In other words, if someone tells me "A implies B", and I say "fair enough, but look - A is false". They say "but if A was true, B would be true too!", to which I reply "maybe so, but perhaps if A was true B would be false - since A is false, it's impossible to tell".

If A implies B, and A is false then "If A was true then B would be true" is still true - that is the definition of A implies B. "Perhaps if A was true B would be false" is a contradiction of our assumption that A implies B. We do know that if A was true, then B would also be true. "since A is false, it's impossible to tell" <-- that statement is incorrect.

edit: Consider phlips' raincoat example. I tell you that "If it will rain tomorrow, then I will wear a raincoat." Now suppose it does not rain tomorrow - then you say "Perhaps if it had rained today, you would not have worn a raincoat. Since it didn't rain, it's impossible to tell." But we can tell - I told you that if it would have rained, I would have worn a raincoat, no matter what.