So I ran into the "drinker's paradox" the other day, which compelled me to learn about the nature of the logical material conditional. But when I studied the truth table (below) I got pretty confused.
The first two rows make sense, but I don't understand the justification for the bottom two.
It seems to me that if you want to test the truth of the material conditional, P must be true; otherwise you can't know how its truth affects the truth of Q (after all, P → Q literally means "if P is true, Q is true").
So, if P is false, how would you go about proving that the material conditional is true? And if you can't, then why is it asserted to be true in the truth table? Am I being an idiot? Thanks.