A logic problem

[exile]

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≠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.

 

[michaeledward]

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)

 

[Andrew Green]

Excile,

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

 

[Andrew Green]

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

 

[exile]

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:

 

[tradrockrat]

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

 

[Andrew Green]

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"

 

[Rich Parsons]

Ok I am really confused.

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?

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.

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"

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?

 

[Andrew Green]

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.

 

[Yari]

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....

 

[tradrockrat]

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

 

[CoryKS]

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.

 

[Rich Parsons]

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.

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

 

[DavidCC]

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.

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

 

[exile]

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: