Five Little Logic Problems...

[Monte314]

There is a land where some of the people always lie ("knaves") and some of the people always tell the truth ("knights").

It is not possible to distinguish knaves from knights except by evaluating statements they make. Answer these 5 questions:

i.) Will any inhabitant of this land claim that he is a knave?

ii.) Will any inhabitant of this land claim that both he and his brother are knaves?

iii.) Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B?

iv.) Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B?

v.) Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B?

[PHS Philip]

i) No
ii) Yes, I think, because both halves have to be true. If his brother is a knight, he would say they are both knaves if he is a knave
iii) A is a knight, B is a knave, because if A were a knave, he would be telling the truth, and if B were a knight, A would have to be lying if he's not a knave himself.
iv) A can be a knight or a knave, but B must be a knave. If A is a knight, his brother is a knave, or else A would be lying. If A is a knave, his brother must be a knave, or else exactly one of them would be a knave, and A would have told the truth.
v) I think I'm missing something on this one, but A can be either, and B must be a knight. If A is a knight, both must be knights, otherwise A would be lying. If A is a knave, B is a knight, because if B were a knave part 1 (both are knaves) of the statement would be true.

Fun, but I'm pretty sure I messed at least a few up, as usual.

[Rafael]

i.) Will any inhabitant of this land claim that he is a knave?
No.

ii.) Will any inhabitant of this land claim that both he and his brother are knaves?
Yes, if he's a knave and his brother is a knight.

iii.) Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B?
A is a knight and B is a knave.

iv.) Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B?
If A is a knight, B is a knave.
If A is a knave, B is a knave.

v.) Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B?
If A is a knave, B is a knight.
If A is a knight, B is a knight.

[Erika Redmark]

•Will any inhabitant of this land claim that he is a knave?
No–knights will tell the truth, and knaves will lie and say they're knights.

•Will any inhabitant of this land claim that both he and his brother are knaves?
A knight would clearly not say this. My first thought was that a knave could say it if his brother was a knight, since the whole statement is false; but logically it amounts to saying "I am a knave. My brother is a knave." The second statement is false, and the first is true. He wouldn't say that he himself was a knave. So my answer to this question is also no.

•Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B?
If A is a knight and B is a knave, this works: one of them is a knave and A's statement is true. For A to say this as a knave, the statement "At least one of us is a knave" has to be false…and since A is a knave, this can't be. So it has to be the first scenario.

•Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B?
The scenario where A is a knight and B is a knave also works for this. If A is a knave and B is also a knave, the statement is false. But it implies "My brother is a knave. I am not a knave" (since he can't admit to being a knave). But his brother really is a knave.

•Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B?
If A and B are both knights, this works. For it to work for a knave, A and B have to be different types–B has to be a knight. The whole statement is (I am a knave and my brother is a knave) XOR (I am a knight and my brother is a knight) = (T and F) XOR (T and F) = F XOR F = F. Somehow I don't have the same problem with the knave saying "I am a knave" or "My (knight) brother is a knight", since he says the opposite, and one of them is the case…

Now time to read all the other responses and see what egregious errors I've made. XD

[foroneonly]

i.) Will any inhabitant of this land claim that he is a knave?

No, because it is impossible for a knight to lie thus he could not lie and claim to be a knave and a knave must lie thus cannot say he is a knave

ii.) Will any inhabitant of this land claim that both he and his brother are knaves?

No

iii.) Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B?

A-knight
B-Knave

iv.) Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B?

If A is knight then B is knave
If A is knave then both are knaves.
Therefore B must be a knave and A's type is undetermined.

v.) Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B?

If A is a knight they will both be knights as if A was a knave the statement would be false and would be different types.
If A is a knave then A and B must be different types thus A would be knave and B knight.
Therefore B is always a knight.

My attempt.......

[Icristhus]

1: No. A knight would be unable to lie and claim he is a knave, while a knave would be unable to tell the truth -- that he is a knave.

2: No. In claiming that they are -both- knaves, the one is claiming that he, himself, is a knave. Which goes against the First Rule (seen above). Thus, a knight would be unable to say that he and his brother are knaves, and a knave would be unable to say that he and his brother are knaves, no matter if his brother were a knight (thus forming only half of the lie required).

3: A would be a Knight, and B would be a Knave. Because a Knave can never claim to be a Knave, nor claim another Knave to -be- a Knave (that would be truth). Meanwhile, a Knight can freely expose a Knave.

4: A would again be a Knight, and B a Knave, for the same reasons as above. The Knave would be unable to tell the truth of one being a Knave, and a Knight would be unable to lie and claim Knave status.

5: A is a Knight and B is a Knave... This one confused me for a bit, because if a Knight is saying he may be a Knave, that could be seen as a lie. Meanwhile, a Knave saying he may be a Knave could be a truth. But, considering that Knaves can never claim to be Knaves, and it -is- true that two identical typed people could be either one.. It'd have to be a knight speaking the somewhat vague truth that they are the same type... But then it occurred that maybe A is a Knave, and falsely claiming that both he and his brother are the same type, in the hopes you'll believe they're both Knights with the above line of thought.

The key is the word 'either' making it so that the knave can -imply- knavedom /or/ knighthood, and the knight can do the same. It isn't an outright lie, but it -is- an outright truth: a statement that they could be either one is ultimately true, no matter how you attempt to twist it. Thus, a knave wouldn't be able to say it, but a knight would, which makes the rest of it fall into place as truth.

I might end up regretting sticking to that choice instead of claiming it as a Knave lie.. But eh, this seems slightly more right to me, and it seems like an answer that could go either way.

Besides, pure logic isn't my strong suite, it's just a good way to kill twenty minutes. ;)

[faedra]

i.) Will any inhabitant of this land claim that he is a knave?
No, as knights will tell the truth and knaves will lie.

ii.) Will any inhabitant of this land claim that both he and his brother are knaves?
Yes, if he's a knave and his brother a knight.

iii.) Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B?
A must be a knight and B a knave.

iv.) Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B?
Nothing here about A, but B is definitely a knave.

v.) Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B?
B is a knight but A can be either.

Oh no, I hope the gallons of midnight oil I burnt last night don't come back to haunt me here.

[Icristhus]

Confusing thoughts there, Faedra.. Care to explain how you came to 'em?

Edit: Here's a friend of mine's attempt at those.. I get the feeling she's probably figured it out, while reminding me why I don't like logic -- too damn absolute ;)



She agreed with me on numbers one through three, but then on four and five agreed while offering alternative answers:

4:Either A is a knight and B a Knave, or A and B are both Knaves (the key being that it says 'exactly' one, instead of just 'one')

5:Either they're both Knights, or A is a Knave and B a Knight, since A would be the one lying in the latter case.

Either way, I figure her alternative answers are probably the logically 'correct' ones, even if I disagree about their reliability or applicability. Since this's all conceptual instead of real.. ;)

[Gloed]

Fun, but I'm pretty sure I messed at least a few up, as usual.

i don't think so.

[WiredBrain]

Supposing on that land there are only Knights and Knaves, as the statement says some of the people are knights and some of the people are knaves, which doesn't necessarily imply all people there belong to one of those types.


i) no. If he's a knight, he will admit it. If he's a knave, he will lie.

ii) Yes, in the case this was said by a knave whose brother is a knight.

iii) A must be a knight and B must be a knave. If both were knights, A would have lied, which is not possible. If A were a knave, either being his brother a knave or a knight, he would be telling the truth. so the only case remaining is A being a knight and B being a knave.

iv) A could be a Knight or a Knave, and B must be a Knave. If A were a knight, he told the truth, and if A were a Knave, he lied.

v) A could be a Knight or a Knave, and B must be a Knight.



I hope I have deduced well... maybe I missed something.

[Xenolar]

My answers:


I.) Will any inhabitant of this land claim that he is a knave? No. A knight shall always state the truth by claiming that he is a knight while a knave, being entirely incapable of telling the truth, will never admit to being a knave.

II.) Will any inhabitant of this land claim that both he and his brother are knaves? Yes. A knave may falsely claim that both he and his brother are knaves under the condition that his brother, in truth, is a knight. However, the answer I just provided could be considered circumstantial. If the claim is one statement, such as, "Both my brother and I are knaves," then, yes (as previously stated). However, a case of two independent statements such as, "(1) I am a knave. (2) My brother is a knave as well," cannot occur. However, since the first mentioned scenario is possible, it is only logical to say that, as originally claimed, yes, an inhabitant of this fictional land can claim that both he and his brother are knaves under the condition that the he is a knave and his brother is a knight.

III.) Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B? Under these circumstances, inhabitant A must be a knight while inhabitant B must be a knave. Since a knave must always speak untruths, it would be a logical contradiction for inhabitant A to be a knave. Thus, it can be readily deduced that inhabitant A must be a knight, and consequently, that inhabitant B must be a knave.

IV.) Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B? It is possible for A to be either a knight or a knave. B, on the other hand, must be a knave, regardless of what A is. (If A is a knight, then the situation is a near replica of that presented in question # III. However, unlike the scenario in question III, A might be a knave as well, as long as B is a knave and not a knight [due to the crucial difference between the respective phrases, "at least one of us" and "exactly one of us"]. Ergo, as already mentioned, the identity of A cannot be deduced, while it may be deduced that B is certainly a knave.)

V.) Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B? B must be a knight. If A is a knight, then the statement, "my brother and I are the same type, etc.," is sound and true, thus resulting in the conclusion that B must be a knight as well. If A is a knave, however, then the statement "my brother and I are the same type, etc.," must be entirely false, and as a consequence, B, similarly, must be a knight. Again, the identity of A remains indeterminate.




EDIT:
Supposing on that land there are only Knights and Knaves, as the statement says some of the people are knights and some of the people are knaves, which doesn't necessarily imply all people there belong to one of those types.

Yes, this is something I also realized. All of my answers are based on the assumption that any given person must be either a knight or a knave, yet nothing else.

[Jakalwarrior]

1: no
2: no
3: neither, or B only is a knave
4: Knight + Knave or Knave + Knave only
5: both are knights, or A is a knave and B a knight.


Only had about 2 minutes to put into it! lunch break!!!
*edit, had to make my responses make sense to people other than myself, oops, and 4 needed a correction.

*edit* I got number 3 wrong.. boo! Also, are knaves allowed to half tell the truth? in reference to number two? I am missing if him telling the truth on part of that question would invalidate it or if the statement is to be taken as a whole.

[Ice Wolf]

I hope this is right.


General def.:

A,B,Y = 1 if Knave
A,B,Y = 0 if Knave

1. Y = A XOR A

A B Y
1 1 0
1 0 0
0 1 0
0 0 0

Answer is therefore NO


2. Y = A XOR (A AND B)

A B Y
1 1 0
1 0 1
0 1 0
0 0 0

YES. Knave whose brother is knight.


3. Y = A XOR (A OR B)

A B Y
1 1 0
1 0 0
0 1 1
0 0 0

A = Knight
B = knave.

4. Y = A XOR (A XOR B)

A B Y
1 1 1
1 0 0
0 1 1
0 0 0

A = Knight or Knave
B = Knave.


5. Y = A XOR (NOT(A XOR B)

A B Y
1 1 0
1 0 1
0 1 1
0 0 0

A = NOT B

They are the opposite

[Homini Lupus]

As usually I try a shot


I)No
II) Yes: if he's a knave and his brother a knight he can say that since it would be a lie
III) They can't be both knights (they would lie); they can't be both knaves (he would say the truth) => they are a knight and a knave, and the knight is the speaking one.
IV) They may be both knaves OR a knight and a knave like III
V) They may be both knights OR a knave and a knight, and the knave is speaking

[SimplyOtter]

:laugh: Shameless INFP!

i.) NO

ii.) NO

iii.) A is KNIGHT, B is KNAVE

iv.) A is KNIGHT, B is KNAVE

v.) A is KNAVE, B is KNIGHT
[/QUOTE]

[Monte314]

So far, most of you agree with Monte.

[Erika Redmark]

Also, are knaves allowed to half tell the truth? in reference to number two? I am missing if him telling the truth on part of that question would invalidate it or if the statement is to be taken as a whole.

That was my question about this, too. If you read my answer it's clear what I ended up deciding to go with, but I might be dead wrong. XD

[Noehelia]

As you said these are only statements, we can not ask questions and get answers.

i.) Will any inhabitant of this land claim that he is a knave?
No, all knights tell the truth and they will say that they are knights, all knaves lie so they will claim that they are knights.

ii.) Will any inhabitant of this land claim that both he and his brother are knaves?
Yes, in case a knave (the person who will make the statement) has a brother knight. (I assume that this can happen if knaves and knights are not races so brothers can be different, however this need clarification).

iii.) Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B?
A is a knight and B is a knave.

iv.) Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B?
It can't be one answer. It might be two knaves but it can also be A is knight and B is knave.

v.) Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B?
It can't be one answer. Either both are knights or A is a knave and B is a knight

[Monte314]

That was my question about this, too. If you read my answer it's clear what I ended up deciding to go with, but I might be dead wrong. XD

The statement, "1 is an odd number OR 2 is an odd number" is TRUE.
The statement, "1 is an odd number AND 2 is an odd number" is FALSE.

In a sense, the first is only "half true", but statements connected by OR are TRUE if and only if at least one of the simple statements is TRUE.

In the same sense, the second is also "half true", but statement connected by AND are TRUE if and only if every one of the simple statements is TRUE.

It should be pointed out also that in propositional logic, the OR is inclusive, not exclusive. That is, it means "one or the other or both". An exclusive or means "one or the other, but not both".

[Noehelia]

I do not think that we can say that a statement is half true when there is no question involved. There can be situations that a question asks only that "half true" answer. We are just used in the social context where questions are implied.

[Erika Redmark]

The statement, "1 is an odd number OR 2 is an odd number" is TRUE.
The statement, "1 is an odd number AND 2 is an odd number" is FALSE.

In a sense, the first is only "half true", but statements connected by OR are TRUE if and only if at least one of the simple statements is TRUE.

In the same sense, the second is also "half true", but statement connected by AND are TRUE if and only if every one of the simple statements is TRUE.

Well, to say the second statement, you have to say a true statement, "1 is an odd number"…

An interesting thing I just thought of is whether the presuppositions of a knave's statements would be allowed to be true, like the presupposition of "It wasn't John who stole the money" is that someone else stole the money…if it's true that John stole the money, would a knave be allowed to say "It wasn't John who stole the money", presupposing that the money was indeed stolen, which it was?

…What I mean to say is, darn it! I picked wrong! XD

[Ice Wolf]

Monte:

Is there any possibility to give us peek into solutions? Pleaseeeeee.

[Monte314]

Monte:

Is there any possibility to give us peek into solutions? Pleaseeeeee.

I've found that when I post solutions, it kills the thread. I'll watch for a day or two, and when people quit posting I'll put my solutions up.

[HackerX]

There is a land where some of the people always lie ("knaves") and some of the people always tell the truth ("knights").

At no point is it mentioned that all the people in the land are either knaves or knights. There's no clarification over whether or not any other occupant must lie or tell the truth.

Thus:


It is not possible to distinguish knaves from knights except by evaluating statements they make. Answer these 5 questions:

i.) Will any inhabitant of this land claim that he is a knave?

Possibly. Neither knaves or knights could claim they were a knave, but any other person could (but be lying)


ii.) Will any inhabitant of this land claim that both he and his brother are knaves?


First answer holds true. A could be anybody, and B could be either anybody or a knave.


iii.) Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B?

See first answer. (seeing the pattern?)


iv.) Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B?

Either see the first answer, or:
A is anybody, B is a knave.
A is a knave B is a knave.
A is a knight, B is a knave.

What A is, isn't relevent to the answer (and there's a logic term for that that I can't think of atm)


v.) Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B?


See answer 1 first.
A is anybody, B is anybody/knave/knight.
A is knave, B is a knight
A is a knight, B is a knight.

See answer 4, first term is irrelevent



;)

[SirJac]

i.) Will any inhabitant of this land claim that he is a knave?

No

Knight: Knight (truth)
Knave: Knight (lie)

ii.) Will any inhabitant of this land claim that both he and his brother are knaves?

Knight with Knave brother: No (truth)
Knight with Knight brother: No (truth)
Knave with Knight brother: Yes (lie) <---
Knave with Knave brother: no (lie)

iii.) Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B?

Knight with Knave brother: Yes (truth) <---
Knight with Knight brother: No (truth)
Knave with Knight brother: No (lie)
Knave with Knave brother: No (lie)

Person A is a knight, person B is a Knave

iv.) Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B?

Knight with Knave brother: Yes (truth) <---
Knight with Knight brother: No (truth)
Knave with Knight brother: No (lie)
Knave with Knave brother: Yes (lie) <---

Don't know what person A is, person B is a knave

v.) Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B?

Knight with Knave brother: No (truth)
Knight with Knight brother: Yes (truth) <---
Knave with Knight brother: Yes (lie) <---
Knave with Knave brother: No (lie)

Don't know what person A is, person B is a knight.

[Antares]

i.) Will any inhabitant of this land claim that he is a knave?

Yes. It was given that SOME are knaves and SOME are knights. The knaves would not admit that they are knaves because they always (using the proper definition of the word 'always') lie. They can't tell a shred of truth; and have never been honest in their lives. The knights can't claim that they are knaves becaue they always tell the truth.

But just because some are knaves and some are knights doesn't mean that they aren't any 'middles'. After all, the question doesn't explicitly communicate that some are A and the rest are B, or that half are knaves and half are knights. In the face of vagueness, it is illogical to make assumptions.

ii.) Will any inhabitant of this land claim that both he and his brother are knaves?

Yes. The people who fall in the middle can claim that they are both knaves (although I don't see the point in doing that). A real knave can also do that only if his brother is not a knave. Given that, it would be a lie to say that both are knaves. Of course, a knave can't claim that if both are knaves, since he'd be telling the truth. A knight can't do that, obviously, even if his brother IS a knave, because 'both' would include him as well, and since he's not a knave, he can't say so.

iii.) Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B?

A can be knight. He is a knight if his brother is a knave; granted, he's vague, but it's still the truth. In short, if A is a knight, then B must be a knave (how A would know that is another problem, but it's largely irrelevant in the context of this problem. Theoretically, if knights can only speak the truth, then they would speak the truth whether they know it or not. but let's not go there. We don't need this to be too complex). Someone who is not a knave or a knight can say the same, since they're always able to lie when they want to. Regardless what type B is, since A have no restrictions. They can say whatever they want, unlike the other two types.

iv.) Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B?

A is a knight if B is a knave. If A is a knave, then B can't be a knave. If A is other, then you can deduce nothing.

v.) Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B?

Again, the 'other's are irrelevant. If A is a knave, then B can't be a knave. If A is a knight, then B has to be a knight. The knight would be vague here, but vagueness doesn't equate not telling the truth.

There is a land where some of the people always lie ("knaves") and some of the people always tell the truth ("knights").

The problem I see with this puzzle is that it's vague. Are all the inhabitants either knaves or knights? I see with many of the responses, that the underlying assumption is that there can only be knaves or knights, but the problem does not state that. Just because they are not mentioned doesn't mean they don't exist. I'm utilising my Se here. It comes in useful when I want to split hairs.

[HackerX]

i.) Will any inhabitant of this land claim that he is a knave?

Yes. It was given that SOME are knaves and SOME are knights. The knaves would not admit that they are knaves because they always (using the proper definition of the word 'always') lie. They can't tell a shred of truth; and have never been honest in their lives. The knights can't claim that they are knaves becaue they always tell the truth.

But just because some are knaves and some are knights doesn't mean that they aren't any 'middles'. After all, the question doesn't explicitly communicate that some are A and the rest are B, or that half are knaves and half are knights. In the face of vagueness, it is illogical to make assumptions.

ii.) Will any inhabitant of this land claim that both he and his brother are knaves?

Yes. The people who fall in the middle can claim that they are both knaves (although I don't see the point in doing that). A real knave can also do that only if his brother is not a knave. Given that, it would be a lie to say that both are knaves. Of course, a knave can't claim that if both are knaves, since he'd be telling the truth. A knight can't do that, obviously, even if his brother IS a knave, because 'both' would include him as well, and since he's not a knave, he can't say so.

iii.) Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B?

A can be knight. He is a knight if his brother is a knave; granted, he's vague, but it's still the truth. In short, if A is a knight, then B must be a knave (how A would know that is another problem, but it's largely irrelevant in the context of this problem. Theoretically, if knights can only speak the truth, then they would speak the truth whether they know it or not. but let's not go there. We don't need this to be too complex). Someone who is not a knave or a knight can say the same, since they're always able to lie when they want to. Regardless what type B is, since A have no restrictions. They can say whatever they want, unlike the other two types.

iv.) Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B?

A is a knight if B is a knave. If A is a knave, then B can't be a knave. If A is other, then you can deduce nothing.

v.) Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B?

Again, the 'other's are irrelevant. If A is a knave, then B can't be a knave. If A is a knight, then B has to be a knight. The knight would be vague here, but vagueness doesn't equate not telling the truth.



The problem I see with this puzzle is that it's vague. Are all the inhabitants either knaves or knights? I see with many of the responses, that the underlying assumption is that there can only be knaves or knights, but the problem does not state that. Just because they are not mentioned doesn't mean they don't exist. I'm utilising my Se here. It comes in useful when I want to split hairs.

See my answer :)

[Monte314]

See my answer :)

I must agree that I did not phrase this well.

The intention is that all the islands inhabitants are either knights or knaves.

Sorry.

[Nanashi]

There is a land where some of the people always lie ("knaves") and some of the people always tell the truth ("knights").

It is not possible to distinguish knaves from knights except by evaluating statements they make. Answer these 5 questions:

Will any inhabitant of this land claim that he is a knave?
No. The knave will not reveal the truth about herself, and the knight will not deceive you.

Will any inhabitant of this land claim that both he and his brother are knaves?No, by the definition of being truthful, the knight would not be a knave AND could not, incorrectly, tell you she was, and the knave would not admit it.[/QUOTE]

Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B?
If they were both liars, the liar would be telling the truth, so that's not true.It is still true to say that 'at least one' is a knave and have there be only one, so the other could be the knave, and the speaker could be the knight(just b/c we sd at least one doesn't mean there has to be more than one, so the statement could still be given by a truth-teller), but the statement cannot be false.The liar cannot admit something true, so (I didn't realize this 'till I typed it out), A must be the non-liar. B must be the liar.
Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B?If A is a knave, A and B are liars. If A is a knight, B is a liar.

Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B?A is a liar and B is a knight, or A and B are knights.

[HackerX]

I must agree that I did not phrase this well.

The intention is that all the islands inhabitants are either knights or knaves.

Sorry.

Yeah, teachers pulls stuff like this all the time. Oh bother, I didn't realise that I worded it wrong, too bad so sad you should have known what I meant.

Ultimately, the people who are learning better, and who end up as better workers, are the ones who see the problem as a whole, and not just a little logic problem with 2 factors.

Also, it would suck to live in this land, no damsels :(

[Antares]

I must agree that I did not phrase this well.

The intention is that all the islands inhabitants are either knights or knaves.

Sorry.

Mon dieu! I'd have to change my answer then.

i.) Will any inhabitant of this land claim that he is a knave?

No.

ii.) Will any inhabitant of this land claim that both he and his brother are knaves?

A knave can do that if his brother is a knight. A knight can't do that.

iii.) Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B?

If:

A = knight B = knave

iv.) Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B?

If:

A = knight B = knave
A = knave B = knave

v.) Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B?

A = knave B = knight
A = knight B = knight

[Monte314]

Probably the best way to attack a problem like this is to create a "truth table". This is done by systematically listing the possible combinations in a table, and looking at the outcomes for each case. For example, the truth table for iv. is (sorry, there's no easy way to insert a table):

Statement..A........B........"Exactly one of us is a knave."......Would A say this?
1............. knight..knight...lie.............................. .............NO
2..............knight..knave...truth.............. .........................Yes
3..............knave..knight...truth.............. .........................NO
4..............knave..knave...lie................. ..........................Yes

Here we see that the statement heard could be uttered only in cases 2 and 4, showing that A can be either a knight of knave, but B must be a knave.

If there are N variables, this approach requires a table that has a number of rows equal to 2 to the Nth power. The knight-knave problem has two variables (what type is A, and what type is B), so it requires 4 rows.

If a problem has 10 variables, it will require 2^10 = 1,024 rows. If it has 30 variables, it will require over 1 BILLION rows! Obviously a human would not be able to handle a problem of this size... but a computer will have no difficulty. In particular, a computing language has been created to solve constraint satisfaction problems such as this one: PROLOG (which stands for "LOGic PROgramming").

Here is Monte's solution to the problem:

i.) Will any inhabitant of this land claim that he is a knave?

No. For a knave to say this would be the truth, and for a knight to say this would be a lie.

ii.) Will any inhabitant of this land claim that both he and his brother are knaves?

Yes. This will happen when you are talking to a knave whose brother is a knight (but under no other conditions).

iii.) Suppose an inhabitant A says about himself and his brother B: "At least one of us is a knave." What type is A and what type is B?

By building a truth table, it is seen that A is a knight, and B is a knave.

iv.) Suppose A instead says: "Exactly one of us is a knave." What can be deduced about A, and what can be deduced about B?

By building a truth table, it is seen that A can be either a knight or a knave, but B must be a knave.

v.) Suppose A instead says: "My brother and I are the same type: either both knaves or both knights." What can be deduced about A and B?

By building a truth table, it is seen that A can be either a knight or a knave, but B must be a knight.