The following statement (p → q) → [(~p → q) → q] is
(p → q) → [(~p → q) → q]
Given statement is a tautology.
Let's analyze the given logical statement: . We need to determine if it is a tautology (always true), a fallacy (always false), or equivalent to one of the given implications.
Step 1: Understand the Components
The statement is an implication where the antecedent is and the consequent is .
Step 2: Construct the Truth Table
We'll evaluate the truth value for all possible combinations of p and q (True or False).
| p | q | ~p | p → q | ~p → q | (~p → q) → q | (p → q) → [(~p → q) → q] |
|---|---|---|---|---|---|---|
| T | T | F | T | T | T | T |
| T | F | F | F | T | F | T |
| F | T | T | T | T | T | T |
| F | F | T | T | F | T | T |
Step 3: Analyze the Result
The final column, which represents the entire statement, is True for all possible truth values of p and q. Therefore, the statement is a tautology.
Step 4: Check the Options
Since it is a tautology, it is not a fallacy. Let's quickly check if it's equivalent to the other options by comparing their truth tables. Both and are not tautologies (they are False in some cases), confirming our result.
Final Answer: The statement is a tautology.
Tautology: A compound statement that is always true, regardless of the truth values of its components. Common examples include (Law of Excluded Middle).
Logical Implication (→): is False only when p is True and q is False; it is True otherwise.
Negation (~): The negation of a statement has the opposite truth value.