A simple logic problem

[bonanova]

In the land of Truth-tellers, Liars and Alternators, every citizen either always answers truthfully T,

always answers falsely L, or alternates true and false answers A. You encounter, and find yourself

extremely attracted to, a citizen. Before exploring a possible relationship, however, you wonder

whether you'll be dealing with a T, L or A. You may ask a number of yes/no questions to satisfy

your curiosity.

How many questions will it take?

[ujjagrawal]

In maximum 2 questions

1. Are you a alternator ?

2. Are you a Truth-teller?

based upon the answers -

If respective answers are - no, yes -> Truth-teller

yes, yes -> Liar

yes,no or no,no -> Alternator

[witzar]

Two questions:

1. Is 2+2=4?

2. Is 2+2=4?

[bonanova]

Both right.

witzar's solution has the following

In the case of an Alternator, you know the parity of the alternation.

This is a great piece of information, should there be continued dialog.

[翁馬克]

There is a

...or do I misunderstand it?

Let the person we are talking to be an Alternator and assume he is in the state where his next answer is the truth.

1. Are you an alternator? Answer: Yes

(Now, his next answer will be false.)

2. Are you a truth-teller? Answer: Yes.

According to ujjagrawal, he would now be considered a Liar, but he is an Alternator!

[WitchOfDoubt]

One question.

"Let's suppose that truthtellers are the opposite parity of liars. If I were to ask someone of the opposite parity of you whether they were a truthteller, what would they say - yes or no?"

A truthteller says "yes". A liar says "No." An alternator can't answer one way or the other, because the question is ill-defined for them.

[omthkkr]

Answer is 2 - 1) Are you the alternator? 2) Are you the alternator? Yes & Yes = Liar Yes & No or No & Yes = Alternator No & No = Truth teller

Edited May 25, 2012 by omthkkr

[bonanova]

One question.

"Let's suppose that truthtellers are the opposite parity of liars. If I were to ask someone of the opposite parity of you whether they were a truthteller, what would they say - yes or no?"

A truthteller says "yes". A liar says "No." An alternator can't answer one way or the other, because the question is ill-defined for them.

I'm not sure why that question is Ill-defined for an Alternator.

Let's do the thought experiment of first asking the the Alternator two questions 

1. Is 2 + 2 = 4?

2. Is 2 + 2 = 4?

And suppose his answers were Yes and No; then his next answer would be truthful. Thus a person, also an Alternator, of the opposite parity, call him an OPP, would next answer falsely, and if asked about being a T would answer Yes. So the truthful answer of the Alternator being questioned would be that the OPP would say Yes.

If the thought experiment answers had been No and Yes, then the Alternator's next answer will be false. The OPP would then next answer truly, and when asked about being a T would answer No. The false answer of the Alternator would then be that the OPP would answer Yes.

Thus an Alternator answers the proposed question just as a T would, and the question can't discriminate between the two, even if asked twice.

Two things can be noted: we must presume an Alternator remembers the truth value of his last statement, else he doesn't know how to answer any question at all. Given that, he can determine how to answer every question, alternately emulating a T or L.

[bonanova]

There is a

...or do I misunderstand it?

Let the person we are talking to be an Alternator and assume he is in the state where his next answer is the truth.

1. Are you an alternator? Answer: Yes

(Now, his next answer will be false.)

2. Are you a truth-teller? Answer: Yes.

According to ujjagrawal, he would now be considered a Liar, but he is an Alternator!

You're right.

[phil1882]

confuse the hell out 'em!

"how would the lair respond to the following question, "are you an alternator?""

it would be quite interesting seeing what the liar says.

no matter what he says he is telling the truth.

[bonanova]

confuse the hell out 'em!

"how would the lair respond to the following question, "are you an alternator?""

it would be quite interesting seeing what the liar says.

no matter what he says he is telling the truth.

So we have a question: How would a Liar respond if he were asked about being an Alternator? The truthful answer to this question is the Liar would say Yes. A Liar would not give the truthful answer; he would say the Liar would say No.

You may be confusing the case that a Liar would say that a Liar will tell the truth with the case that a Liar would tell the truth. Simply ask a Liar if he tells the truth. His answer will be Yes. But that is not a truthful answer.

[phil1882]

i think you're missing my point

once again the question is:

how would the liar respond to the question "are you an alternator?"

now the liar himself is in a bind. if he says yes, he's telling the truth, that's how the liar would respond

if he says no, he's still telling the truth, that's how the liar would respond.

if he says something like "I don't know" or even remains completely silent, that's his honest responce to the question.

as far as i can see, the liar has no way out, but the truth.

Edited May 28, 2012 by phil1882

[phil1882]

this above paradox reminds me of an episode of samari jack, where this two headed beast poses the classic liar-truth-teller puzzler.

"one of us tells only truths" one head says;

"while the other only lies" the other head says;

"with one question" one of the heads says

"can you determine the truth telling head?" the other completes.

the samari makes what i feel to be the correct argument, the beast itself cannot pose this question with it being a contradiction.

either both heads are honest, or more likely both heads are deceptive.

[bonanova]

i think you're missing my point

once again the question is:

how would the liar respond to the question "are you an alternator?"

now the liar himself is in a bind. if he says yes, he's telling the truth, that's how the liar would respond

if he says no, he's still telling the truth, that's how the liar would respond.

if he says something like "I don't know" or even remains completely silent, that's his honest responce to the question.

as far as i can see, the liar has no way out, but the truth.

You haven't said who is being asked the "how would a liar respond" question. Is the puzzle solver supposed to respond? Or is a liar being asked, and the puzzle solver being asked how the liar would respond? You have left that point unclear.

I think you're getting trapped by pronouns. Give names to all the correspondents. Who asks which question of whom, then write down the truthful responses. When you've done that, you won't be unclear as to how a liar will respond.

Again, a Liar if asked, will say he is telling the truth.

But his statement that he is truthful is not truthful; he is a liar after all.

All good liars represent themselves as being truthful.

[phil1882]

oh i see, i thought i did make that clear.

i'm asking that question of the unknown person.

lets call the four possible person states T, F, At, and Af.

i ask the person: "how would the liar respond to the question, "are you an alternator?""

now if they are state T a lair would resond yes, therefore the person who is state T will answer yes.

a person who is At would also respond yes for the same reason

a person who is Af would respond no; the reasoning is fairly clear, a liar would respond yes, but Af has to lie.

but if the person is state F; then he is in a bind. no matter how he responds it would be an honest one.

i could have put any question in the secondary quotes, and the person in state F would still be in the same bind.

[bonanova]

oh i see, i thought i did make that clear.

i'm asking that question of the unknown person.

lets call the four possible person states T, F, At, and Af.

i ask the person: "how would the liar respond to the question, "are you an alternator?""

now if they are state T a lair would resond yes, therefore the person who is state T will answer yes.

a person who is At would also respond yes for the same reason

a person who is Af would respond no; the reasoning is fairly clear, a liar would respond yes, but Af has to lie.

but if the person is state F; then he is in a bind. no matter how he responds it would be an honest one.

i could have put any question in the secondary quotes, and the person in state F would still be in the same bind.

Your quote:

i ask the person: "how would the liar respond to the question, "are you an alternator?""

Assuming it's the case that ambiguity in stating the puzzle is not what is meant to create a paradox, let's assign names to some people. Then who is being asked what is unambiguous.

Let P be a symbol that represents the person whose identity is to be determined.

Among the population of this land are the following representative citizens:

1. Ted is a Truth-teller.
   
2. Larry and Louie are Liars.
   
3. Alex T is an Alternator in the T next-response state.
   
4. Andy F is an Alternator in the F next-response state.
   

That should give us enough people to proceed.

The premise is that a liar, let's say it was Larry, is asked the question "are you an Alternator?"

We know that Larry, not being an Alternator, but being a Liar, will respond Yes.

You say this puts Larry in a bind, but there is no bind.

He must answer, and his answer must be not truthful.

Yes satisfies both of his requirements as a card-carrying Liar.

Your objection seems to add the requirement that Larry must not only answer falsely, but also answer incorrectly: Larry must give a response that is both false and incorrect for a liar. Since it correct for a liar to give a false response, that addition makes it impossible for any liar to answer any question.

As stated here, the liar's only responsibility is to respond to being an Alternator. That is a simple question with a simple answer and, as a Liar, his response must be affirmative. Every response from every Liar, to every question that has truth value, must be to negate the truth. Compliance with that requirement is correctness, not truthfulness.

From a semantic point of view, all declarative statements carry the implicit promise: "I state that {declarative statement} is true. The liar simply puts a false statement within the brackets. That is, the Liar says, "I state that {false} is true. Observing that protocol does not change a false statement into a true statement.

We then ask P: "how would [Larry] respond to [that question]?"

Here are the options:

If P happened to be Ted: Ted would answer that Larry will say Yes.

If P happened to be Alex T: Alex T would answer that Larry will say Yes.

If P happened to be Andy F: Alex F would answer that Larry will say No.

If P happened to be Louie: Louie would answer that Larry will say No.

Closing the book on this question, we see that it discriminates Ts from Ls,

but leaves As of both types uncertain.

We note that Larry says Yes, and Louie says No. This is not a paradox.

They are different people, and they are asked different questions.

[phil1882]

well my question is how would larry himself respond to the "how would [larry] respond to [question]?"

again i feel its ambigious. no matter how he responds he has to tell the truth, even though he is a liar.

which throws the whole thing into a blender.

[bonanova]

well my question is how would larry himself respond to the "how would [larry] respond to [question]?"

again i feel its ambigious. no matter how he responds he has to tell the truth, even though he is a liar.

which throws the whole thing into a blender.

I think the problem unravels like this:

Larry's response to the first question would be: "Yes, I am an Alternator."

Larry's response to the second question would be: "Larry would say 'No, I am not an Alternator'."

Both responses are untruthful.

The second response is the one you receive when you ask the question.