Why does Montresor feel he has the right to take justice into his own hands?

Montresor is insane, but you should also consider his family's motto that no one harms a Montresor without being punished. Even his family's crest is of a snake biting a heel, so acts of revenge run in his family. We don't know whether other people in his family took revenge to the extent that Montresor does against Fortunato, but he believes he has the right to take justice in his own hands partly because of his family's  name, but mostly because of his madness. He has taken revenge to the extreme, and the reader isn't sure whether Fortunato even did anything to Montresor. Because he's insane, Montresor may have just imagined that Fortunato had insulted him.