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Chapter 3: Problem 54

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

Short Answer

Expert verified
To know if the proposition is a tautology, a contradiction or neither, you need to evaluate the final column of your truth table. If all values are true, it is a tautology. If all values are false, it is a contradiction. If there are some true and some false values, it is neither a tautology nor a contradiction.

Step by step solution

01

Understanding Logical Operators

First, it's important to understand the logical operators being used in the expression. The \(\rightarrow\) symbol stands for 'implies', and can be interpreted as follows: false implies anything is true, and true implies false is false. The \(\sim\) symbol stands for 'not', \(\wedge\) stands for 'and'. With these meanings, we can start creating the truth table.
02

Creating a Truth Table

Create a truth table to explore all potential combinations of true/false values for \( p, q, \) and \( r \). This will result in a table with 8 rows (2^3) for each possible combinations of conditioned if-then statements.
03

Fill the Truth Table

Fill out the important columns of the truth table, apply the rules of the individual operations to calculate the final value of the expression for each row.
04

Evaluate the Truth Table

Based on the truth table, evaluate whether the statement is a tautology (all true), a contradiction (all false), or neither (some true, some false).

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Most popular questions from this chapter

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. "I do know that this pencil exists; but I could not know this if Hume's principles were true. Therefore, Hume's principles, one or both of them, are false."

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

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.

Determine whether each argument is valid or invalid. All \(A\) are \(B\), all \(B\) are \(C\), and all \(C\) are \(D\). Thus, all \(A\) are \(D\).

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.
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