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Chapter 3: Problem 19

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If we close the door, then there is less noise. There is less noise. \(\therefore\) We closed the door.

Short Answer

Expert verified
The given argument is invalid and represents the logical fallacy known as 'Affirming the Consequent'. This error happens when the consequent of an if-then statement is affirmed, and the antecedent is then inferred.

Step by step solution

01

Identification

Start by identifying the different statements. Here, 'we close the door' is represented as P and 'there is less noise' is represented as Q.
02

Symbolic Translation

Translate the given statements into symbolic form. Here, 'If we close the door, then there is less noise.' translates as \(P \rightarrow Q\). 'There is less noise.' is simply Q.
03

Formation of the Argument

Now we will form the symbolic representation of the argument. The two premises and the conclusion of the given argument can be written as:\n 1. \(P \rightarrow Q\) (If we close the door, then there is less noise.)\n 2. Q (There is less noise.)\n 3. \( \therefore P\) (Therefore, we closed the door.)
04

Evaluation

Evaluate the argument using a truth table or by comparing it to standard forms. Upon analysis, this argument can be identified as Affirming the Consequent which is an invalid form of argument.

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Most popular questions from this chapter

Use Euler diagrams to determine whether each argument is valid or invalid. All professors are wise people. Some professors are actors. Therefore, some wise people are actors.

Use Euler diagrams to determine whether each argument is valid or invalid. All clocks keep time accurately. All time-measuring devices keep time accurately. Therefore, all clocks are time-measuring devices.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If I'm tired, I'm edgy. If I'm edgy, I'm nasty. \(\therefore\) If I'm tired, I'm nasty.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If some journalists learn about the invasion, the newspapers will print the news. If the newspapers print the news, the invasion will not be a secret. No journalists learned about the invasion. \(\therefore\) The invasion was a secret.

Use a truth table to determine whether the symbolic form of the argument is valid or invalid. $$ \begin{aligned} &p \leftrightarrow q \\ &\underline{q \rightarrow r} \\ &\therefore \sim r \rightarrow \sim p \end{aligned} $$
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