TedEd: Can you solve the troll’s paradox riddle?

A TedEd by Dan Finkel

You and your brother discover a doorway into another realm full of paradoxical creatures. But an evil troll has captured them all and is threatening to kill them. You demand it let them go, so it offers you a challenge. If you say something true, I will let them go. You are about to say it is a troll, but it grabs your brother and says that it will let him go if you say something false. Your statement must be a single sentence, and since it hates paradoxes, it cannot be a paradox. This seems impossible, but you can use coercive logic to free everyone. Coercive logic was first invented by the logician Raymond Smullyan. The trick is to “come up with a statement whose truth truth or falseness depends on what you want the troll to do.” But you still have to be careful. If you said things like, you will free the paradoxes and my brother, he could say that’s false, I’m only going to free your brother. But if you say, you will free my brother, something interesting happens. If the statement is false, then the troll would have to free your brother, but then the statement would be true, creating a paradox. The troll hates paradoxes and would never consciously create one, so the statement must be true. Then the troll will free your brother, and also the creatures because the statement is true. The creatures promise you a treasure for freeing them - if you can reach it (it’s at the top of an infinite staircase).

Bonus riddle: There are two boxes. One has treasure, the other has poison gas. One opens if you say a true statement, and the other opens to a false statement. What do you say to get the treasure and escape alive?