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Chapter 3: Problem 34

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 am tired or hungry, I cannot concentrate. I can concentrate. \(\therefore\) I am neither tired nor hungry.

Short Answer

Expert verified
By examining the truth table, it will be seen whether there are any instances where the premises are true and the conclusion is false. If there are no such instances, then the argument is valid; otherwise, it is invalid.

Step by step solution

01

Define Propositions

Define 'I am tired' as proposition \(T\), 'I am hungry' as proposition \(H\) and 'I can concentrate' as proposition \(C\). Then the first statement translates to '\(T \lor H \rightarrow \lnot C\)', the second statement translates to \(C\) and the conclusion translates to '\(\lnot T \land \lnot H\)'.
02

Construct Truth Table

A truth table with eight rows (since there are three variables) is constructed. Each possible combination for the truth values of \(T\), \(H\), and \(C\) is considered, and the truth values for the premises and the conclusion are calculated using the rules of propositional logic.
03

Judge Validity

After filling out the truth table, look for rows where the premises (first and second statement) are true and the conclusion is false. If such a row exists, the argument is invalid. If no such rows exist (i.e., every time the premises are true, the conclusion is also true), then the argument is valid.

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Most popular questions from this chapter

Write an original argument in words for the transitive reasoning form.

Use Euler diagrams to determine whether each argument is valid or invalid. All actors are artists. Sean Penn is an actor. Therefore, Sean Penn is an artist.

Exercises 59-60 illustrate arguments that have appeared in cartoons. Each argument is restated below the cartoon. Translate the argument into symbolic form and then determine whether it is valid or invalid. If you do not know how to read, you cannot read War and Peace. If you cannot read War and Peace, then Leo Tolstoy will hate you. Therefore, if you do not know how to read, Leo Tolstoy will hate you.

Conservative commentator Rush Limbaugh directed this passage at liberals and the way they think about crime. Of course, liberals will argue that these actions [contemporary youth crime] can be laid at the foot of socioeconomic inequities, or poverty. However, the Great Depression caused a level of poverty unknown to exist in America today, and yet I have been unable to find any accounts of crime waves sweeping our large cities. Let the liberals chew on that. (See, I Told You So, p. 83) Limbaugh's passage can be expressed in the form of an argument: If poverty causes crime, then crime waves would have swept American cities during the Great Depression. Crime waves did not sweep American cities during the Great Depression. \(\therefore\) Poverty does not cause crime. (Liberals are wrong.) Translate this argument into symbolic form and determine whether it is valid or invalid.

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.
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