CHAPTER 10

INDUCTIVE REASONING

Consider these two arguments:

These arguments are of course not valid. Lee might be among the 7% of Chinese who can digest lactose.1 Snow might fall in Jakarta this winter due to unusual changes in global weather. But despite the fact that the arguments are invalid, their conclusions are more likely to be true than false given the information in the premises. If the premises are indeed true, it would be rational for us to be highly confident of the conclusion, even if we are not completely certain of their truth. In other words, it is possible for the premises of an invalid argument to provide strong support for its conclusion. Such arguments are known as inductively strong arguments. We might define an inductively strong argument as one that satisfies two conditions:

Let us elaborate on this definition a bit more:

• Recall that a valid argument can have false premises. The same applies to an inductively strong argument. The two arguments given earlier remain inductively strong, even if Lee is not Chinese, or it turns out that it snowed in Jakarta last year.
• When we say the conclusion is highly likely to be true given that the premises are true, it does not mean “it is highly likely for the conclusion and the premises to be true.” Consider this argument:

It is surely plausible that at this very moment, there are people eating bread and there are people eating rice somewhere in the world. This makes it highly likely for the premise and the conclusion to be true. But the argument is not inductively strong because the fact that someone is eating bread gives us no reason to believe that someone is eating rice. There is no evidential connection between them, which is what is required when the conclusion is highly likely to be true given that the premise is true. What we should do is imagine a situation in which the premises are true, and then ask ourselves how likely it is that the conclusion is true in the same situation.

10.1 INDUCTIVE STRENGTH

Although inductively strong arguments are invalid, they are indispensable for science and everyday life. We often have to make predictions about the future based on past experience. Our past experience can never logically guarantee that our predictions are correct, but they can tell us what is more likely to happen. Our lives would be completely paralyzed if we did not plan our actions on the basis of probability As Bishop Butler (1692–1752) said in a famous quote, “Probability is the very guide of life.”

When we describe an argument as inductively strong, we are saying that although the premises of the argument do not logically entail the conclusion, the premises nonetheless provide strong support for the conclusion. The inductive strength of the argument is a measure of the degree of support that is provided. Unlike validity, inductive strength is not an all-or-nothing matter. An argument is either valid or not valid, and there is no such thing as a partially valid argument. In contrast, the inductive strength of an argument is a matter of degree, as can be seen in this example:

Intuitively, whether the premise supports the conclusion depends crucially on the value of the variable x. If x is 100%, the argument is obviously deductively valid. If x is 99.999%, then the argument is invalid but inductively very strong. If x is 70%, the argument is still strong but less so. If x is 10%, then the premises are too weak to support the conclusion. We might represent inductive strength as a gradient, with deductive validity being the limiting case:

We can give a mathematical definition of inductive strength in terms of the conditional probability of the conclusion given the premises. Inductive strength will then vary from 0 to an upper limit of 1, which corresponds to deductive validity. Suppose we have an argument with premises P1, P2…, Pn and conclusion C. The inductive strength of the argument is then the conditional probability of the conclusion given the conjunction of all the premises, or in mathematical notation:

As an illustration, consider this argument:

Since Shannon has bought only one ticket, she will lose the lottery as long as any of the remaining 999 tickets is chosen. The conditional probability of the conclusion given all the facts about the lottery is therefore 0.999, which is very high, and so this is an inductively strong argument. On the other hand, if we change the conclusion of the argument to “Shannon will win the lottery,” the inductive strength of the argument will be a rather low 0.001.

Of course, since it is often difficult to be precise about probability, the exact inductive strength of many arguments will be difficult if not impossible to ascertain. Suppose we offer our friend a birthday present, but when she opens it she frowns. This is a good reason to think that she does not like the present. In such typical situations, there is no need to calculate the numerical inductive strength of the inference, and it is not even clear whether it can be done. Nevertheless, our conclusion is justified because we are able to make an approximate but accurate qualitative judgment about the likelihood of the conclusion given the evidence. We are still using inductive reasoning and applying the same principles.

10.2 DEFEASIBILITY OF INDUCTIVE REASONING

Inductive strength is a matter of degree; validity is not. Another difference is that inductive strength is defeasible, but not validity. Adding new premises to a valid argument will not make it invalid. If all Chinese have lactose intolerance and Lee is Chinese, then it follows that Lee has lactose intolerance. Our conclusion will not change by additional information such as Lee is a chain-smoking philosopher with peculiar sleeping habits. However, new premises can increase or decrease the inductive strength of an argument. Consider this argument:

This argument as it stands is inductively strong, since it is rather unlikely for someone to survive such a fall. But suppose we discover some new information:

Now the argument becomes weaker than before because it is less clear that Regina must die from the fall. After all, there are cases where people managed to survive after falling from tall buildings. But wait, there is more to come:

Now the situation is again different and the argument is stronger, perhaps even stronger than in the beginning when we are told only that Regina has fallen. As we can see, the inductively strength of an argument can change quite radically depending on new information. This illustrates a major difference between mathematics and the empirical sciences. Mathematics uses deductive reasoning to discover the logical consequences of definitions and axioms. Ideally, a good mathematical proof has to be a sound (and hence valid) argument. So if the proof is done correctly, new discoveries cannot change the proof into an invalid argument. However, science also relies on defeasible inductive reasoning. For example, noting that all penguins observed so far cannot sustain flight, we conclude that no penguin can fly. But this conclusion might turn out to be wrong if we discover a new species of flying penguins tomorrow. Old evidence providing strong support for a theory might fail to do so when new evidence comes in.

10.3 CASES OF INDUCTIVE REASONING

There are different types of inductive reasoning. Here are some main ones:

• Induction based on statistics: We rely on statistics to make generalizations about groups of things, and to make predictions about particular cases. For example, we might have seen lots of spiders, and they all produce silk, and so we conclude this is true of all spiders, including those which have not been observed. (See Chapter 17 for more about statistics.)
• Induction based on analogy: These are arguments where two objects A and B are very similar, and so we conclude that something that is true of A ought to be true of B as well. Supppose a chemical is discovered to be toxic to mice. By analogy we suspect it will be harmful to human beings as well, given the biological similarities between the two. This is again a form of induction since the conclusion does not logically follow. (See Chapter 21.)
• Induction based on inference to the best explanation: Very often we do not have enough evidence to prove that something must be true. Sometimes the evidence can also be conflicting and point to different conclusions. What we can do is to consider the alternative theories available and pick the one that on balance has the most evidence supporting it, all things considered. For example, when we leave our home we might notice that the street is wet. This might be because it has just rained, but it is also possible that somebody has just washed the street. But you notice that some cars passing by are also wet,, so you conclude it is most likely that it rained.

10.4 DEDUCTIVE AND INDUCTIVE ARGUMENTS?

This paragraph is a more technical discussion about the proper use of terminology. You can skip it if you want. In this book, we treat deductive validity and inductive strength as two standards with which to evaluate arguments. However, some critical thinking textbooks use the distinction to classify all arguments as either deductive arguments and inductive arguments. This approach is however problematic because it is not clear how invalid arguments are to be classified. Some authors think that if an invalid argument is intended to be valid but in fact is not then it is a (bad) deductive argument, and when an invalid argument is intended to be inductively strong but fails to be so then it is a (bad) inductive argument. The problem with this view is that very often people give arguments without thinking whether the arguments are supposed to be valid or strong. Of course, we might say that a deductive argument is simply a valid argument, and an inductive argument is an inductively strong argument. But then it is no longer the case that every argument is either deductive or inductive. (But as we shall see later on, every good argument is either deductively valid or inductively strong.)

EXERCISES

10.1 In the novel A Study in Scarlet, detective Sherlock Holmes explained how he came to the conclusion that a certain doctor came from Afghanistan:

Here is a gentleman of the medical type, but with the air of a military man. Clearly an army doctor, then. He has just come from the tropics, for his face is dark, and that is not the natural tint of his skin, for his wrists are fair. He has undergone hardship and sickness, as his haggard face says clearly. His left arm has been injured: He holds it in a stiff and unnatural manner. Where in the tropics could an English army doctor have seen much hardship and got his arm wounded? Clearly in Afghanistan.

Do you think Holmes uses inductive or deductive reasoning here?

10.2 Are the following statements true or false?

a) If you have two sound arguments, and you take the premises of one argument and add to the second argument without changing the conclusion, would the new argument be (a) valid or (b) sound?

b) If you have two inductively strong arguments, and you take the premises of one argument and add to the second argument without changing the conclusion, would the new argument still be inductively strong?

10.3 For each argument below, determine whether it is valid. If it is not, see if you can come up with additional information that would weaken the inductive strength of the argument.

a) There is not much snow this year, and the ski resorts have never done very well whenever there was little snow, so the ski resorts will not do well this year.

b) The ski resorts must have at least 10 feet of snow before they are allowed to open, and this winter the resorts all have less than 10 feet of snow, and so they will not be allowed to open.

c) In the past, the ski resorts have not done very well whenever there is little snow, but since there is so much snow this year, the ski resorts will do very well indeed.

d) The ski resorts are doing great this year, although they were not doing very well five years ago. So it is not the case that the ski resorts have always done well.

10.4 Consider this argument:

Now consider the following statements and think about how they might affect the inductive strength of the argument when added individually as a premise: (1) They still love each other deeply. (2) They sometimes have violent physical fights. (3) Britney has just made an appointment with a divorce lawyer.

1 A person with lactose intolerance lacks the enzyme lactase, which is needed for proper metabolization of lactose, a sugar present in milk and other dairy products.