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Chapter 3: Problem 42

Use a truth table to determine whether each statement is a tautology, a self- contradiction, or neither. \((p \vee q) \wedge(\sim p \wedge \sim q)\)

Short Answer

Expert verified
The statement \((p \vee q) \wedge (\sim p \wedge \sim q)\) is a self-contradiction. For each possible interpretation of \(p\) and \(q\), the truth value of the statement is false.

Step by step solution

01

Set up the truth table

The first step is to set up the truth table. With two propositions \(p\) and \(q\), there are four possible combinations of truth values: T, T; T, F; F, T; F, F (T - True, F - False).
02

Calculate the truth values of sub-expressions

The next step is to calculate the truth values of sub-expressions \(p \vee q\) and \(\sim p \wedge \sim q\). \(p \vee q\) is true if \(p\) is true or \(q\) is true. \(\sim p \wedge \sim q\) is true when both \(p\) and \(q\) are false.
03

Calculate the truth values of the whole expression

Then, using the truth values of sub-expressions calculated in the previous step, calculate the truth values of the whole expression. The given expression \((p \vee q) \wedge (\sim p \wedge \sim q)\) is true when both \(p \vee q\) and \(\sim p \wedge \sim q\) are true.
04

Conclusion

Finally, observe the truth values of the whole expression in the truth table. If all truth values are T, the statement is a tautology. If all truth values are F, the statement is a self-contradiction. If there is a mix of T and F, the statement is neither.

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Most popular questions from this chapter

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 at the beach, then I swim in the ocean. If I swim in the ocean, then I feel refreshed. \(\therefore\) If I am at the beach, then I feel refreshed.

Use Euler diagrams to determine whether each argument is valid or invalid. All writers appreciate language. All poets are writers. Therefore, all poets appreciate language.

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 he was disloyal, his dismissal was justified. If he was loyal, his dismissial was justified. \(\therefore\) His dismissal was justified.

Determine whether each argument is valid or invalid. No \(A\) are \(B\), no \(B\) are \(C\), and no \(C\) are \(D\). Thus, no \(A\) are \(D\).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I made Euler diagrams for the premises of an argument and one of my possible diagr芒ms did not illustraate the conclusion, so the argument is invalid.
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