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

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Multiple Choice

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

Explanation:
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.

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