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Chapter 3: Problem 33

Construct a truth table for the given statement. \((p \vee q) \wedge(\sim p \vee \sim q)\)

Short Answer

Expert verified
The truth values for the statement \((p \vee q) \wedge(\sim p \vee \sim q)\) for various combinations of truth values for \(p\) and \(q\) are: when both are true, the statement is true; when both are false, the statement is true; when \(p\) is true and \(q\) is false, the statement is false; when \(p\) is false and \(q\) is true, the statement is also false.

Step by step solution

01

List all possible combinations of truth values for \(p\) and \(q\)

Since there are two variables, \(p\) and \(q\), there are four possible combinations of truth values for these variables: both can be true, both can be false, \(p\) can be true and \(q\) false, \(p\) can be false and \(q\) true.
02

Calculate the truth values for the statements \((p \vee q)\) and \((\sim p \vee \sim q)\) for each combination

The OR operator (\(\vee\)) is true if either or both of its operands are true, and false otherwise. So, for each combination of truth values for \(p\) and \(q\), calculate the truth value for \((p \vee q)\). The NOT operator (\(\sim\)) flips the truth value of its operand, so calculate the truth values of \(\sim p\) and \(\sim q\), then calculate the truth value for \((\sim p \vee \sim q)\) using these.
03

Calculate the truth value for the entire statement \((p \vee q) \wedge(\sim p \vee \sim q)\) for each combination

The AND operator (\(\wedge\)) is true if both of its operands are true, and false otherwise. So, for each combination of truth values for \(p\) and \(q\), calculate the truth value for the entire statement \((p \vee q) \wedge(\sim p \vee \sim q)\) based on the truth values for the constituents \((p \vee q)\) and \((\sim p \vee \sim q)\) already calculated in step 2.

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Most popular questions from this chapter

Use Euler diagrams to determine whether each argument is valid or invalid. All humans are warm-blooded. No reptiles are warm-blooded. Therefore, no reptiles are human.

In this section, we used a variety of examples, including arguments from the Menendez trial, the inevitability of Nixon's impeachment, Spock's (fallacious) logic on Star Trek, and even two cartoons, to illustrate symbolic arguments. a. From any source that is of particular interest to you (these can be the words of someone you truly admire or a person who really gets under your skin), select a paragraph or two in which the writer argues a particular point. (An intriguing source is What Is Your Dangerous Idea?, edited by John Brockman, published by Harper Perennial, 2007.) Rewrite the reasoning in the form of an argument using words. Then translate the argument into symbolic form and use a truth table to determine if it is valid or invalid. b. Each group member should share the selected passage with other people in the group. Explain how it was expressed in argument form. Then tell why the argument is valid or invalid.

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

Write an original argument in words that has a true conclusion, yet is invalid.

Use the standard forms of valid arguments to draw a valid conclusion from the given premises. If I vacation in Paris, I eat French pastries. If I eat French pastries, I gain weight. Therefore, ...
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