The following statement `(p to q) to [(~p to q) to q]` is

Options

A. a fallacy
B. a tautology
C. equivalent to `~p toq `
D. equivalent to `p to ~q`

Explanation

To determine whether the statement \((p \to q) \to [(\neg p \to q) \to q]\) is a tautology, we will construct a truth table. Here are the steps to solve the problem:

### Step 1: Identify the components of the statement
The statement consists of:
- \(p\)
- \(q\)
- \(\neg p\) (not p)
- \(p \to q\) (p implies q)
- \(\neg p \to q\) (not p implies q)

### Step 2: Create the truth table
We will create a truth table with columns for \(p\), \(q\), \(\neg p\), \(p \to q\), \(\neg p \to q\), and the final expression \((p \to q) \to [(\neg p \to q) \to q]\).

| \(p\) | \(q\) | \(\neg p\) | \(p \to q\) | \(\neg p \to q\) | \((\neg p \to q) \to q\) | \((p \to q) \to [(\neg p \to q) \to q]\) |
|-------|-------|------------|--------------|-------------------|---------------------------|-------------------------------------------|
| 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 | F | T |

### Step 3: Fill in the truth values
1. **For \(p\) and \(q\)**: List all combinations of truth values (T for true, F for false).
2. **Calculate \(\neg p\)**: The negation of \(p\).
3. **Calculate \(p \to q\)**: This is true unless \(p\) is true and \(q\) is false.
4. **Calculate \(\neg p \to q\)**: This is true unless \(\neg p\) is true and \(q\) is false.
5. **Calculate \((\neg p \to q) \to q\)**: This is true unless \((\neg p \to q)\) is true and \(q\) is false.
6. **Calculate the final expression**: \((p \to q) \to [(\neg p \to q) \to q]\).

### Step 4: Analyze the final column
In the final column, we see that all entries are true (T). This means that the statement \((p \to q) \to [(\neg p \to q) \to q]\) is true for all possible truth values of \(p\) and \(q\).

### Conclusion
Since the final expression is always true, we conclude that the statement is a tautology.