GATE Mechanical 2017 Set 2 | GA | Question: 9

Question

All people in a certain island are either 'Knights' or 'Knaves' and each person knows every other person's identity. Knights never lie, and Knaves ALWAYS lie.

$P$ says "Both of us are Knights". $Q$ says "None of us are Knaves".

Which one of the following can be logically inferred from the above?

• A. Both $P$ and $Q$ are knights.
• B. $P$ is a knight; $Q$ is a Knave.
• C. Both $P$ and $Q$ are Knaves.
• D. The identities of $P, Q$ cannot be determined.

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Answer 1

There are $4$ possible cases :

$(A)$ Both $P$ and $Q$ are knights

It means both statements "Both of us are Knights" and "None of us are Knaves" are true which is not contradicting our assumption. So, It is a possible case.

$(B)$ $P$ is knight and $Q$ is knave

It means statement "Both of us are Knights" is true but it is contradicting our assumption. So, It is NOT a possible case.

$(C)$ $P$ is knave and $Q$ is knight

It means statement "Both of us are Knights" is false and statement "None of us are Knaves" is true but this statement by $Q$ is contradicting our assumption . So, It is NOT a possible case.

$(D)$ $P$ is knave and $Q$ is knave

It means statement "Both of us are Knights" is false which means at least one should be knave which is not contradicting our assumption and statement "None of us are Knaves" is false which means at least one should be knave which is also not contradicting our assumption. So, It is a possible case.

So, It is possible that both $P$ and $Q$ are knights and it is also possible that both $P$ and $Q$ are knaves. So, we can't identify them.

Hence, Answer is D.

Comments

@ankitgupta.1729 nicely explained.

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nice explantion

Answer 2

Option D is the correct answer. 

P	Q	Inference
Knight	Knight	This case is possible.
Knight	Knave	This case is not possible.
Knave	Knight	This case is not possible.
Knave	Knave	This case is possible.

So, either both are Knights or both are Knaves.

Answer 3

We have two valid scenarios:

1. Everyone is a Knight: If they are both Knights, they are telling the truth about being Knights. This works.

2. Everyone is a Knave: If they are both Knaves, their claim "We are Knights" is a lie. Since Knaves always lie, this also works.

Because both Case 1 and Case 4 are logically valid, we cannot be certain which one is the reality. Therefore, the specific identities cannot be determined.

Correct Option: D