To make a very serious, solemn pledge or oath, especially that one is telling the truth. The person or thing following "on" is typically someone or something that one loves or respects. Janet has sworn on her mother's grave that she wasn't the one to betray me, and I believe her. I swear on my children that I will get revenge on the people that wronged you. You have to swear on the Bible when you testify in court.