{9} II
Arguments by Example
Some arguments offer one or more examples in support of a generalization.
Women in earlier times were married very young. Juliet in Shakespeare’s Romeo and Juliet was not even fourteen years old. In the Middle Ages, thirteen was the normal age of marriage for a Jewish woman. And during the Roman Empire, many Roman women were married at age thirteen or younger.
This argument generalizes from three examples—Juliet, Jewish women in the Middle Ages, and Roman women during the Roman Empire—to “many” or most women in earlier times. To show the form of this argument most clearly, we can list the premises separately, with the conclusion on the “bottom line”:
Juliet in Shakespeare’s play was not even fourteen years old.
Jewish women during the Middle Ages were normally married at thirteen.
Many Roman women during the Roman Empire were married at age thirteen or younger.
Therefore, women in earlier times were married very young.
It is helpful to write short arguments in this way when we need to see exactly how they work.
When do premises like these adequately support a generalization?
One requirement, of course, is that the examples be accurate. Remember Rule 3: start from reliable premises! If Juliet wasn’t around fourteen, or if most Roman or Jewish women weren’t married at thirteen or younger, then the argument is much weaker. If none of the premises can be supported, there is no argument at all. To check {10} an argument’s examples, or to find good examples for your own arguments, you may need to do some research.
But suppose the examples are accurate. Even then, generalizing from them is a tricky business. The rules in this chapter offer a short checklist for assessing arguments by example.
A single example can sometimes be used for the sake of illustration. The example of Juliet alone might illustrate early marriage. But a single example offers next to no support for a generalization. Juliet alone may just be an exception. One spectacularly miserable billionaire does not prove that rich people in general are unhappy. One great meal at a new restaurant in town does not necessarily mean that its whole menu is first-rate. More than one example is needed.
NO:
Solar power is widely used.
Therefore, renewable energy is widely used.
Solar power is one form of renewable energy, but only one. What about others?
YES:
Solar power is widely used.
Hydroelectric power has long been widely used.
Windmills were once widely used and are becoming widely used again.
Therefore, renewable energy is widely used.
The “Yes” version may not be perfect (Rule 11 returns to it), but it certainly is more energetic, so to speak, than the “No” version.
{11} In a generalization about a small set of things, the strongest argument should consider all, or at least many, of the examples. A generalization about your siblings should consider each of them in turn, for instance, and a generalization about all the planets in the solar system can do the same.
Generalizations about larger sets of things require picking out a sample. We cannot list all women in earlier times who married young. Instead, our argument must offer a few women in earlier times as a sample of all women in earlier times. How many examples are required depends partly on how representative they are, a point the next rule takes up. It also depends partly on the size of the set being generalized about. Large sets usually require more examples. The claim that your town is full of remarkable people requires more evidence than the claim that, say, your friends are remarkable people. Depending on how many friends you have, even just two or three examples might be enough to establish that your friends are remarkable people; but, unless your town is tiny, many more examples are required to show that your town is full of remarkable people.
Even a large number of examples may misrepresent the set of things being generalized about. Do all insects bite, for example? Sure, we can think of lots of insects that do, like mosquitoes and black flies, and naturally those are the examples we think of first. After all, we are bugged by them! We may have to consult a biology textbook or a good online source to remember how many kinds of insects there are that don’t bite—which is most of them, actually: moths, praying mantis, ladybugs, (most) beetles, and so on.
Likewise, a large number of examples of ancient Roman women establishes very little about women generally, since ancient Roman women are not necessarily representative of other women in earlier times. If we want to make a sweeping claim about women in earlier times, the argument needs to consider women from other early times and from other parts of the world as well.
{12} It is easy to overlook how unrepresentative—often wildly unrepresentative—our personal “samples” often are. Actually, very few if any of us really know a representative sample of other people. Yet we constantly generalize about other people as a group, such as when we make claims about “human nature,” or even how our town might vote in the next election.
NO:
Everyone in my neighborhood favors the School Bond. Therefore, the School Bond is sure to pass.
This argument is weak because single neighborhoods seldom represent the voting population as a whole. A well-to-do neighborhood may favor a cause unpopular with everyone else. Student wards in university towns regularly are carried by candidates who do poorly elsewhere. Besides, we do not always have good evidence even about the views held in a specific neighborhood. The set of people eager to display their political preferences to the world in yard signs, for example, is unlikely to be a representative cross-section of the neighborhood as a whole.
A good argument that “The School Bond is sure to win” requires a representative sample of the entire voting population. It is not easy to construct such a sample. In fact, it usually takes professional help, and even professional pollsters regularly predict elections incorrectly. Telephone pollsters used to call landlines, for example, because cell phone numbers are not as publicly accessible; but only certain demographic groups still have landlines, and they are increasingly unrepresentative.
In general, look for the most accurate cross-section you can find of the population being generalized about. If you want to know what students think about some subject at your university, don’t just ask your friends or generalize from what you hear in class. Unless you have quite a range of friends and take a wide range of classes, your personal sample is very unlikely to accurately mirror the whole student body. Similarly, if you want to know what people in other countries think about the United States, don’t just ask foreign tourists—for they, of course, are the ones who chose to come here. A careful look {13} at a diverse range of foreign media should give you a more representative picture.
When your sample is people, an even more basic point is that no one should be able to self-select for it. This immediately disqualifies almost all online or mail-in polls to which individuals can decide whether to respond or not. Again, the set of people who are willing or eager to express their opinions is almost certainly not representative of the whole population, but are the people more likely to have strong opinions—or a lot of time on their hands. It may be interesting to know what that group thinks anyway, but not because they necessarily speak for anyone but themselves.
To persuade you that I am a first-rate archer, it is not enough to show you a bull’s-eye I have made. You should ask (politely, to be sure), “Yes, but how many times did you miss?” Getting a bull’s-eye in one shot tells quite a different story than getting a bull’s-eye in, say, a thousand, even though in both cases I genuinely do have a bull’s-eye to my name. You need a little more data.
Leon’s horoscope told him that he would meet a vivacious new stranger, and lo and behold he did! Therefore, horoscopes are reliable.
Dramatic as such an example may be, the problem is that we are only looking at one case in which a horoscope came true. To properly evaluate this evidence, we need to know something else as well: how many horoscopes didn’t come true. When I survey my classes, we can usually find a few Leons out of twenty or thirty students. It’s a fun moment. But the other nineteen or twenty-nine horoscopes go nowhere. A kind of prediction that comes true only once out of twenty or thirty tries is hardly reliable—it’s just lucky once in a while. It may have some dramatic successes, like my archery, but its success rate may still be abysmal.
To evaluate the reliability of any argument featuring a few vivid examples, then, we need to know the ratio between the number of {14} “hits,” so to speak, and the number of tries. It’s a question of representativeness again. Are the featured examples the only ones there are? Is the rate impressively high or low?
This rule is widely applicable. Today, many people live in fear of crime, or constantly attend to stories of shark attacks, terrorism, or other dramatic events. Of course these things are awful when they occur, but the probability of any of them actually happening to any given individual—say, the shark attack rate—is extremely low. Crime rates continue to go down.
No doubt we are preoccupied with the exceptions because they are constantly featured on TV and in the news. This does not mean that they are actually representative. The same goes, by the way, for desired things, like winning the lottery. Any individual’s chance of winning—that is, the winning rate—is so low as to be basically nil, but we seldom see the hundreds of thousands of losers, just the one or few winners raking in the money. So we wildly overestimate the background rates, and imagine that with the next lottery ticket purchase, we may be the one. Save your money, friends. Background rates make all the difference!
You cannot “prove anything with numbers”! Some people see numbers—any numbers—in an argument and conclude from that fact alone that it must be a good argument. Statistics seem to have an aura of authority and definiteness (and did you know that 88 percent of doctors agree?). In fact, though, numbers take as much critical thinking as any other kind of evidence. Don’t turn off your brain!
After an era when some athletic powerhouse universities were accused of exploiting student athletes, leaving them to flunk out once their eligibility expired, college athletes are now graduating at higher rates. Many schools are now graduating more than 50 percent of their athletes.
{15} Fifty percent, eh? Pretty impressive! But this figure, at first so persuasive, does not really do the job it claims to do.
First, although “many” schools graduate more than 50 percent of their athletes, it appears that some do not—so this figure may well exclude the most exploitative schools that really concerned people in the first place.
The argument does offer graduation rates. But it would be useful to know how a “more than 50 percent” graduation rate compares with the graduation rate for all students at the same institutions. If it is significantly lower, athletes may still be getting the shaft.
Most importantly, this argument offers no reason to believe that college athletes’ graduation rates are actually improving, because no comparison to any previous rate is offered! The conclusion claims that the graduation rate is now “higher,” but without knowing the previous rates it is impossible to tell.
Numbers may offer incomplete evidence in other ways too. Rule 9, for example, tells us that knowing background rates may be crucial. Correspondingly, when an argument offers rates or percentages, the relevant background information usually must include the number of examples. Car thefts on campus may have doubled, but if this means that two cars were stolen rather than one, there’s not much to worry about.
Another statistical pitfall is over-precision:
Every year this campus wastes 412,067 paper and plastic cups. It’s time to switch to reusable cups!
I’m all for ending waste too, and I’m sure the amount of campus waste is huge. But no one really knows the precise number of cups wasted—and it’s extremely unlikely to be exactly the same every year. Here the appearance of exactness makes the evidence seem more authoritative than it really is.
Be wary, also, of numbers that are easily manipulated. Pollsters know very well that the way a question is asked can shape how it is answered. These days we are even seeing “polls” that try to change people’s minds about, say, a political candidate, just by asking loaded questions (“If you were to discover that she is a liar and a cheat, how would that change your vote?”). Then too, many apparently “hard” statistics are actually based on guesswork or extrapolation, such as {16} data about semi-legal or illegal activities. Since people have a major motive not to reveal or report things like drug use, under-the-counter transactions, hiring illegal aliens, and the like, beware of any confident generalizations about how widespread they are.
Yet again:
If kids keep watching more TV at current rates, by 2025 they’ll have no time left to sleep!
Right, and by 2040 they’ll be watching thirty-six hours a day. Extrapolation in such cases is perfectly possible mathematically, but after a certain point it tells you nothing.
Counterexamples are examples that contradict your generalization. No fun—maybe. But counterexamples actually can be a generalizer’s best friends, if you use them early and use them well. Exceptions don’t “prove the rule”—quite the contrary, they threaten to disprove it—but they can and should prompt us to refine it. Therefore, seek out counterexamples early and systematically. It is the best way to sharpen your own generalizations and to probe more deeply into your theme.
Consider this argument once again:
Solar power is widely used.
Hydroelectric power has long been widely used.
Windmills were once widely used and are becoming widely used again.
Therefore, renewable energy is widely used.
The examples here certainly do help to show that many renewable energy sources are widely used: sun, wind, and rain. However, as soon as you start thinking about counterexamples instead of just more examples, you may find that the argument somewhat overgeneralizes.
{17} Are all renewables widely used? Look up the definition of “renewable energy” and you will find that there are other types as well, such as the tides and geothermal energy (the internal heat of the earth). And these, for better or worse, are not so widely used. They aren’t available everywhere, for one thing, and may be difficult to harness even when available.
When you think of counterexamples to a generalization that you want to defend, then you need to adjust your generalization. If the renewable energy argument were yours, for instance, you might change the conclusion to “Many forms of renewable energy are widely used.” Your argument still hits the high points, so to speak, while it acknowledges limits and the possibility for improvement in some areas.
Counterexamples should prompt you to think more deeply about what you actually want to say. For example, maybe your interest in arguing about renewables is to try to show that there are ready and workable alternatives to the usual non-renewable sources. If that is your aim, you don’t necessarily need to argue that all renewables are widely used. It is enough that some are. You might even urge that the ones that are less widely used be better developed.
Or, instead of arguing that every renewable source is or could be widely used, you might really want to be arguing that every (or most every?) place has at least some renewable source available to it, though there may be different sources in different places. This is a quite different and more subtle claim than the original, and gives your thinking some interesting room to move. (Might this argument have counterexamples too? I leave that question for you.)
Ask yourself about counterexamples when you are assessing others’ arguments as well as evaluating your own. Ask whether their conclusions might have to be revised and limited, or rethought in more subtle and complex directions. The same rules apply both to others’ arguments and to yours. The only difference is that you have a chance to correct your overgeneralizations yourself.