Question

If $ P $ is false, then what is the value of the proposition $ (P \rightarrow Q) \rightarrow R $?

• A. \( P \)
• B. \( Q \)
• C. \( R \)
• D. True

Hint:

In an implication \( P \rightarrow Q \), if \( P \) is false, the implication is always true regardless of \( Q \).

Answer 1

The Correct Option is C

The proposition \( (P \rightarrow Q) \rightarrow R \) is a conditional statement. In logic: - \( P \rightarrow Q \) is true if either \( P \) is false or \( Q \) is true. - Since \( P \) is false, the statement \( P \rightarrow Q \) is true (because a false hypothesis always makes the implication true).
Thus, the whole expression \( (P \rightarrow Q) \rightarrow R \) depends on \( R \). Since \( (P \rightarrow Q) \) is true, the value of the proposition is equivalent to the value of \( R \).
Therefore, the value of the proposition is \( R \).