The statement "Lions live in Africa" is true. Therefore, believing it qualifies as knowledge, but not believing it is does not.
The statement "Lions live in Norway" is false. Therefore, neither believing it nor disbelieving it qualify as knowledge.
However, if you do not believe that statement, then you're effectively saying that lions do NOT live in Norway, which is true. Therefore, not believing this false statement implies that you know lions don't live in Norway. Wouldn't this option therefore qualify as knowledge?
Similarly, consider the statement "Lions do not live in Africa."
It's another false statement. So wouldn't NOT BELIEVING it qualify as knowledge?
On top of the answers given, I discovered another problem: Accepting negated beliefs as indicative of the opposite is simply a sloppy process that opens the doors to a variety of problems. If you don't believe polar bears do not live in Greenland, then we might surmise that you believe they DO live in Greenland (a true statement).
But consider this statement:
Pam dyed her hair purple in the morning.
If you don't believe that statement, then what DO you believe - that she dyed her hair orange? She dyed in the evening, or she didn't dye it at all?