Just to clarify, Case 3 and Case 4 must have flawed reasoning in order to reconcile my proof with the author's.
I have been having a problem with a particular riddle from Raymond Smullyan and I can't seem to reconcile my proof with his solution. I am more inclined to think I am wrong, but to myself my proof looks quite convincing. Maybe you'll be able to point out the flaw in my logic.
Background: There are natives on the Isle of Questioners that are either of Type A or Type B. People of Type A may only ask questions to which the correct answer is "Yes" such as "Does 2 + 2 = 4?". People of Type B may only ask questions to which the correct answer is "No" such as "Does 2 + 2 = 5?". Furthermore there are people who have wandered onto the island who are either sane or insane. People who are sane are completely sane and believe all true things, they will always answer honestly and accurately. People who are insane are completely insane and believe all untrue things, they will always answer honestly and inaccurately. And all parties involved have perfect knowledge of the universe, but whether they are correct or incorrect about it depends on their sanity.
Problem: You meet a native and a sane or insane person named Thomas. The native asks Thomas, "Do you believe I am the type who could ask you whether you are insane?" What can be deduced about the native, and what can be deduced about Thomas?
My (failed?) attempt at a solution: There are four cases: the native is Type A and Thomas is sane, the native is Type A and Thomas is insane, the native is Type B and Thomas is sane, and the native is Type B and Thomas is sane. (In Case 1 and 2 my proof follows the author's almost exactly. Smullyan simply says of Case 3 and 4, "The only way out of the contradiction is that the native must be of Type B rather than Type A" but I still reach a contradiction.)
Case 1: Suppose that the native is Type A. Then the answer to the question is "Yes" and Thomas must believe that the native can ask whether he is insane. Now suppose that Thomas is sane, which means that Thomas must be correct in his belief. So the correct answer to the question "Are you insane?" when posed to Thomas must be "Yes" since the native could ask the question. Thus Thomas is insane as well as sane and this is a contradiction. The first case is impossible.
Case 2: Suppose that the native is Type A. Then the answer to the question is "Yes" and Thomas must believe that the native can ask whether he is insane. Now suppose Thomas is insane, which means that Thomas must be incorrect in his belief. So the correct answer to the question "Are you insane?" when posed to Thomas must be "No" since the native could not ask the question. (Thomas would indeed say "No" but be incorrect because he is insane and believes he is sane.) Thus Thomas is sane as well as insane and this is a contradiction. The second case is impossible.
Case 3: Suppose that the native is Type B. Then the answer to the question is "No" and Thomas must believe that the native can not ask whether he is insane. Now suppose Thomas is sane, which means that Thomas must be correct in his belief. So the correct answer to the question "Are you insane?" when posed to Thomas must be "Yes" since the native could not ask the question. Thus Thomas is insane as well as sane and this is a contradiction. The third case is impossible.
Case 4: Suppose that the native is Type B. Then then answer to the question is "No" and Thomas must believe that the native can not ask whether he is insane. Now suppose Thomas is insane, which means that Thomas must be incorrect in his belief. So the correct answer to the question "Are you insane?" when posed to Thomas must be "No" since the native could ask the question. Thus Thomas is sane as well as insane and this is a contradiction. The fourth case is impossible.
Can you see a flaw in my logic? I've tried to re-approach this problem and poke holes in my own proof, but I can't reach any other conclusion other than this encounter between the native and Thomas must have been impossible.
Best Answer
We can analyze this question using boolean algebra. Let $p$ represent whether the native is type A (true) or type B (false), and let $q$ represent whether Thomas is sane (true) or insane (false).
The statement "Can the native ask the question 'Is Thomas insane?'" is logically equivalent to $p\oplus q$, where $\oplus$ is XOR or exclusive or.
The statement "Does Thomas believe that the native can ask the question 'Is Thomas insane?'" is logically equivalent to $(p\oplus q)\Leftrightarrow q$, which is equivalent to $\neg p$. Here, $\Leftrightarrow$ represents "if and only if".
The statement "Can the native ask 'Do you believe I am the type who could ask you whether you are insane?'" is logically equivalent to $\neg p\Leftrightarrow p$, which is always false.