Why is $a \implies b$ is true when $a$ is false

Question

I understand that:

$True \implies True$, is true.

$True \implies False$, is False.

But why is it that

$False \implies True$, is True.

and

$False \implies False$, is True.

If $a$ is false I don't understand how we can say $a \implies b$ is true.

Answer 1

I was taught to think of implications like contracts. For example, say I told you "if you wash my car, I'll pay you ten dollars." Then the only way I could end up lying - the only way this statement could be false - is if I break my contract to you, and I stiff you after you wash my car. If you don't wash my car, I never lied to you, whether or not I end up paying you, and the contract always holds if the first part of the conditional is false.

Answer 2

What do you think of the statement "If 638,544 is divisible by 14, then it is divisible by 7"?

You probably already agreed that this statement was true before realizing that it was of the form $F \Rightarrow F$.