The “If p then q” implication is true when both p and q are true. So, if this implication is true, when is it false? It is false when the same true p implies the opposite of the same true q (the law of excluded middle). After all, the same true p cannot imply the false q while at the same time implying the true q, can it?
What about the other two implications where p is false, that is “If the false p, then the true q”, and “If the false p, then the false q”? And the answer is: why should we even bother? They are not relevant to our original implication “If the true p, then the true q”. They are different statements. There is nothing in our original statement about q being false.
Our conditional is a kind of minimal conditional. It captures only those scenarios that are strictly relevant, leaving others out of consideration - “leaving others out of consideration” but not jumping the gun and consigning them to falsehood. But if we choose to probe, it might be instructive to think of the “If p then q” implication like this: truth can imply only truth, falsehood can imply anything.
So, suppose someone says:
You can roll a seven on a single roll of a dice.
You roll out your eyes and think: “Aha, we are not on the same planet,” or “One of us excluding me is away with the fairies.” But to humour them, you say:
Sure, if a coin has three sides, then you can roll a seven on a single roll of a dice.
You have just shown them under what sort of circumstances (a world of three-sided coins) their sentence is true!
The sentence:
You can roll a six on a single roll of a dice.
is certainly true. If it is true in the demanding world of two-sided coins, it must be even more true in the less demanding world of three-sided coins. So, the sentence:
If a coin has three sides, then you can roll a six on a single roll of a dice.
is true too.