Part I. Introduction to Logic and Proofs

Chapter 1



What is Logic?

. . . it is undesirable to believe a proposition when there is no ground whatsoever for supposing it is true.

Bertram Russell (1872–1970), British philosopher

On the Value of Scepticism



For our purposes, logic is the business of deciding whether or not a deduction is valid; that is,

deciding whether or not a particular conclusion is a consequence of particular assumptions (or

“hypotheses”). Here is one possible deduction:

Hypotheses:

(1) It is raining heavily.

(2) If you do not take an umbrella, you will get soaked.

Conclusion: You should take an umbrella.

(The validity of this particular deduction will be analyzed in section 1B.)

This chapter discusses some basic logical notions that apply to deductions in English (or

any other human language, such as French). Later, we will translate deductions from English

into mathematical notation.

1A. Assertions and deductions

In logic, we are only interested in sentences that can figure as a hypothesis or conclusion of a

deduction. These are called assertions: an assertion is a sentence that is either true or false.

(If you look at other textbooks, you may find that some authors call these propositions or

statements or sentences, instead of assertions.)

You should not confuse the idea of an assertion that can be true or false with the difference

between fact and opinion. Often, assertions in logic will express things that would count as

facts—such as “Pierre Trudeau was born in Quebec” or “Pierre Trudeau liked almonds.” They

can also express things that you might think of as matters of opinion—such as, “Almonds are

yummy.”

EXAMPLE 1.1.

• Questions The sentence “Are you sleepy yet?”, is not an assertion. Although you

might be sleepy or you might be alert, the question itself is neither true nor false. For

this reason, questions will not count as assertions in logic. Suppose you answer the

question: “I am not sleepy.” This is either true or false, and so it is an assertion in the

logical sense. Generally, questions will not count as assertions, but answers will. For

example, “What is this course about?” is not an assertion, but “No one knows what

this course is about” is an assertion.

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1. What is Logic?

• Imperatives Commands are often phrased as imperatives like “Wake up!,” “Sit up

straight,” and so on. Although it might be good for you to sit up straight or it might

not, the command is neither true nor false. Note, however, that commands are not

always phrased as imperatives. “If you sit up straight, then you will get a cookie” is

either true or false, and so it counts as an assertion in the logical sense.

• Exclamations “Ouch!” is sometimes called an exclamatory sentence, but it is neither

true nor false. We will treat “Ouch, I hurt my toe!” as meaning the same thing as “I

hurt my toe.” The “ouch” does not add anything that could be true or false.



Throughout this text, you will find practice problems that review and explore the material

that has just been covered. There is no substitute for actually working through some problems,

because mathematics is more about a way of thinking than it is about memorizing facts.

EXERCISES 1.2. Which of the following are “assertions” in the logical sense?

1) England is smaller than China.

2) Greenland is south of Jerusalem.

3) Is New Jersey east of Wisconsin?

4) The atomic number of helium is 2.

5) The atomic number of helium is π.

6) I hate overcooked noodles.

7) Overcooked noodles are disgusting.

8) Take your time.

9) This is the last question.

We can define a deduction to be a series of hypotheses that is followed by a conclusion.

(The conclusion and each of the hypotheses must be an assertion.) If the hypotheses are true

and the deduction is a good one, then you have a reason to accept the conclusion. Consider

this example:

Hypotheses:

There is coffee in the coffee pot.

There is a dragon playing bassoon on the armoire.

Conclusion: Pablo Picasso was a poker player.

It may seem odd to call this a deduction, but that is because it would be a terrible deduction.

The two hypotheses have nothing at all to do with the conclusion. Nevertheless, given our

definition, it still counts as a deduction—albeit a bad one.

1B. Two ways that deductions can go wrong

Consider the deduction that you should take an umbrella (on p. 3, above). If hypothesis (1)

is false—if it is sunny outside—then the deduction gives you no reason to carry an umbrella.

Even if it is raining outside, you might not need an umbrella. You might wear a rain poncho

or keep to covered walkways. In these cases, hypothesis (2) would be false, since you could go

out without an umbrella and still avoid getting soaked.

Suppose for a moment that both the hypotheses are true. You do not own a rain poncho.

You need to go places where there are no covered walkways. Now does the deduction show

you that you should take an umbrella? Not necessarily. Perhaps you enjoy walking in the rain,

and you would like to get soaked. In that case, even though the hypotheses were true, the

conclusion would be false.



1. What is Logic?



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For any deduction, there are two ways that it could be weak:

1) One or more of the hypotheses might be false. A deduction gives you a reason to

believe its conclusion only if you believe its hypotheses.

2) The hypotheses might fail to entail the conclusion. Even if the hypotheses were true,

the form of the deduction might be weak.

The example we just considered is weak in both ways.

When a deduction is weak in the second way, there is something wrong with the logical

form of the deduction: hypotheses of the kind given do not necessarily lead to a conclusion of

the kind given. We will be interested primarily in the logical form of deductions.

Consider another example:

Hypotheses:

You are reading this book.

This is an undergraduate textbook.

Conclusion: You are an undergraduate student.

This is not a terrible deduction. Most people who read this book are undergraduate students.

Yet, it is possible for someone besides an undergraduate to read this book. If your mother

or father picked up the book and thumbed through it, they would not immediately become

an undergraduate. So the hypotheses of this deduction, even though they are true, do not

guarantee the truth of the conclusion. Its logical form is less than perfect.

A deduction that had no weakness of the second kind would have perfect logical form. If its

hypotheses were true, then its conclusion would necessarily be true. We call such a deduction

“deductively valid” or just “valid.”

Even though we might count the deduction above as a good deduction in some sense, it is

not valid; that is, it is “invalid.” The task of logic is to sort valid deductions from invalid ones.



1C. Deductive validity

A deduction is valid if and only if its conclusion is true whenever all of its hypotheses are

true. In other words, it is impossible for the hypotheses to be true at the same time that the

conclusion is false. Consider this example:

Hypotheses:

Oranges are either fruits or musical instruments.

Oranges are not fruits.

Conclusion: Oranges are musical instruments.

The conclusion of this deduction is ridiculous. Nevertheless, it follows validly from the

hypotheses. This is a valid deduction; that is, if both hypotheses were true, then the conclusion

would necessarily be true. For example, you might be able to imagine that, in some remote

river valley, there is a variety of orange that is not a fruit, because it is hollow inside, like a

gourd. Well, if the other hypothesis is also true in that valley, then the residents must use the

oranges to play music.

This shows that a deductively valid deduction does not need to have true hypotheses or

a true conclusion. Conversely, having true hypotheses and a true conclusion is not enough to

make a deduction valid. Consider this example: