Problem 38 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

Based on the values in the truth table, the logical statement \([(p \vee q) \wedge \sim q] \rightarrow p\) is classified as a tautology/self-contradiction/neither.

Step 1: Setup the Truth Table

Start by setting up the truth table with four columns: one for \(p\), one for \(q\), one for the clause \((p \vee q) \wedge \sim q\) and one for \(p\). Fill each of these columns with both possibilities for binary values: true (T) and false (F). There will be 4 rows under each column because we have two propositions \(p\) and \(q\) and each can take two possible values (T,F).

Step 2: Evaluate the Clause (p ∨ q) ∧ ¬q

Evaluate \((p \vee q) \wedge \sim q\) for each row. Here, \(\vee\) represents 'OR' operation, \(\wedge\) represents 'AND' operation, and \(\sim\) represents 'NOT' operation. Fill in the results in the third column of the truth table.

Step 3: Evaluate the Statement \([(p \vee q) \wedge \sim q] \rightarrow p\)

Now we use the calculated values of \((p \vee q) \wedge \sim q\) from Step 2 and the given values of \(p\) to evaluate \([(p \vee q) \wedge \sim q] \rightarrow p\). '→' represents 'IF - THEN' operation. If the clause is true and \(p\) is false then the result false. In all other cases, the result is true. Fill in these values in the fourth column of the truth table.

Step 4: Classify the Statement

Now we observe the fourth column. If it consists all 'T's, then the statement is a tautology. If it consists all 'F's, then the statement is a contradiction. If it includes both 'T's and 'F's, then the statement is neither a tautology nor a contradiction.

Key Concepts

Tautology

In logic, a tautology is a statement that is always true, regardless of the truth values of its components. This means that in every possible scenario, the statement cannot be false. The concept of a tautology plays a vital role in logical deduction and reasoning, providing a foundational understanding of truth in logical systems.

To determine if a statement is a tautology, we use a truth table to consider all possible truth values of the individual components in the statement. If the final column of the truth table, which represents the overall statement, consists exclusively of 'True' (T) values across all possible combinations of inputs, the statement is considered a tautology.

In the example given, we would analyze \(\left(p \vee q\right) \wedge \sim q\right) \rightarrow p\) within a truth table to ascertain whether it's a tautology. If, after filling in the truth table, the entire column corresponds to the statement is true in every row, it confirms that our statement is indeed a tautology. This validates that no matter the truth values of \(p\) and \(q\), the implication is always true.

Logical Connectives

Logical connectives, also known as logical operators, are symbols used in logical expressions to connect propositions or statements to form more complex statements. The most common logical connectives are 'AND' (\(\wedge\)), 'OR' (\(\vee\)), 'NOT' (\(\sim\)), and 'IMPLIES' (\(\rightarrow\)).

• AND (\(\wedge\)) - This connective is true if both propositions it joins are true, and false otherwise.
• OR (\(\vee\)) - This connective is true if at least one of the propositions it connects is true.
• NOT (\(\sim\)) - This unary connective inverts the truth value of the proposition it precedes.
• IMPLIES (\(\rightarrow\)) - This connective is false only if the proposition on the left (antecedent) is true and the proposition on the right (consequent) is false; in all other cases, it is true.

In our exercise, we encounter these logical connectives when interpreting \(\left(p \vee q\right) \wedge \sim q\) as part of the truth table analysis. Understanding how these connectives operate and how they alter the truth value of combined propositions is essential for accurately completing a truth table and for logical reasoning in general.

Conditional Statements

A conditional statement, often symbolized by \(\rightarrow\), is a logical connective that represents an 'IF - THEN' relationship between two propositions. The initial proposition is called the antecedent, and the proposition that follows is called the consequent. The truth of a conditional statement is determined by the relationship between the truth values of its antecedent and consequent.

A critical aspect of conditional statements is that they are only false when the antecedent is true, and the consequent is false. In all other scenarios—when both are true, when both are false, or when the antecedent is false—the conditional statement is true. This can be counterintuitive because a statement with a false antecedent and a true or false consequent can still be true overall.

In the context of our exercise, we evaluate the statement \(\left(p \vee q\right) \wedge \sim q\right) \rightarrow p\) by considering the truth values of the constituent conditional statement. If the resulting truth table for the conditional statement shows a 'True' result in every possible scenario, except for the case where \(\left(p \vee q\right) \wedge \sim q\right)\) is true and \(p\) is false, this demonstrates the application of the conditional statement in logical deduction.