of the premises remains in question. If it is valid and all of the premises are true, then we know that the argument is sound and that therefore the conclusion must be true as well.

On the other hand, because inductive arguments can go from strong to weak with the addition of more information, there are more questions to consider regarding the connection between the premises and conclusion. In addition to considering the truth of the premises and the strength of the connection between the premises and conclusion, you must also consider whether relevant information has been left out of the premises. If so, the argument may become either stronger or weaker when the relevant information is included.

Later in this chapter we will see that many arguments combine both inductive and deductive elements. Learning to carefully distinguish between these elements will help you know what questions to ask when evaluating the argument.

A Closer Look: Doesn’t Induction Mean Going From Specific to General?

A common misunderstanding of the meanings of induction and deduction is that deduction goes from the general to the specific, whereas induction goes from the specific to the general. This definition is used by some fields, but not by logic or philosophy. It is true that some deductive arguments go from general premises to specific conclusions, and that some inductive arguments go from the specific premises to general conclusions. However, neither statement is true in general.

First, although some deductive arguments go from general to specific, there are many deductive arguments that do not go from general to specific. Some deductive arguments, for example, go from general to general, like the following:

All S are M.

All M are P.

Therefore, all S are P.

Propositional logic is deductive, but its arguments do not go from general to specific. Instead, arguments are based on the use of connectives (and, or, not, and if . . . then). For example, modus ponens (discussed in Chapter 4) does not go from the general to the specific, but it is deductively valid. When it comes to inductive arguments, some—for example, inductive generalizations—go from specific to general; others do not. Statistical syllogisms, for example, go from general to specific, yet they are inductive.

This common misunderstanding about the definitions of induction and deduction is not surprising given the different goals of the fields in which the terms are used. However, the definitions used by logicians are especially suited for the classification and evaluation of different types of reasoning.

For example, if we defined terms the old way, then the category of deductive reasoning would include arguments from analogy, statistical syllogisms, and some categorical syllogisms. Inductive reasoning, on the other hand, would include only inductive generalizations. In addition, there would be other types of inference that would fit into neither category, like many categorical syllogisms, inferences to the best explanation, appeals to authority, and the whole field of propositional logic.

The use of the old definitions, therefore, would not clear up or simplify the categories of logic at all but would make them more confusing. The current distinction, based on whether the premises are intended to guarantee the truth of the conclusion, does a much better job of simplifying logic’s categories, and it does so based on a very important and relevant distinction.

6.2 Choosing Between Induction and Deduction

You might wonder why one would choose to use inductive reasoning over deductive reasoning. After all, why would you want to show that a conclusion was only probably true rather than guaranteed to be true? There are several reasons, which will be discussed in this section. First, there may not be an available deductive argument based on agreeable premises. Second, inductive arguments can be more robust than deductive arguments. Third, inductive arguments can be more persuasive than deductive arguments.

Availability

Sometimes the best evidence available does not lend itself to a deductive argument. Let us consider a readily accepted fact: Gravity is a force that pulls everything toward the earth. How would you provide an argument for that claim? You would probably pick something up, let go of it, and note that it falls toward the earth. For added effect, you might pick up several things and show that each of them falls. Put in premise–conclusion form, your argument looks something like the following:

My coffee cup fell when I let go of it.

My wallet fell when I let go of it.

This rock fell when I let go of it.

Therefore, everything will fall when I let go of it.

When we put the argument that way, it should be clear that it is inductive. Even if we grant that the premises are true, it is not guaranteed that everything will fall when you let go of it. Perhaps gravity does not affect very small things or very large things. We could do more experiments, but we cannot check every single thing to make sure that it is affected by gravity. Our belief in gravity is the result of extremely strong inductive reasoning. We therefore have great reasons to believe in gravity, even if our reasoning is not deductive.

All subjects that rely on observation use inductive reasoning: It is at least theoretically possible that future observations may be totally different than past ones. Therefore, our inferences based on observation are at best probable. It turns out that there are very few subjects in which we can proceed entirely by deductive reasoning. These tend to be very abstract and formal subjects, such as mathematics. Although other fields also use deductive reasoning, they do so in combination with inductive reasoning. The result is that most fields rely heavily on inductive reasoning.

Robustness

Inductive arguments have some other advantages over deductive arguments. Deductive arguments can be extremely persuasive, but they are also fragile in a certain sense. When something goes wrong in a deductive argument, if a premise is found to be false or if it is found to be invalid, there is typically not much of an argument left. In contrast, inductive arguments tend to be more robust. The robustness of an inductive argument means that it is less fragile; if there is a problem with a premise, the argument may become weaker, but it can still be quite persuasive. Deductive arguments, by contrast, tend to be completely unconvincing once they are shown not to be sound. Let us work through a couple of examples to see what this means in practice.

Consider the following deductive argument:

All dogs are mammals.

Some dogs are brown.

Therefore, some mammals are brown.

As it stands, the argument is sound. However, if we change a premise so that it is no longer sound, then we end up with an argument that is nearly worthless. For example, if you change the first premise to “Most dogs are mammals,” you end up with an invalid argument. Validity is an all-or-nothing affair; there is no such thing as “sort of valid” or “more valid.” The argument would simply be invalid and therefore unsound; it would not accomplish its purpose of demonstrating that the conclusion must be true. Similarly, if you were to change the second premise to something false, like “Some dogs are purple,” then the argument would be unsound and therefore would supply no reason to accept the conclusion.

In contrast, inductive arguments may retain much of their strength even when there are problems with them. An inductive argument may list several reasons in support of a conclusion. If one of those reasons is found to be false, the other reasons continue to support the conclusion, though to a lesser degree. If an argument based on statistics shows that a particular conclusion is extremely likely to be true, the result of a problem with the argument may be that the conclusion should be accepted as only fairly likely. The argument may still give good reasons to accept the conclusion.

Fields that rely heavily on statistical arguments often have some threshold that is typically required in order for results to be publishable. In the social sciences, this is typically 90% or 95%. However, studies that do not quite meet the threshold can still be instructive and provide evidence for their conclusions. If we discover a flaw that reduces our confidence in an argument, in many cases the argument may still be strong enough to meet a threshold.

As an example, consider a tweet made by President Barack Obama regarding climate change.

A tweet by Barack Obama that reads, “Ninety-seven percent of scientists agree: #climate change is real, man-made and dangerous. Read more: OFA.BO/gJsdFp.”

Twitter/Public Domain

Although the tweet does not spell out the argument fully, it seems to have the following structure:

A study concluded that 97% of scientists agree that climate change is real, man-made, and dangerous.

Therefore, 97% of scientists really do agree that climate change is real, man-made, and dangerous.

Therefore, climate change is real, man-made, and dangerous.

Given the politically charged nature of the discussion of climate change, it is not surprising that the president’s argument and the study it referred to received considerable criticism. (You can read the study at http://iopscience.iop.org/1748–9326/8/2/024024/pdf/1748 –9326_8_2_024024.pdf.) Looking at the effect some of those criticisms have on the argument is a good way to see how inductive arguments can be more robust than deductive ones.

One criticism of Obama’s claim is that the study he referenced did not say anything about whether climate change was dangerous, only about whether it was real and man-made. How does this affect the argument? Strictly speaking, it makes the first premise false. But notice that even so, the argument can still give good evidence that climate change is real and man-made. Since climate change, by its nature, has a strong potential to be dangerous, the argument is weakened but still may give strong evidence for its conclusion.

A deeper criticism notes that the study did not find out what all scientists thought; it just looked at those scientists who expressed an opinion in their published work or in response to a voluntary survey. This is a significant criticism, for it may expose a bias in the sampling method (as discussed in Chapters 5, 7, and 8). Even granting the criticism, the argument can retain some strength. The fact that 97% of scientists who expressed an opinion on the issue said that climate change is real and man-made is still some reason to think that it is real and man-made. Of course, some scientists may have chosen not to voice an opposing opinion for reasons that have nothing to do with their beliefs about climate change; they may have simply wanted to keep their views private, for example. Taking all of this into account, we get the following argument:

A study found that 97% of scientists who stated their opinion said that climate change is real and man-made.

Therefore, 97% of scientists agree that climate change is real and man-made.

Climate change, if real, is dangerous.

Therefore, climate change is real, man-made, and dangerous.

This is not nearly as strong as the original argument, but it has not collapsed entirely in the way a purely deductive argument would. There is, of course, much more that could be said about this argument, both in terms of criticizing the study and in terms of responding to those criticisms and bringing in other considerations. The point here is merely to highlight the difference between deductive and inductive arguments, not to settle issues in climate science or public policy.

Persuasiveness

A final point in favor of inductive reasoning is that it can often be more persuasive than deductive reasoning. The persuasiveness of an argument is based on how likely it is to convince someone of the truth of its conclusion. Consider the following classic argument:

All Greeks are mortal.

Socrates was a Greek.

Therefore, Socrates was mortal.

Is this a good argument? From the standpoint of logic, it is a perfect argument: It is deductively valid, and its premises are true, so it is sound (therefore, its conclusion must be true). However, can you persuade anyone with this argument?

Imagine someone wondering whether Socrates was mortal. Could you use this argument to convince him or her that Socrates was mortal? Probably not. The argument is so simple and so obviously valid that anyone who accepts the premises likely already accepts the conclusion. So if someone is wondering about the conclusion, it is unlikely that he or she will be persuaded by these premises. He or she may, for example, remember that some legendary Greeks, such as Hercules, were granted immortality and wonder whether Socrates was one of these. The deductive approach, therefore, is unlikely to win anyone over to the conclusion here. On the other hand, consider a very similar inductive argument.

Of all the real and mythical Greeks, only a few were considered to be immortal.

Socrates was a Greek.

Therefore, it is extremely unlikely that Socrates was immortal.

Again, the reasoning is very simple. However, in this case, we can imagine someone who had been wondering about Socrates’s mortality being at least somewhat persuaded that he was mortal. More will likely need to be said to fully persuade her or him, but this simple argument may have at least some persuasive power where its deductive version likely does not.

Of course, deductive arguments can be persuasive, but they generally need to be more complicated or subtle in order to be so. Persuasion requires that a person change his or her mind to some degree. In a deductive argument, when the connection between premises and conclusion is too obvious, the argument is unlikely to persuade because the truth of the premises will be no more obvious than the truth of the conclusion. Therefore, even if the argument is valid, someone who questions the truth of the conclusion will often be unlikely to accept the truth of the premises, so she or he may be unpersuaded by the argument. Suppose, for example, that we wanted to convince someone that the sun will rise tomorrow morning. The deductive argument may look like this:

The sun will always rise in the morning.

Therefore, the sun will rise tomorrow morning.

One problem with this argument, as with the Socrates argument, is that its premise seems to assume the truth of the conclusion (and therefore commits the fallacy of begging the question, as discussed in Chapter 7), making the argument unpersuasive. Additionally, however, the premise might not even be true. What if, billions of years from now, the earth is swallowed up into the sun after it expands to become a red giant? At that time, the whole concept of morning may be out the window. If this is true then the first premise may be technically false. That means that the argument is unsound and therefore fairly worthless deductively.

The inductive version, however, does not lose much strength at all after we learn of this troubling information:

The sun has risen in the morning every day for millions of years.

Therefore, the sun will rise again tomorrow morning.

This argument remains extremely strong (and persuasive) regardless of what will happen billions of years in the future.

6.3 Combining Induction and Deduction

You may have noticed that most of the examples we have explored have been fairly short and simple. Real-life arguments tend to be much longer and more complicated. They also tend to mix inductive and deductive elements. To see how this might work, let us revisit an example from the previous section.

All Greeks are mortal.

Socrates was Greek.

Therefore, Socrates was mortal.

As we noted, this simple argument is valid but unlikely to convince anyone. So suppose now that someone questioned the premises, asking what reasons there are for thinking that all Greeks are mortal or that Socrates was Greek. How might we respond?

We might begin by noting that, although we cannot check each and every Greek to be sure he or she is mortal, there are no documented cases of any Greek, or any other human, living more than 200 years. In contrast, every case that we can document is a case in which the person dies at some point. So, although we cannot absolutely prove that all Greeks are mortal, we have good reason to believe it. We might put our argument in standard form as follows:

We know the mortality of a huge number of Greeks.

In each of these cases, the Greek is mortal.

Therefore, all Greeks are mortal.

This is an inductive argument. Even though it is theoretically possible that the conclusion might still be false, the premises provide a strong reason to accept the conclusion. We can now combine the two arguments into a single, larger argument:

We know the mortality of a huge number of Greeks.

In each of these cases, the Greek is mortal.

Therefore, all Greeks are mortal.

Socrates was Greek.

Therefore, Socrates was mortal.

This argument has two parts. The first argument, leading to the subconclusion that all Greeks are mortal, is inductive. The second argument (whose conclusion is “Socrates was mortal”) is deductive. What about the overall reasoning presented for the conclusion that Socrates was mortal (combining both arguments); is it inductive or deductive?

Sometimes a simple deductive argument needs to be combined with a persuasive inductive argument to convince others to accept it.

The crucial issue is whether the premises guarantee the truth of the conclusion. Because the basic premise used to arrive at the conclusion is that all of the Greeks whose mortality we know are mortal, the overall reasoning is inductive. This is how it generally works. As noted earlier, when an argument has both inductive and deductive components, the overall argument is generally inductive. There are occasional exceptions to this general rule, so in particular cases, you still have to check whether the premises guarantee the conclusion. But, almost always, the longer argument will be inductive.

A similar thing happens when we combine inductive arguments of different strength. In general, an argument is only as strong as its weakest part. You can think of each inference in an argument as being like a link in a chain. A chain is only as strong as its weakest link.

6.4 Reasoning About Science: The Hypothetico–Deductive Method

Science is one of the most successful endeavors of the modern world, and arguments play a central role in it. Science uses both deductive and inductive reasoning extensively. Scientific reasoning is a broad field in itself—and this chapter will only touch on the basics—but discussing scientific reasoning will provide good examples of how to apply what we have learned about inductive and deductive arguments.

At some point, you may have learned or heard of the scientific method, which often refers to how scientists systematically form, test, and modify hypotheses. It turns out that there is not a single method that is universally used by all scientists.

In a sense, science is the ultimate critical thinking experiment. Scientists use a wide variety of reasoning techniques and are constantly examining those techniques to make sure that the conclusions drawn are justified by the premises—that is exactly what a good critical thinker should do in any subject. The next two sections will explore two such methods—the hypothetico–deductive method and inferences to the best explanation—and discover ways that they can improve our understanding of the types of reasoning used in much of science.

The hypothetico–deductive method consists of four steps:

Formulate a hypothesis.

Deduce a consequence from the hypothesis.

Test whether the consequence occurs.

Reject the hypothesis if the consequence does not occur.

Although these four steps are not sufficient to explain all scientific reasoning, they still remain a core part of much discussion of how science works. You may recognize them as part of the scientific method that you likely learned about in school. Let us take a look at each step in turn.

Step 1: Formulate a Hypothesis

A hypothesis is a conjecture about how some part of the world works. Although the phrase “educated guess” is often used, it can give the impression that a hypothesis is simply guessed without much effort. In reality, scientific hypotheses are formulated on the basis of a background of quite a bit of knowledge and experience; a good scientific hypothesis often comes after years of prior investigation, thought, and research about the issue at hand.

You may have heard the expression “necessity is the mother of invention.” Often, hypotheses are formulated in response to a problem that needs to be solved. Suppose you are unsatisfied with the performance of your car and would like better fuel economy. Rather than buy a new car, you try to figure out how to improve the one you have. You guess that you might be able to improve your car’s fuel economy by using a higher grade of gas. Your guess is not just random; it is based on what you already know or believe about how cars work. Your hypothesis is that higher grade gas will improve your fuel economy.

Of course, science is not really concerned with your car all by itself. Science is concerned with general principles. A scientist would reword your hypothesis in terms of a general rule, something like, “Increasing fuel octane increases fuel economy in automobiles.” The hypothetico–deductive method can work with either kind of hypothesis, but the general hypothesis is more interesting scientifically.

Step 2: Deduce a Consequence From the Hypothesis

Your hypothesis from step 1 should have predictive value: Things should be different in some noticeable way, depending on whether the hypothesis is true or false. Our hypothesis is that increasing fuel octane improves fuel economy. If this general fact is true, then it is true for your car. So from our general hypothesis we can deduce the consequence that your car will get more miles per gallon if it is running on higher octane fuel.

It is often but not always the case that the prediction is a more specific case of the hypothesis. In such cases it is possible to infer the prediction deductively from the general hypothesis. The argument may go as follows:

Hypothesis: All things of type A have characteristic B.

Consequence (the prediction): Therefore, this specific thing of type A will have characteristic B.

Since the argument is deductively valid, there is a strong connection between the hypothesis and the prediction. However, not all predictions can be deductively inferred. In such cases we can get close to the hypothetico–deductive method by using a strong inductive inference instead. For example, suppose the argument went as follows:

Hypothesis: 95% of things of type A have characteristic B.

Consequence: Therefore, a specific thing of type A will probably have characteristic B.

In such cases the connection between the hypothesis and the prediction is less strong. The stronger the connection that can be established, the better for the reliability of the test. Essentially, you are making an argument for the conditional statement “If H, then C,” where H is your hypothesis and C is a consequence of the hypothesis. The more solid the connection is between H and C, the stronger the overall argument will be.

In this specific case, “If H, then C” translates to “If increasing fuel octane increases fuel economy in all cars, then using higher octane fuel in your car will increase its fuel economy.” The truth of this conditional is deductively certain.

We can now test the truth of the hypothesis by testing the truth of the consequence.

Step 3: Test Whether the Consequence Occurs

Your prediction (the consequence) is that your car will get better fuel economy if you use a higher grade of fuel. How will you test this? You may think this is obvious: Just put better gas in the car and record your fuel economy for a period before and after changing the type of gas you use. However, there are many other factors to consider. How long should the period of time be? Fuel economy varies depending on the kind of driving you do and many other factors. You need to choose a length of time for which you can be reasonably confident the driving conditions are similar on average. You also need to account for the fact that the first tank of better gas you put in will be mixed with some of the lower grade gas that is still in your tank. The more you can address these and other issues, the more certain you can be that your conclusion is correct.

In this step, you are constructing an inductive argument from the outcome of your test as to whether your car actually did get better fuel economy. The arguments in this step are inductive because there is always some possibility that you have not adequately addressed all of the relevant issues. If you do notice better fuel economy, it is always possible that the increase in economy is due to some factor other than the one you are tracking. The possibility may be very small, but it is enough to make this kind of argument inductive rather than deductive.

Step 4: Reject the Hypothesis If the Consequence Does Not Occur

We now compare the results to the prediction and find out if the prediction came true. If your test finds that your car’s fuel economy does not improve when you use higher octane fuel, then you know your prediction was wrong.

Does this mean that your hypothesis, H, was wrong? That depends on the strength of the connection between H and C. If the inference from H to C is deductively certain, then we know for sure that, if H is true, then C must be true also. Therefore, if C is false, it follows logically that H must be false as well.

In our specific case, if your car does not get better fuel economy by switching to higher octane fuel, then we know for sure that it is not true that all cars get better fuel economy by doing so. However, if the inference from H to C is inductive, then the connection between H and C is less than totally certain. So if we find that C is false, we are not absolutely sure that the hypothesis, H, is false.

For example, suppose that the hypothesis is that cars that use higher octane fuel will have a higher tendency to get better fuel mileage. In that case if your car does not get higher gas mileage, then you still cannot infer for certain that the hypothesis is false. To test that hypothesis adequately, you would have to do a large study with many cars. Such a study would be much more complicated, but it could provide very strong evidence that the hypothesis is false.

At best, the fuel economy hypothesis will be a strong inductive argument because there is a chance that something other than higher octane gas is improving fuel economy. The more you can address relevant issues that may impact your test results, the stronger your conclusions will be.

It is important to note that although the falsity of the prediction can demonstrate that the hypothesis is false, the truth of the prediction does not prove that the hypothesis is true. If you find that your car does get better fuel economy when you switch gas, you cannot conclude that your hypothesis is true.

Why? There may be other factors at play for which you have not adequately accounted. Suppose that at the same time you switch fuel grade, you also get a tune-up and new tires and start driving a completely different route to work. Any one of these things might be the cause of the improved gas mileage; you cannot conclude that it is due to the change in fuel (for this reason, when conducting experiments it is best to change only one variable at a time and carefully control the rest). In other words, in the hypothetico–deductive method, failed tests can show that a hypothesis is wrong, but tests that succeed do not show that the hypothesis was correct.

This logic is known as falsification; it can be demonstrated clearly by looking at the structure of the argument. When a test yields a negative result, the hypothetico–deductive method sets up the following argument:

If H, then C.

Not C.

Therefore, not H.

You may recognize this argument form as modus tollens, or denying the consequent, which was discussed in the chapter on propositional logic (Chapter 4). This argument form is a valid, deductive form. Therefore, if both of these premises are true, then we can be certain that the conclusion is true as well; namely, that our hypothesis, H, is not true. In the specific case at hand, if your test shows that higher octane fuel does not increase your mileage, then we can be sure that it is not true that it improves mileage in all vehicles (though it may improve it in some).

Contrast this with the argument form that results when your fuel economy yields a positive result:

If H, then C.

C.

Therefore, H.

This argument is not valid. In fact, you may recognize this argument form as the invalid deductive form called affirming the consequent (see Chapter 4). It is possible that the two premises are true, but the conclusion false. Perhaps, for example, the improvement in fuel economy was caused by a change in tires or different driving conditions instead. So the hypothetico –deductive method can be used only to reject a hypothesis, not to confirm it. This fact has led many to see the primary role of science to be the falsification of hypotheses. Philosopher Karl Popper is a central source for this view (see A Closer Look: Karl Popper and Falsification in Science).

A Closer Look: Karl Popper and Falsification in Science

Karl Popper, a 20th-century philosopher of science, put forth the idea that unfalsifiable claims are unscientific.

Karl Popper, one of the most influential philosophers of science to emerge from the early 20th century, is perhaps best known for rejecting the idea that scientific theories could be proved by simply finding confirming evidence—the prevailing philosophy at the time. Instead, Popper emphasized that claims must be testable and falsifiable in order to be considered scientific.

A claim is testable if we can devise a way of seeing if it is true or not. We can test, for instance, that pure water will freeze at 0°C at sea level; we cannot currently test the claim that the oceans in another galaxy taste like root beer. We have no realistic way to determine the truth or falsity of the second claim.

A claim is said to be falsifiable if we know how one could show it to be false. For instance, “there are no wild kangaroos in Georgia” is a falsifiable claim; if one went to Georgia and found some wild kangaroos, then it would have been shown to be false. But what if someone claimed that there are ghosts in Georgia but that they are imperceptible (unseeable, unfeelable, unhearable, etc.)? Could one ever show that this claim is false? Since such a claim could not conceivably be shown to be false, it is said to be unfalsifiable. While being unfalsifiable might sound like a good thing, according to Popper it is not, because it means that the claim is unscientific.

Following Popper, most scientists today operate with the assumption that any scientific hypothesis must be testable and must be the kind of claim that one could possibly show to be false. So if a claim turns out not to be conceivably falsifiable, the claim is not really scientific—and some philosophers have gone so far as to regard such claims as meaningless (Thornton, 2014).

Karl Popper and Falsification

Learn more about Karl Popper's criterion of falsifiability in this video.

Critical Thinking Questions

Karl Popper argues that only hypotheses that can be tested and falsified are scientific. Do you agree?

In addition to being unscientific, Popper states that unfalsifiable claims tell us nothing and do not allow us to learn from our mistakes. Can you make an argument against Popper's?

As an example, suppose a friend claims that “everything works out for the best.” Then suppose that you have the worst month of your life, and you go back to your friend and say that the claim is false: Not everything is for the best. Your friend might then reply that in fact it was for the best because you learned from the experience. Such a statement may make you feel better, but it runs afoul of Popper’s rule. Can you imagine any circumstance that your friend would not claim is for the best? Since your friend would probably say that it was for the best no matter what happens, your friend’s claim is unfalsifiable and therefore unscientific.

In logic, claims that are interpreted so that they come out true no matter what happens are called self-sealing propositions. They are understood as being internally protected against any objections. People who state such claims may feel that they are saying something deeply meaningful, but according to Popper’s rule, since the claim could never be falsified no matter what, it does not really tell us anything at all.

Other examples of self-sealing propositions occur within philosophy itself. There is a philosophical theory known as psychological egoism, for example, which teaches that everything everyone does is completely selfish. Most people respond to this claim by coming up with examples of unselfish acts: giving to the needy, spending time helping others, and even dying to save someone’s life. The psychological egoist predictably responds to all such examples by stating that people who do such things really just do them in order to feel better about themselves. It appears that the word selfish is being interpreted so that everything everyone does will automatically be considered selfish by definition. It is therefore a self-sealing claim (Rachels, 1999). According to Popper’s method, since this claim will always come out true no matter what, it is unfalsifiable and unscientific. Such claims are always true but are actually empty because they tell us nothing about the world. They can even be said to be “too true to be good.”

Popper’s explorations of scientific hypotheses and what it means to confirm or disconfirm such hypotheses have been very influential among both scientists and philosophers of scientists. Scientists do their best to avoid making claims that are not falsifiable.

If the hypothetico-deductive method cannot be used to confirm a hypothesis, how can this test give evidence for the truth of the claim? By failing to falsify the claim. Though the hypothetico–deductive method does not ever specifically prove the hypothesis true, if researchers try their hardest to refute a claim but it keeps passing the test (not being refuted), then there can grow a substantial amount of inductive evidence for the truth of the claim. If you repeatedly test many cars and control for other variables, and if every time cars are filled with higher octane gas their fuel economy increases, you may have strong inductive evidence that the hypothesis might be true (in which case you may make an inference to the best explanation, which will be discussed in Section 6.5).

Experiments that would have the highest chance of refuting the claim if it were false thus provide the strongest inductive evidence that it may be true. For example, suppose we want to test the claim that all swans are white. If we only look for swans at places in which they are known to be white, then we are not providing a strong test for the claim. The best thing to do (short of observing every swan in the whole world) is to try as hard as we can to refute the claim, to find a swan that is not white. If our best methods of looking for nonwhite swans still fail to refute the claim, then there is a growing likelihood that perhaps all swans are indeed white.

Similarly, if we want to test to see if a certain type of medicine cures a certain type of disease, we test the product by giving the medicine to a wide variety of patients with the disease, including those with the least likelihood of being cured by the medicine. Only by trying as hard as we can to refute the claim can we get the strongest evidence about whether all instances of the disease are treatable with the medicine in question.

Notice that the hypothetico–deductive method involves a combination of inductive and deductive reasoning. Step 1 typically involves inductive reasoning as we formulate a hypothesis against the background of our current beliefs and knowledge. Step 2 typically provides a deductive argument for the premise “If H, then C.” Step 3 provides an inductive argument for whether C is or is not true. Finally, if the prediction is falsified, then the conclusion—that H is false—is derived by a deductive inference (using the deductively valid modus tollens form). If, on the other hand, the best attempts to prove C to be false fail to do so, then there is growing evidence that H might be true.

Therefore, our overall argument has both inductive and deductive elements. It is valuable to know that, although the methodology of science involves research and experimentation that goes well beyond the scope of pure logic, we can use logic to understand and clarify the basic principles of scientific reasoning.

6.5 Inference to the Best Explanation

You may feel that if you were very careful about testing your fuel economy, you would be entitled to conclude that the change in fuel grade really did have an effect. Unfortunately, as we have seen, the hypothetico–deductive method does not support this inference. The best you can say is that changing fuel might have an effect; that you have not been able to show that it does not have an effect. The method does, however, lend inductive support to whichever hypothesis withstands the falsification test better than any other. One way of articulating this type of support is with an inference pattern known as inference to the best explanation.

As the name suggests, inference to the best explanation draws a conclusion based on what would best explain one’s observations. It is an extremely important form of inference that we use every day of our lives. This type of inference is often called abductive reasoning, a term pioneered by American logician Charles Sanders Peirce (Douven, 2011).

Suppose that you are in your backyard gazing at the stars. Suddenly, you see some flashing lights hovering above you in the sky. You do not hear any sound, so it does not appear that the lights are coming from a helicopter. What do you think it is? What happens next is abductive reasoning: Your brain searches among all kinds of possibilities to attempt to come up with the most likely explanation.

One possibility is that it is an alien spacecraft coming to get you (one could joke that this is why it is called abductive reasoning). Another possibility is that it is some kind of military vessel or a weather balloon. A more extreme hypothesis is that you are actually dreaming the whole thing.

Notice that what you are inclined to believe depends on your existing beliefs. If you already think that alien spaceships come to Earth all the time, then you may arrive at that conclusion with a high degree of certainty (you may even shout, “Take me with you!”). However, if you are somewhat skeptical of those kinds of theories, then you will try hard to find any other explanation. Therefore, the strength of a particular inference to the best explanation can be measured only in relation to the rest of the things that we already believe.

Sherlock Holmes often used abductive reasoning, not deductive reasoning, to solve his mysteries.

This type of inference does not occur only in unusual circumstances like the one described. In fact, we make inferences to the best explanation all the time. Returning to our fuel economy example from the previous section, suppose that you test a higher octane fuel and notice that your car gets better gas mileage. It is possible that the mileage change is due to the change in fuel. However, as noted there, it is possible that there is another explanation. Perhaps you are not driving in stop-and-go traffic as much. Perhaps you are driving with less weight in the car. The careful use of inference to the best explanation can help us to discern what is the most likely among many possibilities (for more examples, see A Closer Look: Is Abductive Reasoning Everywhere?).

If you look at the range of possible explanations and find one of them is more likely than any of the others, inference to the best explanation allows you to conclude that this explanation is likely to be the correct one. If you are driving the same way, to the same places, and with the same weight in your car as before, it seems fairly likely that it was the change in fuel that caused the improvement in fuel economy (if you have studied Mill’s methods in Chapter 5, you should recognize this as the method of difference). Inference to the best explanation is the engine that powers many inductive techniques.

The great fictional detective Sherlock Holmes, for example, is fond of claiming that he uses deductive reasoning. Chapter 2 suggested that Holmes instead uses inductive reasoning. However, since Holmes comes up with the most reasonable explanation of observed phenomena, like blood on a coat, for example, he is actually doing abductive reasoning. There is some dispute about whether inference to the best explanation is inductive or whether it is an entirely different kind of argument that is neither inductive nor deductive. For our purposes, it is treated as inductive.

A Closer Look: Is Abductive Reasoning Everywhere?

Some see inference to the best explanation as the most common type of inductive inference. A few of the inferences we have discussed in this book, for example, can potentially be cast as examples of inferences to the best explanation.

For example, appeals to authority (discussed in Chapter 5) can be seen as implicitly using inference to the best explanation (Harman, 1965). If you accept something as true because someone said it was, then you can be described as seeing the truth of the claim as the best explanation for why he or she said it. If we have good reason to think that the person was deluded or lying, then we are less certain of this conclusion because there are other likely explanations of why the person said it.

Furthermore, it is possible to see what we do when we interpret people’s words as a kind of inference to the best explanation of what they probably mean (Hobbs, 2004). If your neighbor says, “You are so funny,” for instance, we might use the context and tone to decide what he means by “funny” and why he is saying it (and whether he is being sarcastic). His comment can be seen as either rude or flattering, depending on what explanation we give for why he said it and what he meant.

Even the classic inductive inference pattern of inductive generalization can possibly be seen as implicitly involving a kind of inference to the best explanation: The best explanation of why our sample population showed that 90% of students have laptops is probably that 90% of all students have laptops. If there is good evidence that our sample was biased, then there would be a good competing explanation of our data.

Finally, much of scientific inference may be seen as trying to provide the best explanation for our observations (McMullin, 1992). Many hypotheses are attempts to explain observed phenomena. Testing them in such cases could then be seen as being done in the service of seeking the best explanation of why certain things are the way they are.

Take a look at the following examples of everyday inferences and see if they seem to involve arriving at the conclusion because it seems to offer the most likely explanation of the truth of the premise:

“John is smiling; he must be happy.”

“My phone says that Julie is calling, so it is probably Julie.”

“I see a brown Labrador across the street; my neighbor’s dog must have gotten out.”

“This movie has great reviews; it must be good.”

“The sky is getting brighter; it must be morning.”

“I see shoes that look like mine by the door; I apparently left my shoes there.”

“She still hasn’t called back yet; she probably doesn’t like me.”

“It smells good; someone is cooking a nice dinner.”

“My congressperson voted against this bill I support; she must have been afraid of offending her wealthy donors.”

“The test showed that the isotopes in the rock surrounding newly excavated bones had decayed X amount; therefore, the animals from which the bones came must have been here about 150 million years ago.”

These examples, and many others, suggest to some that inference to the explanation may be the most common form of reasoning that we use (Douven, 2011). Do you agree? Whether you agree with these expanded views on the role of inference or not, it clearly makes an enormous contribution to how we understand the world around us.

Form

Inferences to the best explanation generally involve the following pattern of reasoning:

X has been observed to be true.

Y would provide an explanation of why X is true.

No other explanation for X is as likely as Y.

Therefore, Y is probably true.

One strange thing about inferences to the best explanation is that they are often expressed in the form of a common fallacy, as follows:

If P is the case, then Q would also be true.

Q is true.

Therefore, P is probably true.

This pattern is the logical form of a deductive fallacy known as affirming the consequent (discussed in Chapter 4). Therefore, we sometimes have to use the principle of charity to determine whether the person is attempting to provide an inference to the best explanation or making a simple deductive error. The principle of charity will be discussed in detail in Chapter 9; however, for our purposes here, you can think of it as giving your opponent and his or her argument the benefit of the doubt.

For example, the ancient Greek philosopher Aristotle reasoned as follows: “The world must be spherical, for the night sky looks different in the northern and southern regions, and that would be the case if the earth were spherical” (as cited in Wolf, 2004). His argument appears to have this structure:

If the earth is spherical, then the night sky would look different in the northern and southern regions.

The night sky does look different in the northern and southern regions.

Therefore, the earth is spherical.

It is not likely that Aristotle, the founding father of formal logic, would have made a mistake as silly as to affirm the consequent. It is far more likely that he was using inference to the best explanation. It is logically possible that there are other explanations for southern stars moving higher in the sky as one moves south, but it seems far more likely that it is due to the shape of the earth. Aristotle was just practicing strong abductive reasoning thousands of years before Columbus sailed the ocean blue (even Columbus would have had to use this type of reasoning, for he would have had to infer why he did not sail off the edge).

In more recent times, astronomers are still using inference to the best explanation to learn about the heavens. Let us consider the case of discovering planets outside our solar system, known as “exoplanets.” There are many methods employed to discover planets orbiting other stars. One of them, the radial velocity method, uses small changes in the frequency of light a star emits. A star with a large planet orbiting it will wobble a little bit as the planet pulls on the star. That wobble will result in a pattern of changes in the frequency of light coming from the star. When astronomers see this pattern, they conclude that there is a planet orbiting the star. We can more fully explicate this reasoning in the following way:

That star’s light changes in a specific pattern.

Something must explain the changes.

A large planet orbiting the star would explain the changes.

No other explanation is as likely as the explanation provided by the large planet.

Therefore, that star probably has a large planet orbiting it.

The basic idea is that if there must be an explanation, and one of the available explanations is better than all the others, then that explanation is the one that is most likely to be true. The key issue here is that the explanation inferred in the conclusion has to be the best explanation available. If another explanation is as good—or better—then the inference is not nearly as strong.

Virtue of Simplicity

Another way to think about inferences to the best explanation is that they choose the simplest explanation from among otherwise equal explanations. In other words, if two theories make the same prediction, the one that gives the simplest explanation is usually the best one. This standard for comparing scientific theories is known as Occam’s razor, because it was originally posited by William of Ockham in the 14th century (Gibbs & Hiroshi, 1997).

A great example of this principle is Galileo’s demonstration that the sun, not the earth, is at the center of the solar system. Galileo’s theory provided the simplest explanation of observations about the planets. His heliocentric model, for example, provides a simpler explanation for the phases of Venus and why some of the planets appear to move backward (retrograde motion) than does the geocentric model. Geocentric astronomers tried to explain both of these with the idea that the planets sometimes make little loops (called epicycles) within their orbits (Gronwall, 2006). While it is certainly conceivable that they do make little loops, it seems to make the theory unnecessarily complex, because it requires a type of motion with no independent explanation of why it occurs, whereas Galileo’s theory does not require such extra assumptions.

Therefore, putting the sun at the center allows one to explain observed phenomena in the most simple manner possible, without making ad hoc assumptions (like epicycles) that today seem absurd. Galileo’s theory was ultimately correct, and he demonstrated it with strong inductive (more specifically, abductive) reasoning. (For another example of Occam’s razor at work, see A Closer Look: Abductive Reasoning and the Matrix.)

A Closer Look: Abductive Reasoning and the Matrix

One of the great questions from the history of philosophy is, “How do we know that the world exists outside of us as we perceive it?” We see a tree and we infer that it exists, but do we actually know for sure that it exists? The argument seems to go as follows:

I see a tree.

Therefore, a tree exists.

This inference, however, is invalid; it is possible for the premise to be true and the conclusion false. For example, we could be dreaming. Perhaps we think that the testimony of our other senses will make the argument valid:

I see a tree, I hear a tree, I feel a tree, and I smell a tree.

Therefore, a tree exists.

In The Matrix, we learn that our world is simulated by machines, and although we can see X, hear X, and feel X, X does not exist.

However, this argument is still invalid; it is possible that we could be dreaming all of those things as well. Some people state that senses like smell do not exist within dreams, but how do we know that is true? Perhaps we only dreamed that someone said that! In any case, even that would not rescue our argument, for there is an even stronger way to make the premise true and the conclusion false: What if your brain is actually in a vat somewhere attached to a computer, and a scientist is directly controlling all of your perceptions? (Or think of the 1999 movie The Matrix, in which humans are living in a simulated reality created by machines.)

One individual who struggled with these types of questions (though there were no computers back then) was a French philosopher named René Descartes. He sought a deductive proof that the world outside of us is real, despite these types of disturbing possibilities (Descartes, 1641/1993). He eventually came up with one of philosophy’s most famous arguments, “I think, therefore, I am” (or, more precisely, “I am thinking, therefore, I exist”), and from there attempted to prove that the world must exist outside of him.

Many philosophers feel that Descartes did a great job of raising difficult questions, but most feel that he failed in his attempt to find deductive proof of the world outside of our minds. Other philosophers, including David Hume, despaired of the possibility of a proof that we know that there is a world outside of us and became skeptics: They decided that absolute knowledge of a world outside of us is impossible (Hume, 1902).

However, perhaps the problem is not the failure of the particular arguments but the type of reasoning employed. Perhaps the solution is not deductive at all but rather abductive. It is not that it is logically impossible that tables and chairs and trees (and even other people) do not really exist; it is just that their actual existence provides the best explanation of our experiences. Consider these competing explanations of our experiences:

We are dreaming this whole thing.

We are hallucinating all of this.

Our brains are in a vat being controlled by a scientist.

Light waves are bouncing off the molecules on the surface of the tree and entering our eyeballs, where they are turned into electrical impulses that travel along neurons into our brains, somehow causing us to have the perception of a tree.

It may seem at first glance that the final option is the most complex and so should be rejected. However, let us take a closer look. The first two options do not offer much of an explanation for the details of our experience. They do not tell us why we are seeing a tree rather than something else or nothing at all. The third option seems to assume that there is a real world somewhere from which these experiences are generated (that is, the lab with the scientist in it). The full explanation of how things work in that world presumably must involve some complex laws of physics as well. There is no obvious reason to think that such an account would require fewer assumptions than an account of the world as we see it. Hence, all things considered, if our goal is to create a full explanation of reality, the final option seems to give the best account of why we are seeing the tree. It explains our observations without needless extra assumptions.

Therefore, if knowledge is assumed only to be deductive, then perhaps we do not know (with absolute deductive certainty) that there is a world outside of us. However, when we consider abductive knowledge, our evidence for the existence of the world as we see it may be rather strong.

How to Assess an Explanation

There are many factors that influence the strength of an inference to the best explanation. However, when testing inferences to the best explanation for strength, these questions are good to keep in mind:

Does it agree well with the rest of human knowledge? Suggesting that your roommate’s car is gone because it floated away, for example, is not a very credible story because it would violate the laws of physics.

Does it provide the simplest explanation of the observed phenomena? According to Occam’s razor, we want to explain why things happen without unnecessary complexity.

Does it explain all relevant observations? We cannot simply ignore contradicting data because it contradicts our theory; we have to be able to explain why we see what we see.

Is it noncircular? Some explanations merely lead us in a circle. Stating that it is raining because water is falling from the sky, for example, does not give us any new information about what causes the water to fall.

Is it testable? Suggesting that invisible elves stole the car does not allow for empirical confirmation. An explanation is stronger if its elements are potentially observable.

Does it help us explain other phenomena as well? The best scientific theories do not just explain one thing but allow us to understand a whole range of related phenomena. This principle is called fecundity. Galileo’s explanation of the orbits of the planets is an example of a fecund theory because it explains several things all at once.

An explanation that has all of these virtues is likely to be better than one that does not.

A Limitation

One limitation of inference to the best explanation is that it depends on our coming up with the correct explanation as one of the candidates. If we do not think of the correct explanation when trying to imagine possible explanation, then inference to the best explanation can steer us wrong. This can happen with any inductive argument, of course; inductive arguments always carry some possibility that the conclusion may be false even if the premises are true. However, this limitation is a particular danger with inference to the best explanation because it relies on our being able to imagine the true explanation.

This is one reason that it is essential to always keep an open mind when using this technique. Further information may introduce new explanations or change which explanation is best. Being open to further information is important for all inductive inferences, but especially so for those involving inference to the best explanation.