The phrase "sware up and down" means to make a solemn vow or pledge, usually in a forceful or assertive manner. Antonyms for this phrase would include words that convey doubt, uncertainty, or hesitation. Such antonyms might include "waffle," "dodge," "equivocate," "hem and haw," or "beat around the bush." Some other possible antonyms could include "waver," "fluctuate," "stutter," "mumble," or "stammer." Whatever the word, antonyms for "sware up and down" would express a much softer, less confident tone, suggesting a lack of conviction or commitment to one's words.