The way it ties into philosophical implication can be confusing.
This cleared it up for me - forget math! Just remember what you thought when you first heard about implication.
In common sense implication (let’s think of A => B as if A, then B) B must be true if A is true.
Think about it. I tell you the sky will explode if birds chirp.
A => B form is If birds chirp, then the sky will explode.
A = birds chirp
B = sky explodes
If birds do indeed chirp, then the sky MUST EXPLODE.
In other words, if you told me that birds chirped and the sky did not explode (True => False), I would tell you that you are lying.
T | F | F
Back to the math. Common sense wise it is impossible for A to be true without B being true. It’s a lie if that’s the case. Always. So, in math, we say that statement is logically FALSE.
But what about the others?
T | T | T
Is easy. It follows the common sense rules. If A is true, and B is true, then you listened to me, and I was right. Everything checks out!
F | T | T
??? This says birds did not chirp, but the sky did explode! How is this true?
Common sense - we simply say that when birds chirp, the sky will explode. We don’t say that birds chirping is the ONLY thing that causes the sky to explode. A nuclear bomb could hit the moon for all we know! I could not say you are lying.
So, mathematicians say ‘that could be true, so it’s true’.
F | F | T
If you told me birds did not chirp and the sky did not explode, I couldn’t tell you that you are lying either, so that checks out!
Hopefully that clears it up. This revelation cleared it right up for me.