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

Options

A. ` p vee (p wedge q ) `
B. ` (p vee q ) wedge (p wedge q ) `
C. ` (p wedge q ) to p `
D. ` (p vee q ) to ( p vee q ) `

Explanation

To solve the question "Which one of the following statement is neither a tautology nor a fallacy?", we need to analyze the options provided (though they are not explicitly listed in the question). We will determine whether each statement is a tautology (always true), a fallacy (always false), or neither.

### Step-by-Step Solution:

1. **Understanding Tautology and Fallacy**:
- A **tautology** is a statement that is true in every possible interpretation. For example, \( P \lor \neg P \) (either P is true or P is not true).
- A **fallacy** is a statement that is false in every possible interpretation. For example, \( P \land \neg P \) (P is true and P is not true).

2. **Analyzing the Statements**:
- Let's denote the statements as \( A \), \( B \), \( C \), and \( D \) (assuming these are the options).
- We will evaluate each statement to see if it is a tautology, a fallacy, or neither.

3. **Evaluating Statement A**:
- Assume \( A = P \lor Q \) (where \( P \) and \( Q \) are propositional variables).
- Construct a truth table:
- \( P = T, Q = T \) → \( A = T \)
- \( P = T, Q = F \) → \( A = T \)
- \( P = F, Q = T \) → \( A = T \)
- \( P = F, Q = F \) → \( A = F \)
- Since \( A \) is true in 3 cases and false in 1 case, it is neither a tautology nor a fallacy.

4. **Evaluating Statement B**:
- Assume \( B = P \land Q \).
- Construct a truth table:
- \( P = T, Q = T \) → \( B = T \)
- \( P = T, Q = F \) → \( B = F \)
- \( P = F, Q = T \) → \( B = F \)
- \( P = F, Q = F \) → \( B = F \)
- Since \( B \) is true in 1 case and false in 3 cases, it is neither a tautology nor a fallacy.

5. **Evaluating Statement C**:
- Assume \( C = P \to Q \).
- Construct a truth table:
- \( P = T, Q = T \) → \( C = T \)
- \( P = T, Q = F \) → \( C = F \)
- \( P = F, Q = T \) → \( C = T \)
- \( P = F, Q = F \) → \( C = T \)
- Since \( C \) is false in 1 case and true in 3 cases, it is neither a tautology nor a fallacy.

6. **Evaluating Statement D**:
- Assume \( D = P \lor Q \to P \lor Q \).
- This is a tautology because it states that if \( P \lor Q \) is true, then \( P \lor Q \) is true.
- Hence, this is a tautology.

7. **Conclusion**:
- From the analysis, statements \( A \) and \( B \) are neither tautologies nor fallacies, while \( C \) is also neither, and \( D \) is a tautology.
- Therefore, the correct answer is that options \( A \), \( B \), and \( C \) are neither tautologies nor fallacies.