logic – Question about a puzzle in ch. 2 of “to mock a mockingbird”

In chapter 2, the author introduced Nelson Goodman Principle, which says if you ask someone who either always lies or always tells the truth: “Are you the type who could claim A is true?”, he/she will always tell the truthfulness of A. So for problem 3, the given correct answer is “Is Arthur truthful?”, which is supposed to be equivalent to the sentence formulated with Nelson Goodman Principle: “Are you the type who could claim that you are Arthur?”

However, it is really hard for me see the equivalence of the two formulations. I mean, the consequences of the two formulations are the same. But on the surface, I couldn’t tell the equivalence just by transforming the text from one to the other. Can anyone help me to understand it? Thank you!

PS: Ch. 2 Problem 3 in the book: you meet two identical twins, one of whom is Arthur. One of the two always lies. The other always tells the truth. You have no idea who is who or who lies who doesn’t. You are only allowed to ask one question with 3 words to either of the two in order to find out who is Arthur.