Knights Or knaves?
You find yourself on the island of knights and knaves, where every inhabitant is of one of two types: a knight who always tell the truth , or a knave who always lie .
You come across two inhabitants, Artemis and Hera. Hera says: "If I am a knight, so is Artemis."
What type is Artemis and what type is Hera?
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3 solutions
Moderator note:
A slightly easier argument is to consider if Hera is a knight of knave.
Case 1:
Hera is a knight.
Then, from the statement, it follows that Artemis is a knight too.
Case 2:
Hera is a knave.
Then, we know that the statement is false. However, the only way for the statement "If P, then Q" to be false, is for P to be true and Q to be false. This tells us that P = "Hera is a knight" is a true statement, which contradictions the assumption that Hera is a knave. Thus, this case is not possible.
In conclusion, only case 1 is possible, so both of them are knights.
I think this problem is interesting for showing certain paradoxes in logic. Of course we can prove using propositional calculus that both of them must be knights, even more clearly :
The sentence Hera stated is in the form "If P then Q". By the law of implication, this is equivalent to "not P or Q". Suppose Hera is a knave. Then she always lies. So the sentence "not P or Q" is false. The negation of a false sentence is true. "not(not P or Q)" is, by De Morgan's law, equivalent to "P and not Q". So Hera is a Knight and Artemis is not a knight (Artemis is a knave). But this is a contradiction since we first supposed Hera is a knave.
But this sounds counter-intuitive. We can easily think of ourselves facing two liars in the street and one of them saying that if he is not a liar then the other is not a liar too. This apparent paradox maybe can be explained by one or more of this points:
- Translating natural languages into formal languages is not so easy. For example, when one say an "or" in a sentence in English: sometimes it can be better translated to a "xor" or even a "nand" in propositional logic.
- Material implication is an unsolved problem in philosophy of logic. From wikipedia: "Formal logic has shown itself extremely useful in formalizing argumentation, philosophical reasoning, and mathematics. The discrepancy between material implication and the general conception of conditionals however is a topic of intense investigation".
- Language and the metalanguage: Maybe this "if-then" connective is not the material implication but some sort of logical implication. If this is a logical implication, the negation of this must be a "not necessarily true", "not always true".
- Self-reference: when Hera says "If I am a knight..." she is saying "if I always tell the truth...". So if she is a knight and always tells the truth the sentence "if I always tell the truth..." is true itself. This is a self-reference statement, and self-reference statements are famous for causing paradoxes.
- Maybe the language of propositional logic is not the best for doing this here. It's a limited system. Maybe we better use predicate calculus or modal logic. In predicate calculus for example, the negation of "for all x, P(x)" is "there exist x such that not P(x)". "There exist the POSSIBILITY of Hera being a knight and Artemis being a knave".
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Thank you very much for sharing your point of view!
I love your take on several viewpoints. The first time I encountered the use of conditional logic, I thought the people who were discussing it crazy for talking nonsense. It's funny to reflect back on that now.
Artemis is a knight taking both scenarios, but Hera can be either.
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If Hera is a knight, it turns out Artemis is also a knight. And that is what Hera said ("If I am a knight, so is Artemis"). Then we know what she said is true, so Hera must be a knight, and, as you said, so is Artemis.
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To add on to what you're saying, Hera is not saying that she IS a knight, she is saying "if" she were a knight, he would be also. Therefore if she is not a knight, she would still be telling the truth (if Artemis were a knight OR a knave), making it impossible. So she is a knight either way, making him one also.
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@Jack Hillier – Yes, I totally agree with that. Thank you for sharing your opinion with us.
Circular logic
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@Georgi Lyubenov – Please, explain what you mean exactly.
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@Elena Gomes – Can you see it? Jack Hillier has the right solution to this problem, equal as challenge master note. and I would like you to read this . just missing the truth table of the biconditional which is ( A ⟺ B is true) if and only if ( A = B = True or A = B = False )
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@Guillermo Templado – I don't see what the biconditional truth table has to do with all this...
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@Elena Gomes – biconditional table doesn't have to do anything here, here is the conditional truth table... A ⇒ B ... If A then B ..
However, if she is a knave, then it follows that artemis is also a knave...you need to look at both sides before claiming one is perfect
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well, that's really a tricky problem i thought the same till i've read the comments above. The trick there is that.. IF.. cuz it's not like "if and only if hera is a knight then arthemis is too" it says "if i am a knight , so is arthemis" if we suppose hera is a knave, the statement she said, becomes true either if arthemis is a knight or a knave, since hera is not a knight, arthemis' state is not conditioned anymore.
or... we can take the 4 possible cases... 1 hera - knight and arthemis knight 2 hera - knight and arthemis knave 3 hera - knave and arthemis knight 4 both are knaves
the 2nd one we already know it's not posible.. and 1st one we also know it's definetly posible... so we have to analise 3rd one and 4th one..
in 3rd and 4th one hera is a knave and she says "if I am a knight, so is arthemis" and since she is not a knight in any of this cases, the statement 'doesn't care about arthemis' it is a true statement anyway
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That is exactly the case! Thank you for sharing your own point of viewing the problem.
She can't be a knave, because we've shown she would be telling the truth.
I understand that if Hera is is a knight, then Artemis is also a knight. But isn't it possible for Artemis to be either knight or knave if Hera is a knave?
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The thing we have to realize here is that Hera cannot be a knave, because:
1) As you said, if we suppose Hera is a knight, then Artemis is a knight. Which means nothing more or less than "If I (Hera) am a knight, so is Artemis. So Hera told the truth! She must be a knight.
2) If a knave said "If I am a knight, then x", then that statement must be a lie.
The ONLY way for such a statement to be false is for "I am a knight" to be true, and for "x" to be false. In ANY other scenario the statement is true.
This means that if the speaker is a knave, then they must be a knight, since the first part of the conditional is necessarily true for the statement to be false.
We have two different ways to reach a contradiction if we start supposing the speaker is a knave. The speaker is a knight.
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Ok, that makes sense. Thank you for the explanation.
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@A Former Brilliant Member – You're very welcome. Thank you for asking. I really appreciate your questions.
There is an issue here. You are taking into account only one scenario when solving this problem. In fact, there are two. The wording is the following : "If I am a knight, so is Artemis." (If P then Q) - conditional statement. This says nothing about the actual truth or falsehood of the sentence.
For this reason, the alternative scenario is that Hera is not a knight but a knave. If Hera is a knave then everything they say is false. Therefore, in this scenario, Artemis must be a knight. In this situation, we cannot decide which is which.
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If she was a knave, then she would be telling the truth, because, as we've shown her statement is true. Therefore she can't be a knave.
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Yes, Hera can't be a knave if the statement it's true. But you are missing my point.
The way you phrased the question leaves room for the two possibilities I mentioned.
However, we haven't proved that it is the case that Hera it's a knight since this can not result from a conditional statement. The question is: Can a knave utter the same statement? Yes! Because a knave can not say only one thing: that is the fact that he is a knave (contradiction).
If the the wording would be different like in: "I am a knight, so is Artemis" then the answer that both are knights is correct since the opposite situation would result in a logical contradiction.
An alternative way of wording that would ensure the response that both are knights is to change if with iff (If and only if).
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@Sava Sergiu – If you like to look at problems from the "p" and "q" perspective, we can explain this case as follows:
The proposition "if p, then q" is FALSE ONLY when two things happen simultaneously: p is true, and q is false. The proposition "if p, then q" is TRUE in any other scenario.
Now, if it WAS the case that our speaker, Hera, is a knave, then the first part of her statement would be false ("if I am a knight,...") because she is actually not a knight. That is, we have "if p, then q" where "p" is false. As said above, the whole proposition is true in this case!, because it is not the case that p is true and q is false.
As we see, if we suppose Hera is a knave, we end up with a knave telling the truth. This contradiction allows us to conclude Hera can't be a knave, she must be a knight.
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@Elena Gomes – "That is, we have "if p, then q" where "p" is false. As said above, the whole proposition is true in this case!, because it is not the case that p is true and q is false."
Why it is the statement true in this case? You provided no reason. Like I stated, if you take into account the hypothesis that the Hera can be a knave and says: "If am a knight, the Artemis is a knight". If Hera is a knave, then Artemis must be a knight!
But this is not a strict biconditional statement . Only in the case in which you would formulate the statement like: "If and only if I am a knight, then Artemis is also a knight", would leave no room for doubt and the answer that both are knights is correct.
Nota bene : It is irrelevant to point out to a truth table evaluation here since we are interested in the sentences separately (i.e.,"I am a knight", "Artemis is also a knight") based on the problem's assumptions.
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@Sava Sergiu – I did provide a reason to state what you quoted!
I explained that the statement "if p, then q" is false ONLY when p is true AND q is false.
And I said that the same statement is true IN ANY OTHER CASE. Which particularly means it is TRUE when p is false, despite q being true or false.
If a knave did say "If p, then q" then it necessarily follows that p is true and q is false. However, in this case our speaker said "if I am a knight, then q". If she is a knave, then she is a knight (because, if she is a knave, p is true!), which leads us to a contradiction.
Hence, the speaker can't be a knave.
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@Elena Gomes – Yes, the only way that a "If p then q" statement is false, is for "p" to be true and "q" to be false.
Suppose I said "If I am female, then the world is flat". This is always a (vacuously) true statement, because I (Calvin) am not female. The only way for the statement to be false, is for the person who said it to be female, and for the world to be flat.
@Elena Gomes – if I am a knight - how can this statement be false? If Hera is a knave and lying, then saying 'If I am a knight' is neither a false nor is it the true. Isn't this like saying that any statement that someone makes which talks about something that is not true can be refuted, because the condition is not true? For example, since I am not a cat any statement I make about me being a cat is automatically wrong? "If I were cat then I would like fish". By the above argument this is wrong because I am not a cat.
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@Leslie Munday – Not at all. By the above argument, your statement ("if I were a cat, I would love fish") is true! Since it's not the case that you are a cat and you don't like fish.
Editing this because I notice that some of the text got a little convoluted - not changing the intention at all
If Hera is a knave then Artemis can be anything they want knight or knave. Hera = Knave implies that the statement is false, therefore Hera=Knight implies that Artemis = knave. But Hera <> knight therefore this statement is irrelevant, because it tells us nothing about Artemis.
Restating the problem under the assumption that Hero=Knave we get: .. a knight who always tell the truth, or a knave who always lies. You come across two inhabitants, Artemis and Hera. Hera is a Knave who says "If I am a knight, so is Artemis.". But because you are a knave, what you're statement actually meant was "If I am a knight, then Artemis is a knave". But since you are not a knight, telling me what Artemis is if you were gives me no information".
So where does the logic of the problem fall down? I think it is in the following statement: "P to be true and Q to be false. This tells us that P = "Hera is a knight" is a true statement". The word 'if' is missing. It should read "if P= 'Hera is a knight' is a true statement. In the solution I propose "P= 'Hera is a knight' is not a true statement."
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See the Challenge Master note that I wrote up to the solution. It explains the issue that you're having. In particular, I disagree wtih
But because you are a knave, what you're statement actually meant was "If I am a knight, then Artemis is a knave"
The only way that a knave could have said "If I am a knight, then Artemis is a knight" is for the conditional statement to be true. If the conditional statement was false, then the statement would be vacously true.
For example "If I had a blue mini cooper, then I am a prince", could only be false IF I had a blue mini cooper, and I am not a prince. Otherwise, if I didn't have a blue mini cooper, it wouldn't matter if I was a prince or not, the statment would still be true.
Reading through the following comments (btw I have no idea how my comment got up here), I am reminded of the physicist who using logic and mathematics proved that nothing travels faster than the speed of light. But how do your equations explain entanglement (an observed communication between 2 particles that is instantaneous)? The physicist replies that is impossible, something else is going on, but I don't know what it is. (I've possibly been watching too many documentaries about Einstein.)
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I agree with Calvin Lin. You're mistaking the logical meaning of the words "if...,then...", and the way they actually work.
I suggest you read the other comments, where some people have had similar issues, and look carefully at the answers we've given to them.
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@Elena Gomes – I keep reading that 'If I was a knight then Artemis is a knight' is a true statement. If correct then I agree that Hera is a knight, but I cannot find any proof for this to be so. It 'can' be a true statement, but that does not mean that it is. "If I was a knight then Artemis is a knave' could also be a true statement?
What I think is happening is that there the If .. then statement is being interpreted differently. I do not see how it is that because this statement could be true, means that Hera is telling the truth.
Another idea using coins. If Hera states 'if I toss this coin and it lands on heads then Artemis is a knight' could be a true statement. Does this make Hera a knight because they told the truth?
Or have I gone onto another track?
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@Leslie Munday – First of all, thank you very much for not giving up on this, I really appreciate your questions.
I must say the statement you proposed sets up a very different situation, exactly because the speaker is not talking about her own quality of knight or knave. That statement could be either true or false.
In our case, "If I am a knight, so is Artemis" is NECESSARILY TRUE. I know you told me you've read this lots of times, but think about it this way:
If "I (Hera) am a knight" is true, what happens then? You conclude "Artemis is also a knight" is also a true statement, don't you? You could rewrite your whole argument as "If Hera is a knight, Artemis is a knight".
What I want to make clear now is that Hera said something you can logically deduce from the information the problem gives you. That means she is saying something that you proved yourself to be true! Therefore she's telling the truth.
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@Elena Gomes – Thanks for indulging my desire to understand the argument. I think that I have found the best explanation so far of what the confusion is.
"The ONLY way for such a statement to be false is for "I am a knight" to be true, and for "x" to be false. In ANY other scenario the statement is true."
I see that in order to make the statement, 'I am a knight' Hera must be a knight, but that's not what they said. The statement made is 'If I am a knight'.
Once the word 'if' is inserted, anything is possible. The statement becomes purely hypothetical, so they can say 'if I were a cat ..' and they are neither lying nor telling the truth. Same with 'if I am a knight', the statement is neither truth, nor is it a lie.
In fact one might argue that by saying 'if X..' then then speaker is implying that X is not true, otherwise they would simply have said X. So in this case X='I am a knight', is probably not true at all.
My guess is (someone else pointed this out) that it is the word 'if' that is confusing.
The ambiguous word is "if". "If I am" doesn't define Hera as either. Also, if Hera is a knave, it doesn't mean Artemis is not a knight. Watch this one.. "If I'm not a knight, neither is Artemis" That would be a lie, so therefore it's true? Or, "If I am a knight, Artemis is not a knight". This could be true as well.
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"If I am not a knight, neither is Artemis" could be said by both a knight or a knave. But I leave the proof to you, because the original problem is already causing enough confusion.
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"If I am a knight, then so is Artemis". That would be true if it were the case.👀However, the personification of Hera "as" a knight is substituted with an ambiguous presumption. " If" is the ambiguous word which masks her identity in a hypothetical scenario. This provokes an assumption- leading the listener to believe that her statement is truth thus fraudulently identifying her as a knight. Identity is distinct from a truthful statement in a hypothetical scenario. Also, even "if" Hera was a knight, Artemis could be, and be called by Hera either a knight or a knave in that hypothetical scenario and either would be truth.The two options of truth make a contradiction- which makes her statement inherently false. Hera- Knave, and unknown for Artemis. The statement begins with a hypothetical proposition.
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@David Nissen – I honestly didn't get a thing of what you said. I would be glad if you could explain me more clearly.
Anyway, I suggest you read the other comments thoroughly and try to think about the problem again.
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- If I am a knight, then so is Artemis. True (3. False
- If I am a knight, then Artemis is a knave. True
- If I am a knight, I could not say Artemis is a knave. False
A good way to burst the illusion of this is to say this yourself: If I am a cop, I can arrest you.. lol
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@David Nissen – But I think the easiest way to see if she's telling the truth or not would be to ask her: "what's my name?" lol.
@David Nissen – You are arguing the part of "so is Artemis", right? My English teacher taught me so is ≈ same as, and the only way to understand Q is to refer back to P.
IF :
P = { Hera = Knight }
THEN :
Q = { Artemis = Hera }
P = true = IPTQ = Q
P = false ≠ IPTQ ==> contradiction
IPTQ= false ≠ P ==> contradiction
.: Hera = Knight = Artemis
"If I'm not a knight, neither is Artemis" statement guarantees the existence of at least a Knight between the pair, no matter who said it.
"If I am a knight, Artemis is not a knight" statement is only possible as a scripted monologue for a Knight Hera, but this time with a knave Artemis as her companion.
I would like to say how much I've enjoyed this discussion, but that I wish there was some logic to the ordering of the replies to the solution.
But he say IF
Heres where i think it should be not enough information. If hera is a knave (always lying) she is lying by saying she is a knight and can be lying about artemis being one as well means both are knaves. Without artemis having any verbal or physical jester to confirm or deny their position you are left with only your own peraonal view of trusting them.
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The thing is, with "if P then Q" both P & Q are not two separable statements, because they are two parts of a whole and they reacted to each other. It's not like you mix one acidic and one base solutions together where you only get salt and water, it's more like you put in two volatile chemicals together and they go KaBOOM!
For "If P then Q", P can either be true or false, and Q also can be either true or false independently of P, but these qualities, of each their (P's & Q's) own personal truth value, are the very essence that will give meaning to the logical statement as a whole.
Not nuf info. Not "A then B" does not imply "A and not B"
But if Hera is a nave then the statement is false and Hera is a nave and Artemis is a knight And so there isn't enough info
It doesn't said if Hera is a liar, she can not make this statement, therefore it's impossible to figure out.
surly if a condition is not met we can decide on a solution... I can't punch someone I can't reach... Like if i was a Knaves and said "If I punch someone, they will bleed" you could test that by making them punch someone, and if they failed they must have lied... However if a knight said the same thing, when they punched someone that person would bleed, therefore can we have this question removed?
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Sorry, I didn't completely understand what you said. Could you please explain me?
Excerpt from your own comment :
....I can't punch someone I can't reach... Like if I was a knave and said "If I punch someone, they will bleed"....
So, to make your example exactly equivalent to the problem at hand, you have to use the same kind of [P] in both [if P then Q]s AND the same kind of speaker (I can see that you used a knave there), alright? In Elena's, "I am a Knight" is an obvious LIE to HER knave, so your "I punch someone" MUST also be a LIE to YOUR knave.
So, can we agree that your knave
1) did NOT punch someone, AND CONSEQUENTLY,
2) did NOT bleed THAT someone, BUT ANYWAY SOMEHOW,
3) your knave STABBED, BLEED OR EVEN KILLED ANOTHER SOMEONE,
just so we are clear that no bloody punches ever happened though some bleeding could have perhaps happened in a separate, buried and forgotten case, okay?
Now, we can go into building YOUR knave :
1) Your knave is a knave, who would rather commit seppuku than telling a truth.
2) Your knave have to lie, no matter what, where, when, who, why, how.
3) Your knave have to lie even when using conditional logic statements.
4) Your knave have to lie using FALSE IF P THEN Q.
5) Your knave have to lie BY this format, which is THE ONLY FORM of false conditional statements, and that is = if (TRUE statement), then (FALSE statement) ==> false if-then ==> a SUCCESSFUL lie (and a good night sleep with sweet dreams).
6) Your knave have mistakenly used if (FALSE statement) format for the first half instead, since s/he hasn't ever punched anyone.
7) Your knave cannot salvage the conditional logic statement into being any lie suitable for a knave worth his salt.
8) Your knave finally told a truth.
9) Your knave will have to die, and stabbed himself/herself and bled to death.
10) All knave are selfish cowards who think of lying as the best self-preservation strategy, so they will think ahead of their speech and would never, ever speak if it would lead to their deaths.
11) Your knave doesn't exist.
I want to share an idea which might help understand the issue. Let's try and simplify the problem. Firstly, Hera isn't working for me, so I'm going to remove Hera from the equation and I'm going to take their place. Now Artemis and I are standing behind a curtain and you as the observer, are on the other side with no knowledge of what is happening on our side of the curtain. You are told that both of us are dressed as either a knave, who will lie to you, or a knight, who will tell you the truth. You can enquire as to which of us is dressed as a knave and which as a knight, and you will receive a statement from me which will allow you to determine which costume both of us are wearing. (Ok, I believe that I am following the problem as stated.)
Now I am going to add a small twist to the issue. I am removing all of the costumes from behind the curtain, with the exception of 1 knave costume and 1 knight costume. (Since you can't observe the other side of the curtain you are unaware as to what is going on.) Now if I put on the knight costume then Artemis has to put on the knave costume. Similarly when I put on the knave costume Artemis is dressed as a knight. So here we go - I am putting on the knave costume, Artemis the knight costume and when ready you are going to make your enquiry.
I speak exactly as stated in the problem, 'If I am a knight, so is Artemis'.
This is obviously a lie, since we only have 1 knight costume between us. But that's ok, because I am dressed as a knave, so I am allowed. to lie.
My changes are within the scope of the problem, and I don't see that any rules have been broken. Yet a knave and a knight have satisfied the criteria of the problem? The only thing wrong would be that because I am dressed as a knave, me saying 'If I am a knight ..' does not make sense.
That's it. You can have Hera back now .. oh, and here are the other costumes that I had hidden away.
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You have introduced a very different scenario. "You" have dressed as a knave, therefore you could never (being dressed as a knave) make such a staement, because a knight and ONLY A KNIGHT can say "if I am a knight, then X" (being X a true proposition).
I don't agree with you when you say: "I speak exactly as stated in the problem, 'If I am a knight, so is Artemis'.
This is obviously a lie, since we only have 1 knight costume between us. But that's ok, because I am dressed as a knave, so I am allowed. to lie."
I understand that between you two there is only one knight, because that's how you distributed the costumes. However, if your statement has to be false (bc you're a knave), the one and only way for "If I am a knight, then X" to be false is for "I am a knight" to be true, and for "X" to be false, as we have explained in other comments. This tells us that if you are a knave, AND make that statement, then you are a knight,
The conclusion here is that the scenario you set up is impossible! If the problem tells you the speaker is a knave, then he could never make a statement of the form "If I am a knight, then X"
Hera : If I am a knight, so is Artemis.
Hera is stating a conditional statement using the conditional logic of "IF P THEN Q".
If P then Q can either be true or false.
Of course, as any other statements, a true conditional statement can only be spoken by a truthful knight while a false conditional statement can only be spoken by a deceptive knave, that's how this kind of logic puzzles work.
Let's suppose that we have a Knight Hera first, and a knave Hera later.
A Knight Hera have to use a TRUE version of IF P THEN Q. Our P here is "I am a Knight" while the Q given is "so is Artemis" that is equivalent to Q = "Artemis is also a Knight".
Right now, and in this supposed case that we've chosen (to have a Knight Hera), we don't know if this supposed Q is true or false, yet, but we know P [P = "I am a Knight"] is true because we supposed her so, and we have to make THIS Hera's IPTQ (short form for If P Then Q) to also be true so that it becomes possible for a Knight Hera to proclaim the true version of that conditional statement freely as her supposed Knight self.
True IPTQ with true P has to have a true Q to be meaningful, otherwise, with a false Q, the full IPTQ will become false and therefore, impossible to match THIS Hera's Knightly personality and speech patterns. So, if we use arrows to guide this storyline, we have :
Knight Hera ---> in possession of true P ---> mission : to save her IPTQ from falsehood ---> true Q collected! ---> HEA!!! yay
In the end, Knight Hera got a true Q for her conclusion, therefore, with the Q given "so is Artemis" that is equivalent to Q = "Artemis is also a Knight" becoming true, this case is closed with Hera & Artemis as 2 Knight buddies.
Now it's time to consider a knave Hera, so everything has to change and return back to its original state.
A knave Hera have to use a FALSE version of IF P THEN Q. Our P here is "I am a Knight" while the Q given is "so is Artemis" that is equivalent to Q = "Artemis is also a Knight".
Right now, and in this supposed case that we've chosen (to have a knave Hera), we don't know if this supposed Q is true or false, yet, but we know P [P = "I am a Knight"] is false because we supposed her to be Hera the knave, and we have to make THIS Hera's IPTQ (short form for If P Then Q) to also be false so that it becomes possible for a Hera to proclaim the false version of that conditional statement freely as her supposed knave self.
It turned out that false IPTQ with false P doesn't exist, since the only way to make up a false IPTQ is by having a true P and a false Q. Therefore, it's impossible for a knave Hera to utter a knavely conditional statement, the same way that any knave can never proudly and openly introduce themselves as a knave to a new friend.
TRUTH TABLE FOR CONDITIONAL LOGIC STATEMENT {IPTQ}
| P | Q | if-then | ANALOGY |
| T | T | T | Following a signboard- / map- / GPS-recommended road AND finally reaching the planned destination |
| T | F | F | Following a signboard- / map- / GPS-recommended road BUT getting lost in the end (in front of a haunted house in some deserted ghost town) ==> the map is lying, most probably a cursed one |
| F | T | T | Taking an alternative route AND SOMEHOW still managing to reach the destination (the map IS truthful; only that it was just showing the most economically straight forward way but other time-consuming or gas-wasting or dark and dangerous roads definitely still exist) |
| F | F | T | Taking an alternative route and getting lost somewhere (the map IS truthful; people not reaching their destination due to their own mistakes / stubbornness / idiocy is of no fault of the map's) |
In essence, this puzzle is asking for a matching truth values between P and IPTQ. As only true P + true Q = true IPTQ exists, the answer is just TRUE Q, whatever the Q is.
Hera said "if I am a Knight, so is Artemis." Let us see some scenes 1 Hera is a knave and Artemis is a knight A) The first part of sentence would be true because a knave always lies so Hera explains herself as a knight B) The second part of the sentence would be wrong because a knave always lies so a knave would not say the truth that Artemis is a knight. 2 Hera and Artemis both are knaves This wouldn't be true because again the we and part dies not fit. 3 Artemis is a knave and Hera is a knight Knights are truthful so Hera wouldn't lie about Artemis being a knight. 4 Both are knights Now that no other case fits, ur is quite clear that both are knights.
Think not enuf info. Statement could be false in which case Hera is a knave and Artemis either. Reason is that statement "A then B" can be false even if both A and B are true.
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The thing is when a knave wants to use if-then statements which are ALSO LIES (we can have both true if-thens and false if-thens, but a knave can only say the false version ones), they have to use TRUE FIRST HALVES after the ifs AND FALSE SECOND HALVES after the thens. For this particular puzzle, in the eyes of a SUPPOSEDLY knave Hera, the first half of the statement that she already said was "I am a Knight", and this would be her run-of-the-mill LIE which she can even mutter in her sleeptalking ON ITS OWN (without the if-then pair inserts), but within the if-then context, using a lie for the first half is a no-go for a liar, since she cannot turn the if-then into a whole meaningful lie with whatever second half she can think of; that the IF (first lie) + THEN (whatever) = (an unsalvageable lie) ==> (an unspeakable truth) for a SUPPOSEDLY knave Hera.
Artemis never said anything so hera and Artemis could both be knave
Let's start supposing what Hera said is actually true. If she told the truth, then she is a knight, and what she said is right. Since we are supposing she is a knight, and what she said must therefore be true ("if I am a knight, so is Artemis"), then Artemis would also be a knight. So if she is a knight, so is Artemis.
We have discovered that if we suppose Hera is a knight, then it follows that so must be Artemis. That means, if Hera is a knight, so is Artemis .
But that is exactly what Hera said! She said: "If I am a knight, so is Artemis", and we now know for sure that that statement is , in fact, true. This means Hera told the truth , therefore she is a knight , and what she said is then true. Since she is a knight and what she said is true, then Artemis is a knight.
The answer is they are both knights.