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Chapter 3: Problem 11

Use a truth table to determine whether the two statements are equivalent. \((p \vee r) \rightarrow \sim q,(\sim p \wedge \sim r) \rightarrow q\)

Short Answer

Expert verified
The two mentioned logical statements are equivalent if their truth values match for every possible combination of truth values of the propositional variables. This information is obtained by creating and evaluating a truth table.

Step by step solution

01

Construct a truth table

Begin by creating a truth table for the propositional variables, \(p, q, r\). Each possible combination of truth values that we can assign for these variables serves as a row in the table.
02

Evaluate the logical expressions

For each row of the truth table, evaluate the truth values of the individual parts of each statement, then put these together to find the truth value of each statement as a whole.
03

Check the equivalency

Now that we have calculated the truth values of both statements for each row of the truth table, we can check whether they are equivalent by comparing these values. The two statements are equivalent if and only if their truth values match for every row in the table.

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Most popular questions from this chapter

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can't use Euler diagrams to determine the validity of an argument if one of the premises is false.

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?

This is an excerpt from a 1967 speech in the U.S. House of Representatives by Representative Adam Clayton Powell: He who is without sin should cast the first stone. There is no one here who does not have a skeleton in his closet. I know, and I know them by name. Powell's argument can be expressed as follows: No sinner is one who should cast the first stone. All people here are sinners. Therefore, no person here is one who should cast the first stone. Use an Euler diagram to determine whether the argument is valid or invalid.
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