Merestil Haye
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Post by Merestil Haye on Oct 31, 2007 3:47:42 GMT -5
The number of tourists visiting the Island persuaded the natives to adopt a new nomenclature for themselves. They started calling themselves Knights (who always tell the truth) and Knaves (who always lie). Firstly they like the alliteration, and secondly they found being called "liars" by outlanders got a little irritating and threatened to disrupt the tourist biz. Fortunately the incidents were hushed up. But I digress...
You've gone to visit the island for the first time. Walking alone along a back path a native of the island passes you in traditional spring getup. You accost him, and ask if he is a Knight or a Knave. He looks at you, grins and says "If I am a Knight, I will eat my hat."
What conclusions can you draw from this?
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Merestil Haye
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Post by Merestil Haye on Oct 31, 2007 3:50:09 GMT -5
I think I'll adopt Dot's rules. Answers by PM please, don't post them in the thread.
The deadline is 8AM GMT, Saturday 3rd November. Extensions will be considered.
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Merestil Haye
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Post by Merestil Haye on Nov 3, 2007 8:17:41 GMT -5
The deadline has passed, and it seems no-one is inspired to deduce anything. Pity.
Perhaps you need a hint?
When is the statement true and when is it false? What follows if it is true? What follows if it is false?
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Merestil Haye
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Post by Merestil Haye on Nov 6, 2007 13:06:25 GMT -5
Another hint.
Whether a statement in the form "If A then B" is true or false (and therefore can be uttered by a knight or a knave) depends on the truth values of the two clauses individually.
Most people would probably agree that if A were true and B were true then the statement "If A then B" is true. They would also agree that if A were true and B false, "if A then B" must be false.
What if A were false though? Is the statement "If A then B" true if B is true? What about if B is false? Any ideas?
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Post by IronHorse on Jun 2, 2009 1:29:46 GMT -5
Well, since the deadline passed quite a while ago, I don't think anyone would mind if I post my answer.
My first assumption was that both would eat the hat. A Knight would have to, because he was telling the truth. And the Knave would to be telling a lie. But then I wasn't sure. So I pulled up my old logic.
If p, then q. == If not q, then p.
Therefore, the following is identical to the main question. If I don't eat my hat, I'm not a knight. If that is said by the Knight, the value is true. And that works with my first assumption. If that is said by the Knave, the value is false. So let's negate so it will be true:
If not not q then not not p If q, then p If I eat my hat, I am a knight. In that situation, he cannot eat his hat.
So the Knight will eat his hat. The Knave will not. And since eating a hat is a pretty icky thing, I'd say the guy was most likely a Knave.
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Post by Høøpy Frøød on Jun 2, 2009 7:01:58 GMT -5
Well, since the deadline passed quite a while ago, I don't think anyone would mind if I post my answer. My first assumption was that both would eat the hat. A Knight would have to, because he was telling the truth. And the Knave would to be telling a lie. But then I wasn't sure. So I pulled up my old logic. If p, then q. == If not q, then p. I believe you meant "If not q, then not p", especially since you present your actual equivalent statement as such. (Underlining and italicizing mine.) You're begging the question a bit here by retroactively applying the hat eating result to the truth table. The only reason we know he's a knave is because after making that statement, he did not eat his hat. But the knave could have eaten his hat and still been a knave. Because he's not a knight. So the results of what he would do if he were a knight is irrelevant. In truth tables, these are frequently referred as "don't cares" and are designated by the symbol "X". (Whereas true and false are "T" and "F" respectively.)
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Merestil Haye
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Post by Merestil Haye on Jun 2, 2009 8:37:57 GMT -5
My word.
What you can deduce is this.
1. The speaker is a Knight. 2. The speaker will eat his hat. (I hope it's tasty.)
When an inhabitant of an Island of Knights and Knaves makes a statement of the form "If I am a Knight then Q", the speaker is a Knight and Q is true.
We have two statements P and Q, neither of which are dependant on the other, and a linking statement "If P then Q" made by someone who either always tells the truth or always lies.
If P is true and Q is true, then "If P then Q" is also true. If P is true and Q is false, then "If P then Q" is false.
I don't think many people would disagree with those., would you?
However, it turns out that if P is false then "If P then Q" is true - whether or not Q is true.1
Therefore a Knave can never, ever make a statement of the form "If I am a Knight then Q."
Now we know that the speaker is a Knight, and that P is true, we know that Q is true (because if it was false "If P then Q" would be false, and we would have a Knight making a false statement.) Therefore the speaker is going to eat his hat.
1I went to look up the discussion in the book I got the question from, and I can't locate the book right now. I'll keep an eye out for it.
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Post by Høøpy Frøød on Jun 2, 2009 10:51:11 GMT -5
However, it turns out that if P is false then "If P then Q" is true - whether or not Q is true. 1Correct. Which is frequently mentioned in computer science courses. There are many books that will document it. I've got a textbook somewhere that mentions it. And that's what I missed. (And really shouldn't have since we studied these constructions in one of my grad-school courses.) If a knave always lies, he could never make a sentence that would logically be proven true. Though, it's a bit of a technicality because only a Knave who is aware of this rather obscure rule in formal logic would realize this. From a layman's perspective, he's unlikely to realize what he's doing.
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Merestil Haye
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Post by Merestil Haye on Jun 2, 2009 14:24:06 GMT -5
And that's what I missed. (And really shouldn't have since we studied these constructions in one of my grad-school courses.) If a knave always lies, he could never make a sentence that would logically be proven true. Though, it's a bit of a technicality because only a Knave who is aware of this rather obscure rule in formal logic would realize this. From a layman's perspective, he's unlikely to realize what he's doing. Unless he too is a logic graduate. Which, given some of the puzzles, shouldn't be a surprise. The whole island is populated by logicians, in fact...
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Post by IronHorse on Jun 3, 2009 6:08:26 GMT -5
Well, it makes sense that I got it wrong, since I know very little about formal logic. I was just trying to apply what I read in a book once. My main reason for attacking this was thinking that the obvious answer must be wrong. I only worked at it while I was typing. I never thought about non-true, non-false values. I made the mistake of assuming that "If not p, then q" would always have the opposite truth value of "if p then q". It seems it does not. The following real-life (if slightly simplified) example makes that plain to me.
If you have a marriage license, you are legally married: true If you do not have a legitimate marriage license, you are legally married: not necessarily false. Common law marriage.
Here's an exalt for fighting my ignorance, even when not on the SDMB.
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Merestil Haye
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Post by Merestil Haye on Jun 3, 2009 6:31:46 GMT -5
My favourite simple one.
You are walking along one of the many splendid tropical beaches on the Island of Knights and Knaves. As you walk you see a native you know lying lazily on the sand; his name is Edwin or Edward, but right now you cannot pin down which.
You ask him his name, and he says "Edward."
What is his name?
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Post by The Real FCOD on Jun 3, 2009 10:49:25 GMT -5
How can you possibly know without more information?
--FCOD
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Post by Høøpy Frøød on Jun 3, 2009 11:54:19 GMT -5
How can you possibly know without more information? --FCOD Maybe it's something along the lines of the fact that given the shear possibility of fake names a Knave could have given you, the odds of them giving you the name Edward as a fake name is slim. After all, you didn't ask if he was Edwin or Edward, you asked him what his name was. If he had answered something like George, you would know he was a knave.
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Merestil Haye
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Post by Merestil Haye on Jun 3, 2009 14:03:50 GMT -5
Indeed.
But there is enough information in the question to give an answer.
Trust me.
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Post by Høøpy Frøød on Jun 3, 2009 14:29:50 GMT -5
Indeed. But there is enough information in the question to give an answer. Trust me. So my playing the odds isn't quite what you're looking for, eh? Okay, so do you literally ask him "What is your name?" and his response is "Edward"?
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Merestil Haye
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Post by Merestil Haye on Jun 3, 2009 14:37:43 GMT -5
Yep.
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Post by IronHorse on Jun 3, 2009 19:19:58 GMT -5
He prefers to go by Ed.
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Merestil Haye
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Post by Merestil Haye on Jun 4, 2009 6:14:19 GMT -5
My favourite simple one. You are walking along one of the many splendid tropical beaches on the Island of Knights and Knaves. As you walk you see a native you know lying lazily on the sand; his name is Edwin or Edward, but right now you cannot pin down which. You ask him his name, and he says "Edward." What is his name? His name is Edwin. When you get a question like this, some of the information needed is hidden in the question. In this case, I told you he was "lying lazily on the sand" which means he was lying. Therefore he is a Knave. You can groan now, if you like.
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Post by Høøpy Frøød on Jun 4, 2009 7:04:35 GMT -5
I prefer the industrious liars myself.
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Post by The Real FCOD on Jun 4, 2009 7:42:24 GMT -5
His name is Edwin. When you get a question like this, some of the information needed is hidden in the question. In this case, I told you he was "lying lazily on the sand" which means he was lying. Therefore he is a Knave. You can groan now, if you like. Shenanigans! --FCOD
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Post by IronHorse on Jun 11, 2009 2:46:01 GMT -5
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Merestil Haye
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Post by Merestil Haye on Jun 11, 2009 11:18:22 GMT -5
*checks the links.*
Oh, Raymond Smullyan. "What is the name of this book" is the book I took the above puzzle from. It progresses from light hearted stuff to quite thought-provoking questions. (And also proved that Dracula is ... well, I won't tell you. You might want to read the book for yourself.
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