A puzzle of identical twins

Following is an excerpt from
5000 B.C. and Other Philosophical Fantasies
by R.Smullyan

Suppose there are two identical twin brothers, one who always lies and the other who always tells the truth. Now, the truth teller is also totally accurate in all his beliefs; all true propositions he believes to be true and all false propositions be believes to be false. The lying brother is totally inaccurate in his beliefs; all true propositions he believes to be false, and all false propositions he believes to be true. The interesting thing is that each brother will give the same answer to the same question. For example, suppose you ask whether two plus two equals four. The accurate truth teller knows that it is and will truthfully answer yes. The inaccurate liar will believe that two plus two does not equal four (since he is inaccurate) and will then lie and say that it does; he will also answer yes.

The situation is reminiscent of an incident I read about in a textbook on abnormal psychology: The doctors in a mental institution were thinking of releasing a certain schizophrenic patient. They decided to give him a test under a lie detector. One of the questions they asked him was, "Are you Napoleon?" He replied, "No." The machine showed that he was lying!

Getting back to the twin brothers, two logicians were having an argument about the following question: Suppose one were to meet one of the two brothers alone. Would it be possible by asking him any number of yes-no questions to find out which one he is? One logician said, "No, it would not be possible because whatever answers you got to your questions, the other brother would have given the same answers." The second logician claimed that it was possible to find out. The second logician was right, and the puzzle has two parts: (1) How many questions are necessary?; and (2) more interesting yet, What was wrong with the first logician's argument? (Readers who enjoy doing logic puzzles might wish to try solving this one on their own before reading further.)

To determine which brother you are addressing, one question is enough; just ask him if he is the accurate truth teller. If he is, he will know that he is (since he is accurate) and truthfully will answer yes. If he is the inaccurate liar, he will believe that he is the accurate truth teller (since he is inaccurate in his beliefs), but then he will lie and say no. So the accurate truth teller will answer yes and the inaccurate liar no to this question.

Now what was wrong with the first logician's argument; don't the two brothers give the same answer to the same question? They do, but the whole point is that if I ask one person, "Are you the accurate truth teller?" and then ask another, "Are you the accurate truth teller?" I am really asking two different questions since the identical word you has a different reference in each case.