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Learning Objectives

After reading this chapter, you should be able to:

1. Define basic key terms and concepts within deductive reasoning.

2. Use variables to represent an argument’s logical form.

3Deductive Reasoning

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3. Use the counterexample method to evaluate an argument’s validity.

4. Categorize different types of deductive arguments.

5. Analyze the various statements—and the relationships between them—in categorical arguments.

6. Evaluate categorical syllogisms using the rules of the syllogism and Venn diagrams.

7. Differentiate between sorites and enthymemes.

By now you should be familiar with how the field of logic views arguments: An argument is just a collection of sentences, one of which is the conclusion and the rest of which, the premises, provide support for the conclusion. You have also learned that not every collection of sentences is an argument. Stories, explanations, questions, and debates are not arguments, for example. The essential feature of an argument is that the premises support, prove, or give evidence for the conclusion. This relationship of support is what makes a collection of sentences an argument and is the special concern of logic. For the next four chapters, we will be taking a closer look at the ways in which premises might support a conclusion. This chapter discusses deductive reasoning, with a specific focus on categorical logic.

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3.1 Basic Concepts in Deductive Reasoning

As noted in Chapter 2, at the broadest level there are two types of arguments: deductive and inductive. The difference between these types is largely a matter of the strength of the connection between premises and conclusion. Inductive arguments are defined and discussed in Chapter 5; this chapter focuses on deductive arguments. In this section we will learn about three central concepts: validity, soundness, and deduction.

Validity

Deductive arguments aim to achieve validity, which is an extremely strong connection between the premises and the conclusion. In logic, the word valid is only applied to arguments; therefore, when the concept of validity is discussed in this text, it is solely in reference to arguments, and not to claims, points, or positions. Those expressions may have other uses in other fields, but in logic, validity is a strict notion that has to do with the strength of the connection between an argument’s premises and conclusion.

To reiterate, an argument is a collection of sentences, one of which (the conclusion) is supposed to follow from the others (the premises). A valid argument is one in which the truth of the premises absolutely guarantees the truth of the conclusion; in other words, it is an argument in which it is impossible for the premises to be true while the conclusion is false. Notice that the definition of valid does not say anything about whether the premises are actually true, just whether the conclusion could be false if the premises were true. As an example, here is a silly but valid argument:

Everything made of cheese is tasty. The moon is made of cheese. Therefore, the moon is tasty.

No one, we hope, actually thinks that the moon is made of cheese. You may or may not agree that everything made of cheese is tasty. But you can see that if everything made of cheese were tasty, and if the moon were made of cheese, then the moon would have to be tasty. The truth of that conclusion simply logically follows from the truth of the premises.

Here is another way to better understand the strictness of the concept of validity: You have probably seen some far? fetched movies or read some bizarre books at some point. Books and movies have magic, weird science fiction, hallucinations, and dream sequences—almost anything can happen. Imagine that you were writing a weird, bizarre novel, a novel as far removed from reality as possible. You certainly could write a novel in which the moon was made of cheese. You could write a novel in which everything made of cheese was tasty. But you could not write a novel in which both of these premises were true, but in which the moon turned out not to be tasty. If the moon were made of cheese but was not tasty, then there would be at least one thing that was made of cheese and was not tasty, making the first premise false.

Therefore, if we assume, even hypothetically, that the premises are true (even in strange hypothetical scenarios), it logically follows that the conclusion must be as well. Therefore, the argument is valid. So when thinking about whether an argument is valid, think about whether it would be possible to have a movie in which all the premises were true but the conclusion was false. If it is not possible, then the argument is valid.

Here is another, more realistic, example:

All whales are mammals. All mammals breathe air. Therefore, all whales breathe air.

Is it possible for the premises to be true and the conclusion false? Well, imagine that the conclusion is false. In that case there must be at least one whale that does not breathe air. Let us call that whale Fred. Is Fred a mammal? If he is, then there is at least one mammal that does not breathe air, so the second premise would be false. If he isn’t, then there is at least one whale that is not a mammal, so the first premise would be false. Again, we see that it is impossible for the conclusion to be false and still have all the premises be true. Therefore, the argument is valid.

Here is an example of an invalid argument:

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Consider the following argument: “If it is raining, then the streets are wet. The streets are wet. Therefore, it is raining.” Is this a valid argument? Could there be another reason why the road is wet?

All whales are mammals. No whales live on land. Therefore, no mammals live on land.

In this case we can tell that the truth of the conclusion is not guaranteed by the premises because the premises are actually true and the conclusion is actually false. Because a valid argument means that it is impossible for the premises to be true and the conclusion false, we can be sure that an argument in which the premises are actually true and the conclusion is actually false must be invalid. Here is a trickier example of the same principle:

All whales are mammals. Some mammals live in the water. Therefore, some whales live in the water.

This one is trickier because both premises are true, and the conclusion is true as well, so many people may be tempted to call it valid. However, what is important is not whether the premises and conclusion are actually true but whether the premises guarantee that the conclusion is true. Think about making a movie: Could you make a movie that made this argument’s premises true and the conclusion false?

Suppose you make a movie that is set in a future in which whales move back onto land. It would be weird, but not any weirder than other ideas movies have presented. If seals still lived in the water in this movie, then both premises would be true, but the conclusion would be false, because all the whales would live on land.

Because we can create a scenario in which the premises are true and the conclusion is false, it follows that the argument is invalid. So even though the conclusion isn’t actually false, it’s enough that it is possible for it to be false in some situation that would make the premises true. This mere possibility means the argument is invalid.

Soundness

Once you understand what valid means in logic, it is very easy to understand the concept of soundness. A sound argument is just a valid argument in which all the premises are true. In defining validity, we saw two examples of valid arguments; one of them was sound and the other was not. Since both examples were valid, the one with true premises was the one that was sound.

We also saw two examples of invalid arguments. Both of those are unsound simply because they are invalid. Sound arguments have to be valid and have all true premises. Notice that since only arguments can be valid, only arguments can be sound. In logic, the concept of soundness is not applied to principles, observations, or anything else. The word sound in logic is only applied to arguments.

Here is an example of a sound argument, similar to one you may recall seeing in Chapter 2:

All men are mortal. Bill Gates is a man. Therefore, Bill Gates is mortal.

There is no question about the argument’s validity. Therefore, as long as these premises are true, it follows that the conclusion must be true as well. Since the premises are, in fact, true, we can reason the conclusion is too.

It is important to note that having a true conclusion is not part of the definition of soundness. If we were required to know that the conclusion was true before deciding whether the argument is sound, then we could never use a sound argument to discover the truth of the conclusion; we would already have to know that the conclusion was true before we

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Interpreting the intention of the person

could judge it to be sound. The magic of how deductive reasoning works is that we can judge whether the reasoning is valid independent of whether we know that the premises or conclusion are actually true. If we also notice that the premises are all true, then we may infer, by the power of pure reasoning, the truth of the conclusion.

Therefore, knowledge of the truth of the premises and the ability to reason validly enable us to arrive at some new information: that the conclusion is true as well. This is the main way that logic can add to our bank of knowledge.

Although soundness is central in considering whether to accept an argument’s conclusion, we will not spend much time worrying about it in this book. This is because logic really deals with the connections between sentences rather than the truth of the sentences themselves. If someone presents you with an argument about biology, a logician can help you see whether the argument is valid—but you will need a biologist to tell you whether the premises are true. The truth of the premises themselves, therefore, is not usually a matter of logic. Because the premises can come from any field, there would be no way for logic alone to determine whether such premises are true or false. The role of logic—specifically, deductive reasoning—is to determine whether the reasoning used is valid.

Deduction

You have likely heard the term deduction used in other contexts: As Chapter 2 noted, the detective Sherlock Holmes (and others) uses deduction to refer to any process by which we infer a conclusion from pieces of evidence. In rhetoric classes and other places, you may hear deduction used to refer to the process of reasoning from general principles to a specific conclusion. These are all acceptable uses of the term in their respective contexts, but they do not reflect how the concept is defined in logic.

In logic, deduction is a technical term. Whatever other meanings the word may have in other contexts, in logic, it has only one meaning: A deductive argument is one that is presented as being valid. In other words, a deductive argument is one that is trying to be valid. If an argument is presented as though it is supposed to be valid, then we may infer it is deductive. If an argument is deductive, then the argument can be evaluated in part on whether it is, in fact, valid. A deductive argument that is not found to be valid has failed in its purpose of demonstrating its conclusion to be true.

In Chapters 5 and 6, we will look at arguments that are not trying to be valid. Those are inductive arguments. As noted in Chapter 2, inductive arguments simply attempt to establish their conclusion as probable—not as absolutely guaranteed. Thus, it is not important to assess whether inductive arguments are valid, since validity is not the goal. However, if a deductive argument is not valid, then it has failed in its goal; therefore, for deductive reasoning, validity is a primary concern.

Consider someone arguing as follows:

All donuts have added sugar. All donuts are bad for you. Therefore, everything with added sugar is bad for you.

Even though the argument is invalid—exactly why this is so will be clearer in the next section—it seems clear that the person thinks it is valid. She is not merely suggesting that maybe things with added sugar might be bad for you. Rather, she is presenting the reasoning as though the premises guarantee the truth of the conclusion. Therefore, it appears to be an attempt at deductive reasoning, even though this one happens to be invalid.

Because our definition of validity depends on understanding the author’s intention, this means that deciding whether something is a deductive argument requires a bit of interpretation—we have to figure out what the person giving the argument is trying to do. As noted briefly in Chapter 2, we ought to seek to provide the most favorable possible interpretation of the author’s intended reasoning. Once we know that an argument is deductive, the next question in evaluating it is whether it is valid. If it is deductive but not valid, we

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making an argument is a key step in determining whether the argument is deductive.

really do not need to consider anything further; the argument fails to demonstrate the truth of its conclusion in the intended sense.

Practice Problems 3.1

Examine the following arguments. Then determine whether they are deductive arguments or not. Click here (https://ne.edgecastcdn.net/0004BA/constellation/PDFs/PHI103_2e/Answers_PracticeProblems3.1.pdf) to check your answers.

1. Charles is hard to work with, since he always interrupts others. Therefore, I do not want to work with Charles in the development committee.

2. No physical object can travel faster than light. An electron is a physical object. So an electron cannot travel faster than light.

3. The study of philosophy makes your soul more slender, healthy, and beautiful. You should study philosophy.

4. We should go to the beach today. It’s sunny. The dolphins are out, and I have a bottle of fine wine. 5. Triangle A is congruent to triangle B. Triangle A is an equilateral triangle. Therefore, triangle B is an

equilateral triangle. 6. The farmers in Poland have produced more than 500 bushels of wheat a year on average for the past 10

years. This year they will produce more than 500 bushels of wheat. 7. No dogs are fish. Some guppies are fish. Therefore, some guppies are not dogs. 8. Paying people to mow your lawn is not a good policy. When people mow their own lawns, they create

self?discipline. In addition, they are able to save a lot of money over time. 9. If Mount Roosevelt was completed in 1940, then it’s only 73 years old. Mount Roosevelt is not 73 years

old. Therefore, Mount Roosevelt was not completed in 1940. 10. You’re either with me, or you’re against me. You’re not with me. Therefore, you’re against me. 11. The worldwide use of oil is projected to increase by 33% over the next 5 years. However, reserves of oil

are dwindling at a rapid rate. That means that the price of oil will drastically increase over the next 5 years.

12. A nation is only as great as its people. The people are reliant on their leaders. Leaders create the laws in which all people can flourish. If those laws are not created well, the people will suffer. This is why the people of the United States are currently suffering.

13. If we save up money for a house, then we will have a place to stay with our children. However, we haven’t saved up any money for a house. Therefore, we won’t have a place to stay with our children.