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
- 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. In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of P, Q, and R ?
- Column-I has statements made by Shanthala; and, Column-II has responses given by Kanishk.
Column-I Column-II P. This house is in a mess. 1. Alright, I won't bring it up during our conversations. Q. I am not happy with the marks given to me. 2. Well, you can easily look it up. R. Politics is a subject I avoid talking about. 3. No problem, let me clear it up for you. S. I don't know what this word means. 4. Don't worry, I will take it up with your teacher.
Identify the option that has the correct match between Column-I and Column-II. - In the following truth table, what does X stand for?
P Q X 1 1 1 1 0 0 0 1 0 0 0 1 A color model is shown in the figure with color codes: Yellow (Y), Magenta (M), Cyan (Cy), Red (R), Blue (Bl), Green (G), and Black (K).
Which one of the following options displays the color codes that are consistent with the color model?
