10.2: Distinguishing Deduction from Induction

When someone says to you, "That's a good argument," you need to figure out what the person means by "good." Arguments are properly evaluated as being good in different ways, most importantly as to whether they are deductively valid, deductively sound, or inductively strong.

Definition

An argument is deductively valid if its conclusion follows with certainty from the premises

Definition

An argument is deductively sound if its conclusion follows with certainty from the premises and all the premises are actually true.

Definition

An argument is inductively strong if its conclusion follows with high probability from the premises.

An argument can also be evaluated as to whether it is understandable for the audience intended to receive it, or whether it addresses the issue under discussion, and so on. However, this section will focus only on validity, soundness, and inductive strength.

By definition, a deductive argument is an argument presented with the intention of being valid or sound. By definition, an inductive argument is one intended to be inductively strong. If the arguer's intentions aren't clear, then it's indeterminate whether the argument is deductive or inductive. It will be one or the other, though—there is no other kind. Here is an example that one speaker might give as a deductive argument but that another might give as an inductive argument:

If she's Brazilian, then she speaks Portuguese. She does speak Portuguese.
So, she is Brazilian.

This would be a deductive argument if its author intended for it to be deductively valid (which it isn't). The argument would be inductive if its author intended that speaking Portuguese be a "sign" or "positive evidence" making it probable that the person is Brazilian, in which case it would have some inductive strength but not a great deal. If you cannot guess the intentions of the arguer, you cannot tell whether you have been given an inductive argument or a deductive one, and you should assess the argument both ways. Then apply the principle of charity for arguments and suppose that the arguer intended the argument to be interpreted in the way in which it gets the best assessment.

Although inductive strength is a matter of degree, deductive validity and deductive soundness are not. In this sense, deductive reasoning is much more cut and dried than inductive reasoning. Nevertheless, inductive strength is not a matter of personal preference; it is a matter of whether the premise ought to promote a higher degree of belief in the conclusion.

(Deductively) valid does not mean "legitimate." When somebody is said to have made a valid criticism of somebody's argument, the word valid is not being used in our technical sense. Using our book’s technical terminology, valid inductive argument and sound inductive argument are not meaningful phrases.

Declarative sentences are nearly always true or false. At least the ordinary declarative sentences are. For example, "It's noon" is a declarative sentence. "What time is it?" is not. A question is neither true nor false. A command is also neither true nor false.

It does not make sense to say an argument is true. Arguments can be good or poor, valid or invalid, sound or unsound, strong or weak, but never true or false.

Outside the classroom, people are not so careful with their use of these terms, so you have to be alert to what they mean rather than only to what they say. From the barest clues, the detective Sherlock Holmes cleverly "deduced" who murdered whom, but actually he made an educated guess. Strictly speaking, he produced an inductive argument, not a deductive one. Charles Darwin, who discovered the process of evolution, is famous for his "deduction" that circular atolls in the oceans are actually coral growths on the top of barely submerged volcanoes, but he really performed an induction, not a deduction.

Aerial view of a coral atoll in the ocean

Exercise \(\PageIndex{1}\)

Assess the quality of the following argument:

Most all wolves are not white. King is a wolf in the San Diego Zoo that we are going to visit tomorrow. So, he is not going to be white.

1. inductively very weak
2. inductively strong
3. deductively valid

Answer

Answer (b). If you had to bet on whether that next wolf is going to be white or not white, the best bet would be that it would not be white. Thus, the conclusion is made probable by the premises, which is a telltale sign of an inductive argument with some strength. Answer (c) would be correct if the word “Most” were “All.”

Are you clear about the difference between being valid and being sound?

Exercise \(\PageIndex{2}\)

Is this argument deductively valid and sound?

The current president of Russia is an Asian, and all Asians are dope addicts, so the president of Russia is a dope addict.

Answer

Yes, this is a valid argument. One of the premises is false, so it is definitely not a sound argument.

At the heart of the notion of deductive validity is the notion that, if an argument's conclusion follows with certainty from its premises, you would be violating the cardinal rule to avoid inconsistency if you were to assert the premises while denying the conclusion. So, for reasons of logical consistency, the following principle holds:

Exercise \(\PageIndex{3}\)

If it can be done, give an example of a valid argument with a false conclusion. If it cannot be done, say why.

Answer

Here is an example: George Washington is from Bangladesh, so he is from Asia since anyone from Bangladesh is also from Asia

Starting from true premises, valid reasoning will never lead you to a false conclusion. The trouble is that invalid reasoning can lead you to a true conclusion and thereby trick you into thinking that the reasoning is really valid. For instance, let’s suppose you know my pet is a dog and not a lion. Therefore, you might accept this invalid argument:

A lion is a cat.
My pet is not a lion.
So, my pet is not a cat.

This argument may look good at first, but it is a terrible one. What the arguer has done here is preach to the already converted—you already believe the conclusion. We have a psychological tendency to cheer for any argument that concludes with what we already agree with, but we lower our standards of logical reasoning when we do. It's an unfortunate tendency that we all need to watch out for.

As we’ve seen in previous chapters and as we will explore in greater depth in later chapters, deciding whether a conclusion follows with probability from the premises is a matter of high art, deep science, and common sense. However, with deductive arguments, the following-from is cut and dried. Consequently, if an argument is deductively valid and there is something wrong with the conclusion, we can be sure there is something wrong with one of the premises, even if you can’t figure out which premise. For example, the following argument seems to be deductively valid:

If something goes away, there has to be a place where it goes. In the morning, night is gone. So it must have gone somewhere.

Do you agree that this is a deductively valid argument? The conclusion is obviously false, isn't it? It’s valid. But if a valid argument has a false conclusion, then it can’t have all its premises be true. In valid reasoning, true premises will always take you to a true conclusion. Would you agree that therefore the argument above must have a faulty premise somewhere? The first premise looks OK, because when a dog goes away there is a place where it goes, even if you can't find it. The dog has to go somewhere, right?

The same if a cat goes away. Yes, for cats and dogs, but still the first premise is not OK. In the morning the night just ceases to be; it doesn't go anywhere. So, the whole trouble begins in the argument's first sentence.

You might think this explanation rather obvious, but it hasn't always been so obvious to people. Many ancient Greeks would have accepted the faulty first premise as common sense. Times have changed. A related, enduring philosophical debate concerns whether there is a place where you go when you die. Well, we cannot settle this deep question here, but it is interesting to think about.

Sound deductive arguments are also called proofs, but so are strong inductive arguments, although in a different sense of the word ‘proof’. All mathematical proofs are deductively sound. Some scientific proofs are, too, but most are merely inductively very strong. When science proves that dinosaurs are extinct, the evidence for this conclusion does not imply the conclusion with certainty, only with very high probability. The scientific proof that Jupiter revolves around the sun doesn't meet the high standard of the mathematical proof that 7 + 5 = 12. Consequently, we can be surer that 7 + 5 = 12 than that dinosaurs are extinct or that Jupiter revolves around the sun. Nevertheless, the revolution is almost certain because the evidence is overwhelming. Ditto that dinosaurs lived millions of years ago and now are extinct. In the later chapters of this book we will examine in greater detail what it takes to produce a scientific proof.

Mathematical proofs that involve what mathematicians call "induction" are actually deductive and not inductive, by our standards. Similarly, in the field of mathematics called probability theory or statistics, the conclusions of mathematical proofs will be probability statements, but these proofs are deductively sound arguments whose conclusions about probability follow with certainty, not with probability. They are about probability, but they are deductive, not inductive. So, the terminology in these areas requires some attention.

When people say to you, "That may be a strong argument, but it's not really a proof," they are probably using proof in the sense of our "deductive proof” or "deductively sound argument." If this is what they are saying, they are making the good point that just because an argument is inductively strong, it need not be deductively sound, although they may not be familiar with this technical terminology that we use to explain what they mean.

It is worth noting that some dictionaries and texts define "deduction" as reasoning from the general to specific and define "induction" as reasoning from the specific to the general. However, there are many inductive arguments that do not have that form, e.g., "I saw her kiss him, so I'm sure she's having an affair."

So far, we have been concentrating on arguments in which a conclusion has followed either from a single statement or from a small group of statements. However, the logical reasoner also has the ability to decide whether a conclusion follows from everything said in a book or in a long article. For example, does it follow from Herman Melville's most famous book Moby Dick that the author believes each person has an inner nature that cannot be changed? Answering this question is much more difficult, and it won't be answered here. We will leave that for your English literature class. An analogous question might be to ask whether, from Albert Einstein's 1916 physics paper introducing the general theory of relativity, it follows that there can be places where gravity is so strong that light could not escape even if there’s nothing else blocking the light’s path.