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Chapter 3: Problem 40

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

Short Answer

Expert verified
This will be determined after the truth table is constructed and analyzed. Depending on the truth values of the logical statement in various conditions, it may be a tautology, a contradiction, or neither.

Step by step solution

01

Build up the fundamental truth table

In this step, the task will be to create the fundamental truth table for p and q, two logical variables which can take on either True (T) or False (F) values. There will be a total of four possible combinations: (T, T), (T, F), (F, T), (F, F).
02

Determine the truth values for intermediate logical expressions

Calculate the intermediate truth values for \(q \rightarrow p\), and \(p \vee \sim q\). For \(q \rightarrow p\), it will be false only when q is true and p is false, otherwise it's true. For \(p \vee \sim q\), it will be true whenever p is true or q is false, and false only when p is false and q is true.
03

Calculate the truth values for the whole expression

Now calculate the truth values for the entire expression \((q \rightarrow p) \rightarrow(p \vee \sim q)\) based on the intermediate truth values calculated in step 2. This expression will be false only when \(q \rightarrow p\) is true and \(p \vee \sim q\) is false.
04

Make a conclusion

If the whole expression is always true, then statement is a tautology. If it's always false, then it's a contradiction. If its truth values vary, then it's neither a tautology nor a contradiction.

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Most popular questions from this chapter

Explain how to use Euler diagrams to determine whether or not an argument is valid.

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 it rains or snows, then I read. I am not reading. \(\therefore\) It is neither raining nor snowing.

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.

Use Euler diagrams to determine whether each argument is valid or invalid. All cowboys live on ranches. All cowherders live on ranches. Therefore, all cowboys are cowherders.

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.
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