The statement `~(pharr ~q)` is

Options

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

Explanation

To solve the statement `~(P ↔ ~Q)`, we will analyze it step by step using a truth table.

### Step 1: Identify the Variables
We have two variables:
- P
- Q

### Step 2: Determine the Negation of Q
We need to find the negation of Q, which is denoted as `~Q`. The truth values for Q will be:
- T (True)
- F (False)

Thus, the truth values for `~Q` will be:
- If Q is T, then `~Q` is F.
- If Q is F, then `~Q` is T.

### Step 3: Create the Truth Table
We will create a truth table with the columns for P, Q, `~Q`, `P ↔ ~Q`, and `~(P ↔ ~Q)`.

| P | Q | ~Q | P ↔ ~Q | ~(P ↔ ~Q) |
|-----|-----|-----|--------|------------|
| T | T | F | F | T |
| T | F | T | T | F |
| F | T | F | T | F |
| F | F | T | F | T |

### Step 4: Fill in the Truth Values
- For `P ↔ ~Q`:
- T ↔ F = F
- T ↔ T = T
- F ↔ F = T
- F ↔ T = F

- For `~(P ↔ ~Q)`:
- Negation of F = T
- Negation of T = F
- Negation of T = F
- Negation of F = T

### Step 5: Analyze the Result
The final column `~(P ↔ ~Q)` has the truth values:
- T, F, F, T

### Conclusion
The statement `~(P ↔ ~Q)` is neither a tautology (always true) nor a contradiction (always false). It is a contingent statement as it can be true or false depending on the truth values of P and Q.