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Topic: Forever Undecided (Read 866 times) |
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Shelter417
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Forever Undecided
« on: Nov 21st, 2008, 11:53am » |
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I'm not sure if this is the correct forum, but here goes. On page 17 of Smullyan's "Forever Undecided," he presents a basic knights/knaves puzzle; a man states "If I am a knight, then so is my wife." The gist of the proof involves assuming that the husband is a knight and extrapolating from that the face that his wife is also a knight; therefore, since it is indeed true that "if [he is] a knight, then his wife is a knight," the man was telling the truth.
Smullyan's "Theorem I" states "Given any proposition p, suppose a native says "If I am a knight, then p." Then the native must be a knight and p must be true."
Later on, Smullyan says "no inhabitant of the knight-knave island can say 'If I'm a knight, then Santa Claus exists." This doesn't seem to make sense. But what if the speaker was lying? Isn't that possibility still there? My reasoning is that it doesn't matter what the situation would be if the speaker was telling the truth; all we have now is his statement.
I'm sure I'm just missing some trivial detail here, so any help would be greatly appreciated.
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teekyman
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Re: Forever Undecided
« Reply #1 on: Nov 21st, 2008, 12:26pm » |
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As Smullyan says, he can't be lying. A conditional statement of the form "if P then Q" is considered True when the premise P is false, or if both the premise P and the conclusion Q are true. It is a *contradiction* (not just a lie) if a knave says a statement that goes "if I am a knight then p".
A knave could never say "If I'm a knight then Santa Claus exists" because that would mean he was telling the truth (false premise), and a knight could never say this either, since that would mean he was lying.
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Shelter417
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Re: Forever Undecided
« Reply #2 on: Nov 22nd, 2008, 1:56pm » |
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on Nov 21st, 2008, 12:26pm, 1337b4k4 wrote:
A knave could never say "If I'm a knight then Santa Claus exists" because that would mean he was telling the truth (false premise), and a knight could never say this either, since that would mean he was lying. |
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I'm sorry; could you clarify that point? What is the false premise? Doesn't the word "if" change things?
Isn't the statement itself ("If I'm a knight, then Santa Claus exists") false?
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towr
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Re: Forever Undecided
« Reply #3 on: Nov 22nd, 2008, 2:17pm » |
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on Nov 22nd, 2008, 1:56pm, Shelter417 wrote:| I'm sorry; could you clarify that point? What is the false premise? |
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That we're dealing with a knave, who by definition cannot tell the truth.
Quote:| Isn't the statement itself ("If I'm a knight, then Santa Claus exists") false? |
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No; it is true if you're not a knight. From a false premise (in this case "I am a knight", when you're not) you can logically conclude anything.
So if a knave says it, he's telling the truth, but if he's telling the truth, then he can't be a knave.
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Shelter417
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Re: Forever Undecided
« Reply #4 on: Nov 23rd, 2008, 5:33am » |
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on Nov 22nd, 2008, 2:17pm, towr wrote:
No; it is true if you're not a knight. From a false premise (in this case "I am a knight", when you're not) you can logically conclude anything.
So if a knave says it, he's telling the truth, but if he's telling the truth, then he can't be a knave. |
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But the statement isn't "I am a knight." The statement is "If I am a knight." Isn't there a difference?
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towr
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Re: Forever Undecided
« Reply #5 on: Nov 23rd, 2008, 7:59am » |
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on Nov 23rd, 2008, 5:33am, Shelter417 wrote:| But the statement isn't "I am a knight." The statement is "If I am a knight." Isn't there a difference? |
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The premise is "I am a knight"; the statement is "if I am a knight then santa clause exists", where the part in red is the premise, and the part in blue is the conclusion of the implication (which is what the statement is).
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iono
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Re: Forever Undecided
« Reply #6 on: Nov 24th, 2008, 5:33pm » |
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Since Santa claus doesn't exist, you are not a knight, which means you told the truth
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So, if I help you, I'll get kicked for ksing, but if I don't, then I'll get kick for not helping...
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rmsgrey
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Re: Forever Undecided
« Reply #7 on: Nov 25th, 2008, 3:35am » |
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The key point of logic here is that any statement of the form "If X then Y" can only be false if X is true and Y is false.
The only way a knave could say "if X then Y" is if X is true and Y is false. When X is "I am a knight", since we're assuming it's a knave saying it, X is automatically false. So no knave can ever say anything of the form "If I am a knight then Y" no matter what Y is.
Seeing that "If X then Y" should be true whenever X is false takes some thought - the way I prefer to think of it is that "If n is even, then n is composite" is true for all integers n>2 - including n=7 (odd, non-composite), n=8 (even, composite) and n=9 (odd, composite), but not n=2 - when it's false.
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teekyman
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Re: Forever Undecided
« Reply #8 on: Nov 25th, 2008, 11:39am » |
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Its possible that you might be confused about the difference between "If I am a knight then I wear armor" and "If one is a knight then one wears armor." The first one is personal, and is about specifically you being a knight and wearing armor. The second one is a general statement about all knights.
Thats why if you are a knave, you can't say:
"If I am a knight, I wear armor" or
"If one is a knight then one wears armor" or even
"If I am a knight then I am the President of the United States" because they are all true.
but you can say "If one is a knight then one is the President of the United States" because that is indeed false.
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| « Last Edit: Nov 25th, 2008, 6:03pm by teekyman » |
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iono
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Re: Forever Undecided
« Reply #9 on: Dec 1st, 2008, 5:58pm » |
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The President can be knighted...
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So, if I help you, I'll get kicked for ksing, but if I don't, then I'll get kick for not helping...
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rmsgrey
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Re: Forever Undecided
« Reply #10 on: Dec 2nd, 2008, 7:39am » |
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on Dec 1st, 2008, 5:58pm, iono wrote:| The President can be knighted... |
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but if there's any knight anywhere in the world who isn't President of the US, then "if one is a knight, one is PotUS" is false.
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iono
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Re: Forever Undecided
« Reply #11 on: Dec 2nd, 2008, 8:52pm » |
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Oh yea 
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So, if I help you, I'll get kicked for ksing, but if I don't, then I'll get kick for not helping...
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