Treasure?

Logic Level 2

You come to an island which is populated by knights who always tell the truth and knaves who always lie. You meet someone on the island and he says,

$\quad \quad$ "Either I am not a knight or there is a treasure on this island."

If the person you have asked knows whether there is a treasure or not, should you search for it?

Impossible to tell Yes No

4 solutions

Jacopo Piccione
Jan 1, 2019

"Either A or B" is equivalent to $A \dot{\lor} B$ , which is true if and only if one and only one between A and B is true. Hence there are two options:

If he's a knight he has to tell the truth. But "I am not a knight" is false, so "There is a treasure on this island" must be true, therefore we should search for it.
If he's a knave he has to lie. But "I am not a knight" is true, so "There is a treasure on this island" must be true, therefore we should search for it.

I found your knave explanation a little confusing (though I understand it now). Here’s my take: If the person is a knave, then the entire statement this false. Thus, we have $\neg(A\dot{\lor}B) \iff (A \land B)\lor (\neg A \land \neg B)$ This means that either both statements are true (in which case there is treasure on the island) or both statements are false (in which case he is a knight, a contradiction).

Jordan Cahn - 2 years, 5 months ago

Yes, I thought it was implicit in $A \dot{\lor} B$ . Anyway, thanks for the clarification!

Jacopo Piccione - 2 years, 5 months ago

If a knave makes this statement, then he is a knight AND there is no treasure on the island. The first part of this is a contradiction.

G Silb - 2 years, 3 months ago

Is this an inclusive "or " or an exclusive?

BAOZHU LIANG - 1 year, 4 months ago

Exclusive use by this answer.

Saya Suka - 3 months ago

This has always confused me. OR and XOR have always been used interchangeably in natural languages in which many puzzles are worded by and no distinctions were mentioned when these ORs are employed. How do you differentiate the two? Do you want 🍵 or ☕? Can I say I want both, the way I can for 🍫 or sugar and 🍼 for my coffee? Or try a new tea recipe with both 🍋 and 🥛 and even 🧊.

Saya Suka - 3 months ago

Rich Person
Jan 5, 2019

It’s impossible for anyone on the island to say “I am not a knight” because: A knight in that case would be lying (but they always speak the truth), A knave would be telling the truth (but they always lie). So the treasure chest is the only option that is true and the original statement says either “I am not a knight” or “there is treasure” but we ruled out the first statement to be impossible so the only thing left is the treasure.

Exactly my thinking

Dennis K - 2 years, 5 months ago

Zhong Si Wei
Jan 1, 2019

If there is a treasure, then the person is a knight

Isn't this right out of a logic book where all the problems are about knights and knaves?

Martha Sapeta - 1 year ago

Saya Suka
Mar 7, 2021

"Either I am not a knight or there is a treasure on this island."

A knave would need to use 2 false info for this OR statement for one to be able to use it (a lie of false OR statement), but a double negative to the Knight would make the speaker a Knight themselves, and a contradiction to our supposition.

Therefore, this OR statement must have been spoken by a Knight, and with the part before the OR already a falsehood, then the only way this statement can stay true is by making the part after it as a truth.

Knight : True OR statement (at least one truth)
"I am not a Knight" ==> False
"There is a treasure on this island" ==> True

Treasure?

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