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Chapter 3: Problem 47

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

Short Answer

Expert verified
The statement \((p \rightarrow q) \leftrightarrow (\sim p \vee q)\) is a tautology, as all outcomes in the final column of the truth table are true.

Step by step solution

01

Setup of Truth Table

First, a truth table is created with columns for p, q, \( \sim p \), \( p \rightarrow q \), \( \sim p \vee q \), and \( (p \rightarrow q) \leftrightarrow (\sim p \vee q) \). The table will have four rows to cover all possible truth value combinations for p and q: TT, TF, FT, FF.
02

Fill in Truth Values

The truth values for p and q are filled in based on the combinations, and the negation \( \sim p \) is calculated. The result of \( p \rightarrow q \) is true except when p is true and q is false. The disjunction \( \sim p \vee q \) is only false when both \( \sim p \) and q are false.
03

Determine Final Outcome

The column for \( (p \rightarrow q) \leftrightarrow (\sim p \vee q) \) is filled in. The biconditional is true only when both sides have the same truth value. If all outcomes are true, then the statement is a tautology. If all outcomes are false, it's a self-contradiction. If it鈥檚 mixed, then it's neither.

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Most popular questions from this chapter

Use Euler diagrams to determine whether each argument is valid or invalid. All physicists are scientists. All scientists attended college. Therefore, all physicists attended college.

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 The Graduate and Midnight Cowboy are shown, then the performance is sold out. Midnight Cowboy was shown and the performance was not sold out. \(\therefore\) The Graduate was not shown.

Draw what you believe is a valid conclusion in the form of a disjunction for the following argument. Then verify that the argument is valid for your conclusion. "Inevitably, the use of the placebo involved built-in contradictions. A good patient-doctor relationship is essential to the process, but what happens to that relationship when one of the partners conceals important information from the other? If the doctor tells the truth, he destroys the base on which the placebo rests. If he doesn't tell the truth, he jeopardizes a relationship built on trust."

Under what circumstances should Euler diagrams rather than truth tables be used to determine whether or not an argument is valid?

Use the standard forms of valid arguments to draw a valid conclusion from the given premises. The writers of My Mother the Car were told by the network to improve their scripts or be dropped from prime time. The writers of My Mother the Car did not improve their scripts. Therefore, ...
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