The following statement `:`
`( p rarr q ) rarr [ ( p rarr q ) rarr q ]` is `:`

Options

A. equivalent to `~ p rarr ~ q`
B. equivalent to `p rarr ~ q `
C. a fallacy
D. a tantalogy

Answer

To determine the logical validity of the statement \(( p \rightarrow q ) \rightarrow [ ( p \rightarrow q ) \rightarrow q ]\), we will construct a truth table and analyze the results step by step.

### Step 1: Define the Variables
Let:
- \( p \) and \( q \) be the two logical variables.

### Step 2: Construct the Truth Table
We will create a truth table for \( p \) and \( q \) to evaluate the expression.

| \( p \) | \( q \) | \( p \rightarrow q \) | \( (p \rightarrow q) \rightarrow q \) | \( (p \rightarrow q) \rightarrow [(p \rightarrow q) \rightarrow q] \) |
|---------|---------|-----------------------|---------------------------------------|---------------------------------------------------------------------|
| T | T | T | T | T |
| T | F | F | T | T |
| F | T | T | T | T |
| F | F | T | F | F |

### Step 3: Evaluate Each Column
1. **Column for \( p \rightarrow q \)**:
- If \( p \) is true and \( q \) is true, \( p \rightarrow q \) is true.
- If \( p \) is true and \( q \) is false, \( p \rightarrow q \) is false.
- If \( p \) is false, \( p \rightarrow q \) is true regardless of \( q \).

2. **Column for \( (p \rightarrow q) \rightarrow q \)**:
- If \( p \rightarrow q \) is true and \( q \) is true, this is true.
- If \( p \rightarrow q \) is false, this is true (since false implies anything).
- If \( p \rightarrow q \) is true and \( q \) is false, this is false.

3. **Final Column for \( (p \rightarrow q) \rightarrow [(p \rightarrow q) \rightarrow q] \)**:
- If \( (p \rightarrow q) \) is true and \( [(p \rightarrow q) \rightarrow q] \) is true, this is true.
- If \( (p \rightarrow q) \) is false, this is true.
- If \( (p \rightarrow q) \) is true and \( [(p \rightarrow q) \rightarrow q] \) is false, this is false.

### Step 4: Analyze the Final Column
From the truth table, we see that the final column has the values:
- T, T, T, F

This means that the statement \(( p \rightarrow q ) \rightarrow [ ( p \rightarrow q ) \rightarrow q ]\) is not a tautology since it evaluates to false in one case.

### Conclusion
The statement is not a tautology.