So yeah, basically from what i've read, and i've checked multiple sources on this, is that A -> B = ¬A V B.
So to really see where the confusion lies, I'll first state where it doesn't. Where A -> B, then this is saying that if A true, then B is true.
Thus 1 -> 1 is true, since this is just confirming the original statement.
1 -> 0 is false, since than A does not imply B.
Now, what I don't get is this:
A -> B | Truth value 0 0 | 1 0 1 | 1
Now, since the truth value lies in the confirmation or falsificion of A implying B, then a value of 1 means, 'yes, A implies B', and 0 means 'no, A does not imply B'. Now whilst this may sound blindingly obvious, this means that if A does not occur, this confirms (or perhaps reconfirms) that A implies B. A implies B given that A does not happen.
So for instance, if A = I am a qualified chef. B = I can cook well. Then if we let A -> B, then if I am not a qualified chef, then by me being an unqualified chef, this affirms the conditional that since I am a qualified chef, thus I can cook well. But I just said that I am NOT a chef.
I think it's the case that A could imply B in the absense of A, whether B occurs or not, but we just don't know. Why then don't we just say that the conditional leads to a partial truth table, where the only conclusions we verify are the one's where the antecedent occurs.
I mean, B is even called a consequent, one due to A, so we can't know about the consequences of A (or lack thereof) given that A does not occur.
And to be honest, I don't know why this in the norm. There's a reason why the way we handle the absense of A utterly confuses those new to logic (including myself).