# Discrete Mathematics: An Active Approach to Mathematical Reasoning

## Section2.3Valid and Invalid Arguments

We've been looking at logical statements, and now we want to be able to put statements together to form logical arguments. Just like with the statements, we are going to be concerned more about the structure of an argument than the specific content.
An argument consists of premises and a conclusion. You must have at least one premise, but can have as many as you like. You have exactly one conclusion.

### General Form of an Argument.

 A B $$\therefore\$$C
Statements A and B are the premises, and statement C is the conclusion. The symbol $$\ \therefore\$$ is read “therefore.”
Since an argument is just a list of statements, we need some structure for what makes a “good” argument.

### Definition2.3.1.

An argument is valid if whenever the premises are true, the conclusion must be true.

### Definition2.3.2.

An argument is invalid if it is possible for the premises to be true and the conclusion false.
It is really important to note that validity of an argument does not depend on the actual truth or falsity of the statements. To decide if an argument is valid, we construct a truth-table for the premises and conclusion. Then we check for whether there is a case where the premises are true and the conclusion false.
Decide whether the following argument is valid or invalid.
 $$p\wedge q$$ $$\therefore\$$$$p$$
We construct a truth-table with a columnn for each premise and conclusion.
Since we are looking for where the premise is true, we only need to look at the first row (in bold). In this case, the conclusion is also true. Thus, whenever to premises are true the conclusion must be true. Hence, the argument is valid.
Decide whether the following argument is valid or invalid.
 $$p\vee q$$ $$\therefore\$$$$p$$
We construct a truth-table with a columnn for each premise and conclusion.
The first three rows all have true premises. However, in the case that $$p$$ is false and $$q$$ is true, the premise is true while the conclusion is false. Thus, it is possible to have true premises and a false conclusion. Hence, the argument is invalid.

### Activity2.3.1.

Use a truth-table to determine if the following argument is valid or invalid.
 $$p\rightarrow q$$ $$\sim q$$ $$\therefore\$$$$\sim p$$

### Activity2.3.2.

Use a truth-table to determine if the following argument is valid or invalid.
 $$p\rightarrow q$$ $$\sim p$$ $$\therefore\$$$$\sim q$$

### Activity2.3.3.

Use a truth-table to determine if the following argument is valid or invalid.
 $$p\ \vee q$$ $$p\rightarrow r$$ $$q\rightarrow r$$ $$\therefore\$$$$r$$
Let's look at a more specific example:
 The sun is purple and the sun sets in the west. Therefore, the sun is purple.
Although the two statements are false, the argument is still valid. It has the form of Example 2.3.3, which we determined was valid. But if we think about the definition of validity, we should be able to see that it would be impossible to have the premise be true while the conclusion is false.

### Activity2.3.4.

If possible, give an example of an argument (in sentences, not variables) that meets the given criteria. If it is not possible, just state that it is not possible.

#### (a)

A valid argument that has false premises and a true conclusion.

#### (b)

An invalid argument that has false premises and a true conclusion.

#### (c)

A valid argument that has true premises and a true conclusion.

#### (d)

An invalid argument that has true premises and a true conclusion.

#### (e)

A valid argument that has true premises and a false conclusion.

#### (f)

An invalid argument that has true premises and a false conclusion.

### Some Common Forms for Valid Arguments.

Modus ponens:
 $$p\rightarrow q$$ $$p$$ $$\therefore\$$$$q$$
Modus tollens:
 $$p\rightarrow q$$ $$\sim q$$ $$\therefore\$$$$\sim p$$
Transitivity:
 $$p\rightarrow q$$ $$q\rightarrow r$$ $$\therefore\$$$$p\rightarrow r$$
We will show that transitivity is a valid argument using a truth-table.
Looking at the rows where both premises are true (in bold), we can see that the conclusion must be true. Thus, the argument is valid.

### Some Common Invalid Arguments.

Converse error:
 $$p\rightarrow q$$ $$q$$ $$\therefore\$$$$p$$
Inverse error:
 $$\sim p\rightarrow q$$ $$p$$ $$\therefore\$$$$\sim q$$
Since it is possible to have a valid argument with a false conclusion, but we'd like our arguments to have true conclusions, we need something more to have a “good” argument.

### Definition2.3.10.

An argument is sound if it is valid and all the premises are true.
Since a valid argument cannot have true premises and a false conclusion, if the premises are actually true, then the argument must have a true conclusion. Note, soundness of an argument does depend on the actual content of the statements.

### Activity2.3.5.

Which or your arguments in Activity 2.3.4 are sound?

#### 1.

The following is the truth-table for an argument with converse error.
 $$p$$ $$q$$ $$p\rightarrow q$$ $$q$$ $$p$$ T T T T T T F F F T F T T T F F F T F F
Which row of the truth-table shows us the argument is invalid?
• Row 1
• To be invalid, you do want to look for true premises, but what should the conclusion be?
• Row 2
• To be invalid, you want to look for true premises.
• Row 3
• In the third row we have true premises and a false conclusion, thus the argument is invalid.
• Row 4
• To be invalid, you want to look for true premises.

#### 2.

Use a truth-table to determine if the following argument is valid or invalid.
 $$p\vee q$$ $$\sim p$$ $$\therefore\$$$$q$$
• The argument is valid.
• The third row of the truth-table is the only one with all true premises. In this case the conclusion is true. Thus it is valid.
• The argument is invalid.
• In the truth-table is it possible to have all the premises be true while the conclusion is false?
• I am unable to determine if the argument is valid or invalid.
• In the truth-table look for all rows in which all the premises are true. Is is possible to have a false conclusion (invalid), or must the conclusion be true (valid)?

#### 3.

True or False: A valid argument can have false premises and a true conclusion.
• True.

• False.

#### 4.

True or False: An invalid argument can have true premises and a true conclusion.
• True.

• False.

#### 5.

True or False: A valid argument can have true premises and a false conclusion.
• True.

• False.

#### 6.

True or False: A sound argument can have false premises and a true conclusion.
• True.

• False.

#### 7.

True or False: A sound argument can have true premises and a false conclusion.
• True.

• False.

### ExercisesExercises

#### 1.

Use a truth-table to determine if the following argument is valid or invalid. Indicate the premises and conclusion on your table. Clearly state your conclusion and explain how your truth-table supports your conclusion.
 $$p$$ $$p\rightarrow q$$ $$\sim q\ \vee r$$ $$\therefore r$$

#### 2.

Use a truth-table to determine if the following argument is valid or invalid. Indicate the premises and conclusion on your table. Clearly state your conclusion and explain how your truth-table supports your conclusion.
 $$(p\ \wedge q)\rightarrow \sim r$$ $$p\ \vee \sim q$$ $$\sim q\rightarrow p$$ $$\therefore \sim r$$

#### 3.

Use a truth-table to show that the following argument is valid. Indicate the premises and conclusion on your table. Explain how your truth-table supports your conclusion.
 $$p \ \vee q$$ $$\sim p$$ $$\therefore q$$

#### 4.

Use a truth-table to show that the following argument (modus tollens) is valid. Indicate the premises and conclusion on your table. Explain how your truth-table supports your conclusion.
 $$p\rightarrow q$$ $$\sim q$$ $$\therefore \sim p$$

#### 5.

Use a truth-table to show that the following argument (also known as proof by cases) is valid. Indicate the premises and conclusion on your table. Explain how your truth-table supports your conclusion.
 $$p\ \vee q$$ $$p\rightarrow r$$ $$q\rightarrow r$$ $$\therefore r$$

#### 6.

Determine whether the following arguments are valid or invalid. If they are invalid, determine if they exhibit the converse error or the inverse error. Rewrite each argument using symbols to help determine validity.
1.  If Jules solved this problem correctly, then Jules obtained the answer $$\pi\text{.}$$ Jules obtained the answer $$\pi\text{.}$$ $$\therefore$$ Jules solved the problem correctly.
2.  If at least one of these two numbers is divisible by 6, then the product of these two numbers is divisible by 6. Neither of these two numbers is divisible by 6. $$\therefore$$ The product of these two numbers is not divisible by 6.

#### 7.

Explain in your own words what distinguishes a valid form of argument from an invalid one.