That's a problem right there. Maybe that made sense to the Greeks, but it definitely doesn't make any sense in the 21st century. "Knowing" falsehoods is something we broadly acknowledge that we all do.
To say that this is not "knowing" is (as another commenter noted) hair-splitting of the worst kind. In every sense it is a justified belief that happens to be false (we just do not know that yet).
The strength of the justification is, I would suggest, largely subjective.
This is one of the best questions ever, not just for philosophers, but for all us regular plebes to ponder often. The number of things I know is very very small, and the number of things I believe dramatically outnumbers the things I know. I believe, but don’t know, that this is true for everyone. ;) It seems pretty apparent, however, that we can’t know everything we believe, or nothing would ever get done. We can’t all separately experience all things known first-hand, so we rely on stories and the beliefs they invoke in order to survive and progress as a species.
Also I don't think this definition fits with people's intuition. At least, certainly not my own. There are times where I realise I'm wrong about something I thought I knew. When I look back, I don't say "I knew this, and I was wrong". I say "I thought I knew this, but I didn't actually know it".