In logic, given P→Q and Q is true, can we conclude P?

In logic, given P→Q and Q is true, can we conclude P?

• A. Yes, P must be true.
• B. P cannot be concluded from Q.
• C. Not necessarily; cannot deduce P.
• D. Q implies P.

Answer: (C)

An implication P→Q says that whenever P happens, Q must happen; but it does not say anything about P when Q is already true. If Q is true, the implication holds whether P is true or false, so you cannot conclude P from P→Q and Q being true. For example, “If it rains, the street will be wet.” The street is wet. It could have rained, or it could be wet for another reason (like a sprinkler). So P may be true or false. To deduce P, you’d need additional information, such as Q→P.

An implication P→Q says that whenever P happens, Q must happen; but it does not say anything about P when Q is already true. If Q is true, the implication holds whether P is true or false, so you cannot conclude P from P→Q and Q being true.

For example, “If it rains, the street will be wet.” The street is wet. It could have rained, or it could be wet for another reason (like a sprinkler). So P may be true or false. To deduce P, you’d need additional information, such as Q→P.