Chapter Thirteen: How to Think About Inductive Logic

The known is finite, the unknown infinite; intellectually we stand on an islet in the midst of an illimitable ocean of explicability. Our business in every generation is to reclaim a little more land, to add something to the extent and solidity of our possessions.

—T. H. Huxley

Our instincts certainly cause us to believe that the sun will rise tomorrow, but we may be in no better a position than the chicken which unexpectedly has its neck wrung.

—Bertrand Russell, The Problems of Philosophy

13.1 Inductive Arguments

Deductive logic, even at its best, can help us to see no more than truths that are already built into an argument’s premises. To use Huxley’s metaphor—it can help us to better understand the small island of knowledge that we find ourselves standing on, but only that island. It is by inductive logic that we can go beyond our premises, that we can expand the size of our island of knowledge by reclaiming land from the surrounding ocean of the unknown. In these chapters, we will cover the most common varieties of inductive arguments.

Most of our arguments have conclusions that go well beyond their premises. Consider these examples:

• Based on my experience of sunrises in the past, I believe that the sun will rise tomorrow—but even if I am right about my experience of the past, it is at least possible that I am wrong about what tomorrow will bring.
• Public opinion organizations interview only a thousand people, yet they draw conclusions about how millions of voters—not just the thousand interviewed—are likely to cast their votes. Even if their data are correct about the opinions of the thousand that were interviewed, they could be mistaken about the millions.
• A detective finds fingerprints, a motive, and an opportunity and concludes that the butler did it. But even if there is nothing wrong with her evidence, it is still possible that she is wrong about the butler.

These arguments are not deductive. Deductive arguments are those in which the premises are intended to make the conclusions certain. Rather, they are inductive arguments, which we define as arguments in which the premises are intended to make their conclusions merely probable. The aim for an inductive argument is not validity but inductive strength, meaning that the premises of the successful argument make the conclusion probable. And, while deductive arguments are either valid or invalid with no middle ground, inductive arguments may be judged as having various degrees of strength—ranging from no support to very weak to fairly strong to very strong.

The difference between induction and deduction has to do with standards for logical success. Put simply, the logical standards for inductive arguments are lower, but there are more of them. They are lower, since inductive arguments aim only for probability, not for certainty. But there are more of them, since in inductive arguments, like deductive arguments, the conclusion must fit the premises; but, unlike deductive arguments, the conclusion must also fit the total available evidence.

13.1.1 Inductive Arguments Aim for Probability, Not for Certainty

Suppose you get on the freeway only to realize that you’re going south when you meant to go north. You see an off-ramp ahead and wonder whether there is an on-ramp that would allow you to return to the freeway in the opposite direction. You reason as follows:

It turns out that you are right, and your trip continues successfully. But, even assuming the truth of your premises, your argument does no better than make probable your conclusion. It surely doesn’t guarantee there will be an on-ramp, for it is possible that this off-ramp is among those not covered by the “almost all” of the first premise.

Another way of saying that the premises do not guarantee the conclusion is to say that the argument is invalid—that it is possible for an argument of this form to have true premises and a false conclusion. We can show this by using the now-familiar two step procedure for producing validity counterexamples. The first step, recall, is to extract the form that the argument apparently relies on; the off-ramp argument is apparently relying on this form:

The second step is to substitute for the variables to produce clearly true premises and a clearly false conclusion. That can readily be done, as follows:

Both premises are clearly true. But the conclusion is clearly false. So the argument is shown to be invalid. It cannot provide certainty.

Validity counterexamples can be given for nearly any inductive argument, no matter how logically strong it is. But inductive arguments can no more be faulted for invalidity than can the fabled soprano Maria Callas be faulted for failure to be enshrined in the Country Music Hall of Fame. Inductive arguments aim to make their conclusions probable, and that is the standard against which they should be judged.

Recall the question posed in Chapter 10 as the best way to think about the logic of any argument: If the premises were true, would that make it reasonable to believe the conclusion? With deductive arguments, it is certainty that makes it reasonable to believe the conclusion; but with inductive arguments, it is probability. It would be nice if we could always reason with certainty. But we have to reason with what we have. The off-ramp argument would have been valid if I had used the premise All freeway off-ramps are paired with on-ramps in the opposite direction, but it would also have been unsound, since the new premise would have certainly been false.

13.1.2 Inductive Arguments Must Fit with the Total Available Evidence

The standards for logical success in inductive arguments are lower. But, at the same time, there are more of them—there are twice as many. Deductive arguments must satisfy only one condition to be logical; they must exemplify some correct form—that is, they must satisfy the correct form condition. For deductive arguments, it must be a deductive form such as affirming the antecedent or the process of elimination. Inductive arguments must also satisfy the correct form condition—that is, they must exemplify some correct inductive form. In this chapter and in the following three, we will cover the four most common forms of inductive arguments:

• Frequency arguments
• Inductive generalizations
• Arguments from analogy
• Explanatory arguments

Once you become familiar with each of these forms, you will find it fairly simple to determine whether you have an inductive or a deductive argument and, when inductive, to determine whether the correct form condition is satisfied (it usually is). As we will see, inductive arguments that make formal mistakes, like deductive arguments that make formal mistakes, typically provide no support to their conclusions.

An inductive argument, however, can satisfy the correct form condition but at the same time be logically defective. This is because inductive arguments must also satisfy a second condition, the total evidence condition. For each inductive form we will cover a customized way of stating this condition. To express it, for now, in general terms, the condition requires that the conclusion fit appropriately with the total available evidence—that is, with all of the beliefs and experiences (i.e., the total evidence) that you personally have (i.e., that are available to you). Do not confuse this condition with the requirement of true premises. The total evidence condition bears on logic and is evaluated on the assumption that the premises are true; that is, this condition is to be considered as part of the process of answering the question, “If the premises were true, would they make it reasonable for me to believe the conclusion?”

Consider again the off-ramp argument:

Suppose this argument has occurred to you when you suddenly think, “Wait a minute—I made the same mistake last week, and when I took the Firestone exit then I was stranded—no way back onto the freeway in either direction.” You still believe that each of the two premises is true, but they clearly no longer make the conclusion probable. Why not? Because there is evidence available to you—your background belief about what you experienced last week—that undermines the conclusion. Even though the argument satisfies the correct form condition, it fails the total evidence condition and is thus, from a logical point of view, extremely weak.

13.1.3 Standard Evaluating Format

The same format should be used for evaluating inductive arguments as for deductive ones. By way of review, this is the format:

As always, direct your evaluation to the reasonable objector over your shoulder. The only new wrinkle is this: under the subheading logic, in your defense of the evaluation always consider whether it satisfies the total evidence condition, and always include a clear defense of that evaluation.

13.2 Frequency Arguments

13.2.1 Correct Form for Frequency Arguments

The off-ramp argument is a frequency argument,^[1] the simplest and most intuitive form of inductive argument. Such arguments attempt to show that a specific item has a property because that sort of thing usually does have the property—or that it does not have a property because that sort of thing usually does not. Because the form (like repetition, simplification, conjunction, and disjunction, as we have seen in preceding chapters) is so simple and intuitive, it is not often explicitly offered. But it does provide a useful starting place for introducing inductive arguments. A frequency argument typically takes either a positive or negative form. The positive form is this:

This is the form of the off-ramp argument.

The negative form is this:

This would be the form of my argument if I had said, “Almost no off-ramps leave you without a way to get back onto the freeway. The Firestone Boulevard exit is an off-ramp, so it’s reasonable to think that it won’t leave me without a way to get back onto the freeway.”

The premise Almost all freeway off-ramps are paired with on-ramps in the opposite direction is a frequency statement^[2] of the sort that is used in the Chapter 10 discussion of frequency probability. These statements take the form n of F are G. The variable n stands for some frequency, or proportion; it may be stated in ordinary language—as with “almost all” in this premise—or as a decimal, a fraction, or a percentage. (We will use the lowercase letters m and n as variables standing for frequencies.) F and G, as always, are predicate letters. In inductive arguments, the predicate in the F position is usually termed the population, while the predicate in the G position is usually termed the property. A frequency statement states that with a certain frequency (n), a certain population (F) has a certain property (G). Here are a few frequency statements:

Almost all marchers for Black Lives Matter are peaceful.
One in twenty US adults now smoke e-cigarettes.
2% of the trumpet players in the top fifteen symphony orchestras are women.
Over half of the population of Europe was wiped out by the Black Death.

Note that none of the following qualifies as a frequency statement:

Many marchers for Black Lives Matter are peaceful.
Millions of US adults now smoke e-cigarettes.
Two of the trumpet players in the top fifteen symphony orchestras are women.
Some 45 million Europeans were wiped out by the Black Death.

The quantities expressed in the last four cases are not proportional; they provide no estimate of how the quantity relates to the total population.

Frequency arguments use a form that is very close to that of a familiar deductive argument—namely, the singular categorical argument of which the Socrates argument is our stock example. Universal categorical statements are frequency statements in which the frequency is either all or none, so All men are mortal does qualify as a frequency statement. But All men are mortal, Socrates is a man, so Socrates is mortal is deductive, not inductive, so we will not categorize it as a frequency argument. The frequency statement in a frequency argument, as noted in the two forms above, cannot express a proportion of 1 or 0 (i.e., of all or none), since we are defining frequency arguments as inductive, and universal statements would make them deductive.^[3]

Note also that the frequency statement in a frequency argument (shown as the first premise in each of the preceding two typical forms) cannot be exactly .5. If it were .5, it would mean that exactly half of the things that are F are also G—but that the other half are not G, making it exactly as probable as not that A is G and thus giving no reason to believe one over the other. An argument with such a premise may still be categorized as a frequency argument, but as one that fails the correct form condition—and thus, that is logically unsuccessful.

There are other ways that frequency arguments can fail the correct form condition. Charlie Hough, a steady but not dominant knuckle-balling major league pitcher, was quoted in the sports pages as saying, “They say most good managers were mediocre players. I should be a helluva manager.” His argument seems to be best clarified as follows:

Its form is this:

The F and G of premise 2 and the conclusion are reversed, creating a problem that is analogous to Chapter 11’s fallacy of singular affirming the consequent. This argument provides no support for the conclusion.

Inductive strength and deductive validity have this in common: if either sort of argument fails the correct form condition, it normally provides no support for its conclusion. But after this they part ways. For even if an inductive argument satisfies the correct form condition, how strong it is remains an open question. How that question is to be closed is determined by how well the argument satisfies the total evidence condition—to which we now turn.

13.2.2 The Total Evidence Condition (1): How High the Frequency Is

For frequency arguments, there are typically two simple components to the total evidence condition: first, the closer the frequency is to 1 or 0, the stronger the argument can be; and second, there must not be strong background evidence against the conclusion.

First, you should consider how high the frequency is. For the positive form of the argument the closer the frequency is to 1, the stronger the argument can be. This establishes a ceiling for the argument’s logical strength. The argument’s logic, for example, can never be judged as stronger than extremely weak if the frequency is a mere .51. It can ultimately be judged as weaker than that, however, since failure to satisfy the second part of the total evidence condition would offset anything gained by this first part. The negative form of frequency argument is the mirror image of the positive. The weakest arguments are just under .5, and the closer to 0 the stronger they can be.

You should be especially careful when evaluating colloquial terms on this criterion. Most, for example, is quite vague—ranging from just over half to practically all. If you are stuck with such a term, it is usually best not to assume that it moves you any more than halfway from .5 to 1—that is, that it represents anything greater than a .75 frequency. So a frequency argument with most cannot usually be judged as any better than “moderately strong.”

13.2.3 The Total Evidence Condition (2): No Strong Undermining Evidence

In addition, there must not be strong undermining evidence—that is, strong background evidence against the conclusion. You must carefully consider whether you know anything else about A that would undermine the argument. This is exactly what happened in our revised version of the off-ramp argument. You reflected on your total evidence, remembered that you had already discovered that this exit was an exception—that is, that it allowed no way of getting back onto the freeway—and so you discounted the initial argument without further ado.

There is a more careful way of stating this part of the total evidence condition for frequency arguments. It is slightly more complicated, but is worth introducing because it will in some cases be helpful. You should consider whether there are other populations to which A belongs that weaken the initial support provided by the argument. Return once again to the off-ramp argument; the Firestone exit, according to the second premise, belongs to the population of freeway off-ramps. But when you reflect on your total evidence, you realize it also belongs to the population of things that you vividly remember as not having an opposing on-ramp. When expressed in this way, it may at first look as though you are in an evaluative logjam. On the one hand, almost all freeway on-ramps do provide an opposing return to the freeway; on the other hand, almost all things that you vividly remember are true. The frequency is about the same in each of the statements, yet one of them points to the truth of the conclusion while the other points to its falsity. It is intuitively clear that the second frequency statement wins—that is, that you are right to trust your memory. Why?

A simple, general procedure provides the solution. When you recognize the undermining evidence, you should consider yet another population, namely, the population that is made up of items belonging to both of the populations in question, and ask with what frequency this new population has the property in question. In the current case, the new, combined population is freeway off-ramps (original population) that you vividly remember as not being paired with opposing on-ramps (undermining population). With what frequency are members of this population likely to still have the property of being paired with an on-ramp in the opposite direction? Almost never. The argument does not satisfy the second part of total evidence condition. The logic portion of your evaluation would look something like this:

The logic of the argument is extremely weak. The argument does satisfy the correct form condition for a frequency argument, but does not satisfy the total evidence condition. Although the frequency (“almost all”) is high enough, I remember that there is no opposing on-ramp for this off-ramp, and my memories of this sort of thing are almost always accurate.

One more example may prove useful. Suppose you are looking for a friend to join you for a skiing trip, and you want to be sure that you do not end up with a partner who will spend the whole time on the bunny slopes. You consider your Irish friend, Joseph Vaughan, but then reason as follows:

This is a frequency argument of the second, negative form. The form is correct and the frequency is extremely low, so, at first glance, you might expect to evaluate it as inductively strong. But then you must ask whether there is any undermining evidence that you know of—that is, whether Vaughan belongs to any populations in which the frequency of experienced downhill skiers is high. To make it easy, suppose you know that he is a former member of an Olympic downhill skiing team. With what frequency are members of both populations—residents of Ireland who have been Olympic downhill skiers—experienced skiers? Always. Your evaluation of the logic of the argument would look something like this:

The logic is extremely weak. The argument does satisfy the correct form condition for a frequency argument, but does not satisfy the total evidence condition. Though the frequency (under 1 percent) is low enough, Vaughan is a former Olympic skier, and no Olympic skier, even if from Ireland, is inexperienced at skiing.

13.3.4 Arguments that Appeal to Authority

Arguments that appeal to authority can often best be understood as frequency arguments. In a text called Thinking Better, by David Lewis and James Greene, we are told this:

Ulrich Neisser, one of the world’s foremost specialists in the field of mental functioning, sets the record straight when he explains: “Human cognitive activity would be more usefully conceived of as a collection of acquired skills than as the operation of a single, fixed mechanism.” In other words, “intelligence” is something we acquire from experience rather than an inborn ability.

The authors are telling us, apparently, that because an expert such as Neisser supports their conclusion, we should accept it. This can be clarified as an argument in which the claim that Neisser is an expert on the relevant subject is charitably paraphrased as a frequency statement.

It is particularly important that you be careful when you specify the population in these sorts of arguments. It is most charitable to suppose that the argument is not depending on the implicit supposition that most of what Neisser says is true, but that most of what Neisser says about mental functioning is true. It is most charitable because a premise that declared Neisser to be an expert in everything would be almost certainly false, while a premise that declared him to be an expert in this particular arena would likely be true.

What can be said about the logic of the argument? It does have correct form for a frequency argument. As for the total evidence condition, the argument is prevented from being any more than moderately strong since the frequency is expressed merely as “most.” But is there undermining evidence? Yes, because even the experts disagree sharply on this question. Some, such as Neisser, believe that intelligence is not innate; many others, however, are persuaded that it is largely innate. If this information is all you have to go on, the next thing to determine is the frequency with which any statement is true when the experts disagree about its truth. (That is, the next thing to determine is the frequency with which a statement is true when it belongs to the population that includes both the things that one set of experts says are true and the things that another set of experts says are false). This frequency is half the time. So it turns out that the argument is logically impotent. The premises, if true, provide no more reason to believe the conclusion than to disbelieve it, given the total evidence condition. (When evaluating appeals to authority that are represented as frequency arguments, it can be very useful to review the discussion of appeals to authority in Chapters 8 and 9.)

13.4 Summary of Chapter Thirteen

Inductive arguments differ from deductive ones in their standards for logical success. In both cases we ask whether true premises would make it reasonable for us to believe the conclusion. But, first, the standards for induction are lower. The aim of an inductive argument is for the premises to make the conclusion probable, while the aim of a deductive argument is for the premises to make the conclusion certain. Second, there are more standards for induction. Like deductive arguments, inductive ones must satisfy the correct form condition. But, unlike deductive arguments, they must also satisfy the total evidence condition. (The conclusion must not only fit the premises, but it must also fit the total available evidence.) There is a different version of this condition for each form of induction, and each will be introduced in the text in subsequent chapters. For now, you need only know that this condition requires that the conclusion accord with the total available evidence—that is, with all of the beliefs and experiences that you personally have.

Logical inductive arguments are not termed valid (they are in fact not valid, nor do they aspire to be so); rather, they are termed strong. And their success is a matter of degree, ranging from no support at all to very weak to fairly strong to very strong.

The simplest and most intuitive inductive form is the frequency argument. It always includes a frequency statement, which states that a specified population (say, dogs) has a certain property (say, being flea-ridden) with a certain frequency (say, 40 percent of the time). There are two parts to the total evidence condition for frequency arguments. First, the higher the frequency, the stronger the logic can be. (If the argument takes the form of denying that something has a property because the frequency in the population is so low, then the reverse holds and the lower the frequency, the stronger the argument.) And, second, you must not have any substantial undermining evidence among your background beliefs and experiences.

Appeals to authority are often aptly interpreted as frequency arguments, since they typically include an implicit premise to the effect that Most of what this expert says about this topic is true.

13.5 Guidelines for Chapter Thirteen

• When asking whether an inductive argument is logically successful, ask the same general question you ask of deductive arguments: If the premises were true, would that make it reasonable to believe the conclusion? In this case, however, expect only probability, not certainty.
• Inductive arguments must satisfy not only the correct form condition, but also the total evidence condition (that is, the conclusion must fit not only the premises, but also the total available evidence). This means that even if an inductive argument is formally correct, you can judge it to be a logical argument only if it also fits appropriately with your background beliefs and experiences.
• When evaluating inductive arguments, continue to follow the standard format for argument evaluation, but be sure to include in the logic section a discussion of the total evidence condition.
• Structure frequency arguments, when it would be loyal to do so, as follows: the first premise is a frequency statement about a property found in a population; the second premise identifies a member of the population; and the conclusion says whether this member has the property.
• In a frequency argument, the frequency establishes a ceiling for the argument’s logical strength. It cannot be higher, but it can be lower.
• When assessing the logic of a frequency argument, one way of considering whether there is undermining evidence is to ask whether A is a member of another population in which the frequency of those that do not have the property in question is high, and then to ask about the frequency of the property in the population of those who are members of both the first and the second population.
• When it is consistent with the principles of loyalty and charity, present arguments that appeal to authority as frequency arguments in which the frequency statement declares that most of what the authority says about a particular subject is true.

13.6 Glossary for Chapter Thirteen

Correct form condition—the logical requirement on any argument that it exemplify some correct form (that its conclusion fit with its premises). Correct deductive form is sufficient for deductive validity. But correct inductive form is sufficient for inductive strength if and only if it is paired with satisfaction of the total evidence condition.

Frequency argument—an inductive argument that takes one of these two forms:

This is the form of the off-ramp argument.

The negative form is this:

These are also called statistical syllogisms, probabilistic syllogisms, myriokranic (that is, thousand-headed) syllogisms, and direct singular inferences.

Frequency statement—a statement of the following form: n of F are G. The variable n stands for some frequency, or proportion, stated in ordinary language (as with almost all) or as a decimal, a fraction, or a percentage. The predicate in the F position is usually termed the population, while the predicate in the G position is usually termed the property. A frequency statement states that with a certain frequency (n), a certain population (F) has a certain property (G). Also called a simple statistical hypothesis.

Inductive argument—an argument in which the premises are intended to make the conclusion probable. Alternatively termed a probabilistic, ampliative, or nondemonstrative argument.

Inductive strength—the measure of an inductive argument’s logical success (contrast with deductive validity) based on how probable the premises make the conclusion. To be logically strong an inductive argument must satisfy both the correct form condition and the total evidence condition. There is a continuum of logical strength, ranging from no support at all to very weak to fairly strong to very strong.

Total available evidence—all of the beliefs and experiences (i.e., the total evidence) that you as the evaluator personally have (i.e., that are available to you).

Total evidence condition—the logical requirement upon any inductive argument that its conclusion fit appropriately with the total available evidence. Do not confuse this condition with the requirement of true premises. The total evidence condition bears on logic and is evaluated on the assumption that the premises are true; that is, this condition is to be evaluated as part of answering the question If the premises were true, would they make it reasonable for me to believe the conclusion?