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?
Analyzing Knights and Knaves Statements
This problem involves a classic logic puzzle type where inhabitants are either Knights (always truthful) or Knaves (always liars). We need to determine the identities of P and Q based on their statements.
Let K denote Knight and V denote Knave.
- Knights always tell the truth.
- Knaves always lie.
P's Statement: "Both of us are knights."
Q's Statement: "None of us are knaves." (This is logically equivalent to "Both of us are knights.")
Let $S_P$ be the statement "P is Knight and Q is Knight". Let $S_Q$ be the statement "P is Knight and Q is Knight". Note that $S_P$ and $S_Q$ are identical statements.
Scenario Analysis
We examine the possible identities of P:
-
Assume P is a Knight (K):
- If P is a Knight, P's statement ($S_P$) must be true.
- $S_P$ = "P is K and Q is K". For this to be true, both P and Q must be Knights.
- So, if P is K, then Q must also be K.
- Now, check Q's statement. Since Q is assumed to be K, Q's statement ($S_Q$) must be true.
- $S_Q$ = "P is K and Q is K". This is true since we concluded P=K and Q=K.
- This scenario (P=K, Q=K) is logically consistent.
-
Assume P is a Knave (V):
- If P is a Knave, P's statement ($S_P$) must be false.
- $S_P$ = "P is K and Q is K". The negation is "P is V or Q is V".
- Since we assumed P is V, this condition ("P is V or Q is V") is satisfied.
- Now, consider Q's possible identities:
- If Q is a Knight (K): Q's statement ($S_Q$) must be true. But $S_Q$ ("P is K and Q is K") is false because P is V. This is a contradiction (Q is K but made a false statement). So, Q cannot be K if P is V.
- If Q is a Knave (V): Q's statement ($S_Q$) must be false. $S_Q$ = "P is K and Q is K". Since P is V and Q is V, the statement "P is K and Q is K" is indeed false. This is consistent (Q is V and made a false statement).
- This scenario (P=V, Q=V) is also logically consistent.
Conclusion
We found two possible and consistent scenarios:
- P is a Knight and Q is a Knight.
- P is a Knave and Q is a Knave.
Since both possibilities exist, we cannot definitively determine the identities of P and Q based solely on the given statements.
Therefore, the correct inference is that the identities of P and Q cannot be determined.
Important Questions from Logical Deduction
- The following observation is made about the scores obtained by 100 students in an exam:
'For each student, there exists another student in the class such that their scores are at most ten marks away.'
If the above statement is false, which one of the following statements is necessarily true? - Consider the following two phonological rules:
Rule 1: Vowel Epenthesis of [ɪ] after sibilant-ending stems
Rule 2: Progressive Voicing Assimilation of the plural affix
Which ONE of the following rule ordering relations apply in the case of regular English pluralization as in ‘horse’ – ‘horses’? Consider a linear arrangement of seven bulbs, each of which can be in the ON or OFF states. The initial configuration of the bulbs is shown in the figure. In every Step, the states of the bulbs are changed based on the following rules:
- Any OFF bulb with exactly one ON neighbor at the end of the previous Step is turned ON.
- Any ON bulb with both neighbors ON at the end of the previous Step is turned OFF.
- The state of any bulb not meeting the conditions above is left unchanged.
The state of bulbs at the end of Step 1 and Step 2 are also shown in the figure.
The number of bulbs which are ON at the end of Step 8 is ______
- Based only on the conversation below, identify the logically correct inference:
“Even if I had known that you were in the hospital, I would not have gone there to see you", Ramya told Josephine. Consider a five-digit number PQRST that has distinct digits P, Q, R, S and T, and satisfies the following conditions:
$P < Q$
$S > P > T$
$R < T$
If integers 1 through 5 are used to construct such a number, the value of P is:
