Problem 47 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

Given the rules for logical connectives, the final truth table will show that each row in the column for \((p \rightarrow q) \leftrightarrow (\sim p \vee q)\) is true, regardless of the truth values of 'p' and 'q'. Hence, the statement \((p \rightarrow q) \leftrightarrow (\sim p \vee q)\) is a tautology.

Step 1: Identify the Components

The statement given is \((p \rightarrow q) \leftrightarrow (\sim p \vee q)\). Here, 'p' and 'q' are simple statements, \(\rightarrow\) stands for 'implies', \(\leftrightarrow\) is the 'if and only if' connective, \(\sim\) represents 'not', and \(\vee\) is the 'or' connective.

Step 2: Construct the Initial Truth Table

The initial truth table will include columns for 'p' and 'q'. Each of these columns will contain four rows, reflecting all possible combinations of truth values for two components: TT, TF, FT, FF.

Step 3: Add Columns for the Parts of the Complex Statement

Next add columns for \(\sim p\), \(p \rightarrow q\), \(\sim p \vee q\), and finally, \((p \rightarrow q) \leftrightarrow (\sim p \vee q)\). The values in these columns are determined based on the truth values of 'p' and 'q' and the logical connectives' definitions.

Step 4: Evaluate the Main Statement

Evaluate the values under the column \((p \rightarrow q) \leftrightarrow (\sim p \vee q)\). If all the entries in this column are 'T', the statement is a tautology. If they are all 'F', it is a self-contradiction. Otherwise, if the entries are a mix of 'T's and 'F's, the statement is neither a tautology nor a contradiction.

Key Concepts

Logical Connectives

Logical connectives are tools that combine, modify, or relate statements within logical propositions. In our exercise, we deal with several connectives:

• Implication (\(\rightarrow\)): Indicates that one statement leads to another. If "p implies q" (\(p \rightarrow q\)), it means whenever p is true, q must also be true.
• Negation (\(\sim\)): Simply flips the truth value of a statement. So, "not p" (\(\sim p\)) is true when p is false, and vice versa.
• Disjunction (\(\vee\)): Represents an "or" situation. "p or q" (\(p \vee q\)) is true if either p, q, or both are true.
• Biconditional (\(\leftrightarrow\)): Expresses "if and only if". It tells us the statements on either side must both be true, or both be false, for the compound statement to be true.

These connectives are foundational in logic, allowing us to build complex expressions from simpler ones.

Tautology

A tautology is a logical statement that is true in every possible situation. The statement holds true irrespective of the truth values of the individual components involved. In our exercise, the complete process of evaluating whether \((p \rightarrow q) \leftrightarrow (\sim p \vee q)\) is a tautology involves:

• Construction of a truth table that displays all possible truth value combinations for the variables p and q.
• Application of logical operations to assess the truth of the entire expression for each combination.

If the truth table column for \((p \rightarrow q) \leftrightarrow (\sim p \vee q)\) results entirely in 'true,' the statement is a tautology by definition. Tautologies are critical in logical proofs because they reveal elements of statements that are always true.

Self-Contradiction

A self-contradiction is a logical statement that is false in every situation. No matter the input truth values of its components, the entire statement will always evaluate as false. This provides a sharp contrast to a tautology. For our present challenge, determining if \((p \rightarrow q) \leftrightarrow (\sim p \vee q)\) is a self-contradiction works through the truth table as follows:

• Examine each possible truth value setup for p and q.
• Assess the resulting value for the statement under all combinations.

If all entries in the truth table's statement column result in 'false,' the expression is a self-contradiction. Recognizing these helps identify statements that are inherently flawed or logically untenable.

Truth Values

Truth values are the building blocks of logical expressions. They are simply the data that tells us whether a statement is true or false. In constructing truth tables, we assign truth values to each part:

• True (T): Indicates the statement is correct.
• False (F): Indicates the statement is incorrect.

For any logical proposition, the truth values can be lined up in different combinations to see how the connectives behave:

• For two statements (like p and q), there are four possible combinations: True/True, True/False, False/True, and False/False.
• This forms the basis for constructing a comprehensive truth table, showing all possible truth values for the expressions being evaluated.

Understanding truth values is essential for analyzing the validity of complex logical statements.