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In to mock a mockingbird, we have the following puzzle: There are four people:

  • A is an accurate truth teller
  • B is an inaccurate truth teller
  • C is an accurate liar
  • D is an inaccurate liar

Smullyan claims that only c will claim to be b when asked "who are you?" But it seems to me like D would also claim to be b - he thinks he's anyone but d and then lies about it, so he could claim to be anyone.

What am I missing?

Here is the full puzzle:

enter image description here

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3  
Could you explain what the qualifier 'inaccurate' means? –  Ian Coley May 12 '13 at 22:04
    
In my answer, I have taken "inaccurate" to mean inconsistent with whatever philosophy governs their behavior, but hopefully @Xodarap can answer. –  Alex Wertheim May 12 '13 at 22:06
    
@Frank: I have pasted a section from the book; unfortunately it's not very clear, which I guess is the problem. –  Xodarap May 12 '13 at 22:28
1  
I’ll quote from an earlier page ($28$): an accurate truthteller ‘is completely accurate in all his judgments; all true propositions he knows to be true, and all false propositions he knows to be false’. Someone who is inaccurate is ‘totally inaccurate in his judgments; all true propositions he believes to be false and all false propositions he believes to be true’. –  Brian M. Scott May 12 '13 at 22:32

1 Answer 1

up vote 3 down vote accepted

Note: in this answer, I have taken "inaccurate" to mean inconsistent with whatever philosophy governs their behavior.

Well, going through them, we see that, when asked "Who are you?":

A will answer A, as he is an accurate truth teller.

B, in order to be inconsistent with the fact that he is a truth teller, can answer with anyone other than B, as he is inaccurate and cannot think he is B, but will tell you whom he truly (but inaccurately) thinks he is.

C is an accurate liar, and so when asked who he is, he will lie, and lie accurately, i.e, he can answer someone other than C. Thus, he can consistently answer that he is B.

D is an inaccurate liar, i.e., he will try to lie about who he is, but because he is inaccurate, in order to be inconsistent with his philosophy, he must name himself when he is asked who he is. If he were to name anyone other than D, then he would be an accurate liar. As the problem statement says, he is both deluded and dishonest, and so he will try to give you inaccurate information but can't - in other words, he tells the truth (without meaning to!)

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2  
will answer seems to be "provable", while your answer seems to be "consistently". There's a difference, as we know. –  Asaf Karagila May 12 '13 at 22:06
    
Right you are. I'll try to edit my answer accordingly. –  Alex Wertheim May 12 '13 at 22:07
    
Suppose d thinks he's a, then lies and says he's b. Why wouldn't that be an inaccurate lie? –  Xodarap May 12 '13 at 22:09
    
It all depends on what inaccurate means, which seems to be the trouble here. I have taken inaccurate to be relative to expected behavior from a liar, but it seems you are taking inaccurate to mean inaccurate information or preconception on the part of the person being asked. Edit: additionally, even if D thinks he is A and lies and says B, that is still an accurate lie. He is still not B. –  Alex Wertheim May 12 '13 at 22:12

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