Dwight and Dave
You happen to be in a village full of knights (who always tell the truth) and knaves (who always lie), but you can't tell which is which.
You approach two people, Dwight and Dave, one of whom is a knight and the other is a knave. One of them makes the following two statements:
"If I am a knight and he is a knave, then I am Dwight and he is Dave.
But If I am a knave and he is a knight, then I am Dave and he is Dwight."
Hmm... knight, knave, Dwight, Dave...
Is the speaker a knight or a knave? Dwight or Dave?
Clarification:
As with all knights and knave problems, treat this question as a
formal logic
question.
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6 solutions
Moderator note:
To clarify:
An if-then statement is false ONLY in the case where the hypothesis is true and the conclusion is false.
A false hypothesis indicates the if-then statement is TRUE.
With "If I am a knight and he is a knave, then I am Dwight and he is Dave" the hypothesis is "I am a knight and he is a knave" and the conclusion is "I am Dwight and he is Dave."
With "If I am a knave and he is a knight, then I am Dave and he is Dwight" the hypothesis is "I am a knave and he is a knight" and the conclusion is "I am Dave and he is Dwight."
Because it is given in the problem one of them is a knight and one of them is a knave, one of the two hypotheses must be false, which means one of the two if-then statements must be true.
The case of a false hypothesis indicating an if-then statement is true can be anti-intuitive, because it allows statements "if pigs can fly, then I am a llama" to be true. Even in this case, though, it's not off from how people use the language in real life: "if the salesman is telling the truth, then I'm a blue-skinned alien" is a statement in the confidence in the veracity of the salesman.
To me, Dwight the Knave is also possible, depending on what you might consider a lie.
The way I see it, a liar always says the oppostite of a true logic statement. A statement is only a statement, when the condition for the statement is true.
When Dwight is Knave, we got the following two statements:
If Dwight = Knight, then speaker would be Dave, but Dwight is not a Knight, therefore the statement is irrelevant. If Dwight = Knave, then speaker would be Dwight. The condition is true, and so is the statement.
Therefore, Dwight is Knave could also be a possibility.
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I think this touches on one of the differences of propositional logic problems (on which knights and knaves are based) vs. written English.
According to propositional logic, if a knave starts off with "If I am a knight, _ _ _ _ . " then this is a TRUE statement, regardless of how you fill in the blank, since he isn't a knight. So, the statement can't really be considered irrelevant since by this definition it is indeed TRUE.
This is different for written English, where there are clearly ways you could make this a "false" statement.
@Calvin Lin thoughts?
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Perhaps this would make more sense in the subjunctive tense. As the question is worded, the knave can say what he likes after the "If I am a knight..." phrase and it would be neither true no false, but rather irrelevant.
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@Malcolm Rich – In the common interpretation of the English language I would agree, but when dealing with propositional logic, as is the case with knights and knaves all statements are either TRUE or FALSE. There isn't the concept of an irrelevant statement.
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@Geoff Pilling – But as the FALSE status of the first statement already makes the whole thing true, whatever comes after that can have no relevance in determining if someone is a liar or not.
We have two propositions which need to be compatible with each other. I will use propositional logic.
First proposition: “If I am a knight and he is a knave, then I am Dwight and he is Dave” .
(A = knight) ˄(B = knave)⇒(A = Dwight)˄(B= Dave)
| Num. | A=Knight˄B=knave | A=Dwight˄B=Dave | Proposition |
| 1 | True | True | True |
| 2 | True | False | False |
| 3 | False | True | True |
| 4 | False | False | True |
Possibilities: N.1 is true then A is a Knight and he’s Dwight, and the proposition is true, compatible with A=Knight.
N.2: If A=Knight and A=Dave then the second part is false and the preposition is false, not compatible with A=knight.
N.3 and #4 are illogical as well because if A were not a knight then the statement would be true, incompatible with A = knave.
Therefore, for the first sentence, the only logic possibility is N.1.
In the same way, for the second statement: “If I am a knave and he is a knight, then I am Dave and he is Dwight” .
(A = knave) ˄(B = knight) ⇒(A = Dave)˄(B= Dwight)
| Num. | A=knave ˄ B=knight | A=Dave˄B=Dwight | Proposition |
| 1 | True | True | True |
| 2 | True | False | False |
| 3 | False | True | True |
| 4 | False | False | True |
N.1 The statement is true, and incompatible with A=knave.
N.2: A=knave and A=Dwight, and proposition false. Compatible with A=knave.
N.3 means A=knight, A=Dwight, and proposition true, compatible with A=knight.
N4 means A=knight, A=Dave, and proposition true compatible with A=knight.
Possibilities N.2,3,4 for the second proposition also work.
The only scenario that works for both propositions is N.1 for the first together with N.3 for the second, which means that the speaker is a knight and he’s Dwight.
But why can't it be DWIGHT THE KNAVE......If you check it does satisfy all the conditions and comes out to be a possible answer......Please if u have any logic then expalin to me.
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If the speaker were Dwight the Knave, then his first sentence would be the truth, and this would be inconsistent with knaves. (Since the "If" part of the sentence isn't satisfied, then we claim that the statement is logically true)
Check out this page on propositional logic.
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(Since the "If" part of the sentence isn't satisfied, then we claim that the statement is logically true)
The only thing I would suggest is that then "If I were" should be changed to "If I am".
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@Peter Byers – I think you mean "am" to "were"
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@Malcolm Rich – I have already made the change from "were" to "am"... Hopefully the wording is all correct now.
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@Geoff Pilling – I think it is the other way round. In the subjunctive "were" case, the knave is still obliged to say what would happen. Like the familiar problem with the two knights with their doors to death, the subjunctive case allows the knave to say what would happen if he just happened to be a truth teller.
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@Malcolm Rich – I think this formulation makes the quote more catchy. In it's simplest form it would be "Were I a Knave, I'd be Dave. Were I a Knight, I'd be Dwight"
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@Malcolm Rich – Or: "I'm Dave if I'm a knave I'm Dwight if I'm a knight"
@Malcolm Rich – That could be a good problem too. But the most immediate point is that the new/current version i.e. "If I am a knight and he is a knave, then I am Dwight and he is Dave. But If I am a knave and he is a knight, then I am Dave and he is Dwight." makes the answer to the problem "Dwight the knight".
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@Peter Byers – I don't see how it can be claimed that a false conditional clause makes a statement true - or even what purpose it serves. I could then conclude that no knave can say "If black is white,...." unless you are saying he can't be a Knave because he has inadvertently made a true statement.
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@Malcolm Rich – Yes, this is correct... According to propositional logic, if the "If" part is false then the statement is true... That's what makes it different from the spoken language. Here is a reference.
I agree with Max. We can say the speaker is Dwight, but cannot conclude anything about whether he is a knight or knave. I am not a logic expert, but according to my understanding of English, if you say "If A, then B", and A is true, then B can be taken to be true. But if A is false, B is unknown (not false). Here are the possibilities as I see them:
We have 4 possible hypotheses about who the speaker (S) is:
1) Dwight the Knight (S = DT = KT) 2) Dave the Knight (S = DV = KT) 3) Dwight the Knave (S = DT = KV) 4) Dave the Knave (S = DV = KV)
We have two conditional statements:
A) If S = KT, then S = DT B) If S = KV, then S = DV
Now, if S = KV, the speaker's statements are lies. If we interpret "lie" to mean the opposite of the truth, that would seem to imply that condition (B) is inverted:
A) If S = KT, then S = DT B') If S = KV, then S = DT
So working through the possibilities, we have:
1) Dwight the Knight (S = DT = KT): A) If S = DT (true), S = DT (true) - consistent with hypothesis B') If S = KV (false), failed condition - no information
2) Dave the Knight (S = DV = KT) A) If S = KT (true), then S = DT (false) - inconsistent with hypothesis B') If S = KV (false), failed condition - no information
3) Dwight the Knave (S = DT = KV) A) If S = KT (false), failed condition - no information B') If S = KV (true), then S = DT - consistent with hypothesis
4) Dave the Knave (S = DV = KV) A) If S = KT (false), failed condition - no information B') If S = KV (true), then S = DT - inconsistent with hypothesis
From this it seems we can exclude hypotheses where S = DV, but cannot exclude S = KT or S = KV. So we know we are talking to Dwight, but don't know if he is a knight or knave.
I've tried to lay this argument out as clearly as possible so the logic experts in the room can pinpoint where this analysis is going wrong!
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Yes, B is unknown, but according to prepositional logic (the basis of knights and knaves problem) the entire "If... then..." statement is considered true, if the "if" part is false.
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Sorry, I'm being thick. In the case of Dwight the knave, the first statement has a failed if condition, so I am unsure what we can conclude from that. If I understand you correctly, you are saying that because the condition failed, the statement as a whole is true, and this is inconsistent with being a knave.
I think this comes back to how we define a liar. As I have interpreted the problem, lying means that the the statement following the "if then" condition is the opposite of the truth. e.g. "I am a Dwight" actually means "I am Dave." However, it's a bit unclear to me how we would extend this to the statement as a whole. What is the opposite of an "if then" statement? Should we interpret a lie as "If not A, then B" And if we do this, do we invert the entire statement, or individual parts of the statement? i.e. "If not A, then not B"
To me this falls back a bit on the paradox of self-referential statements, such as "This statement is false". Such statements defy logical analysis since the language used to express the statement is the subject of the statement. In this case, what parts of a knave's statements can we accept as the independent framework of language used to express information vs. statements that are subject to inversion based on the identity of the speaker?
As the problem is posed, it is not clear that statements should be given their formal-logic meanings rather than their English meanings. Maybe that is supposed to be apparent from the context but it would be nicer to the solver to make it express.
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Good point... I've clarified the question. Thanks for the feedback!
Whether and argument is counterintuitive or not sounds to me like a "get out of jail free" card. And saying this is "formal logic" is unsatisfactory because it doesn't even begin to explain why a Knave cannot begin a statement with "If FALSE, then..." - it's a breach of Free speech and identity politics rolled into one. If we do have a fundamental mathematical rule here, it's not obvious to me. Ignoring the definitions, can the "If false then..." statement ever be proved to be True?
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The Challenge Master has probably explained it best above. (Appended to my original solution)
"If false then true" is the roundabout, longer but reachable route towards a destination. Say you are at the position of 6 o'clock and want to go to the 9 o'clock on a 4-roundabout, you could go closer to the left before reaching it and just keep to the left to reach it the fastest and easiest possible way by traffic's analogous of "if true then true", but you can do it the American way and go against the flow and travel triple the distance plus risk your safety (or life) by doing this and if you're lucky, you'd just pulled off an "if false then true".
WRONG! WRONG! WRONG! I don't know where you get your ideas about logic, but they are WRONG! Both statements effectively say, "Dwight is a Knight, and Dave is a Knave." Since we know that one is a Knight, the other a Knave, either BOTH statements are true, or both are false. It is IMPOSSIBLE that one is true, the other false. If both are true, the speaker is Dwight. If both are false, the speaker is Dave. Since we don't know who the speaker is, we can't know which scenario is correct. Your reasoning is NOT logical.
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How is this wrong? The "if…then" statement is true if the hypothesis is false, so both statements are true regardless of the speaker's identity. Since both statements are true, the speaker has to be a Knight and making him Dwight.
Look, Knights tell the truth okay? So if this was a knave then its not true at all. But if he's a knight(Which he his) then it literally says the answer...
Also i got the answer correct.
To solve these truth-teller problems I replace the phrase "If I am a knight and he is a knave, then I am Dwight and he is Dave. But If I am a knave and he is a knight, then I am Dave and he is Dwight." with " If I am telling the truth (a knight) (disregard: and he is a knave,) then I am Dwight (disregard: and he is Dave). (Disregard: But) If I am am telling the lie (a knave) (disregard:and he is a knight, then) I am Dave (disregard: and he is Dwight)."
But how does that solve the problem?
So far, you have only reached that the claim is equivalent to the claim, "I am Dwight". But what if the person speaking is lying?
Yes, I also used that strategy! I also changed the names because the names were obviously put there to confuse us.
Actually he can't be a Knight because that would make the first statement true.
Both statements point to the speaker being Dwight the knight.
If you look at the first statement: "If I am a knight and he is a knave, then I am Dwight and he is Dave."
If the speaker is actually the knight, and since all knights tell the truth, that means he is what he says to be: a knight named Dwight.
If you look at the second statement: "But If I am a knave and he is a knight, then I am Dave and he is Dwight."
If the speaker is actually a knave, and since all knaves lie, that means he is the opposite of what he says to be. Since in this statement he says he is a knave named Dave, then that means he is actually a knight named Dwight.
Therefore, the speaker must be Dwight the Knight!
This logic doesn't exclude the possibility that the speaker is a knave named Dwight. Just because "I am Dave the knave" is a lie doesn't mean the speaker has to be a knight.
A reply to Geoff. My first thought was exactly what you wrote but is it possible for Dwight to be a knight and Dave to also be one or neither thus making the scenario impossible with the given information? I don't know why its not clear for me but that's why I went ahead and did a whole mess. I started typing my reasoning and found the need to make the table below (I had to do them simultaneously). Here, K ≡ is a knight, Kn ≡ is a knave, S ≡ said the sentences. The pair (x,y) means that the person is x and did y. My whole reasoning is found below the table.
| row 2 col 1 | (K,S) | (Kn,S) | (K, ¬ S ) | (Kn, ¬ S ) |
| Dave | X | X | X | ✓ |
| Dwight | ✓ | X | X | X |
Reasoning Let's suppose that Dave is the knight and he said those 2 statements. Then the first statement is false because he is a knight and therefore he should be Dwight. If Dave is the knave and made those 2 statements, then the first sentence is false, making him a knight by treating the question as a formal logic question. Dwight is the knave and made the 2 statements, then since the first sentence is false, Dwight is a knight. This is not possible.
Suppose that Dave is a knight but did not made those statements. Then Dwight is the knave and made the two. We already know that this is impossible. If Dwight is a knight but did not say those statements, Then Dave is a knave who said them
and we have another impossibility.
Dave is a knave and didn't speak, then Dwight is a knight who did speak. Then ok. Finally, Dwight is a knave who did not speak, then Dave is a knight who did and that's impossible too. Then Dave spoke and he is the knight.
Assume the speaker is a knight. Then statement one says he is Dwight and the other is Dave, and this is true, which is consistent. Statement two then says "If I am a knave..." which in the given situation is false, which by the rules of formal logic, makes the statement true. So the case in which the speaker is a knight is fully consistent, and the answer. If the speaker were a knave, then both his statements would have to be lies, which is inconsistent.
Since the two statements are consistent, he is telling the truth and is therefore, a knight and Dwight.
I think we must account for the whole sentence which is composed of 2 sentences joined by "but" and consider when the whole sentence is true or false, and not just the 2 sentences disjuncted. Doing so the sentence of the speaker can be rephrased as " If I am a knave and he is a knight (P) then I am Dave and he is Dwight (Q) else if I am a knight and he is a knave, then I am Dwight and he is Dave (R) "
A sentence like IF P THEN Q ELSE R is a conjunction of two conditionals and is represented by the formal (P → Q ) ∧ (¬P → R)
I set the truth table and it resulted that the speaker can be Dwight the knight as well as Dave the knave
He is either a knight or a knave.
If he is a knight :
Then from statement one, he is Dwight. And statement two is true.
Bingo! This is a possibility!
Now suppose he is a knave .
Statement one would be true which would contradict the fact that he is a knave.
Therefore, he must be a k n i g h t , and his name must be D w i g h t .