When I visited the Knights and Liars Archipelago, one island I visited was called Liontruth. The tourism had a great influence on the island, so much that the knaves on the island spoke differently from most knaves. They didn't have to follow an alternating pattern, but could tell truths (like knights always do) and lie (like liars always do) in whatever pattern they wanted. The three types of inhabitants are indistinguishable by eye.

If a tourist thinks the local knaves alternate truths and lies how can a knave convince the tourist that he is not a knave?

How can a knave from this island prove himself in one statement without revealing whether he is lying or not?

How can a knight prove himself in one statement?

How can a liar prove himself in one statement?

What single statement can be said by either a knight or liar but not a knave?

(In reply to One last clarification by Tristan)

(1) If a tourist thinks the local knaves alternate truths and lies how can a knave convince the tourist that he is not a knave?

"I am not a knave and have never been a knave". (The knave can only say that if knaves are allowed to mix truths and lies unpredictably. The tourist thinks they have to alternate). 

(2) How can a knave from this island prove himself in one statement without revealing whether he is lying or not?

"My answer to your next question will be a lie." (Only a knave can tell a mixture of truths and lies. So I an islander says this, and his answer to the next question is true, he will have told a lie-truth. Otherwise he will have told a truth-lie.) 

A knave could also prove himself by saying "Next year exactly 100 inches of rain will fall on the island." (No one knows if this is true or false, so neither a knight nor a liar could say it). 

(3) How can a knight prove himself in one statement ?

"If I am not a knight, then this statement is false." (If a knight says this, the antecedent is false and the truth or falsehood of the consequent is irrelevant. But if a liar or a knave says it, the antecedent is true and the consequent is paradoxically neither true nor false. No one on the island can make a statement that is neither a truth nor a falsehood. Hence only a knight can say this.)

(4) How can a liar prove himself in one statement ?

"If I am not a liar, then this statement is false." (If a liar says this, the antecedent is false and the truth or falsehood of the consequent is irrelevant. But if a knight or a knave says it, the antecedent is true and the consequent is paradoxically neither true nor false. No one on the island can make a statement that is neither a truth nor a falsehood. Hence only a liar can say this.)

(5) What single statement can be said by either a knight or liar but not a knave?

"If I am a knave, then this statement is false." (If a knight or a liar says this, the antecedent is false and the truth or falsehood of the consequent is irrelevant. But if a knave says it, the antecedent is true and the consequent is paradoxically neither true nor false. No one on the island can make a statement that is neither a truth nor a falsehood. Hence a knave cannot say this.)

 

 

 

 

 

Edited on December 11, 2004, 12:01 pm

Posted by Penny on 2004-12-11 08:57:26