You are visiting the island described in Example 2.3.14 and have the following e

Question

You are visiting the island described in Example 2.3.14 and
have the following encounters with natives:

a. Two natives A and B address you as follows:
A says: Both of us are knights.
B says: A is a knave.
What are A and B?

b. Another two natives C and D approach you but only C
speaks.
C says: Both of us are knaves.
What are C and D?

c. You then encounter natives E and F.
E says: F is a knave.
F says: E is a knave.
How many knaves are there?

Example 2.3.14 Knights and Knaves The logician Raymond Smullyan describes an island containing two types of people knights who always tell the truth and knaves who always lie. You visit the island and are approached by two natives who speak to you as follows: A says: B is a knight. B says: A and I are of opposite type What are A and B? Solution A and B are both knaves. To see this, reason as follows: Suppose A is a knight. What A says is true. B is also a knight. What B says is true A and B are of opposite types. That's what B said by definition of knight That's what A said. by definition of knight .:. We have arrived at the following contradiction: A and B are both knights and A and B are of opposite type The supposition is false. A is not a knight. A is a knave. by the contradiction rule negation of supposition Raymond Smullyan (born 1919) by climination: It's given that all inhabitants are knights or knaves, so since A is not a knight, A is a knave. What A says is false .:. B is not a knight. Bis also a knave by elimination This reasoning shows that if the problem has a solution at all, then A and B must both be knaves. It is conceivable, however, that the problem has no solution. The problem statement could be inherently contradictory. If you look back at the solution, though, you can see that it does work out for both A and B to be knaves.

Explanation / Answer

b.

If C is a knight, then his statement must be true, and hence he is a knave. This is a contradiction. Therefore,C

is a knave. Since C is a knave, his state- ment must be false. Therefore, D is a knight.

c.

If E and F are both knaves, then E s statement is true, which is a con- tradiction since he is a knave. If E and F are both knights, then E is statement is false, which is a contradiction since he is a knight. Thus, one of E and F is a knight and one is a knave. Therefore, there is one knave

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