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 P and Q 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 and Q cannot be determined.

GATE 2017 · General Aptitude · Statements Follow · medium

Answer: Both KK (both Knights) and NN (both Knaves) are consistent; the identities cannot be determined. Answer: D.

1. Case 1: Both P and Q are Knights (KK): P's statement 'Both are Knights' = TRUE (since both are actually Knights). Knight P tells the truth. Consistent. Q's statement 'None are Knaves' = TRUE. Knight Q tells the truth. Consistent. Case KK is VALID.
2. Case 2: P is Knight, Q is Knave (KN): P's statement 'Both are Knights' = FALSE (Q is actually a Knave). But P is a Knight who must tell the truth. CONTRADICTION. Case KN is INVALID.
3. Case 3: P is Knave, Q is Knight (NK): P's statement 'Both are Knights' = FALSE (P is actually a Knave). Knave P must lie, so the statement must be false. Consistent for P. Q's statement 'None are Knaves' = FALSE (P is a Knave). But Q is a Knight who must tell the truth. CONTRADICTION. Case NK is INVALID.
4. Case 4: Both P and Q are Knaves (NN): P's statement 'Both are Knights' = FALSE (both are Knaves). Knave P must lie, so the statement must be false. Consistent. Q's statement 'None are Knaves' = FALSE (both are actually Knaves). Knave Q must lie. Consistent. Case NN is VALID.
5. Conclusion: two valid cases remain: Both KK and NN are consistent with the given statements. Since two different assignments are valid, the identities of P and Q cannot be uniquely determined.