17 replies to this topic

[bonanova]

araver has it. Nice solve. No lie.

[k-man]

Spoiler for perhaps

the audience is a red herring. Assuming a single audience member who can be truthteller, liar or undecided/human , a single question that uses the truth value of the answer inside the question might do the trick for going away with the whole prize all the time. Higher order logic though, bu then so are the axioms about the three types of people.

If I would ask you if the prize is behind door #1 and your answer would be as truthful as the answer you're gonna give me now, would your answer be yes?

Truth teller says yes/no equivalent to prize being behind door 1/ being before door 2

Liar would give opposite answer to what he would normally evaluate the expression. If a and b then c.
If prize is behind door one and he would lie his answer would be no. So trying to lie about the outcome pf the hypothetical question, he says yes. Similar argument for the other case.

Undecided person is forced by the paradox of the question to evaluate the trith value of the actual answer before evaluating the answer itself. A machine might get syntactically stuck but a person shoul be able to evaluate/decide the truth value / context before evaluating the question itself. Therefore after he/she does that, the evaluation goes a deterministic path. Hence his/her answer would match the same equivalence as the other types discussed above.

Key here is if this is allowed by the OP.
The wording I intepreted would suggest this apparent paradox would qualify.

Then again, I've already decided if I'm gonna lie or tell the truth in this post

Spoiler for That's pretty clever, but it seems like cheating...

Effectively, by stating your question in this fashion you are forcing the random answerer to either tell the truth or lie twice in a row and thus making his answer deterministic, but then that violates his indeterministic nature of randomness.

[bonanova]

Spoiler for That's pretty clever, but it seems like cheating...

 
Effectively, by stating your question in this fashion you are forcing the random answerer to either tell the truth or lie twice in a row and thus making his answer deterministic, but then that violates his indeterministic nature of randomness.

 

Spoiler for Not really how else is there a solution

The responses of T and L are the same, and R acts like one or the other.
It was wrongfully stated by some, in the discussion, that R randomly chooses Yes and No.

To me, the most elegant solution is to ask the question to the entire audience.
It takes only a moment to deduce that a unanimous answer is reliable.

[CaptainEd]

I see k-man's point: if the randomizer is really indifferent to whether his/her answer is true or false, then why would he/she be "forced by the paradox to evaluate the truth value..." Why not just flip the coin? There's no requirement that the randomizer actually knows the truth or falsity of his/her answer.

 

I'm reminded of an old Superboy episode (yes, way old...) where a special kind of Kryptonite made Superboy into Pinocchio--whenever he lied, his nose grew longer. Superboy used this to figure out where the bad guys were hiding, by uttering the words "the bad guys are North of Main Street". When his nose grew longer, he knew that the bad guys were actually South of Main Street. This was an odd theory of "lying"--saying something that doesn't happen to be true, even though you don't know it. But our randomizer could be this way.

 

(Actually, I generally interpret one of these "randomizers" as being more like Maxwell's daemon--he does something pathological, whatever will best thwart your intended theory)

 

Now Bonanova's answer has come in. OK, I can bear that interpretation--the randomizer makes an explicit choice between lying and telling the truth. Having made that choice, he/she figures out what IS the truth, and then tells the truth or lies. Humph.

 

Yes, having a unanimous crowd is very pleasing!

Edited by CaptainEd, 31 January 2013 - 06:07 PM.

[bonanova]

Some solvers were discussing questions to which R could not respond.

What door would you choose in that case?

 

araver's approach is a solution, in that R can not respond erroneously.

[Prime]

Let me offer the following perspective:

Spoiler for Undecidable Proposition

Consider the question “Are you telling the truth?”
If you answer “No” it means that you are lying, from which it follows that you have told the truth. “No” makes the proposition indeterminable in terms of true/false.
On the other hand, the answer “Yes” is consistent with the truth of the statement (if you answer truthfully, then “Yes” is true.) “Yes” is also consistent with the false of the statement (if you answer falsely, then “Yes, I tell the truth” is in fact false.)
Both Truth Teller and Liar cannot answer “No” to the above question, not because they would be acting opposite to their nature, but because that would make their statement undecidable in terms of true/false. Whereas conditions prohibit them from making undeterminable statements. Consequently, while answering with the only possible determinable choice “Yes”, Random may believe he speaks the truth, or lies and his belief would make it so.
In the same way, the only possible determinable answer to the question, “Are you kidding me?” is “No”.

The inner part of the question proposed by Araver states that the prize is behind the door #1, the outer part (evaluated second) asks to equate own statement to the truthfulness of the first evaluation. In this way Araver may rightfully collect the “Best Answer” prize hidden behind one of the doors.

To remove doubts about whether a person would actually make such a statement about the door #1, I would rephrase the question as follows:

Is your answer as honest as the statement: “the prize is behind the door #1”?

Then what you are asking is in effect: “Your statement = X?”, where X can evaluate to True making the only possible answer “Yes”, or False forcing “No” out of everyone.

 

[k-man]

Let me offer the following perspective:

Spoiler for Undecidable Proposition

Consider the question “Are you telling the truth?”
If you answer “No” it means that you are lying, from which it follows that you have told the truth. “No” makes the proposition indeterminable in terms of true/false.
On the other hand, the answer “Yes” is consistent with the truth of the statement (if you answer truthfully, then “Yes” is true.) “Yes” is also consistent with the false of the statement (if you answer falsely, then “Yes, I tell the truth” is in fact false.)
Both Truth Teller and Liar cannot answer “No” to the above question, not because they would be acting opposite to their nature, but because that would make their statement undecidable in terms of true/false. Whereas conditions prohibit them from making undeterminable statements. Consequently, while answering with the only possible determinable choice “Yes”, Random may believe he speaks the truth, or lies and his belief would make it so.
In the same way, the only possible determinable answer to the question, “Are you kidding me?” is “No”.

The inner part of the question proposed by Araver states that the prize is behind the door #1, the outer part (evaluated second) asks to equate own statement to the truthfulness of the first evaluation. In this way Araver may rightfully collect the “Best Answer” prize hidden behind one of the doors.

To remove doubts about whether a person would actually make such a statement about the door #1, I would rephrase the question as follows:

Is your answer as honest as the statement: “the prize is behind the door #1”?

Then what you are asking is in effect: “Your statement = X?”, where X can evaluate to True making the only possible answer “Yes”, or False forcing “No” out of everyone.

 

Thanks, Prime. That's an interesting perspective...So, I've been mulling this over in my head trying to convince myself that everything is correct, so here is a slightly different perspective...

Spoiler for

The question "Are you telling the truth?" can be interpreted in different ways. There could be a context (like an answer to a preceding question) or this question could imply "Do you always tell the truth?". In both these cases the Random can answer "No" without creating a paradox. A liar will still answer "Yes" to this question in any context. In a pure case, without any prior context and without "always" being implied, this question indeed forces everybody to answer "Yes", but it's not a useful question anymore. It becomes a rhetorical question and you already know what the answer will be regardless of who is answering. You cannot gain any knowledge from it.

 

Now, to make it useful, let's add some context to it:

 

Is your answer going to be as truthful as statement X?

 

This forces everybody to perform 2 true/false evaluations. Truthteller and Liar are consistent and deterministic. As we've seen many times in the past, it's not hard to force the Liar to tell the truth.

 

Truthteller: T+T=T.

Liar: L+L=T.

 

But, to force a Random into a deterministic path is not something I've seen before. A Random can produce 4 combinations of truths and lies leading to 2 different outcomes:

 

T+T=T, T+L=L, L+T=L and L+L=T

 

So, now we eliminate some of these combinations by requiring that the truthfullness of both was the same. This eliminates T+L=L and L+T=L as possible answers to the question, just like "No" is an impossible answer to the rhetorical question "Are you telling the truth?". This leaves the Random with only two valid choices that both lead to the truthful answer. Unlike the truthteller and the liar he still has a choice. He can get there by lying twice or telling the truth twice, but other choices have been eliminated.

 

Very nice puzzle and a great solve, Araver!

 

[Prime]

Let me offer the following perspective:

Spoiler for Undecidable Proposition

Consider the question “Are you telling the truth?”
If you answer “No” it means that you are lying, from which it follows that you have told the truth. “No” makes the proposition indeterminable in terms of true/false.
On the other hand, the answer “Yes” is consistent with the truth of the statement (if you answer truthfully, then “Yes” is true.) “Yes” is also consistent with the false of the statement (if you answer falsely, then “Yes, I tell the truth” is in fact false.)
Both Truth Teller and Liar cannot answer “No” to the above question, not because they would be acting opposite to their nature, but because that would make their statement undecidable in terms of true/false. Whereas conditions prohibit them from making undeterminable statements. Consequently, while answering with the only possible determinable choice “Yes”, Random may believe he speaks the truth, or lies and his belief would make it so.
In the same way, the only possible determinable answer to the question, “Are you kidding me?” is “No”.

The inner part of the question proposed by Araver states that the prize is behind the door #1, the outer part (evaluated second) asks to equate own statement to the truthfulness of the first evaluation. In this way Araver may rightfully collect the “Best Answer” prize hidden behind one of the doors.

To remove doubts about whether a person would actually make such a statement about the door #1, I would rephrase the question as follows:

Is your answer as honest as the statement: “the prize is behind the door #1”?

Then what you are asking is in effect: “Your statement = X?”, where X can evaluate to True making the only possible answer “Yes”, or False forcing “No” out of everyone.

 

Thanks, Prime. That's an interesting perspective...So, I've been mulling this over in my head trying to convince myself that everything is correct, so here is a slightly different perspective...

Spoiler for

The question "Are you telling the truth?" can be interpreted in different ways. There could be a context (like an answer to a preceding question) or this question could imply "Do you always tell the truth?". In both these cases the Random can answer "No" without creating a paradox. A liar will still answer "Yes" to this question in any context. In a pure case, without any prior context and without "always" being implied, this question indeed forces everybody to answer "Yes", but it's not a useful question anymore. It becomes a rhetorical question and you already know what the answer will be regardless of who is answering. You cannot gain any knowledge from it.

 

Now, to make it useful, let's add some context to it:

 

Is your answer going to be as truthful as statement X?

 

This forces everybody to perform 2 true/false evaluations. Truthteller and Liar are consistent and deterministic. As we've seen many times in the past, it's not hard to force the Liar to tell the truth.

 

Truthteller: T+T=T.

Liar: L+L=T.

 

But, to force a Random into a deterministic path is not something I've seen before. A Random can produce 4 combinations of truths and lies leading to 2 different outcomes:

 

T+T=T, T+L=L, L+T=L and L+L=T

 

So, now we eliminate some of these combinations by requiring that the truthfullness of both was the same. This eliminates T+L=L and L+T=L as possible answers to the question, just like "No" is an impossible answer to the rhetorical question "Are you telling the truth?". This leaves the Random with only two valid choices that both lead to the truthful answer. Unlike the truthteller and the liar he still has a choice. He can get there by lying twice or telling the truth twice, but other choices have been eliminated.

 

Very nice puzzle and a great solve, Araver!

 

I see there is some ambiguity in my example - not deterministic enough.

Let me try again:

Spoiler for Disambiguation

I meant my example as a self-referential question/answer. Classical liar paradox is expressed elegantly as follows:
"This sentence is false."
Where one cannot decide the truthfulness of that sentece.

For this problem:
"Is your answer to this question true?"
Where the answer "No" will make the proposition (answer) undecidable. Truth-teller, Liar, and Random alike must answer "Yes" to that question.
By itself it does not yield any information about the identity of the replier. However, it forces a consistent answer. When you add a proper variable to it, an answer to the question will produce the desired information.
"Is your answer to this question = X?" Where X may be True or False.
When the answer is "Yes", we conclude X = True; when the answer is "No" -- X = False.

There was nothing in the statement of the problem requiring Random to be consistent by being dishonest exactly twice, or being honest all the time. He can be truthful or not as he pleases. However, while Random has a bit more freedom than Liar or Truth-teller, he is still limited to the one or the other. Random is not allowed to produce undecidable propositions.