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Introduction to Proofs An Active Exploration of Mathematical Language

Jennifer Firkins Nordstrom

Section 1.4 Logical Arguments with Connectives

In Section 1.2 we defined valid and and invalid arguments. In Section 1.3 we looked at some common forms for logical statements. Now we want to look at how to determine if an argument made from logical statements is valid or invalid. As before, validity just depends on the structure of the argument, not the specific content.
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Recall from Definition 1.2.2 and Definition 1.2.3, an argument is valid if whenever the premises are true, the conclusion must be true, and an argument is invalid if it is possible for the premises to be true and the conclusion false. Furthermore, validity of an argument does not depend on the actual truth or falsity of the statements.
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To decide if an argument built from logical statements 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.
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Example 1.4.1. Valid Argument.

Decide whether the following argument is valid or invalid.
\(p\wedge q\)
\(\therefore\ \)\(p\)
We construct a truth table.
Table 1.4.2.
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\(p\) \(q\) \(p\wedge q\) \(p\)
T T T T
T F F T
F T F F
F F F F
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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.
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Example 1.4.3. Invalid Argument.

Decide whether the following argument is valid or invalid.
\(p\vee q\)
\(\therefore\ \)\(p\)
We construct a truth table.
Table 1.4.4.
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\(p\) \(q\) \(p\vee q\) \(p\)
T T T T
T F T T
F T T F
F F F F
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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.
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Activity 1.4.1. Validity with a Truth Table.

Use a truth table to determine if the following argument is valid or invalid.
\(p\rightarrow q\)
\(\neg q\)
\(\therefore\ \)\(\neg p\)
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Activity 1.4.2. More Validity with a Truth Table.

Use a truth table to determine if the following argument is valid or invalid.
\(p\rightarrow q\)
\(\neg p\)
\(\therefore\ \)\(\neg q\)
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The following two argument forms are useful for constructing valid arguments.
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Modus ponens:
\(p\rightarrow q\)
\(p\)
\(\therefore\ \)\(q\)
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Transitivity:
\(p\rightarrow q\)
\(q\rightarrow r\)
\(\therefore\ \)\(p\rightarrow r\)
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Example 1.4.5. Validity of Transitivity.

We will show that transitivity is a valid argument using a truth table.
Table 1.4.6. Truth table for transitivity.
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\(p\) \(q\) \(r\) \(p\rightarrow q\) \(q\rightarrow r\) \(p\rightarrow r\)
T T T T T T
T T F T F F
T F T F T T
T F F F T F
F T T T T T
F T F T F T
F F T T T T
F F F T T T
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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.
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Activity 1.4.3. Validity of Modus Ponens.

Use a truth table to show that modus ponens
\(p\rightarrow q\)
\(p\)
\(\therefore\ \)\(q\)
is a valid argument.
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Activity 1.4.4. Even More Validity with a Truth Table.

Use a truth table to determine if the following argument is valid or invalid.
\(p\ \vee q\)
\(p\rightarrow r\)
\(q\rightarrow r\)
\(\therefore\ \)\(r\)
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The following arguments are common invalid arguments. Showing they are invalid is left as an exercise.
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Converse error:
\(p\rightarrow q\)
\(q\)
\(\therefore\ \)\(p\)
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Inverse error:
\(\neg p\rightarrow q\)
\(p\)
\(\therefore\ \)\(\neg q\)
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Exercises Exercises

1.

Use a truth table to show that the following argument (converse error) is invalid. Indicate the premises and conclusion on your table. Explain how your truth table supports your conclusion.
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\(p\rightarrow q\)
\(q\)
\(\therefore p\)
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2.

Use a truth table to show that the following argument (inverse error) is invalid. Indicate the premises and conclusion on your table. Explain how your truth table supports your conclusion.
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\(\neg p\rightarrow q\)
\(p\)
\(\therefore \neg q\)
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3.

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.
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\(p\)
\(p\rightarrow q\)
\(\neg q\ \vee r\)
\(\therefore r\)
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4.

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.
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\((p\ \wedge q)\rightarrow \neg r\)
\(p\ \vee \neg q\)
\(\neg q\rightarrow p\)
\(\therefore \neg r\)
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5.

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.
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\(p \ \vee q\)
\(\neg p\)
\(\therefore q\)
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