1.1: Assertions, Deductions, and Validity

We will begin our discussion of Logic by introducing three basic ingredients: assertions, deductions, and validity.

Here is one possible deduction:

Hypothesis:

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 Example \(1.1.10\) below.)

In Logic, we are only interested in sentences that can be a hypothesis or conclusion of a deduction. These are called “assertions”:

Definition \(1.1.1\).

An assertion is a sentence that is either true or false.

Other Terminology.

Some textbooks use the term proposition or statement or sentence, instead of assertion.

Example \(1.1.2\).

• 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 do not count as assertions in 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 itself is neither true nor false.

• Exclamations “Ouch!” is sometimes called an exclamatory sentence, but it is neither true nor false. so it is another example of a sentence that is not an assertion.

Remark \(1.1.3\).

Roughly speaking, an assertion is a statement of fact, such as “The earth is bigger than the moon” or “Edmonton is the capital of Alberta.” However, it is important to remember that an assertion may be false, in which case it is a mistake (or perhaps a deliberate lie), such as “There are less than 1,000 automobiles in all of Canada.” In many cases, the truth or falsity of an assertion depends on the situation. For example, the assertion “It is raining” is true in certain places at certain times, but is false at others.

In this and the next, which are introductory, we will deal mostly with assertions about the real world, where facts are not always clear-cut. (For example, if Alice and Bob are almost the same height, it may be impossible to determine whether it is true that “Alice is taller than Bob.”) We are taking a mathematical (or scientific) view toward Logic, not a philosophical one, so we will ignore the imperfections of these real-world assertions, which provide motivation and illustration, because our goal is to learn to use Logic to understand mathematical objects (not real-world objects), where there are no grey areas.

Throughout this text, you will find exercises that review and explore the material that has just been covered. There is no substitute for actually working through some problems, because this course, like most advanced mathematics, is more about a way of thinking than it is about memorizing facts.

Exercise \(1.1.4\).

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 \(\pi\).
6. Take your time.
7. This is the last question.
8. Rihanna was born in Barbados.

Definition \(1.1.5\).

A deduction is 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.

Example \(1.1.6\).

Here are two deductions.

1. Hypotheses:
   1. All men are mortal.
   2. Socrates is a man.
      Conclusion: Socrates is mortal.
2. Hypotheses:
   1. The Mona Lisa was painted by Leonardo da Vinci.
   2. Neil Armstrong was the first man on the moon.
      Conclusion: Justin Trudeau went swimming yesterday.

The first of these deductions is very famous (and was discussed by the ancient Greek philosopher Aristotle), but the second one is lame. It may seem odd to even call it a deduction, because the two hypotheses have nothing at all to do with the conclusion, but, given our definition, it does count as a deduction. However, it is is a very poor one, so it cannot be relied on as evidence that the conclusion is true.

We are interested in the deductions that do provide solid evidence for their conclusions:

Definition \(1.1.7\).

A deduction is valid if its conclusion is true whenever all of its hypotheses are true. In other words, it is impossible to have a situation in which all of the hypotheses are true, but the conclusion is false.

The task of Logic is to distinguish valid deductions from invalid ones.

Example \(1.1.8\).

Hypotheses:

Oranges are either fruits or musical instruments.
Oranges are not fruits.

Conclusion:
        Oranges are musical instruments.

The conclusion is ridiculous. Nevertheless, the deduction is valid, because its conclusion follows validly from its hypotheses; 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 logically 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:

Example \(1.1.9\).

Hypotheses:

London is in England.
Beijing is in China.

Conclusion: Paris is in France.

The hypotheses and conclusion of this deduction are, as a matter of fact, all true. This is a terrible deduction, however, because the hypotheses have nothing to do with the conclusion. For example, if Paris declared independence from the rest of France, then the conclusion would be false, even though the hypotheses would both still be true. Thus, it is logically possible to have a situation in which the hypotheses of this deduction are true and the conclusion is false. Therefore, the deduction is not valid.

Example \(1.1.10\).

Recall the deduction that you should take an umbrella (above), and suppose for a moment that both of its hypotheses are true. (Thus, you will get wet if you do not take an umbrella.) Now is it necessarily true that you should take an umbrella? No—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. Thus, the deduction is not valid.

Example \(1.1.11\).

Hypotheses:

You are reading this book.
This is an undergraduate textbook.

Conclusion: You are an undergraduate student.

This is not a terrible deduction, because most people who read this book are undergraduate students. Yet, it is possible for someone besides an undergraduate to read this book. For example, 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. Thus, even though some people might say that the deduction has some value, it is certainly not valid.

Remark \(1.1.12\).

It is important to remember that validity of a deduction is not about the truth or falsity of the deduction’s assertions in the real world. Instead, it is about the form of the deduction, in that the truth of the hypotheses is incompatible with the falsity of the conclusion in every possible world (real or imaginary). Furthermore, a deduction gives you a reason to believe its conclusion only in situations where its hypotheses are true.

Exercise \(1.1.13\).

Which of the following is possible? If it is possible, give an example. If it is not possible, explain why.

1. A valid deduction that has one false hypothesis and one true hypothesis.
2. A valid deduction that has a false conclusion.
3. A valid deduction that has at least one false hypothesis, and a true conclusion.
4. A valid deduction that has all true hypotheses, and a false conclusion.
5. An invalid deduction that has at least one false hypothesis, and a true conclusion.