Geometry (A): READ: Deductive Reasoning Reading

Deductive Reasoning

Learning Objectives

Recognize and apply some basic rules of logic.
Understand the different parts that inductive reasoning and deductive reasoning play in logical reasoning.

Introduction

You began to study deductive reasoning, or logic, in the last section, when you learned about if-then statements. Now we will see that logic, like other fields of knowledge, has its own rules. When we follow those rules, we will expand our base of facts and relationships about points, lines, and planes. We will learn two of the most useful rules of logic in this section.

Direct Reasoning

We all use logic—whether we call it that or not—in our daily lives. And as adults we use logic in our work as well as in making the many decisions a person makes every day.

Which product should you buy?
Who should you vote for?
Will this steel beam support the weight you place on it?
What will be your company’s profit next year?

Let’s see how common sense leads to the two most basic rules of logic.

Example 1

Suppose Bea makes the following statements, which are known to be true.

If Central High School wins today, they will go to the regional tournament.

Central High School does win today.

Common sense tells us that there is an obvious logical conclusion if these two statements are true:

Central High School will go to the regional tournament.

Example 2

Here are two true statements.

$5$ is an odd number.

Every odd number is the sum of an even and an odd number.

Based on only these two true statements, there is an obvious further conclusion:

$5$ is the sum of an even and an odd number.

(This is true, since $5 = 2 + 3$ ).

Example 3

Suppose the following two statements are true.

If you love me let me know, if you don’t then let me go. (A country music classic. Lyrics by John Rostill.)
You don’t love me.

What is the logical conclusion?

Let me go.

There are two statements in the first line. The second one is:

If you don’t (love me) then let me go.

You don’t love me is stated to be true in the second line.

Based on these true statements,“Let me go” is the logical conclusion.

Now let’s look at the structure of all of these examples, using the $p$ and $q$ symbols that we used earlier.

Each of the examples has the same form.

$& p \rightarrow q \\ & p$

conclusion: $q$

A more compact form of this argument, (logical pattern) is:

$& p \rightarrow q \\ & p \\ & \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ & q$

To state this differently, we could say that the true statement $q$ follows automatically from the true statements $p \rightarrow q$ and $p$ .

This reasoning pattern is one of the basic rules of logic. It’s called the law of detachment.

Law of Detachment

Suppose $p$ and $q$ are statements. Then given

$p \rightarrow q$ and $p$

You can conclude

$q$

Practice saying the law of detachment like this: “If $p \rightarrow q$ is true, and $p$ is true, then $q$ is true.”

Example 4

Here are two true statements.

If $\angle{A}$ and $\angle{B}$ are a linear pair, then $m\angle{A} + m\angle{B} = 180^\circ$ .

$\angle{A}$ and $\angle{B}$ are a linear pair.

What conclusion do we draw from these two statements?

$m \angle{A} + m \angle{B} = 180^\circ.$

The next example is a warning not to turn the law of detachment around.

Example 5

Here are two true statements.

If $\angle{A}$ and $\angle{B}$ are a linear pair, then $m \angle{A} + m \angle{B} = 180^\circ.$

$m \angle{A} = 90^\circ$ and $m \angle{B} = 90^\circ.$

What conclusion can we draw from these two statements?

None! These statements are in the form

$p \rightarrow q$

$q$

Note that since $m \angle{A} = 90^\circ$ and $m \angle{B} = 90^\circ$ , we also know that $m \angle{A} + m \angle{B} = 180^\circ$ , but this does not mean that they are a linear pair.

The law of detachment does not apply. No further conclusion is justified.

You might be tempted to conclude that $\angle{A}$ and $\angle{B}$ are a linear pair, but if you think about it you will realize that would not be justified. For example, in the rectangle below $m \angle{A} = 90^\circ$ and $m \angle{B} = 90^\circ$ (and $m \angle{A} + m \angle{B}=180^\circ$ , but $\angle{A}$ , and $\angle{B}$ are definitely NOT a linear pair.

Now let’s look ahead. We will be doing some more complex deductive reasoning as we move ahead in geometry. In many cases we will build chains of connected if-then statements, leading to a desired conclusion. Start with a simplified example.

Example 6

Suppose the following statements are true.

1. If Pete is late, Mark will be late.

2. If Mark is late, Wen will be late.

3. If Wen is late, Karl will be late.

To these, add one more true statement.

4. Pete is late.

One clear consequence is: Mark will be late. But make sure you can see that Wen and Karl will also be late.

Here’s a symbolic form of the statements.

$p \rightarrow q$
$q \rightarrow r$
$r \rightarrow s$
$& p \\ & \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\ & s$

Our statements form a “chain reaction.” Each “then” becomes the next “if” in a chain of statements. The chain can consist of any number of connected statements. Once we add the true $p$ statement as above, we know that the conclusion (the then part) of the last statement is justified.

Another way to look at this is to imagine a chain of dominoes. The dominoes are the linked if-then statements. Once the first domino falls, each domino knocks the next one over, and the last domino falls. $p$ is the tipping over of the first domino. The final conclusion of the last if-then statement is the last domino.

This is called the law of syllogism. A formal statement of this rule of logic is given below.

Law of Syllogism

Suppose $a_1, a_2, \ldots, a_{n-1}$ , and $a_{n}$ are statements. Then given that $a_1$ is true and that you have the following relationship:

$& a_1 \ \rightarrow \ a_2\\ & a_2 \ \rightarrow \ a_3\\ & \vdots\\ & a_{n-1} \ \rightarrow \ a_n$

Then, you can conclude

$a_{1} \rightarrow a_{n}$

Example 6

Use the Law of Syllogism to draw a conclusion from the following true statements.

If a number is prime, then it does not have repeated factors.

If a number does not have repeated factors, then it is not a perfect square.

You have two true conditionals where the conclusion of one is the hypotheses of the other. You can use the Law of Syllogism to draw the following conclusion.

If a number is prime, then it is not a perfect square.

Inductive vs. Deductive Reasoning

You have now worked with both inductive and deductive reasoning. They are different but not opposites. In fact, they will work together as we study geometry and other mathematics.

How do these two kinds of reasoning complement (strengthen) each other? Think about the examples you saw earlier in this chapter.

Inductive reasoning means reasoning from examples. You may look at a few examples, or many. Enough examples might make you suspect that a relationship is true always, or might even make you sure of this. But until you go beyond the inductive stage, you can’t be absolutely sure that it is always true.

That’s where deductive reasoning enters and takes over. We have a suggestion arrived at inductively. We then apply rules of logic to prove, beyond any doubt, that the relationship is true always. We will use the law of detachment and the law of syllogism, and other logic rules, to build these proofs.

Lesson Summary

Do we all have our own version of what is logical? Let’s hope not—we wouldn’t be able to agree on what is or isn’t logical! To avoid this, there are agreed-on rules for logic, just like there are rules for games. The two most basic rules of logic that we will be using throughout our studies are the law of detachment and the law of syllogism.

Points to Consider

Rules of logic are universal; they apply to all fields of knowledge. For us, the rules give a powerful method for proving new facts that are suggested by our explorations of points, lines, planes, and so on. We will structure a specific format, the two-column proof, for proving these new facts. In upcoming lessons you will write two-column proofs. The facts or relationships that we prove are called theorems.

Last modified: Monday, May 10, 2010, 2:13 PM