If I am a knight
You find yourself in a village that has knights (who always tell the truth) and knaves (who always lie), and come across an individual who says,
"If I am a knight, then I always lie."
Is this person a knight or a knave?
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7 solutions
I disagree that because the speaker is not a knight any statement starting with "if I were a knight..." must then be true, as this construction causes us to enter a world where this if statement was true much more similar to a supposition than a standard if statement. For example, when I say "If I were 18, I would be legally allowed to drink alcohol" that's clearly a false statement even though I am not 18 and so by your logic it would then be true. I would agree with however if the speaker had instead said "If I am a knight, I always lie." and then because the first part is untrue the sta
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*statement as a whole is true.
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OK, I've updated the problem as per your suggestion... Thanks for your feedback! Do you think it looks cleaner now?
Yeah... This question does hit upon one of the most hotly debated and notoriously counter intuitive part of propositional logic vs the spoken language.
As quoted from "http://criticalthinkeracademy.com":
The English language expression “If A then B” is used in a variety of ways, and not all of them map onto the meaning of the conditional as defined in propositional logic.
“If A then B” is FALSE when the antecedent “A” is true and the consequent “B” is false; for all other combinations of truth values, the conditional is TRUE.
I guess this brings up the all important question: Should I mention, somehow, that the knights and knaves are "propositional logicians"?
Thoughts @Calvin Lin ?
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I am familiar with that slight pecularity of propositional logic, I guess I would argue that "If I were a knight..." has a truth value that does not depend on the realities of the actual world, so that "If I were a knight" is not false simply because the speaker is not actually a knight and is thus always true. This is to differentiate from the similar but distinct statement "If I am a knight" whose truth value does depend on outside reality
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@Anthony Holm – Good point, lemme think about that for a bit...
Like you mentioned, that is how the truth table of the conditional is defined. In mathematics, if you believe a false statement, then any conclusion is true (even statements that contradict each other). This allows us to prove by contradiction .
We can add that they are "propositional logicians", which is the implicit setup for such knight/knave problems (unless otherwise defined).
I think it's, well, unreasonable too.
To say that "If I were 18, I would be legally allowed to drink alcohol" is false is to say that whoever said this turns out to be 18 and he/she would not be legally allowed to drink alcohol. In other words, we have to assume that there's a person who is 18 that is not allowed to drink alcohol. For the statement "If I am a knight, then I always lie," if it's false, we have to assume that someone who said this is a knight and not always lies. So if a knave says this, then he would have to be a knight, a contradiction. I hope this helps.
The if makes it possible to not be true but you were correct because the statement ,I always lie , makes it impossible to be said by a night or a knave.
Knaves are incapable of truth. They are not incapable of contradicting themselves. If the speaker is a Knave, and if he doesn't KNOW that you know the rules of Knights and Knaves, then he is lying, perfectly consistent with Knavery. Internal contradiction does NOT exclude lying.
Why can we not examine the sentence as a whole instead of breaking it down into parts? He says if he were a knight he would always lie. Which is false because knight's cannot lie. Therefore the statement is false. False statement = knave. We know that knight's cannot lie. Probably everyone on the island does. Feels like a poor example of demonstrating this principle.
"Formal logic" is very limiting. I think a whole nother class is needed for cases where the 'if' part is false. Formal logic is lazy and just dumps these cases into the 'true' category.
Looking at the bigger picture (assuming the village ONLY has knights and knaves) you could deduce that the individual is a knave.
Disagree... if he was a knave, he couldn't say "if i'm a knight, i would tell the truth" which is actually tru. So he would lie and say "if i'm a knight, i would lie"
EDIT: Do not pay attention to this solution, the problem has been changed.
NOTE: This is not an actual solution.
I do not think that the answer is correct. If the statement were "If I am a knight, then I always lie" then the answer would be correct. But since the statement is "If I were a knight, I'd always lie" the answer is not correct, as propositional logic does not apply to this statement.
The statement "If I were 6, I'd be able to drive" is false even though I am not 6, as the "I were" indicates suppositional logic. "I were" indicates suppositional logic, as it implies wishful thinking; it would not have been proper grammar otherwise.
The problem needs to be changed to "If I am a knight, then I always lie."
Thanks. I see that @Geoff Pilling has updated the problem statement.
Ok, thanks.
It puzzles me, and I'd like to object.
The statement is conditional and is not personal. I read it as “If someone is a knight, then he/she is a liar”. It is obviously false no matter who is saying it. So the teller is a liar.
Even if we apply the sentence personally (and we know the person is a liar) then it means “If I am a knight (which I'm not, so it does not apply to me), then I would be always lying (which is not true, but you know that I'm not a knight and I'm supposed not to tell the truth anyway)”.
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Yes, the answer ìs counterintuitive, but that doesn'the mean it's wrong. With the old problem statement, you would be correct. But now you are wrong. "If I am a knight, then I always lie" is a simple conditional statement, and it must be evaluated as such. The only way it can be false is if the knight doesn't always lie. When a knave is speaking, then it is always true, because if the conditional is false, the whole statement is true by default. You cannot just change "I" to "someone" and expect the new statement to be logically equivalent.
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"if the conditional is false, the whole statement is true by default" Can you explain this?
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@Oksana Oriekhovska – That is one of the rules of propositional logic: p -> q = !p or q.
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@Siva Budaraju – And what's the sense in it?
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@Oksana Oriekhovska – If you assume a false statement, then anything can be proven to be true. For example, consider the statement "If 1 + 1 = 3 , then Brilliant is worth a zillion dollars".
We will show that Brilliant is worth a zillion dollars under the assumption that
1
+
1
=
3
.
Proof by contradiction. Suppose not, that Brilliant is not worth a zillion dollars.
Then, since
1
+
1
=
2
and
2
=
3
, we reached a contradiction.
Thus, the premise is wrong and so Brilliant is worth a zillion dollars.
Hence we can conclude that, "If 1 + 1 = 3 , then Brilliant is worth a zillion dollars" is a true statement.
Of course, it is equally true that "If 1 + 1 = 3 , then Brilliant is not worth a zillion dollars".
The statement, "If I am a knight, then I always lie" is a false statement because the states (being a knight and always lying) are mutually exclusive. A knave is defined by being incapable of making a true statement. This statement fulfills that requirement; therefore, the speaker must be a knave.
If one wishes to treat the statement as two separate statements, then the words, "I always lie" would be true if he is a knave and a knave cannot tell the truth, and they would be a lie if he was a knight and knights cannot lie. But we can dispense with that entirely because they are not two separate statements. They form two phrases within a single statement, and only one need be false to render the statement false. If one requires every element of speech to be false to qualify the speaker as a knave, he would be incapable of coherent speech.
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Yes, the answer is counterintuitive, but that doesn't mean it's wrong. With the old problem statement, you would be correct. But now you are wrong. "If I am a knight, then I always lie" is a simple conditional statement, and it must be evaluated as such. The only way it can be false is if a knight is speaking. When a knave is speaking, then it is always true, because if the conditional is false, the whole statement is true by default.
I wish that were true. Makes sense... Dunno why logicians change it.
I think I understand the problem here. No, it isn't that the logic is counterintuitive. It's that the words have a different definition to logicians than they do to the rest of us. From the perspective of the english language speaker who is uninitiated into the world of logic, your solution is utter nonsense. A statement is false if any part of it is false. I can't persuade the jury that I am innocent merely by changing the word order in which I answer a conditional question. This answer only makes sense when the words if and then represent are markers for a game of logic that allows for such inanities as falacious truths.
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Right, but we really need a logician to ask for an explanation of how this works. Then...
I love the ridiculous example. It made me laugh a lot.
And I agree. So agree
Formal logic is stupidly limiting. It makes no distinction between 'If I am' , 'If I was', and 'If I were'. Furthermore when the 'if' part is false it lazily dumps the whole thing into the true category without any further analysis.
These problems are either lifted from Smullyan or mimicking the types of problems in those books.
Smullyan, however, describes how to handle if...then conditionals and then does several different simple problems to illustrate the point. His books are teaching formal logic in an easy and fun way.
When the problems are taken out of that context then they become trick questions. No one unfamiliar with Smullyan or unfamiliar with propositional logic will read "if I am a knight, then I always lie" as being true for a knave. Because in English that sentence means "knights always lie". Which is a false statement for everyone, including knaves. And only knaves can speak a falsehood. Therefore that person is a knave.
Essentially tricking people by withholding vital information is not the way to engage people with logic and maths.
Yeah, I agree with you. Another thing which bothers me is that it is clearly written that villagers only have Knaves and Knights. There cannot be anyone who belongs to neither of these categories but the answer says this guy does then it means the question provided us with false information. Seriously, does Brilliant.org ever checks the questions people are posting?
Andy Gregory, and where is that context where the answer given by author will be the right one (He can be neither a knight nor a knave)?
I am not really convinced with this solution . Okay he can not be a knight since knights always tell the truth but this statement is false so it’s a lie and knaves always lie🌚.
If he is a knight, then he would always lie according to what he said. This cannot happen. If he were a knave, then he would be a knight according to what he said. This cannot happen either. The individual is neither.
Knights,as we know, never lie. But do not just assume he is a knave. In logic, 'If (False) then q' is true. So a knave he cannot be.
Not that I agree with prepositional logic, myself
This is a sentance, therefore it must be taken as a whole unless specified otherwise (since it would be non-standard english), which it was not. Therefore, the statement as a whole can not be spoken by a knight but can be spoken by a knave. We are told only that there are knights and knaves, implying that a speaker must be one or the other in which case it must be a knave, if we allow that implication to be false, then it could be anyone able to tell a lie, including knaves but not knights.
Let's have another look at the statement. "If I'm a knight, then I always lie." The person wouldn't be a knight if he was saying he's a knight, but there's never a statement of him saying whether or not he's a knight, or for that matter him making any comment on himself. Yes, he sure could be a knight, or a knave, but there's no definitive statement of him saying anything in regard. Hence he could be a knight, or a knave, or neither.
Nope. I'm confused
If Knight (true) ==> then lie (false) ==> False Statement ==> Not a Knight
If Knight (false) ==> then lie (true) ==> True Statement ==> Not a knave
If he is a knight, then the second part of the statement
would be a contradiction, and since knights always tell the truth, he can't be a knight.
If he is a knave, then the sentence,
will always be a logically true statement no matter how you fill in the rest, since he is not a knight. And since knaves never tell the truth he can't be a knave.
Therefore, he can’t be a knight or knave .