On a the island of Smullyan the inhabitants are either knigh

On a the island of Smullyan, the inhabitants are either knights or knaves, where knights always tell the truth and knaves always lie. You encounter two people, A and B. Determine, if possible, what A and B are if each of them says the following. Justify your responses.

a) A says ”The two of us are both knights” and B says ”A is a knave.”

b)  A says ”I am a knave or B is a knight” and B says nothing

c) A says ”I am a knight” and B says ”I am a knight.”

Solution

Solution: Given that on the island of Smullyan, the inhabitants are either knights or knaves, where knights always tell the truth and knaves always lie.

a) A says ”The two of us are both knights” and B says ”A is a knave.”

b)  A says ”I am a knave or B is a knight” and B says nothing

c) A says ”I am a knight” and B says ”I am a knight.”

According to (b) A says ”I am a knave or B is a knight” and B says nothing means A tells the truth.

If A tells the truth, then A is a knight and B is a knave. But according to (c) A says ”I am a knight” and B says ”I am a knight.”

So A tells a lie because (b) and (c) are not possible simultaneously and B tells the truth that means B is a knight.

So from (b) and (c), we can conclude that B is a knight.

According to (a) A says ”The two of us are both knights” and B says ”A is a knave.” and

(b)  A says ”I am a knave or B is a knight” and B says nothing, we can say that A is a knave.

Also according to (b)  A says ”I am a knave or B is a knight” and B says nothing, we can see that

A is telling the truth, it means A is a knight.

So we have A is both knight and knave, which is not possible since A is either knight or knave

Here we can say that B is a knight.