The History Paper
Your argument was valid and conclusion was correct.
You expressed your answer, with the logic that followed it.
“Well done.” Plato folded his arms across his chest. “Are you ready for the next one?”
You swallowed hard, growing increasingly nervous, before voicing you were.
“Alright, the next one is a little harder: The Knights and Knaves. This is a slightly different version:
On the island of Knights and Knaves, every inhabitant is either a knight or a knave. Knights always tell the truth. Knaves never tell the truth; any sentence uttered by a knave is false. One day I went to the island of knights and knaves and encountered an inhabitant who said, "Either I am a knave or else two plus two equals five." What should you conclude?”
You thought for a moment and looked over at Chase who was, quite literally, sitting on the edge of his seat. Narrowing your eyes, you confidently answered, “I would conclude you are lying. You never would have encountered an inhabitant who said so.”
Without showing any emotion, Plato asked a vital question: “why do you say so?”
“If a knight said it, it would be false, and a knight cannot say anything that’s false. If a knave said it, it would be true, and they cannot say anything true. Therefore, the only logical conclusion is that you – the one who ‘experienced’ this – must be lying.”
Plato couldn’t help but smile. “Very good. I have to admit I am thoroughly enjoying this.”
“Thank you. What is the next one?”
Plato chuckled slightly. “This one has been said to be the hardest puzzle in existence. Do you think you can solve it?”
“What have I got to lose? I’ve already got 2 right.” You had a gut feeling you knew the puzzle he was going to present.
“Very well.” Plato took a deep breath. “This one was posed by Boolos:
Three gods, A, B, and C, are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is completely random. You must determine the identities of A, B, and C by asking three yes-or-no questions, and each question must be posed to exactly one god. The gods understand English but will answer all questions in their own language. In their unknown language, the words for "yes" and "no" are "da" and "ja," in some order. You do not know which word means which. What questions do you ask to identify the gods? There are a few other important things to note: a single god may be asked more than one question, questions are allowed to depend on the answers to earlier questions, and the nature of Random's response should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he speaks truly; if tails, falsely.”
You smiled; you were right, you did know the puzzle. It had been a bonus question on the midterm of one of your philosophy courses. Of course, no one got the answer, but everyone badgered your professor until he explained the answer.
You gave the answer he had given you: “The first question I would ask would be addressed to god A: does 'da' mean 'yes' if and only if you are True and if and only if B is Random? Now supposing A answered with ‘ja’, (because if A said ‘da’ you would already have the identities: A as True, B as Random, and C as False) this makes B either True or False.” You took a deep breath. “The second question would be for god B: does ‘da’ mean 'yes' if and only if 3 plus 2 equals 5? We know of course 3 plus 2 does equal 5, so if B answers with ‘da’ we know B is True. If B answers with ‘ja’ we know B is False. Let’s suppose B answers ‘da’. For my final question, I would ask god B: does ‘da’ mean yes if and only if A is Random? Since B is True, the response would give you the answer to the other’s identities; if B says ‘da’ then A is Random, B is True, and by process of elimination, C is False; If B says ‘ja’ then C must be Random, so A must be False.” You beamed, very clearly proud of yourself.
Chase looked at you completely dumbstruck. “How on earth did you know that?” You smiled at him.
“I have to admit, I am impressed.” Plato tilted his head. “I have the same question as Chase – how did you know that?”
“I have good reasoning skills.”
“Come on.” Chase gave you a fed-up look.
“The puzzle was given as a bonus question on an exam,” you admitted.
Plato laughed. “I would love to take back my initial offer – especially if I had known you’ve seen the puzzle before, but no matter.” He stood. “You have adequately proven yourself. I will fetch the pomegranate seeds.”
Plato was smiling when he came back with the seeds in a small velvet bag with a draw string. He looped, then finally tied, the string around the bag. “It’s odd giving this to somewhere. I’ve had them for quite some time.”
“It is greatly appreciated,” you reassured him ad he handed you the bag.
“Be careful with them. And absolutely do not lose it.”
You nodded. “Of course.”
And with that, you were rushed out of his house.
You expressed your answer, with the logic that followed it.
“Well done.” Plato folded his arms across his chest. “Are you ready for the next one?”
You swallowed hard, growing increasingly nervous, before voicing you were.
“Alright, the next one is a little harder: The Knights and Knaves. This is a slightly different version:
On the island of Knights and Knaves, every inhabitant is either a knight or a knave. Knights always tell the truth. Knaves never tell the truth; any sentence uttered by a knave is false. One day I went to the island of knights and knaves and encountered an inhabitant who said, "Either I am a knave or else two plus two equals five." What should you conclude?”
You thought for a moment and looked over at Chase who was, quite literally, sitting on the edge of his seat. Narrowing your eyes, you confidently answered, “I would conclude you are lying. You never would have encountered an inhabitant who said so.”
Without showing any emotion, Plato asked a vital question: “why do you say so?”
“If a knight said it, it would be false, and a knight cannot say anything that’s false. If a knave said it, it would be true, and they cannot say anything true. Therefore, the only logical conclusion is that you – the one who ‘experienced’ this – must be lying.”
Plato couldn’t help but smile. “Very good. I have to admit I am thoroughly enjoying this.”
“Thank you. What is the next one?”
Plato chuckled slightly. “This one has been said to be the hardest puzzle in existence. Do you think you can solve it?”
“What have I got to lose? I’ve already got 2 right.” You had a gut feeling you knew the puzzle he was going to present.
“Very well.” Plato took a deep breath. “This one was posed by Boolos:
Three gods, A, B, and C, are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is completely random. You must determine the identities of A, B, and C by asking three yes-or-no questions, and each question must be posed to exactly one god. The gods understand English but will answer all questions in their own language. In their unknown language, the words for "yes" and "no" are "da" and "ja," in some order. You do not know which word means which. What questions do you ask to identify the gods? There are a few other important things to note: a single god may be asked more than one question, questions are allowed to depend on the answers to earlier questions, and the nature of Random's response should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he speaks truly; if tails, falsely.”
You smiled; you were right, you did know the puzzle. It had been a bonus question on the midterm of one of your philosophy courses. Of course, no one got the answer, but everyone badgered your professor until he explained the answer.
You gave the answer he had given you: “The first question I would ask would be addressed to god A: does 'da' mean 'yes' if and only if you are True and if and only if B is Random? Now supposing A answered with ‘ja’, (because if A said ‘da’ you would already have the identities: A as True, B as Random, and C as False) this makes B either True or False.” You took a deep breath. “The second question would be for god B: does ‘da’ mean 'yes' if and only if 3 plus 2 equals 5? We know of course 3 plus 2 does equal 5, so if B answers with ‘da’ we know B is True. If B answers with ‘ja’ we know B is False. Let’s suppose B answers ‘da’. For my final question, I would ask god B: does ‘da’ mean yes if and only if A is Random? Since B is True, the response would give you the answer to the other’s identities; if B says ‘da’ then A is Random, B is True, and by process of elimination, C is False; If B says ‘ja’ then C must be Random, so A must be False.” You beamed, very clearly proud of yourself.
Chase looked at you completely dumbstruck. “How on earth did you know that?” You smiled at him.
“I have to admit, I am impressed.” Plato tilted his head. “I have the same question as Chase – how did you know that?”
“I have good reasoning skills.”
“Come on.” Chase gave you a fed-up look.
“The puzzle was given as a bonus question on an exam,” you admitted.
Plato laughed. “I would love to take back my initial offer – especially if I had known you’ve seen the puzzle before, but no matter.” He stood. “You have adequately proven yourself. I will fetch the pomegranate seeds.”
Plato was smiling when he came back with the seeds in a small velvet bag with a draw string. He looped, then finally tied, the string around the bag. “It’s odd giving this to somewhere. I’ve had them for quite some time.”
“It is greatly appreciated,” you reassured him ad he handed you the bag.
“Be careful with them. And absolutely do not lose it.”
You nodded. “Of course.”
And with that, you were rushed out of his house.