In the kingdom of Boolistan, every inhabitant is either a Knight, Knave or Normal. Knights can only make true statements, Knaves can only lie, and Normals must either tell the truth or lie.
Warmup: The local tavern only allows Normals (no one can relax around Knights and Knaves). What can a Normal say to prove their identity?
Challenge: Only knights can dine at King Arthur's Round Table. What can a Knight say to prove their identity?
Remarks: In conventional logic, where every statement is either true or false, the challenge is impossible (since Normals can say anything). To make this doable, we allow circular self-referential statements, like the famous example, "this statement is false". Formally, a circular self-referential statement is an equation of the form $$ s = f(x_1,\dots,x_n,s) $$ where $x_1,\dots,x_n$ are grounded logical propositions (like "I am a Knave"), $f$ is a Boolean function, and $s$ is a Boolean variable. We say that such a statement is True if setting $s=$ True makes the equation hold, and similarly say it is False if $s=$ False is a solution. This means some such statements are both True and False, while others are neither. For example, "this statement is false" would be the equation $s=\neg s$, which has no solutions, so is neither True nor False. On the other hand, "this statement is true" would be $s=s$, which is both True and False.
We then allow knights to say any True statement, Knaves to say any False statement, while Normals can say a statement as long as it is True or False or both.