I cannot understand the context and formulation of these problems.



> 
> The inhabitants ask only questions answerable by yes or
> no. Each inhabitant is one of two types, A and B. Those of
> type A ask only questions whose correct answer is yes; those
> of type B ask only questions whose correct answer is no. For
> example, an inhabitant of type A could ask, "Does two plus
> two equal four?" But he could not ask whether two plus two
> equals five. An inhabitant of type B could not ask whether
> two plus two equals four, but he could ask whether two plus
> two equals five, or whether two plus two equals six.
> 
> 
> 


It follows from this that no native can ask if he belongs to type B, since the correct answer would make it the wrong question for him to ask.


Later, a twist is introduced:



> 
> A STRANGE ENCOUNTER
> The next three exchanges I witnessed on the Island of Questioners were the most bizarre >of all! Three patients from one of the insane asylums of Chapter 3 escaped and decided >to pay a visit to the island. We recall that a patient from one of
> these asylums could be sane or insane and that the sane ones
> are totally accurate in all their beliefs, and the insane ones
> totally inaccurate in all their beliefs. We also recall that the
> patients, whether sane or insane, are always truthful; they
> never make statements unless they believe them to be true.
> 
> 
> 


Problem 11:



> 
> On the day after their arrival, one of the patients, whose
> name was Arnold, met a native of the island. The native
> asked him, "Do you believe I am of type B?"
> What can be deduced about the native, and what can be
> deduced about Arnold?
> 
> 
> 


Solution given in the book:



> 
> This strikes me as the funniest problem of this chapter,
> since nothing can be deduced about the native who asked the
> question; but as to Arnold, though he never opened his
> mouth (as far as we know), he must be insane I The fact is that
> no native could ask a sane person whether he believes the
> native to be of type B, because asking a sane person whether
> he believes such-and-such to be the case is tantamount to
> asking whether such-and-such really is the case, and no native can ask whether he is of >type B. So no native X could ask
> a sane person whether he believes X is of type B.
> On the other hand (and we need this fact for a subsequent
> problem), any native X could ask an insane person whether
> he believes X is of type B, because asking that of an insane
> person is tantamount to X asking whether X is of type A,
> which, as we have seen, any native X can do.
> 
> 
> 


So we can assume natives can ask questions if the answer fits their type, even if that answer is incorrect?



R. Smullyan's "Lady or the tiger", Ch. 5, Island of Questioners, problems 11-12
**Q:** *So we can assume natives can ask questions if the answer fits their type, even if that answer is incorrect?*


No, it is stated clearly that "Those of type A ask only questions whose correct answer is yes; those of type B ask only questions whose correct answer is no."


It all works out: Arnold is insane, he is "totally inaccurate in all his beliefs". If the inhabitant is of type A, the correct answer by Arnold to the question "Do you believe I am of type B? is "Yes", since that is indeed what Arnold believes. Similarly, if the the inhabitant is of type B, the correct answer is "No".


But if Arnold is sane, the answers would be "No" and "Yes", respectively, which is not allowed. So Arnold must be insane.


The key thing here is that the question is not "Am I of type B", but "*Do you believe* I am of type B".


