Like Ragtime Mouse, False, and its counterpart, True, are not regular enemies and are not designed to have their HP depleted. Attacking False indicates that the player answers False to the question, and the player can hit it even if the attack misses. If the player attacks all opponents with a group-cast ability then the battle will end as if the Ragtime Mouse is attacked and it will be as if the player has not answered the question.
Of the sixteen potential questions asked, half of the answers are False. The answer to the final and most important question is not False.