# A logic problem



## Andrew Green (Jun 16, 2007)

Stolen from elsewhere (yes, there is a answer):


You are the most eligible bachelor in the kingdom, and as such the King has invited you to his castle so that you may choose one of his three daughters to marry. The eldest princess is honest and always tells the truth. The youngest princess is dishonest and always lies. The middle princess is mischievous and tells the truth sometimes and lies the rest of the time. 

As you will be forever married to one of the princesses, you want to marry the eldest (truth-teller) or the youngest (liar) because at least you know where you stand with them. 

The problem is that you cannot tell which sister is which just by their appearance, and the King will only grant you ONE yes or no question which you may only address to ONE of the sisters. What yes or no question can you ask which will ensure you do not marry the middle sister?


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## Touch Of Death (Jun 16, 2007)

Andrew Green said:


> Stolen from elsewhere (yes, there is a answer):
> 
> 
> You are the most eligible bachelor in the kingdom, and as such the King has invited you to his castle so that you may choose one of his three daughters to marry. The eldest princess is honest and always tells the truth. The youngest princess is dishonest and always lies. The middle princess is mischievous and tells the truth sometimes and lies the rest of the time.
> ...


I know the answer only when refering to Parots. Now that they are princesses, it puts a whole new spin. LOL
Sean 
PS what do you think? LOL


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## Andrew Green (Jun 16, 2007)

I know it too, but don't want to give anything away quite yet


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## MA-Caver (Jun 16, 2007)

Gotta be the most obvious question for ANY woman... 

Are you fat?

ok, ok I was just kidding... also thought it over and realized that what if they all answer "no?" 

how about... Would you mind being poor?


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## tradrockrat (Jun 16, 2007)

"Are you the middle sister?"

The oldest must say no, the youngest must say yes.  The middle sister can say either, but that just excludes her and the other that answered the same as her.


am I right?

Oh crap - just realized only one has to answer - -be back later...  lol


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## tradrockrat (Jun 16, 2007)

Ask them all a bunch of questions that are NOT yes or no answers - how about THEM apples... lmao


logic tells me that no yes or no question can ever satisfactorily tell you whom is whom if you can only ask it once to one person.  sorry.

whats the answer?


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## terryl965 (Jun 16, 2007)

Is it ask the one that will tell you the truth and add every single one in a single question Right.


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## Andrew Green (Jun 16, 2007)

No, you get one question, to one girl, and you don't know which of the girls is which.


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## tellner (Jun 17, 2007)

You ask any one of them "Are you the middle daughter? No funny stuff because if I find out you're lying we're moving to a split level pig sty in Lower Slobbovia which doesn't have an extradition treaty with your daddy's kingdom."


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## michaeledward (Jun 17, 2007)

You need to ask the yes or no question to *one *daughter but *about* a different daughter. This will allow you to eliminate two. I'm still struggling to get an answer (actually, I struggled for about 5 minutes yesterday, and then gave up).

Let's assume the daughters have blonde, red, and brown hair, respectively. 

You ask the redhead "Is the blonde haired daughter the middle daughter?"


Think along those lines ... and you may have more success.


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## tradrockrat (Jun 17, 2007)

michaeledward said:


> You need to ask the yes or no question to *one *daughter but *about* a different daughter. This will allow you to eliminate two. I'm still struggling to get an answer (actually, I struggled for about 5 minutes yesterday, and then gave up).
> 
> Let's assume the daughters have blonde, red, and brown hair, respectively.
> 
> ...



but with only one question to one daughter, you always run the risk of getting the middle daughter who can answer either way.

I still don't see it.


EDIT:  OK called in favor from a friend - she says this - ask a question that excludes the princess who is answering - you WILL NOT pick the sister answering.  My friend love logic riddles, but didn't have the actual answer (at least not that she'd admit).


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## shesulsa (Jun 17, 2007)

crap ... not enough caffeine for this


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## Rich Parsons (Jun 17, 2007)

Andrew Green said:


> Stolen from elsewhere (yes, there is a answer):
> 
> 
> You are the most eligible bachelor in the kingdom, and as such the King has invited you to his castle so that you may choose one of his three daughters to marry. The eldest princess is honest and always tells the truth. The youngest princess is dishonest and always lies. The middle princess is mischievous and tells the truth sometimes and lies the rest of the time.
> ...




Given:

All three sisters look a like.

Sister 1 (S1) = S2 = S3 in looks


Truthfulness
S1 = Truth

S3 = Lies = ~Truth

S2 = Mischievous

Age
S1 > S2 > S3 in age


If you ask a sister if she is the oldest then youngest will answer Yes. Then the Oldest will answer Yes and then middle child will given different answers depending. 

If you ask the question, Are you the oldest who always tells the truth?

S1 will answer Yes
S2 can answer with anything
S3 will answer Yes

If you ask the question Are you not the Oldest who tells the truth?

~ = Not

Read ~(Oldest & Truth) = ~Oldest & ~Truth = ~Oldest & Lies


S1 will answer No
S2 will answer with anything
S3 will answer No to the Oldest part and No to the Lies. 

So How about Are you the youngest who tells the truth? 

S1 would not be able to answer for she is not the youngest
S2 anything
S3 would be able to answer yes - but S2 coudl still answer yes so not enough




Still thinking


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## Andrew Green (Jun 17, 2007)

Micheal Edwards is on the right sort of path...


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## exile (Jun 17, 2007)

Awraight. First, I call on Andrew (or any of the admins) to verify that (i) I have not logged in to this thread since last night before 11 or so, enough to read the problem and notice that no correct answers were given (or attempted, in most cases ), and (ii) that I have spent no more time in this thread at this moment than was required to upload the following text. 

Second, I've not encountered this variant of the problem before, but I'm familiar with some simpler versions (of which the path-in-the-woods problem is the most familiar version). The point is that a question must be designed which allows you to infallibly sort the compulsive truth-teller and the compulsive liar's examples together, as vs. the unpredictable sisters. My take on the answer---arrived at after cominng back from a hellish week of family visits, in the hope of a relaxing evening instead of several hours of preoccupied distraction involving a logic problem that my family has zero interest in (thanks very much Andrew :EG---is the following, based on the assumptionn that the sisters know which of them has which habits so far as truthful responses to yes/no questions is concerned:

*Does the truth of your answer to any particular question Q coincide with the truth value of the statement `A true inference is possible here' which your sisters can infer with respect to that question?*

Here are the possible outcomes:

(i) you ask the truth-teller. She always tells the truth, and her sisters know it. The truth value of her answer is always 1, and her sisters, knowing that, know that the truth value of her answer is always 1;  hence her sisters can invariably infer the true answer, so that the value of `A true inference is possible here' is 1; therefore she must answer Q `yes'.

(ii) you ask the liar. She never tells the truth, and her sisters know it. The truth value of her answer A is always 0, and her sisters, knowing that, know that the truth value of A is always 0, therefore can always infer that the true answer is invariabley  ~A; therefore the answer to `Is a true inference is possible here' is yes, i.e, the value of `A true inference is possible here' is 1; therefore  the truth value of her answer (0 by hypothesis) and the the truth value of `A true inference is possible here' (1) do not coincide, therefore a truthful answer would be `no'. But since this sister invariably lies, her answer must be `yes'.

(iii) you ask the human one (i.e., the one who sometimes lies and sometimes tells the truth... are we allowe to editorialize here? :lol Let's assume

a. She decides to tell the truth. That means the value of her answer is 1, but since her sisters know that she is not always truthful, the value of `A true inference is possible here' is 0; therefore a truthful answer (which ex hypothesi she is giving) is `no', since the two value (1 and 0) do not coincide.

b. She decides to lie. That means that the value of her answer is 0, and since her sisters know that she is not always truthful, the value of `A true inference is possible here', is 0; therefore the value of her answer and the truth value of the statement coincide. A true answer is yes. But ex hypothesi, she's lying. Therefore she says `no' in answer to the question.

Short summary: the eternal liar and eternal truth-teller both answer yes to this question. The one you don't want to marry is the one who answers `no', since, whether she's lying or telling the truth in answer to this particular question, the anwswer she gives, `no', distinguishes her from her sisters.

Am I warm?


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## Lisa (Jun 17, 2007)

exile said:


> Awraight. First, I call on Andrew (or any of the admins) to verify that (i) I have not logged in to this thread since last night before 11 or so, enough to read the problem and notice that no correct answers were given (or attempted, in most cases ), and (ii) that I have spent no more time in this thread at this moment than was required to upload the following text.
> 
> Second, I've not encountered this variant of the problem before, but I'm familiar with some simpler versions (of which the path-in-the-woods problem is the most familiar version). The point is that a question must be designed which allows you to infallibly sort the compulsive truth-teller and the compulsive liar's examples together, as vs. the unpredictable sisters. My take on the answer---arrived at after cominng back from a hellish week of family visits, in the hope of a relaxing evening instead of several hours of preoccupied distraction involving a logic problem that my family has zero interest in (thanks very much Andrew :EG---is the following, based on the assumptionn that the sisters know which of them has which habits so far as truthful responses to yes/no questions is concerned:
> 
> ...



Can you repeat that in English please? :lol2:


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## exile (Jun 17, 2007)

Lisa said:


> Can you repeat that in English please? :lol2:



_English???_ You are having the want  I say the logic theorem she be the English??? So yes, now I try to say what she want to say, the logic theorem...

The terms of the problem are: you get one question, only one, and it has to be of the yes/no variety. Sigh... that limits your choices a lot. And it's worse than that. If you ask a question which separates the truth teller from the liar, that doesn't help with the third sister, because her answer might be the truthful one, in which case it'll be the same as that of the truth teller. And you don't get a second question to sort those two out. If she decides to lie, her answer will be the same as that of the compulsive liar sister. And again, you don't get to sort the two out with a followup question.

The only way you can get at the true/false sister, so far as I can see, is to ask a question which, whether she's telling the truth or lying, results in a different answer than the compusive truth-telling sister and compulsive liar will give. Yes? So there are two parts to this trick (I _think_):

Part 1: figure out a question that both the truth-teller and the liar will give the same answer to. Now one thing about them (something the original question shoves in your face): they're at least consistentyou `know where you stand' with them. And the point is, the sisters know who tells the truth, who lies, and who does both. So the strategy is, ask the question _about the sisters and what they know from each other's  answers_. The question has to allow us to group the two `extreme' sisters togetherwell, since all the sisters know what the true answer is from either the truth-teller's or the liar's answers (because they know who is who), make your one shot at the answer a question which, no matter which of the sisters you ask, will yield the same result for the liar and the truth teller. 

So here's a question Q. And the truth teller will always answer Q truthfully (A = 1) and the liar will always answer Q falsely (A = 0). But in both cases,  the sisters all know how to get at the truth. Take the truth-teller's answer's literally, but flip the value of the liar's answers. In one case, a literal answer is necessary, in the other, a flip is necessary. Think of my bolded question as, `can you take my answer literally?' The truth teller says, truthfully, yes you can take my answer literally. The liar saysbecause she's a liaryes, you can take my answer literally. So for both of them, the answer is `yestake my answer literally'. 

Part 2: Make sure that the true/false sister has to answer differently from the other two, whether or not she's telling the truth. And here the idea is, you have to play on the fact that unlike the first two, the third sister's answers can't automatically be used as pointers to the truth, because no one knows (including her sisters) what she's doing in any given answer. 

That means, if she answers the bolded question truthfully, she has to admit that no, her truthful answer _cannot_ be taken as a literal truthful answer to the question Q, because her sisters know she might be lying, hence they cannot answer the statement `we can infer the correct answer' with a `yes'. So even though her answer is truthful, her sisters cannot make the right inference automatically; hence: is she telling the truthyes!Can her sisters know that for sure?no! Therefore her truthful answer tow whether the truthfulness of her answer and the truthfulness of `your sisters know what the truth is from your answer' are identical has to be `no': you cannot take my truthful answer to the particular question you're asking literally as what my sisters can say about the answer to the question.

But if she's lying when she answers, it means that the truth value of her answer (false) is the same as the truth value of the statement that her literal answer to the question is truthfulnamely, uh-uh. Under no circumstances do we, or her sisters,  know what she's up to, hence we can't  conclude that what she says is true. So the statement that we can would be false, just as her answer to Q is false. So the two overlap, and when she's lying, she knows that the two overlap. But because she's lying, she can't answer Q truthfully, by saying yes, the two overlap. On the contrary: by assumption, since she's lying, she has to claim that the two _don't_ overlap. In other words, she has to answer the statement contained in Q with a resounding `no'.

So the true/false sister always has to answer the bolded question with a big fat `no'. And the other two must answer it `yes'. And that's how you can tell 'em apart...

... does that help at all??


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## Andrew Green (Jun 17, 2007)

exile said:


> So the true/false sister always has to answer the bolded question with a big fat `no'. And the other two must answer it `yes'. And that's how you can tell 'em apart...




The the middle sister will be able to answer either "yes" or "no".  The other two will answer based on the fact that they always tell the truth, or always lie.


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## Lisa (Jun 17, 2007)

exile said:


> _English???_ You are having the want  I say the logic theorem she be the English??? So yes, now I try to say what she want to say, the logic theorem...
> 
> <snip>
> 
> ... does that help at all??



No.


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## exile (Jun 17, 2007)

Andrew Green said:


> The the middle sister will be able to answer either "yes" or "no".  The other two will answer based on the fact that they always tell the truth, or always lie.



Wait a mo. The middle sister has a question Q, the bolded question I gave. A `yes' answer to that question means that the truth of her answer to any question (including the one she's answering) is the same as the truth value of the claim that her sisters can tell for sure if her answer is truthful. But her sisters can't tell for sure if her answer is truthful, 'cause she can always lie. So a `yes' answer means that the truth value of her answer (`true') is the same as the value of the claim that her sisters know what's going on (`false'). Since, on hypothesis a, she's answering truthfully, this outcome entails that her answer (the two values match) is falsewhich is a contradiction to the assumption on this scenario that she's answering truthfully. So she _can't_ answer `yes' on this scenario (=hypothesis a). And on hypothesis b., when her answer to the question is false, her `yes' answer' would truethe falseness of her answer and the falseness of the claim that her sisters know what's going on match exactly. But since by hypothesis she's got to be lying this time, she can't give the true answer, which is `yes' (the two truth-values do match, this time). The lying answer is `no'. In other words, for the true/false sister, when she tells the truth, her answer to the question is `no' and when she lies, the answer to the question is also `no'. So she can't answer `yes' in either case w/o violating the assumed truth, or untruth of her answer.


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## exile (Jun 17, 2007)

Lemme amplify this analysis of the third sister:

*Her answer to Q.... her sisters KNOW... truth of claim of matchup... response to Q*

true (=1).................false (=0)..................0 (=no)................0 (true, ex. hyp)

false (=0)................false (=0)..................1 (=yes)..............0 (false, ex. hyp)

In both cases, the answer she gives is no, reflecting either the truth of the non-matchup, or the false claim that there is a non-matchup. In the case of her intention to tell the truth, `no' is the truthful answer as to whether her sisters' ability to make the correct inference from her response (false) corresponds to the truth-value of her response (true). 1&#8800;0, eh? In the case of her false answer, `no' is the lie that there is a difference between her sister's ability to make the right inference (which is a false claim) and the truth of her answer (which is also false). 0=0, so the true answer to Q is`yes'; therefore, in order to lie, as supposed by this scenario, she must answer `no'. Repeat: in the latter case, both her answer and the claim that her sisters know what's going on wrt to her answer are false, hence the answer to Q should be `yes'. But since she's lying (by assumption), her actual answer will be `no'. Hence, to satisfy the assumptions that she only has two choices, to lie or tell the truth, and that she knows what the truth is even in the case where she lies in answer to Q, and to conform to the usual definitions of `lie' and `tell the truth', she has to answer `no' in both cases.


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## michaeledward (Jun 17, 2007)

I think the question has to ask about the variable ... which is the middle sister. Something like ... 

.... If I ask the redhead if she is the middle sister, will she tell me the truth?

I haven't figured out the answers yet. But it will be something like ... 

if the answer is YES marry the one you ask the question of 
if the answer is NO marry the redhead

(or vice versa)


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## Andrew Green (Jun 17, 2007)

Excile, 

It's a impressive little piece of logic, but she could still lie in a different way  and answer "Yes"


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## Andrew Green (Jun 17, 2007)

Here's a hint, Knowing the *oldest* tells the truth, and the *youngest* lies does matter


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## exile (Jun 17, 2007)

Andrew Green said:


> Excile,
> 
> It's a impressive little piece of logic, but she could still lie in a different way  and answer "Yes"



OK, I'm game.... tell me. How can she lie and answer `yes' to question Q, and have it be consistent with the constraints? 

Let me strip this down to its essentials. Show me where I've made a misstep.

Let S' = "your sisters can tell the true answer on the basis of your response".

Q  = is (forall Kw (the truth of your assertion that the answer to Kw is true) = S') true?, with <&#8212;> indicating biequivalence (where Kw is the set of all yes/no questions); a corollary of this is that 

Q = is ((the truth of X's answer to Q)= S') true?

There are two initial premises:

*Premise 1: Assume [[S']] = 0* (this is _invariably_ true for the true/false sister, since there is no way her nontelepathic sisters know whether or not she's telling the truth.)

Then [[S']] = 0 

So the question is,

Is (the truth value of X's answer to Q = 0) true?

*Premise 2: Assume that there are two possible answers to a yes/no questions, a true answer and a false answer.* (this is a simplification but the appropriate extension is irrelevant to the proof).

If the middle sister is asked the question, then she is X. If she tells the truth, then Q is,

(Is [[X truthfully answers Q]] = [[0]] true?)

 <&#8212;>

is [[1 = 0]] true?

Since 1 =/= 0, the truthful answer X gives is `no'.

If the middle sister lies, then Q is,

(Is [[X falsely answers Q]] = [[0]]) <&#8212;> Is [[0 = 0]]

(since the question is stated so that the truthfulness of the sister's answer is a component of the structure of the question).

Since 0 = 0, the truthful answer to the question is `yes'. But since X will (by the definition of lying) answer ¬`yes'  (the question itself entails that she answers Q falsely), if the truthful answer to the question is `yes', then 
she _must_ answer `no' (or she's not lying!).

Thus, whether she answers truthfully or falsely, the sister in question's answer&#8212;the answer of the sister the truth of whose answer cannot be predicted&#8212;_must_ be `no'.

OK. So show me what I've made a misstep in this proof... :wink1:


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## tradrockrat (Jun 17, 2007)

here's the best I can come up with - 

Ask ANY sister (call her sister1) the following question realizing that you WILL NOT CHOOSE HER.  This way you have already narrowed it down to two choices:

Ask sister 1 if sister 2 is a bigger liar than sister 3.

sister 1 will not be picked so all you have to worry about is sister 2 or 3

middle sister gets asked?  doesn't matter cause either is ok - so follow this rule - I hope 

If the answer is yes, pick sister 2  - honest sis just told you that 2 is the liar, and liar sis just told you that 2 is the honest one

If the answer is no, pick sister 3 - honest sis just told you that three is the liar and liar just told you that 3 is the honest.

my brain says this works - does yours?

Just so everyone knows - I just wasted an entire day and had to get a clue from a much smarter friend to start me down this path - please don't post anymore of these!!!!! lol


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## Andrew Green (Jun 17, 2007)

I think you got it.

Or worded differently:

"Is sister B older then sister C?"

If A is the liar, doesn't matter, both are good.

If A is telling the truth go with the one she tells you is the youngest, the older is the middle one.

If A is lieing, then go with the one she says is youngest, who is really the oldest with the other being the middle.

So always choose "No"


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## Rich Parsons (Jun 17, 2007)

Ok I am really confused.



Andrew Green said:


> Stolen from elsewhere (yes, there is a answer):
> 
> 
> You are the most eligible bachelor in the kingdom, and as such the King has invited you to his castle so that you may choose one of his three daughters to marry. The eldest princess is honest and always tells the truth. The youngest princess is dishonest and always lies. The middle princess is mischievous and tells the truth sometimes and lies the rest of the time.
> ...




The second paragraph states that you do not want to be involved or married with the middle Sister who osmetimes lies and sometimes tells the truth. 



Andrew Green said:


> I think you got it.
> 
> Or worded differently:
> 
> ...




You first statement "If A is the liar, doesn't matter, both are good." implies it is ok to be married with the Middle Sister. A contradiction between possible answers and the givens. 

The second statement "rings" true.  

The third is similiar to the second. 

The problem with this is how does one know if they are telling the truth or telling a lie?


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## Andrew Green (Jun 18, 2007)

Rich Parsons said:


> You first statement "If A is the liar, doesn't matter, both are good." implies it is ok to be married with the Middle Sister. A contradiction between possible answers and the givens.



My bad, wrote that wrong.

If A is the half and half one it doesn't matter, either C or D are ok.

If A is the Liar she will tell you the oldest, the good choice, is the youngest of the other two.  Pick the youngest.

If A is the truth teller she will tell you the liar, the better of the remaining two, is the youngest of the remaining two.  Pick the Youngest.

So to avoid the middle one ask one about the other two, take which ever they say is the youngest.


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## Yari (Jun 18, 2007)

OK, I've ust decieded that in any future games concenring princess and the such, I'm going to be a monk......

/yari

Sorry for the little off-thread....


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## tradrockrat (Jun 18, 2007)

There were two answers!?!?!?!  Geez that makes me feel twice as slow... lol


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## CoryKS (Jun 18, 2007)

I'm going to cut the Gordian Knot here and point out that a logical man would not marry a princess in the first place.  Too high-maintenance.  So there is no logical answer to this riddle.


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## Rich Parsons (Jun 18, 2007)

Andrew Green said:


> My bad, wrote that wrong.
> 
> If A is the half and half one it doesn't matter, either C or D are ok.
> 
> ...



Well then I accept the answer presented with the given statement that there was a misunderstand or misstatement of the givens.   :lol:


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## DavidCC (Jun 19, 2007)

exile said:


> Lemme amplify this analysis of the third sister:
> 
> *Her answer to Q.... her sisters KNOW... truth of claim of matchup... response to Q*
> 
> ...


 
Where are you going to find 3 princesses that can understand your question?


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## exile (Jun 19, 2007)

DavidCC said:


> Where are you going to find 3 princesses that can understand your question?



Now we're in the realm of cognitive psychology, not logic, and the likely answer here is: nowhere. But that problem infects even the (very clever) elimination-algorithm response: if you ask a particularly dumb princess, `which of these two is the bigger liar' and she answers on the basis of which of them is taller or heavier... well, you got the same problem! 

I'm assuming that each of the princesses knows whether she's telling the truth or lying in answer to the question, and that the subpropositions which the question Q is about (the first being, is your answer truthful, yea or nay?, and the second, can anything be infallibly inferred from your answer to any question, yea or nay?) require information which each princess has available (she knows whether or not she's lying, and she knows that her sisters are fully familiar with her utterance-habits so far as truth (or not) is concerned). If she's too thick to answer correctly, so be it. But again, if she thinks `who's the bigger liar....?' refers to how big-boned her sisters are...   then we're back in the same boat, eh? :wink1:


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