Which one of the following statement is neither a tautology nor a fallacy?

Options

A. `pvv(p^^q)`
B. `(pvvq)^^(p^^q)`
C. `(p^^q)to p`
D. `(pvvq)to(pvvq)`

Explanation

To determine which of the given statements is neither a tautology nor a fallacy, we will analyze each statement using truth tables. A tautology is a statement that is always true, while a fallacy is a statement that is always false.

### Step-by-Step Solution:

1. **Identify the Statements**:
We need to evaluate the following statements:
- Statement 1: \( P \land Q \)
- Statement 2: \( P \lor (P \land Q) \)
- Statement 3: \( (P \land Q) \implies P \)
- Statement 4: \( (P \lor Q) \implies P \)

2. **Create a Truth Table**:
Since we have two variables \( P \) and \( Q \), we will have \( 2^2 = 4 \) combinations of truth values. The combinations are:
- \( P = T, Q = T \)
- \( P = T, Q = F \)
- \( P = F, Q = T \)
- \( P = F, Q = F \)

We will evaluate each statement for these combinations.

3. **Evaluate Statement 1: \( P \land Q \)**:
| P | Q | \( P \land Q \) |
|-------|-------|------------------|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |

- Result: True only when both \( P \) and \( Q \) are true. This is **not a tautology** (not always true) and **not a fallacy** (not always false).

4. **Evaluate Statement 2: \( P \lor (P \land Q) \)**:
| P | Q | \( P \land Q \) | \( P \lor (P \land Q) \) |
|-------|-------|------------------|---------------------------|
| T | T | T | T |
| T | F | F | T |
| F | T | F | F |
| F | F | F | F |

- Result: True when \( P \) is true. This is **not a tautology** and **not a fallacy**.

5. **Evaluate Statement 3: \( (P \land Q) \implies P \)**:
| P | Q | \( P \land Q \) | \( (P \land Q) \implies P \) |
|-------|-------|------------------|-------------------------------|
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | T |

- Result: Always true. This is a **tautology**.

6. **Evaluate Statement 4: \( (P \lor Q) \implies P \)**:
| P | Q | \( P \lor Q \) | \( (P \lor Q) \implies P \) |
|-------|-------|-----------------|------------------------------|
| T | T | T | T |
| T | F | T | T |
| F | T | T | F |
| F | F | F | T |

- Result: Not always true and not always false. This is **not a tautology** and **not a fallacy**.

7. **Conclusion**:
The statements that are neither tautology nor fallacy are:
- Statement 1: \( P \land Q \)
- Statement 2: \( P \lor (P \land Q) \)
- Statement 4: \( (P \lor Q) \implies P \)

### Final Answer:
The statements that are neither tautology nor fallacy are:
- \( P \land Q \)
- \( P \lor (P \land Q) \)
- \( (P \lor Q) \implies P \)