Answer: Tuesday Teaser #11
There are many ways to answer the first three, but here are my favourite answers:
1. “I am not a knight.”
2. “I am a knave.”
3. “If someone asks me whether I’m a normal, I’ll answer ‘Yes’.”
Clearly if a knight or a knave were to say any of 1, 2 or 3, they would have to be lying or telling the truth respectively.
The second of the puzzle is much harder and relies on what method of semantics are used.
4. “If I am not a knight, then this statement is false.”
5. “If I am not a knave, then this statement is false.”
6. “If I am not a knight, then I am not a knave, and this statement is false.”
Let’s work through 4 to see why it’s a satisfying answer. If said by a knight, then the statement is true (under the ordinary rules of logic) since the implication is vacuous. If said by a knave, we arrive at a curious paradox because it has to be a lie (by definition) and hence must also be true (because “this statement is false” must be a lie). Note that it therefore cannot be said by a knave, because the statement would not be a lie (it’s a paradox so is neither true nor false).
Likewise, if said by a normal it could be true (in which case ”this statement is false” must be true) or it could be a lie (in which case ”this statement is false” must be a lie).
At this point, it depends on what your interpretation of the rules are as to whether this is an answer or not. If we assume that everything a normal says is either true or false, then it couldn’t be said by a normal (as it would be neither true nor false) and hence can only have been said by a knight. If, on the other hand, we allow normals to say paradoxical statements, we have no way of discerning whether the statement was said by a knight or a normal. If this is our interpretation, there is no way of answering the final three parts of the puzzle.
