Question

If p→q is true and q is true, then p is:

a. Always true
b. Never true
c. Sometimes true
d. None of the above

Answer 1

The correct answer is c. Sometimes true because the conditional statement p→q being true does not guarantee the truth of p when q is true, as other factors could cause q to be true.

Step-by-step explanation:

If p→q is true and q is true, then p can only be confirmed as true if the logical statement was a biconditional, where the truth of q would imply the truth of p. However, since the statement is a conditional and not a biconditional, p can be either true or false without impacting the truth value of the conditional p→q. Therefore, the correct answer is c. Sometimes true.

To illustrate this, consider an example where p represents 'It is raining,' and q represents 'The ground is wet.' It is possible for the ground to be wet because of reasons other than rain, such as a sprinkler system. Therefore, the ground being wet does not necessarily mean it has rained. This example supports the notion that p does not have to be true even if q is true, and the original conditional statement p→q remains true.