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Chapter 3: Problem 35

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

Short Answer

Expert verified
After creating and analyzing the truth table, depending upon the final column, one can affirm if the given statement \([(p \rightarrow q) \wedge \sim q] \rightarrow \sim p\) is a tautology, self-contradiction or neither.

Step by step solution

01

Establishing the Truth Table

First, draw a truth table with columns for p, q, \(p \rightarrow q\), \(\sim q\), \((p \rightarrow q) \wedge \sim q\), and \((p \rightarrow q) \wedge \sim q \rightarrow \sim p\). Then fill in all possible combinations of truth values for p and q, which are True (T) and False(F).
02

Determine \(p \rightarrow q\)

Now calculate the values of \(p \rightarrow q\), which is false only when p is true and q is false, otherwise it is true.
03

Calculate \(\sim q\)

Calculate the values of \(\sim q\), which should be the logical opposite of the values in the q column.
04

Derive \((p \rightarrow q) \wedge \sim q\)

Now, derive the truth values of \((p \rightarrow q) \wedge \sim q\) corresponding to the conjunction of \(p \rightarrow q\) and \(\sim q\). It will be true only when both of these expressions are true.
05

Deduce \((p \rightarrow q) \wedge \sim q) \rightarrow \sim p\)

Finally calculate the values of \((p \rightarrow q) \wedge \sim q \rightarrow \sim p\), which would be false only when \((p \rightarrow q) \wedge \sim q\)) is true and \(\sim p\) is false. Otherwise, it will be true.
06

Evaluating the Expression

After calculating the values, look at the final column. If it contains only true values then the original statement is a tautology. If it only contains false values then it's a self-contradiction. If it contains a mix of true and false values 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 dancers are athletes. Savion Glover is a dancer. Therefore, Savion Glover is an athlete.

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 an argument is in the form of the fallacy of the inverse, then it is invalid. This argument is invalid. \(\therefore\) This argument is in the form of the fallacy of the inverse.

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'm at the beach, then I swim in the ocean. If I swim in the ocean, then I feel refreshed. \(\therefore\) If I'm not at the beach, then I don't feel refreshed.

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, ...

No animals that eat meat are vegetarians. No cat is a vegetarian. Felix is a cat. Therefore,,\(.\) a. Felix is a vegetarian. b. Felix is not a vegetarian. c. Felix eats meat. d. All animals that do not eat meat are vegetarians.
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