False and.... False?!?!
On an island, everyone is either a knight or a knave. Knights only tell the truth (only make true statements), and knaves always lie (only make false statements).
There exists a legal statement such that neither it nor its negation can be said by a knight.
Is this true or false ?
Note :
By a legal statement is meant a sentence which does not lead to inconsistency . For example: ‘ 1 + 1 = 2 ‘ and ‘This sentence is more than 5 characters long‘ and ‘I am a knight‘ are legal statements (which, in this case, can also be said by a knight), while ‘This sentence is false‘ and ‘This sentence is true‘ are not legal statements.
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1 solution
Why does the negation of a statement have to be false?
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The negation of a statement P is defined to be the statement which is true if and only if P is false (and therefore false if and only if P is true). So basically, that's the definition of negation! And since S is true, the negation of S is false.
If S is true, then it can be said by the knight (as a true statement) and not said by the by the knight (by its content). Doesn't this lead to an inconsistency, making it an illegal statement?
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We have given a knight the property that he can only say true statements. In other words: If a knight makes a statement, it's true. From which does not directly follow: If a statement is true, it can be said by a knight. What has been shown by this problem is that the truth of a statement is indeed not equivalent with the possibility that a knight can say it!
Can you formulate this problem in terms of mathematical objects? Because often when we allow statements to refer to themselves, everything goes wrong, so I am not sure if your problem can be well defined.
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I think something like this should do:
Let K P be the proposition that a knight can say P , then K P ⟹ P . If K P does not lead to inconsistency we say that a knight can say P . If P is wrong, it follows ¬ K P and we conclude that a knight can not say P . Now, we have S ≡ ¬ K S , therefore K S ⟹ ¬ K S , from which we conclude ¬ K S ∧ S and thus ¬ ( K ¬ S ) (from ¬ ¬ S ).
'Forever Undecided: A Puzzle Guide to Godel' (Raymond Smullyan, 1987) features this whole idea of self reference with knights and knaves (that's basically what most of the book is about) and asks very similar questions to this one. That's why I truly believe that such a question makes sense. I myself am not a mathematician however and I do see the possibility that I am completely wrong on this. Please correct me if I am wrong.
The negative of S is " This sentence can be said by a knight". That phrase can obviously be said by a knight.
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The negation of S is "The sentence <This sentence can not be said by a knight.> can be said by a knight.".
This is a paradox and not a legitimate answer. Paradoxes have no answer
The negation of this statement is this sente5can be said by a knight, which a knight can obviously say. When you negate the statement, you lose traces to the original statement
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That is not correct. Read up the meaning of negation. What you refer to is not negation but something else. I have linked the definition of negation in my problem to avoid this misconception. If you have read the first few sentences of the wikipedia article consider the following:
We have the statement "This statement can not be said by a knight". What you propose as a negation is "This sentence can be said by a knight.". Now, both of these are always true. Thus, one can not be the negation of the other one. Now consider my proposed negation. Each of the statements is true if and only if the other one is false, thus, they are negations of each other. Furthermore, I have shown this formally.
Look Matt a statement is either true or its false. If it's TRUE then a knight can say it. If its false its negation must be true so a knight can say the negation. What you wrote is a proposition that refers to itself which is subtle and gets theoretically complicated. If anyone is gonna sit down and write down a logical theory , you'll soon find out that's quite illegal
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I can relate to the feeling that there has to be something wrong with this. But in fact, it's fine. There are ways to look at these kind of propositions and you can show that neither this statement nor its negation can be said by a knight. More formally, in every possible world (or equivalently every model of the axioms), this statement has a definite truth value, but in no possible world a knight can say it.
The claim you made that every true statement can be said by a knight can simply not be proven and is in fact not true.
I will show that both the following statement (call it S ) and its negation ( ¬ S ), can not be said by a knight:
from which follows that the solution to the problem is 'True'.
Assume that S can be said by a knight. Then S would be false, since it asserts that it can not be said by a knight. This contradicts with the assumption that knights can only make true statements. Thus, we get a contradiction from which we conclude that the assumption that S can be said by a knight is false - in other words, that S can not be said by a knight. Since this is precisely what S asserts, we have shown that S is true .
Since S is a true statement, its negation ¬ S is false and can thus also not be said by a knight (because knights can only say true statements).
More formally : Let K P be the proposition that a knight can say P , then K P ⟹ P . If P is false, it follows ¬ K P and we conclude that a knight can not say P . Now, the sentence becomes S ≡ ¬ K S , therefore K S ⟹ ¬ K S , from which we conclude ¬ K S ∧ S and thus ¬ ( K ¬ S ) (from ¬ ¬ S ).
Further remarks : We have given a knight the property that he can only say true statements. In other words: If a knight makes a statement, it's true. From which does not directly follow: If a statement is true, it can be said by a knight. What has been shown by this problem is that the truth of a statement is indeed not equivalent with the property that a knight can say it!
Further implications : One way to interpret this result very roughly and informally is that a 'reasoner' can never know all true statements and only those. You can read more on this here (Doxastic logic) .