Inspired by Elena Gomes
You find yourself on the island of knights and knaves, where every inhabitant is of one of two types: a knight who always tells the truth, or a knave who always lie.
You know that Artemis is a knight.
Which of the island's inhabitants can say that "If I am a knave, then Artemis is a knight"?
Note: You are not an inhabitant of the island.
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2 solutions
But that again makes the person knave, following his statement "if I am a knave" part, and that contradicts the answer. So, answer should be nobody in the island.
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A knight saying the statement " If I am a knave, then Artemis is a knight" does not mean the knight would become a knave. Look at the statement carefully, " if ".
Hence, if he/she is not a knave, then according to the said true statement, Artemis is either a knight or not a knight. There is no contradiction as we know that Artemis is a knight and therefore the statement is still true.
It most certainly does not. In formal logic, "If P then Q" is true if - and ONLY if - either Q is true OR P is false (or both). And since Artemis is a knight no matter what, then he's a knight even if the speaker is a knave - which he isn't, but oh well. :-)
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Just to fix something. Isn't the statement "If P then Q" still true even when P is false and Q is true? If P is false, Q can either be true or false and the statement would still be true.
In short, the statement is false only when P is true and Q is false.
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@Kenneth Choo – You make reason, it is like this. I think Whitney was joking... or if we are kidding and "talking for speaking", everybody can say it...
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@Guillermo Templado – I most certainly was NOT joking. Please reread my statement and then tell me what was confusing about it. Knights cannot lie, knaves cannot tell the truth.
@Kenneth Choo – That is true. I did not use the exclusive "or". Hm, maybe I should change that.
This problem is called vacuously true!. You can view reports if you are interested ... and truth tables ... "if I a am a knave" is false the whole statement is true
To realize that Knaves can't say that is more or less self-evident but what confused me was the sentence 'if ... then' . I wondered if it's possible that someone says 'if non-sense then' before the true information part 'Arthemis is a Knight'. There is no substance at all in the 'if ... then' part of the sentence.
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That's not how a logician thinks, though. If my name is John, then yours is Leif, even though my name is not John.
"If God exists then He isn't all powerful" because if he existed and were all powerful could create a stone that he could not lift and then he wouldn't be all powerful because he couldn't lift it(This is a paradox)... I think, there is "much" substance in the "if... then" and I'm going to say you why. Ok, The implication A ⇒ B is equivalent tautologically to ~ A ∨ B and this is false ⟺ A is true ∧ B false. This is like this because in maths, when we want to prove A ⇒ B , this implication will only have some sense if the hypothesis A being true,then the thesis B is true, and in this case, if A is false, the thesis B doesn't matter .
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I meant 'if ... then' has no substance in the given sentence. The sentence "If my car is blue then Arthemis is a Knight" can't change anything. "If I'm dead then Arthemis is a Knight" is harder to understand.
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@Leif Hoving – Yup, I understand you, and from this point of view, you are right, you make all the reason....
A stone that God cannot lift is nothing, because He can do anything. There is a solution to any paradox, you know.
Note that the opposite statement "If Artemis is a knight then I am a knave" (the other side of the bi-conditional) cannot be said by anyone.
If the knave said it, it would be true, because Artemis is a knight and the speaker is a knave, contradicting the speaker's knave-ness.
If the knight said it, it would be false because Artemis is a knight but the speaker is not a knave, and knights do not make false statements.
In the actual puzzle, once the hypothesis is out, the conclusion does not matter.
This problem is flawed beyond conception. A knight can never say "" If I am a Knave, the the other is a knight""" It's an Assuming the Consequent Fallacy. And Knights always say the truth.
Now, if it's a knave, look what happens :
If the other guy is a knight, then the knave will be telling a truth. Which is impossible.
If the other guy is a knave, then the knave will be telling a truth. Which is impossible.
So NO ONE can say it !!!
I thought you guys were supposed to be smart !!
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The thing is, he is not saying he IS a knave, just that IF he's a knave. It doesn't matter whether he's a knight OR a knave, NEITHER one changes Artemis' status. It's as true as saying, "If the United States doesn't exist, then Artemis is a knight." Because he IS, no matter what!
Your argument is flawed beyond conception, and you're assuming that you're the only smart one here. I'll call that "assuming the consequent fallacy". I'll just copy my previous explanation here:
A knight saying the statement " If I am a knave, then Artemis is a knight" does not mean the knight would become a knave. Look at the statement carefully, " if ". Hence, if he/she is not a knave, then according to the said true statement, Artemis is either a knight or not a knight. There is no contradiction as we know that Artemis is a knight and therefore the statement is still true.
I recommend you consult a text on basic logic, for example A Beginner's Guide to Mathematical Logic - Smullyan ( Reasonably price Dover Edition, you are clearly not up to his primary tome, First Order Logic, yet) or the text used (in the 1970's at least) in the philosophy course at SF State Logic: Techniques of Formal Reasoning (Kalish and Montague 1964) to get these basic forms under control.
Very simple.
If P then Q statements are only untrue if Q is false, correct?
Then this means that the statement is true, regardless, since P doesn't actually imply anything.
This means that knaves can't say it.
The island doesn't have any jokers; therefore, only knights can say it.
Yes, that island doesn't have any jokers
1 "If I a am a Knave then Arthemis is a Knight" is equivalent (tautologically) to "If Arthemis is not a knight then I'm not a Knave". and then the hypothesis "If Arthemis is not a knight" is false and hence the whole stament is true, and the only people that are able to say truth in this island are the knights. This problem is a little tricky.
( A ⇒ B is false ) ⟺ ( A is true and B is false ) 2 A ⇒ B is tautologically equivalent to ¬ A ∨ B , and since B is true because B = Arthemis is a Knight , the whole statement is true, and this can only be said by knights.