This way of distinguishing deduction and induction shows why that distinction is important. Since deductive arguments are intended to be valid, it is fair to criticize them for being invalid. In contrast, the fact that an inductive argument is invalid is no criticism at all, because it is not intended to be valid. To criticize an inductive argument for being invalid is just as inappropriate as criticizing a rugby ball for failing as a football (or soccer ball), when the rugby ball was never intended for use in that other game.
Although this notion of deduction is common among philosophers and logicians, others conceive of deduction very differently. Some people say that induction rises from particulars to generalizations. This characterization is inaccurate, because some inductive arguments run in the reverse direction, as we will see.
Another potential source of confusion is Sir Arthur Conan Doyle, who described his fictional detective, Sherlock Holmes, as a master of the science of deduction because Holmes could draw conclusions from minor observations that others overlooked. In one story, Holmes glimpses a man on the street and immediately pegs him as “an old soldier . . . served in India . . . Royal Artillery.” How could he tell so much so quickly? “Surely, answered Holmes, “it is not hard to say that a man with that bearing, expression of authority, and sun-baked skin, is a soldier, is more than a private, and is not long from India . . . . He had not the cavalry stride, yet he wore his hat on one side, as is shown by the lighter skin on that side of his brow. His weight is against his being a sapper [a soldier who works on fortifications]. He is in the artillery.”1 These inferences are amazing, but are they deductive? Well, the arguments are clearly not valid, because it is possible that the man is an actor playing the part of an old artilleryman in India. Since their invalidity is so obvious, it is unlikely that anyone as smart as Holmes would have intended them to be valid. So these arguments are not deductive by our definition. That does not mean that the arguments are no good. Their brilliance is the point of the incident in the story. Still, instead of being a master of deduction, Holmes is a master of induction—in the philosophical sense of these terms.
WHAT’S SO GREAT ABOUT DEDUCTION?
Why did Conan Doyle misleadingly describe Sherlock Holmes as a master of deduction instead of induction? Perhaps to heap the highest possible praise on Holmes’s reasoning. Many people assume that deduction is somehow better than induction. The comparisons among arguments (I)–(V) should already make us skeptical of this assumption, but it is worth asking why so many people believe it.
One reason for preferring deduction might be that it seems to achieve certainty by ruling out all possibilities. A valid argument excludes any possibility of a false conclusion when its premises are true. Another apparent advantage of deduction is that validity is indefeasible in the sense that if an argument is valid, then adding an extra premise can never make it invalid. (Just try it with argument (II).) Addition cannot invalidate validity.
These features of deduction seem desirable if you want certainty. Unfortunately, you can’t always get what you want, according to philosophers Mick Jagger and Keith Richards. The appearance of certainty in deductive arguments is an illusion. The conclusion of a valid argument is guaranteed only if its premises are true. If its premises are not true, then a valid argument shows nothing. Hence, when we cannot be certain of its premises, a deductively valid argument cannot create certainty about its conclusion.
An argument’s validity does rule out the option of believing the premises and denying the conclusion, but you still have several alternatives: You can either accept the conclusion or deny a premise. In argument (II) above, you can deny the conclusion that Noel speaks Portuguese as long as you give up either the premise that Noel is Brazilian or the other premise that all Brazilians speak Portuguese. The argument cannot tell you whether its own premises are true, so it cannot force you to accept its conclusion as long as you are willing to give up one of its premises.
This point is ossified in the adage: “One person’s modus ponens is another person’s modus tollens.” Recall that modus ponens is the argument form “If x, then y; x; so y,” whereas modus tollens is the argument form “If x, then y; not y; so not x.” In modus ponens, the antecedent x is accepted, so the consequent y is also accepted. But in modus tollens, the consequent y is rejected, so the antecedent x is also rejected. The conditional “If x, then y” cannot tell us whether to accept its antecedent x and then apply modus ponens or instead to deny its consequent y and then apply modus tollens. Similarly, a valid argument cannot tell us whether to accept its premises and then accept its conclusion or instead to reject its conclusion and then also reject one of its premises. As a result, the valid argument by itself cannot tell us whether to believe its conclusion.
We cannot easily give up either premise if both premises are certain or justified. However, all that shows is that the real force of a valid argument comes not from its validity but from the justifications for its premises. If my only reason to believe that all Brazilians speak Portuguese is that all Brazilians whom I know speak Portuguese, then it is hard to see why valid argument (II) is any better than invalid argument (III). The only real difference is that the uncertainty in argument (II) is about its first premise, whereas the uncertainty in argument (III) is about the relation of its premises to its conclusion. Neither form of argument avoids uncertainty. They simply locate that uncertainty in different places.
For these reasons, we need to give up our quest for certainty.2 One way to curtail this impossible dream is to turn from deductive arguments to inductive arguments. Inductive arguments are not intended to be valid or certain. They do not try or pretend to rule out every contrary possibility. They admit to being defeasible in the sense that further information or premises can turn a strong inductive argument into a weak one. All of this might seem disappointing, but it is actually invigorating. The realization that more information could make a difference motivates further inquiry. A recognition of uncertainty also brings humility and openness to contrary evidence and competing positions. These are advantages of inductive arguments.
HOW STRONG ARE YOU?
Since inductive arguments by definition do not aim at validity, what do they aim at? The answer is strength. An inductive argument is better if its premises provide stronger reasons for its conclusion. Satisfied? I hope not. You should be asking, “But what is strength? It is a relation between premises and conclusion, but how can we tell when one reason or argument is stronger than another? And what makes it stronger?”
No answer has achieved consensus. The notion of inductive strength is still highly controversial, but one natural way to think about strength is as probability. On this view, the strength of an inductive argument is (or depends on) the conditional probability of its conclusion, given its premises. An inductive argument is stronger when the probability of its conclusion, given its premises, is higher.
To understand this standard of strength, we need to learn a little about conditional probability. Imagine an area of India where it rains one out of five days in general, but it rains four out of five days during monsoon season. What is the probability that it will rain there on Gandhi’s birthday? That depends on the date of Gandhi’s birthday. If you have no idea when Gandhi’s birthday is, it is reasonable to estimate this probability as one out of five or 0.20. But suppose you discover that Gandhi’s birthday is during the monsoon season in this area of India. With that extra information, it now becomes reasonable to estimate the probability of rain on Gandhi’s birthday as four out of five or 0.80. This new figure is the conditional probability of rain on Gandhi’s birthday in this area, given that his birthday is during the monsoon season in that area.
The application to inductive arguments is straightforward. Consider this argument:
Our parade will occur on Gandhi’s birthday in that area.
Therefore, it will rain on our parade.
This argument is neither valid nor deductive, so it makes sense to evaluate it by t
he inductive standard of strength. The premise by itself gives no information about when Gandhi’s birthday is, so the conditional probability of the conclusion, given the premise, is 0.20. That argument is not very strong, since it is more likely than not that it won’t rain there then, given only the information in the premises. But now let’s add a new premise:
Our parade will occur on Gandhi’s birthday in that area.
Gandhi’s birthday is during monsoon season in that area.
Therefore, it will rain on our parade.
The argument is still not valid, but it is stronger, because the conditional probability of the conclusion, given the premise, has risen to 0.80. The extra information in the new premise increases the probability. All of this is common sense. If you do not know when Gandhi’s birthday is, the first argument is not a strong reason to reschedule the parade. But when someone adds, “That’s during monsoon season!” then it makes sense to reschedule the parade, unless you like walking in the rain.3
HOW DO I INDUCE THEE? LET ME COUNT THE WAYS
What is in the grab bag of inductive arguments? Let’s reach deep into the bag and see what comes out.
Imagine that you want to open a restaurant, and you have chosen a location in Edinburgh, but you have not yet decided whether to serve Ethiopian food or Turkish food, your chef’s two specialties. The success of the restaurant depends on how many people in the neighborhood like each kind of food. To answer this crucial question, you ask random people in the neighborhood and discover that 60% like Turkish food but only 30% like Ethiopian food. You conclude that these same percentages hold throughout the whole neighborhood. This inference is a statistical generalization that argues from premises about the small sample that you tested to a conclusion about a larger group. Such generalizations are inductive arguments because they are not intended to be valid. The tested sample clearly might not match the whole neighborhood.
Next you need to test items for your menu. You decide to try them out on friends and neighbors, but you do not want to test Turkish food on people who do not like it, since they won’t come to your restaurant anyway. You wonder whether your neighbor to the south of your restaurant likes Turkish food. You don’t know anything special about him, so you conclude that he probably has a 60% chance of liking Turkish food. This argument can be called a statistical application, because it applies a generalization about the whole population to an individual. It is inductive, because it is clearly not valid. It could underestimate the probability if, for example, your neighbor happens to be Turkish.
Finally, your restaurant opens, but nobody shows up. Why not? The explanation cannot be that people in the neighborhood do not like Turkish food, since 60% do. The explanation cannot be that your prices are too high or that your dishes taste bad, because potential customers do not know your prices or quality yet. The explanation cannot be lack of advertising, because you have big banners, a fancy website, and advertisements in local papers. Then you hear that someone has been spreading rumors that your restaurant is filled with cockroaches. Who? Nobody else would have a motive, so you suspect the owner of the older restaurant across the street. This conclusion is supported by an inference to the best explanation. It is also an inductive argument, because its premises give some reason to believe your conclusion, but your suspicions could still be wrong.
Although discouraged, you regain hope when you remember the story of another Turkish restaurant that had a rough first month but then later became extremely popular as soon as people tried it. That other restaurant is a lot like yours, so you conclude that your restaurant will also probably take off soon. This argument from analogy is inductive, because it is clearly not valid but does give some reason for hope.
Luckily, your restaurant turns into a huge success. Customers pile in. What attracts them to your restaurant? To find out, you lower your prices a little, but that has no effect on turnout. Then you check your records to see which dishes customers order more often, but nothing sticks out. Your curiosity is piqued, so you drop items off your menu one by one and observe changes in the clientele. There is a big drop in customers when you take kokoreç off the menu. Kokoreç consists of lamb or goat intestines wrapped around seasoned hearts, lungs, and kidneys. You had no idea that local people like offal so much, but your experiment supports the conclusion that this dish is what causes people to come to your restaurant. This causal reasoning is inductive, because it is possible that something else is the cause, so the argument is not valid, but it still gives you some reason to believe its conclusion. Accordingly, you put kokoreç back on your menu.
All goes well until your restaurant is robbed. The only witness reports that the robber drive off in a Fiat. Only a small percentage (2%) of cars in Edinburgh are Fiats, so the witness’s report is surprising, and you wonder whether to trust it. You and the police estimate that this witness in these lighting conditions will identify a Fiat correctly around 90% of the time and will misidentify another kind of car as a Fiat around 10% of the time. That sounds pretty good, but then (using Bayes’s theorem) you calculate that the probability of this report being accurate is less than one in six.4 It is five times more likely that the witness misidentified another car as a Fiat. This argument exemplifies reasoning about probability.
This story could go on, but it already includes six kinds of inductive arguments: statistical generalization, statistical application, inference to the best explanation, argument from analogy, causal reasoning, and probability. Each of these forms of argument is common in many areas of everyday life. Each has its own standards and can be performed well or poorly. Each has special fallacies associated only with it. Instead of surveying them all, I will focus on a few of the most important kinds of inductive argument.5
HOW CAN DATES AND POLLS GO SO WRONG?
Profiling and stereotypes are anathema to many people. Police are supposed to choose whom to stop or arrest by observing what those people do instead of what they look like or where they are. In everyday life, many people aspire to Martin Luther King’s vision: “I have a dream that my four little children will one day live in a nation where they will not be judged by the color of their skin, but by the content of their character.”6 We all hope to be treated as individuals rather than as members of groups.
Despite these hopes and dreams, all of us often use stereotypes about groups to predict how other individuals will act. Marketing experts use generalizations about groups to predict which customers will buy their products, as with our Turkish restaurant. Doctors use risk factors—which include group membership—to recommend medications and operations. Insurance agents charge individual clients on the basis of whether they belong to groups that cost insurers expensive payments. Universities decide which applicants to admit on the basis of their grades. We hope that these professionals will not judge customers, patients, clients, or applicants by the color of their skin, but they also do not base their decisions on the content of their character. They can’t, because they do not know the content of their character.
In many contexts, it is hard to see how we could do without stereotypes. If I do not know someone at all but I need to make a fast decision, then the only information I can use is what I can observe quickly. For example, if a stranger in a public bar talks casually with me for a few minutes and then offers to buy me a drink or dinner, then I need to decide whether to trust this stranger. What is he up to? As we saw, Sherlock Holmes might be able to induce a great deal about this stranger, but most of us have no choice but to rely on a few inaccurate generalizations based on our limited experience. We all do it, whether or not we accept the stranger’s offer.
These cases depend on arguments up and down. First, they generalize up from premises about a sample of a group to a conclusion about the group as a whole. Second, they apply the resulting generalization back down to a conclusion about the individual. These two stages can be described as generalization and application.
Generalization
Each of these form
s of argument introduces numerous complexities and complications. Even the most sophisticated reasoning of this sort can go badly wrong. Just recall the surprising mistakes made by political polls in the Brexit vote in the United Kingdom and also the 2016 presidential election in the United States. In those cases, even professional statisticians with tons of data were way off base. To avoid such errors and to fully understand statistical generalizations and applications, we all need to take several courses in statistics and probability, and then we need to gather big data of high quality. Who has the time? Luckily, a simple example can illustrate a few common methods and mistakes without going into technical detail.
Imagine that you are seeking a male life partner who will play golf with you, and you are curious about online dating websites. You go onto one site, randomly pick ten potential dates, and ask each of them how often he played golf in the last six months. Only one of them reports having played golf at all in the last six months. You reason that only 10% of your sample played golf in the last six months, so around 10% of people who use online dating services play golf. This argument is a statistical generalization, because it runs from a premise about a sample (the ten you asked) to a conclusion about the whole group (people who use online dating sites).
On the next day, someone else who uses the site contacts you. You decide not to reply, because you reason like this: “This person uses an online dating website, and only 10% of online dating website users play golf, so this person probably does not play golf—or, more precisely, there is only a 10% chance that this person played golf in the last six months.” This argument is a statistical application because it applies premises that include a generalization about the whole group to a conclusion about this particular user.
Think Again: How to Reason and Argue Page 14