Knights and Knaves Recall the Knights and Knaves puzzle we saw in class. In that fictional world there are two types of people. Knights - who always tell the truth, and Knaves - who always lie. It is impossible to distinguish them by appearance, but only by the truth of their statements. 1. One day a traveller was wandering around the island of Knights and Knaves, when he encountered two local inhabitants, P and Q. The traveller asked: "Is any of you a knave?" P replied: "At least one of us is a knave". Can you find out what P and Q are? If so, what are they? If not, explain why not, and what other information you would need to know. A replied: "If B is a knave, then I am a knave too". What are A and B? 2. Later on, the traveller met two other locals, A and B. He asked whether either of them is a knight Logical Identities Simplify the following propositions. Show all steps of your solutions.

P and Q cannot be determined without asking further questions. A is a knave and B is a knight

Explanation:

QUESTION 1

One day a traveller was wandering around the island of Knights and Knaves, when he encountered two local inhabitants, P and Q. The traveller asked: "Is any of you a knave?" P replied: "At least one of us is a knave". Can you find out what P and Q are? If so, what are they? If not, explain why not, and what other information you would need to know.

SOLUTION TO QUESTION 1

No, we cannot find out what P and Q are, we require more information.

We cannot determine if the reply of P is true or false, since we cannot determine the truth of the statement we cannot tell which is a knave or a knight, so we need further information.

To get further information we need to ask a question. And the question the traveler needs to ask P is “am I a knight?”. If he says yes, it means he told a lie so P is a knave, but if he says NO, it means he told the truth so he is a knight.

QUESTION 2

A replied: "If B is a knave, then I am a knave too". What are A and B? 2. Later on, the traveler met two other locals, A and B. He asked whether either of them is a knight Logical Identities Simplify the following propositions. Show all steps of your solutions.

I think the sentences in this part of the question was misplaced. The correct order should be’

Later on, the traveler met two other locals, A and B. He asked whether either of them is a knight, A replied: "If B is a knave, then I am a knave too". What are A and B?  Logical Identities Simplify the following propositions. Show all steps of your solutions.

SOLUTION TO QUESTION 2

The use of the word “if” in A reply indicates a condition in his reply. This means if B is a knight, then he will be a knight too. That means A can claim to be a knave or a knight depending on who B is. Since knights are known to tell the truth, they cannot claim who they are not. Since A can claim who is he is not, it means A can tell a lie, so A is a knave. B can either be a knight or A knave, we cannot tell as he did not confirm or deny A claims. But from A’s reply “If B is a knave, then I am a knave too” we could tell that B is a knight. If A’s replay was true, we could say that B is a knave, but we have established that A is a knave so he could not tell the truth.

Answer A is a knave and B is a knight

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