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Chapter 3: Problem 37

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

Short Answer

Expert verified
The statement \([(p \vee q) \wedge p] \rightarrow \sim q\) is neither a tautology nor a self-contradiction, but is contingent, based upon constructing and analyzing the truth table.

Step by step solution

01

Identify Variables

The statement contains two variables: \(p\) and \(q\). Create columns in your truth table for each of these individual variables. Each variable will have two possible truth values: true (T) or false (F). Since there are two variables, there will be four total combinations of truth values.
02

Construct Truth Table for Connectives

In the given statement, the logical connectives used are: disjunction (\(\vee\)), conjunction (\(\wedge\)), and implication (\(\rightarrow\)). We also see the use of negation (\(\sim\)). Create columns for each of these connectives. For disjunction, the truth value is T if either \(p\) or \(q\) is T. For conjunction, the truth value is T if both \(p\) and \(q\) are T. For implication, the truth value is F only when the first statement is T and the second is F. For negation, the truth value is simply the opposite of q.
03

Fill Out the Truth Table

The next step is to fill out the truth table. Start by listing all possible combinations of truth values of the individual variables. For two variables there are four combinations: TT, TF, FT, FF. Use these values to fill out the truth values of the connectives.
04

Check for Tautology or Self-contradiction

Once the truth table is completed, the type of the statement can be determined. A statement is a tautology if all outcomes in the final column are T. It is a self-contradiction if all outcomes in the final column are F. If neither, the statement is considered contingent.

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