The Logician at the Crossroads: Newsletter Challenge (January 2020)

If I ask each of the other 2 natives, will I be guaranteed at least one false answer? The answer to this question from 2 natives will give the information needed to tell which is a truth telling native...

Here's my analysis:

Call the three natives T (truthteller), L (liar), and R (random)

Both T and L can be tricked into giving the correct answer. Asking either T or L the following question would indicate the path to the village:

Q1: "Would you say 'yes' if I asked if the left path leads to the village?"

Both T and L would respond "yes" if the left path led to the village and "no" if the right path led to the village. (For the sake of argument, assume "left" is the correct path.)

The problem is that if you ask R any question, you get a random answer, no information. The first native you ask could be T, L, or R. But if the first native is R, the second will be T or L, which we can get the truth out of. But if the second is R, it will invalidate the first.

Ask native #1, Q1.

So if you ask the above question to one native, you have 2/3 chance of getting 100% truth (T or L) and 1/3 chance (R) of getting 50% truth, or 5/6 chance of getting the true path.

You still have one question left. If you ask the same question (Q1) from native #2, you have no more information because native #1 or native #2 could be R.

I feel that there ought to be a way (or they wouldn't ask), but I'm going to say no.

If you could ask 3 questions, ask all 3 natives Q1: "Would you say 'yes' if I asked if the left path leads to the village?"

At least 2 answers will match, and that's your answer. The 3rd answer will either match or not, depending on what "Random" comes up with.

What you want to know is which path leads to the village. You don't care which native is T, L, or R.

Interesting that adding one additional question could add so much context. Good analysis!

Just ask one question: Do you know they're serving free beer in the village?

Don't worry about the answers; just follow everybody.

You don't have to ask any questions, you pull out a gun and say we are going to take the shortest path to the village, if it's not the shortest path, I'm going to whack all three of yous, now lead the way....

Binge watching the Sopranos

Well he can choose right or left, he has a 50% chance of being right without any questions being answered...Without knowing which native is which, the answers are useless..so he must determine which is either the liar or which is the truth teller and then ask if the left path is the shortest way...so with the first question this must be determined....

I would think the more heavily traveled road would lead to the nearest village...so I would ignore the natives and use my own judgement....

Don't know whether it's relevant, but do the 3 natives know each other, so know which is which? Might affect the answer.

Just to be sure - you say two questions to one native - does that mean you can only ask each question once, and the same native must answer both questions?

Not quite -- she can ask each question only once, but the second question could be directed at a different native than the first. So the two yes-or-no questions can (and probably should) each be directed at two different natives.

Very clever. The first question identifies the first native as R or non-R, so you can direct the path question to a non-R. Two questions determine the path. Good answer!

Well reasoned...however it must be considered that the liar, deceptive by nature, is on to your plan, and might throw a monkey wrench into the mix, if the opportunity should present itself...but still GA

… and so the logician, having solved the crossroads puzzle, went on her way. As she was approaching the outskirts of the village, she passed an old man sitting by the side of the road under the shade of a tree. He hailed her. "Greetings, m'lady, what brings you to our village?"

"I wanted to find out what the people are like, because I was thinking of buying a house and moving here."

"What were the people like where you come from?" asked the old man.

She made an ugly face. "I have never seen gathered together in one place such a bunch of back-stabbing, dishonest, double-crossing malicious vermin! I'm so glad to get out of there."

"Well," replied the old man, "That's exactly the kind of people you are going to find here. I suggest you find somewhere else to live."

The next day, another traveller, having also solved the crossroads puzzle, passed by where the old man sat. This traveller too was thinking of moving to the village, and the old man again asked what the people were like in the place the traveller last lived."

"Oh, they were wonderful! So kind, honest, generous... I miss them so much. What a blessing it was to have known them!"

"Well," replied the old man, "That's exactly the kind of people you are going to find here. I wish you all the best, and I hope you find a place and settle here real soon."

(adapted from a forgotten source)

So true...know thyself

A variation on the story.

The hiker, looking for a hotel to stay the night, seeing an old man sat on a farm gate at the crossroads, asked him if the road ahead led to a hotel, and if so, the quickest way there.

" Yes!... first left about a mile. Then along the road a mile to the farm gate. Over the gate and across the field, though the woods to the ditch, and turn left to follow the ditch until you come the road. Then left again to the cross roads - and ask!"

An hour later, the traveler, scratched, muddy and tired, reached the crossroads - wondering to if there would be anybody to ask. Then to see an old man sat on a gate - and surprised - it was the same old man.

Thus somewhat annoyed , "...You've sent me in a 4 mile cross-country circle - and there's no hotel!.."

"Yes there is! - I answered your question correctly. The hotel is just over there behind those trees, 50 yards on the right."

I was thinking of questions along similar lines - ie, ask the natives a question about each other - rather than the route - but assuming "....each question directed to just one native..." meant the same native had to answer both questions - I got stuck - hence my post in #11.

Meanwhile, if any native can be asked to answer the second question, then in #12 you have done it ..... great!

The Randomizer (R) is really the wrench in the works, so I reasoned that the first question should be to obtain information, not about the route, but about the natives - to separate R from non-R. After this insight, I wondered if the first question could be framed to jam the circuits of the truth-teller and the falsifier - to render them silent. This suggested that the circuit-jamming first question should be about how the randomizer would respond to a given question.

Of course, my first question in #12, "Will the randomizer agree that the left path is correct?" needn't have referred to the route, since it is not intended to obtain information about the route. It could, for example, have been, "Will the randomizer agree that 2+2=4?" I just wanted the question to be consistent with the puzzle's theme.

Very good, but I have one note:

If this second question ("Would you say 'yes' if...") is to have any meaning, it requires a metacognition (concience?) on the part of the natives (at least the T and L ones). They should be able to know what they are about to answer and elaborate on it (either propagating it as it is, or reversing it). This of course is taken for granted by humans. I only noticed and mentioned this detail to warn anyone that could attempt to generalize the problem, replacing the villagers with any sort of mundane electronic devices that are just programmed to either tell something random, or tell this or that specific road.

Metacognition for the second question? I don't know. When Garry Kasparov played Deep Blue, he said that it seemed as if he were playing a human, which would have been a violation of the rules - but IBM denied any human intervention. I give credit to Rixter for the second question. For the first question it helped me to think of the natives as embodiments of logic circuits, one being a random flip-flop, the other two with fixed input to output instructions (if this, then that). This suggested the possibility of a question that would jam the two fixed input to output circuits, but not the flip-flop circuit, and by that means identify which was which.

I edited both of these comments to be on-topic rather than off, because they're providing a good discussion -- I hope you two don't mind.

Now that the answer is provided, keep in mind that this puzzle was written by Martin Gardner in the mid-1950s, at the latest. Given the prevailing racial sentiments at the time, Gardner probably chose "natives" precisely to generalize the problem, as you say -- they function as human versions of a flip-flop or electrical switch with two poles.

Gardner actually takes it a step further in the original puzzle and has the natives speak in their original tongues, of which the logician only understands "yes" and "no," making metacognition a moot point. Cf. this:

https://blogs.ams.org/mathgradblog/2013/09/06/gardners-tribes/

Great link, thanx!

I still wonder though, how a mere electronic device could answer more elaborate questions, like: "What would you answer if I asked ...", if not already programmed to answer such questions. It is obviously not the same like asking: "Is the left road the correct one?", which can be hard-wired even in the most stupid hardware.

I mean, it is only given that the two of the natives (machines or humans) can know which is the correct answer (then either provide it as it is or reversed). To assume that they can answer any other introspective question (i.e. something that has to first answer to him/her/itself before producing the final answer to the questioner) requires a capability that is either present due to conscious thinking, or otherwise pre-programmed. In the latter case, we have to assume that the programmer already anticipated such question to turn up.

So, as far as I understand (please correct me if I'm wrong), one has to assume that there is some "hidden" intelligence in the problem. Either due to the nature of the natives themselves (being humans) or due to some cool programmer hidden behind the bushes. Being humans is a rather trivial assumption, so we can relax our mathematical rigor a bit and move along. So I don't claim that the solution is wrong. I just mentioned that we cannot go much further without this assumption.

I think you have a point. If the solution assumes self-awareness of the entities being questioned, is it a mathematical problem? Could the human entities in the problem be replaced with machines programmed to simulate the needed level of self-awareness so that they would reliably respond to the type of self-reflective questions posed in the solution? Could the problem and the solution be precisely replicated in a logic circuit, and if not, how do we categorize it?

Human self-awareness is itself imperfect. When autistic savant Daniel Tammet is asked how he accomplishes his amazing feats of calculation, he goes on about numbers having different shapes, colours, textures - which gives us no insight to how he does what he does. How can he possibly know, in an instant, that a certain date fifty years ago fell on a Wednesday? He doesn't seem to know himself how he does it. It is as if the answer comes to him from outside the realm of mathematics, above and beyond mundane calculation.

I wish we had some robust understanding about what is intelligence and consciousness. We have reached the point to emulate quite some cognitive functions by using neural networks, but I'm afraid, still the NNs cannot EXPLAIN how they infer what they infer.

I guess (or I wish), with the modern driverless cars technology there are some improvements on this department, but we still cannot tell how much of the human intelligence we can emulate, and if we can ever really TRUST the machines. (Hmmm, 'trust' requires some mathematical rigor... but is intelligence just the result of a deterministic and mathematically describable algorithm?)

Answer is posted -- the solution is slightly different but I'd say canadianslidewinder's creative answer takes first prize this month. Good job!

But, why would a compulsive liar be feel compelled to tell the truth, under any circumstances?...

Here is a possible circumstance in which a compulsive liar might tell the truth:

If the liar knows that the other person knows he is a liar, and since the liar always wants to deceive, he might tell the truth to fool the other person. For example, he might say, "Yes, the left path leads to the village.", when the left path actually does lead to the village. The traveller, hearing this, and knowing that a liar has given him directions, takes the other path, and gets lost. "Ha, ha!" laughs the liar, who has deceived the traveller with the truth.

Germaine to this, in the crossroads problem, the liar's compulsion to lie must be strictly interpreted as a machine input/output response. If it is interpreted with a human element, as actually being a compulsion to deceive, no solution stated in this blog is watertight. An alert liar, whose compulsion to deceive takes priority over the compulsion to lie, would recognize the question, "Would you say yes, if I asked if the left path is correct?" as a trap, and would deliberately tell the truth about what he would say if asked if the left path is correct, thereby avoiding the trap and causing the traveller to choose the wrong path - to the liar's triumph and delight.

This is surely what Solar Eagle meant when he commented that, " the liar, deceptive by nature, is on to your plan, and might throw a monkey wrench into the mix."

It seems like the logician could only gather meaningful info from the question if the logician already knew the questions' correct answer before asking said question...

The self-reflective questions posed by the logician in the solution make my head hurt.

As Rixter explained in #2: If the logician asks, "Would you say 'yes' if I asked if the left path leads to the village?", both the truth-teller and the liar would respond "yes" if the left path leads to the village, and both would respond "no" if the right path leads to the village. So the answer yes or no provides the logician with the information needed to make the correct choice, whether the question is posed to T or L.

In my example in #31 though, if the liar, to avoid the trap and deceive the logician, doesn't follow the rules of the game and answers truthfully "no" to the question, "Would you say 'yes' if I asked if the left path leads to the village?" (that is, the liar tells the truth about the lie he would tell about whether the left path leads to the village), the logician will conclude from the 'no' answer that the right path leads to the village - which, because the liar has told the truth, is the wrong conclusion. (whew!)

You can avoid the self reflective question in the following way.

I'll start by describe the "two natives" answer.

First get the two natives to stand, one at each of the two possible roads. Ask either the question "Does the man standing at the road to the village tell the truth?" If he answers "Yes" you take his road; "No" you take the other road.

Although it's more difficult for us to understand why this always works: for the liar and truth teller there is no difficulty or "self reflection". (I know that the liar does have to know that he's a liar, and be forced to lie about it, but he's much less likely to contrive to decieve anyway.)

To extrapolate to the three natives you need to use the same trick as in the given solution to determine that one of them is either liar or truth teller; then position him at one road and both of the others at the other; then ask the known non Random man the slightly changed question "Is there a man at the road to the village who always tells the truth?". Again if he answers "Yes." you take his road; "No." the other.

I think there is a flaw in your solution. It works for the two natives and the three natives problem if the truthteller is by chance the one being asked the question, but if the liar is by chance being asked the question, the framed two natives question and the altered three natives question, both offer the liar the opportunity to throw a wrench into the works by embedding two lies in his response. He can in his 'yes' or 'no' response be lying both about who is on the correct road to the village and about the truthfulness of the other or others.

For example: If by chance, in the three natives problem, the traveller picks the liar to stand alone on one road, and the truthteller and the randomizer to stand together on the other road (this is after the three have been identified as randomizers and non-randomizers as per the given solution). Assuming in this case that the liar is on the correct road to the village, he is asked your question, "Is there a man on the road to the village who always tells the truth?" The liar will respond according to his nature with a complete lie. He will answer as if he is on the wrong road (a lie), and the truthteller and the randomizer are on the correct road (a lie), and upon that lie, and consistent with it, will therefore answer 'no'. The traveller, hearing the 'no' response, foolishly assumes that the liar is telling the truth about who is on the correct road, and is only lying about the truthfulness of who allegedly is on that road, and consequently takes the road on which the truthteller and the randomizer stand (the wrong road), and gets lost.

The liar is answering a question about the person or people on the road to the village: if that is him, he must lie; if it is the other two he must lie.

As far as his "logic" is concerned the positions of the three people are known: not a question.

But implicit in your question is a preface question, which is, "Do you know who is on the correct road to the village? Whether your question is posed to the truthteller or to the liar, both must internally answer this implicit question before vocally responding to your explicit question with respect to the truthfulness of whoever is on the correct road. For both, the truthful answer to the stated implicit question is 'yes'. The assumption of your explicit question is that both the truthteller and the liar will, in the internal dialogue of each, truthfully answer the stated implicit question with a 'yes' and then, both on that foundation of truth, vocalise an answer to your explicit question - the truthteller truthfully, and the liar untruthfully. This is a flawed assumption. The liar is a liar. Possibly in his internal dialogue he will truthfully answer 'yes' to your implicit question, but you cannot expect the liar to stand on that foundation of truth while vocalising the predicted answer your explicit question. Internally, he will say, yes I know who is on the correct path, but I'm going to pretend that that person or persons is on the incorrect path, and my answer to the traveller will be consistent with that lie.

Maybe I can better clarify the problem with your proposed solution than I have. After the traveller positions the natives as you instruct and then asks the question of either L or T, "Is there a man on the road to the village who always tells the truth?" the traveller would need to ask another question to whomever was questioned. That question being, "When you answered the first question, were you being truthful about who is on the correct road to the village?" Unfortunately both L and T will answer 'yes' to this follow-up question so the traveller remains no wiser, and still lost (and in any case a follow-up question is not allowed).

I believe that your "implicit question" objection applies equally to the published answer.

The implicit question would just need to be "Do you know which road leads to the village?".

You are correct that the second (final) question in the published solution ("If I were to ask whether that road leads to the village, would you say yes?") includes the implicit question "Do you know which road leads to the village?", but the subject of the published solution question is still about which is the correct road, and anticipates that the liar, while knowing the answer, will lie about it.

Your proposed final question also includes the implicit question, "Do you know which road leads to the village?" but unlike the published solution question, your proposed question has as its subject the truthfulness of who is standing on the correct road. It has no safeguards against the certainty that the liar, while knowing which is the correct road (the subject of the implicit question), will lie about who is standing on it.

To put it succinctly: The traveller cannot rely upon the liar to be truthful about who is standing on the correct road to the village when the liar answers the traveller's (your) question, "Is there a man who is always truthful standing on the road to the village?"

I now note that your objection also applies to the "two natives one question" problem. I've re-read your original response at #39, and see that you always held that view.

I think that makes it much easier to discuss, and, I'll re-state the solution:-

First get the two natives to stand, one at each of the two possible roads. Ask either the question "Does the man standing at the road to the village tell the truth?" If he answers "Yes" you take his road; "No" you take the other road.

If you really believe that that solution contains a greater opportunity for the liar to deceive than "Would you say yes if I asked if the left path leads to the village?", then, I don't think we are ever going to agree.

The liar is compelled to always lie. That is a specification of the problem. The question, "Would you say yes if asked if the left path leads to the village?" takes the liar's compulsion to lie into account so that the answer given by both the liar and the truthteller will be identical, which makes it impossible for the liar to deceive the traveller.

But in you proposed solution, even as re-stated, the liar can deceive the traveller. The liar will furnish the traveller the predicted response to your question, "Does the man standing at the road to the village tell the truth?" only if the liar is truthful about who is standing at the road to the village. But that would be in conflict with the terms of the problem. The liar, being compelled to lie, will pretend to be on the correct road when he isn't, or on the incorrect road when he is. His yes or no response will be consistent with that lie, and will differ from the truthteller's answer, thereby providing the traveller with no useful information.

To illustrate my point, here are the scenarios:

If, the liar is by chance asked your proposed question, "Does the man standing at the road to the village tell the truth?" and the liar is by chance standing on the correct road, he will pretend (as he is compelled to do) that not he, but the truthteller is standing on the correct road, and will answer 'no'. If the truthteller is by chance asked your question and the truthteller is by chance standing on the correct road, he will answer 'yes'.

If the liar is by chance asked your question and is by chance standing on the wrong road, he will pretend (as he is compelled to do) that he is standing on the correct road, and will answer 'yes'. If the truthteller is by chance asked the question and is by chance standing on the wrong road, he will answer 'no'.

As I wrote earlier, the traveller obtains no useful information from this. Your proposed question does not guard against the liar's certain deception.

Take the example of the liar standing on the road to the village: the traveller asks him if the man standing at the road to the village tells the truth.

If the liar answers "yes", he has clearly told the truth which is against the rules of his tribe.

I think you are getting mixed up, but I nevertheless think I understand what you are getting at, and it has helped me understand the fundamental problem with your proposed question. The question puts the liar in an impossible conflict with his tribe rules: If he pretends to be on a road that he's not (as in my illustration), that is a lie, and in accord with his tribe rules, but his yes or no answer consistent with that implicit unspoken lie can actually result in him uttering a truth with respect to who is actually on the correct road, and that is against his tribe rules.

On the other hand, if he answers either yes or no as you predict, these will be lies and therefore in accord with his tribe rules, but whether the lie is a yes or a no, it will be a surface lie, dependant on him holding to the implicit unspoken truth of whether or not he is on the correct road, and that is against his tribe rules.

Therefore, however the liar answers your question he will always be in violation of his tribe rules because his answer will always either 1) Contain an implicit lie and an explicit truth; or 2) Contain an explicit lie and an implicit truth.

The only means by which the liar can avoid this unresolvable conflict is to remain silent in response to your question. This has some usefulness. If the traveller happens to ask the question of the truthteller, the truthteller will answer yes or no, and the traveller can confidently take the correct road to the village. If the question is posed to the liar, his responding silence will identify him as the liar, but no further questions are allowed, so the traveller will still be lost. Your question therefore gives the traveller a fifty percent chance of learning which is the correct road to the village.

Further to my #49 post, here is an illustration that shows the impossible conflict into which your proposed question places the liar. It requires a temporary suspension of the problem's question limit to make the point.

Let us suppose the liar is standing on the road to the village, and is asked, "Are you standing on the road to the village?" The liar will of course lie and answer 'no', in accordance with his tribe rules.

Then suppose he is asked your proposed question, "Does the man on the road to the village tell the truth?"

How is the liar to answer this? He has already stated by his response to the first question that he is not standing on the road to the village, but according to your explanation of your solution in #38, you predict that the liar will answer in accordance with the actuality of his being on the village road, although as shown here, if asked directly, he will and must deny that he is standing on the village road.

Your question has placed the liar in a dilemma. To be consistent with his denial that he is on the village road, he must pretend that the truthteller, and not he, is on the village road. Then, to be consistent with this lie, the liar must answer 'no', the man on the village road (the falsely placed truthteller) does not tell the truth. Unfortunately, although this 'no' is a proper lie within the liar's framework of lies, it turns out to be a true statement with respect to the actuality of the liar being on the road to the village. 'No', the liar is clearly not a man who tells the truth, and he is standing on the road to the village. This true statement is in violation of his tribe's rules. To the liar's frustration, by following tribal rules in answering your question, he has broken tribal rules. It is a catch-22 for the liar.

As I concluded in #49, and as I hope I have more clearly explained here, the liar's only possible response to your question in order to avoid the inevitable conflict with his tribe's rules, is silence.

To whom it may concern: Rather than hide behind your 'Off Topic' rating of my #51 post, if you see a flaw in my reasoning, please explain.

You were correct that in my earlier post, I got mixed up (I think I forgot which tribe I was in).

I have balanced the off topic for the post above with an on topic. Someone obviously has an opinion: it would be better if they weighed in.

I still don't understand why my "implicit question" is more compelling to you than your own.

If the implicit question is "Does the left road lead to the village", then the fact that it is the same as one of the explicit questions in "Would you say yes if I asked if the left path leads to the village?" just adds to the liars dillema.

Incidentally, there's another implicit question that will really put the cat amongst the pidgeons (now that you've let it out of the bag):

"Are you a liar?"

Normally it is easier to let the cat out of the bag than to get the cat back into the bag, but in this case, the published solution question elicits the same yes or no answer whether posed to the liar or the truthteller, so the implicit question, "Are you a liar?" is irrelevant, and that particular cat can go right back into the bag. As long as the traveller can determine the correct road, whether the specific native questioned is a liar or no, is of no consequence.

But the implicit question, "Are you standing on the road to the village?" contained within your proposed question, is completely relevant. Implicitly, the liar is compelled to answer 'no' if he is on the road to the village, in order to lie and thus be in compliance with his tribe's rule -and that, as I explained in #51, is where everything starts to go wrong in your proposed solution.

Although in the published solution the implicit question, "Are you a liar?" remains unanswered and irrelevant when the traveller goes on her way, in your proposed solution, if the traveller by chance questions the liar, the self-conflicting logic of any yes or no answer to your proposed question will force the liar into silence, and that silence will identify him as the liar. It is still irrelevant, but if the traveller had any curiosity about who is the liar and who is the truthteller, that curiosity would be satisfied. Unfortunately, she would still be lost.

Thankyou for the 'on topic' rating of my earlier post. Also, there is no shame in briefly losing one's bearings in this problem. It's tricky.

I had less understanding in earlier posts in this discussion of the flaw in your proposed solution. I sensed there was something wrong, but I didn't see the catch-22 situation into which it placed the liar. I was aware of that only from my #49 post onward. In consideration of my new understanding, the actual implicit question within your proposed question that really throws the liar into a dilemma is, "Are you standing on the road to the village?"

Assuming, for the sake of illustration that the liar is standing on the road to the village, by his implicit 'no' answer (a proper lie) to the implicit, "Are you standing on the road to the village?", he is implicitly changing places with the truthteller, putting the truthteller instead of him on the road to the village, and so when he must answer the explicit question, "Does the man on the road to the village tell the truth?" it creates the conflict I described in #51. Your proposed question concerns itself with which road is the village road, who is standing on the village road, and the truthfulness of who is standing on the village road. This is not a problem if by chance the truthteller is the native being asked the question, but if by chance the liar is chosen, the several true/false variables will result in an inevitable conflict. His only viable response to avoid violation of tribe rules, is silence.

As you point out, the posted solution question also contains an implicit question. So let's temporarily suspend the question limit and do the same thought experiment as I did in #51. We'll assume for the experiment that the left road leads to the village and we'll ask the liar the implicit question, "Does the left road lead to the village?" The liar will answer 'no' (a proper lie). Then we ask the explicit question, "Would you say yes if asked if the left road leads to the village?" The liar's 'no' answer to the implicit first question creates no dilemma for the liar. Nothing has changed. His location on or off the village road is not the subject of either the implicit or the explicit question, so there is no possibility for confusion or conflict resulting from the liar's necessary lie about where he is standing. The liar simply answers 'yes' to the explicit second question, which is a proper lie, a lie consistent with his first lie, and a lie consistent with reality. There is no catch-22 for the liar, and by the 'yes' answer the traveller now knows which is the correct road.

"he is implicitly changing places with the truthteller,"

No he is not. The question "Are you standing on the road to the village?" Does not involve the other person at all. If the liar was the only person there the true answer would be "YES", and the liars answer would be "NO".

And the question "Does the man on the road to the village tell the truth?" Only involves the other man if the person you ask is not on the road to the village.

So we can agree that if the person asked is the truth teller then there is no problem, and, if you ask the liar and he happens to be on the wrong road then there is no problem.

The only possible problem arises if the liar is on the road to the village, and, he is the man you ask.

When you ask the question "Does the man on the road to the village tell the truth?", the correct answer is "NO", and the liar must lie, and say "YES".

As soon as you start to introduce implicit questions, you get into all sorts of problems. Certainly with my question, but, with the "Would you say yes if asked if the left road leads to the village?" question the liar must first decide if he is a liar, and whether or not he would acknowledge that to himself. His dilemma is much greater.

"As soon as you start to introduce implicit questions, you get into all sorts of problems."

My point is that you introduce implicit questions in order to reveal and avoid problems. For example, if you were to ask, "Are we going to use that winch to lift that load?", the implicit question, "Does the winch have sufficient capacity to lift that load?" must be answered in order to avoid problems. With respect to the crossroads problem and as I showed in #55, asking and answering the implicit question, "Does the left road lead to the village?" causes no logical conflict in the posted solution. However, asking and answering the implicit question, "Are you standing on the road to the village?" does cause a logical conflict in your proposed solution.

I point out that in your proposed solution, the liar's answer to the implicit question results in him implicitly changing places with the truthteller, and you write that you disagree. But to be consistent with and to maintain the described set-up of your proposed solution - one native on one road and one native on the other road - then when the liar lies and answers 'no', he is not on the road to the village, then he must be on the other road. If the liar is on the other road, then the truthteller, to preserve your described set-up, must be on the village road. This is a lie, and not consistent with reality, but it is consistent with the liar's denial that he is standing on the road to the village. However, since it is not in accord with the reality of the liar actually being on the road to the village, when he answers your explicit question it gives rise to the self-contradictory logic as explained in #51.

You assert that if the liar happens to be on the wrong road, there is no problem with your proposed solution. This is incorrect. The same logical inconsistencies arise when he is on the wrong road and is by chance asked your proposed question.

Again, as in #51, temporarily remove the question limit and ask the liar the implicit question, "Are you on the road to the village?" The liar must answer 'yes'. To maintain consistency with that lie, and to maintain consistency with your described set-up (one native on one road and one native on the other), the falsehood of the liar on the road to the village and the truthteller on the other road (that is, their places exchanged) - is presented to the traveller.

Then your explicit question, "Does the man on the road to the village tell the truth?" is posed to the liar. To be consistent with the lie that he is now on the road to the village, the liar answers 'yes'. This is a proper lie within the framework of the liar's lies, but identical to the example in #51, this lie turns out to be a true statement with respect to the actuality of the truthteller being on the road to the village. 'Yes' the truthteller is a man who tells the truth, and he is standing on the road to the village. This true statement is in violation of the liar's tribal rules. Once again, to the liar's frustration, by following tribal rules in answering your proposed question, he has broken tribal rules. His only recourse, to avoid this catch-22, is silence.

In your proposed solution, if the traveller by chance asks the truthteller your question, she will learn which road leads to the village. Otherwise, she remains lost.

There is no difference (in logical perception) between the two implicit questions "Does the left road lead to the village?" and "Are you standing on the road to the village?": they are both matters of fact.

If the liar is on the road to the village and he is asked "Does the man on the road to the village tell the truth?" Then the other road and other person do not enter into the thought process at all.

If the liar is on the wrong road and is asked the question "Does the man on the road to the village tell the truth?" Then the subject of the question is the truth teller and, the liar does not need to consider their own position at all.

You still haven't addressed the real problem with the "Would you say yes if I asked if the left path leads to the village?" I claim that this question contains the intrinsic question "Are you a liar?" (Or perhaps "Am I a liar?") To me, intrinsic is much more compelling than implicit as a problem with the solution.

On reflection it is possible to use a "double question" without resulting in the introspection of the liar: "If I asked him if the left road leads to the village would he say yes?" Then of course if the answer is "NO" you take the left road and "YES" the other road.

If the Logician at the Crossroads problem isn't going to descend into a hopeless tangle of logical conflicts and paradoxes, we have to assume that in the liar's internal dialogue he would be truthful with himself and answer the question, "Am I a liar?" with a 'yes' (I emphasize that this would only occur internally. It would never be vocalized.) For the same reason we have to assume that he would be truthful with members of his own tribe (Suppose the tribal leader were to ask him, "Are you complying with our rule to always lie?"). The rule to always lie must be taken as a rule that applies only to interactions with those outside the tribe; and to prevent the liar from going mad, does not apply to his internal answer to the question, "Am I a liar?"

The two implicit questions, "Does the left road lead to the village?" (relating to the published solution); and, "Are you standing on the road to the village?" (relating to your proposed solution), are, as you state, matters of fact. However (this is getting tiresome) the liar's necessary lie to the implicit question about whether the left road leads to the village does not result in any logical conflict with his necessary lie to the question, "Would you say yes if asked if the left road leads to the village?"

In your proposed solution, as I have exhaustively explained, the liar's necessary lie to the implicit question about whether he is standing on the road to the village, does result in a logical conflict. You cannot acknowledge that the liar would lie if asked directly if he is standing on the road to the village, and then pretend, as you have been doing, that he would answer your question, "Does the man on the road to the village tell the truth?" with no deception with respect to whether he is that man on the road to the village. You also cannot pretend, as you have been doing, that the liar's necessary deception about where he is standing, and his consequent answer to your question, is going to contain no deception about the actual position of the both the liar and the truthteller. You explicitly describe the set-up of your proposed solution as a positioning of one native on one road and one native on the other road. It logically follows that the liar's implicit and necessary deception about where he is standing is going to result in a deception about where both natives are standing. Again, as I have exhaustively explained, this results in logical conflicts, regardless of which road the liar is on.

I know it is easy to become attached to an idea, particularly if that idea is your own, but it is time to abandon your proposed solution. Its flaws have been dragged out into the open, examined, and found to be real. The cat is out of the bag and won't be put back in.

PS: Your "double question" alternative question appears to have merit, although I reserve the right to change this first opinion.

As I think I've indicated, I do accept that your objection to my solution does have some merit, but, your explanation contains errors, which I have explained exhaustively, and, repeat at ii.) and iii.) below.

i.) There are four possibilities; if the truth teller is asked the question there can be no problem.

ii.) If the truth teller is at road to the village when the liar is asked the question, then, the liar's position is irrelevant, and, there can be no problem.

iii.) If the liar is on the road to the village when the liar is asked the question, then, the truth tellers position is irrelevant.

And

iv.) "If the Logician at the Crossroads problem isn't going to descend into a hopeless tangle of logical conflicts and paradoxes, we have to assume that in the liar's internal dialogue he would be truthful with himself and answer the question, "Am I standing on the road to the village" with a 'yes'.

Which seems to resolve the final position.

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v.) I have deliberately used your own statement of my original assumption, but, accept that it is an assumption that could be argued against.

vi.) However if you do drop this assumption then the liars introspection which results from the "double inversion" question is diabolical.

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"For the same reason we have to assume that he would be truthful with members of his own tribe. "

I don't think you can assume that: I would have assumed the opposite, but, it's irrelevant.

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I took your "double question" in #59 as a variation on the published solution question, "Would you say yes if asked if the left road leads to the village?" Your "double question" is interesting and seems viable, but since any answer to it is not dependent on which road either native is on, it is unclear how it relates to your proposed solution, as you seem to assert in your point (VI).

Addressing your point (IV): The question, "Am I a liar?" is an introspective question to which the liar must internally answer with a truthful 'yes' to avoid collapse of the entire "Logician at the Crossroads" structure, but there is nothing introspective about your inserted question, "Am I standing on the road to the village?" Whether he is or isn't on the village road is not a question of his state of mind. It is a question about the reality of his physical position. The liar accepts that he is a liar, and in his interactions with those outside his tribe he will lie if asked which road he is on. If he is standing on the village road of course he will say to himself, "I am on the village road.", but he will lie about that and also lie about whether the truthteller is on the village road, to anyone who asks directly. He will also, to comply with tribe rules, answer any question to which the answer is dependant on which road he or the truthteller is on, as if they are standing on a road they are not (For example: his answer to your proposed solution question).

I agree with your point (I) that if the truthteller is asked your question there is no problem.

In your points (II) and (III) you state that regardless of which road the liar is on if asked your question, either the liar's position is irrelevant or the truthtellers position is irrelevant. I disagree. The question of both the native's actual positions, and their positions within the framework of the liar's lies, is completely relevant, as I have explained in past posts, and which I will summarize again in this post

From the simple line of reasoning set out below, all of the logical inconsistencies contained within your proposed solution flow. You have stated that you do not accept and do not agree with this line of reasoning, but you have never presented any argument to explain why you disagree.

1) The traveller doesn't know which of the natives is the liar or the truthteller; and does not know which of two roads is the correct road to the village (the described problem)

2) The two natives are positioned, one on one road and one on the other road (this is the described required set-up in your proposed solution)

3) The traveller by chance chooses the liar and asks the liar the implicit question contained within your proposed solution question, "Are you on the road to the village?" In answering, the liar necessarily lies, in accordance with his tribe rules.

4) The liar's lie about where he is standing places him, by deception, somewhere else. Since a requirement of your proposed solution is that one native is on one road and one native on the other (2), the "somewhere else" that the liar must be standing, by deception, is on the other road.

5) The truthteller is on the other road, therefore if by deception the liar is also on that road, this violates the required set-up described in (2).

6) To be consistent with the liar's lie about which road he is on, and to be consistent with the required set-up described in (2), the liar's lie includes a deception about which road the truthteller is standing on.

7) The full deception, within the framework of the liar's lie and consistent with it, is that the liar and the truthteller trade places.

It is therefore within the context of (7) that the liar answers your proposed question, "Does the man on the road to the village tell the truth?" The answer is dependant on which native is standing on the village road, and from the liar's necessary lie to the implicit question, "Are you on the road to the village?", the deceptive consequences from (3) to (7) flow. The lies are in accordance with his tribe's rules, and your set-up (one native on each road) is retained within the framework of the liar's lie. Regardless of whether the liar is actually standing on the village road or on the other road, he answers your proposed question as if he and the truthteller have traded places.

If you agree and accept the (1) to (7) line of reasoning, then you have arrived at the place where the liar, having lied in compliance with tribe rules, if he lies in answer to your proposed question from within the framework of his lies, his lie turns out to be a true statement with respect to who is actually on the village road. It is the catch-22 into which your proposed question leads him. It is the fundamental problem of your proposed solution. I explain the specifics of the catch-22 in post #51, if the liar is on the village road. A couple of posts later I explain the specifics of the catch-22 if the truthteller is on the village road.

Your statements in points (II) and (III) denying the relevance of the liar's or the truthteller's position in given situations, are not counter-arguments to my explanations of that positional relevance. You are simply making unsupported statements of disagreement. If you don't agree with my (1) to (7) line of reasoning, and if you have no argument to explain your disagreement, then our discussion is at a dead end.

1.) Yes

2.) Yes

3.) No

The "Are you on the road to the village?" question is, in your own words "in the liar's internal dialogue he would be truthful with himself".

Most of the rest of your post seems to depend on that.

Yes, I stand by my statement, "In the liar's internal dialogue he would be truthful with himself."

The liar would have a truthful internal dialogue with himself when the traveller asks him your question. Assume the liar is standing on the road to the village when the traveller asks, "Does the man on the road to the village tell the truth?" In considering how to answer, here is the liar's truthful internal dialogue with himself:

"Am I standing on the road to the village? Yes, I am. Do I tell the truth? No, I don't. Therefore I will answer as if I am not standing on the road to the village. To be consistent with the lie that I am not on the road to the village, I will answer as if I am on the other road and the truthteller is on the road to the village instead of me. So I'll answer 'no' to the traveller's question. But wait! If I answer no to the traveller's question, although that is a proper lie within the framework of my lies, it is actually the truth with respect to me actually being the one on the road to the village! No, I don't tell the truth, and I am standing on the road to the village. What a dilemma! If I comply with tribal rules and lie, I will violate tribal rules and tell the truth. My tribal leader told me that this self-contradictory logic is called a catch-22, and he said if anyone ever asks you a question that puts you into a catch-22 position, the only way to avoid violation of tribal rules is to remain silent. So, that is what I will do. By remaining silent I won't be in violation of tribal rules, and the traveller still won't know which road leads to the village."

It's easy to construct a similar argument for the double inversion question.

As an aside you still seem to be sticking to your assertion that the person on the other road's position matters: it does not. Suppose the truthteller is on the road to the village; the liar could be on the moon when he is asked the question, he would still be forced to lie (with no internal dialogue). If the liar is on the road to the village when he is asked the question, then, neither the truth teller nor the other road are relevant.

Your "double inversion" question posted in #59, "If I asked him if the left road leads to the village would he say yes?", I interpreted as the traveller pointing to one of the natives ("If I asked him..." etc.). It is a variation on the published solution, but sufficiently different that you deserve some applause for formulating it. However, since it has no requirement to position the natives on the two roads, it is a separate and unrelated solution from your proposed solution.

I acknowledge that if the truthteller is on the road to the village, the liar's position is irrelevant. Nevertheless, if he is permitted an internal dialogue (and how can he be denied such?) he will still encounter difficulties, different from those when the liar is on the village road, but from the liar's point of view, unresolvable, as shown in the following truthful internal dialogue of the liar when the traveller asks him your question when the truthteller is on the road to the village:

"I hate it when I am forced to answer yes or no to a question that is actually a question within a question. "Does the man on the road to the village tell the truth?" contains another question, 'Are you the man on the road to the village?'. I would lie about both questions if I were allowed to answer both questions. I'm a liar. That's what I do. This bleeping traveller has put me in a pickle. The truthteller is the guy on the road to the village, so if I were to answer 'no' to the main question, that is a lie, at least on the surface, but contained within that lie would be an unspoken truthful answer to the other question. I would be implicitly saying, 'I'm not the guy on the road to the village. The truthteller is.' This really bugs me."

The liar continues his truthful internal dialogue...

"I wonder what my tribal leader would say about this? Would he accept that my 'no' answer would be a proper explicit lie in compliance with tribal rules, when that lie contains an implicit truth about who is actually on the village road? I honestly don't know. I don't remember him ever saying anything about such a thing. He did warn me that some travellers can pose some very tricky questions. He said long ago two travellers had stopped at the crossroads. One said he was Bertrand Russell. The other introduced himself as Georg Cantor. Luckily, Russell chose the randomizer to ask his questions about set theory. We were completely baffled when Cantor asked, 'If these two roads are of infinite length, is it possible that one could be longer than the other?' My tribal leader said to me, 'I won't lie to you. Sometimes it's very difficult to tell a good airtight lie - one that fully complies with tribal rules. If you are at all uncertain, my advice is this, 'It is better to keep your mouth closed and look like a fool, than to open it and remove all doubt.' "

The liar concludes his truthful internal dialogue...

"I sure don't want to get into any trouble by not telling a proper lie. At the next tribal council meeting I'll ask if it would violate tribal rules if an implicit truth were contained within an explicit lie. I want to hear what the elders have to say on the subject. Maybe they will issue a formal directive to clarify the matter. Until then I better just keep my mouth shut. If I do that, the traveller won't learn the way to the village, but who cares? The main thing is that I won't get into any trouble."

The trouble is that you can easily construct a similar argument for any of the "double questions".

"The trouble is you can easily construct a similar argument for any of the 'double questions'."

No you can't. In the double question of the published solution, "Would you say yes if asked if the left road leads to the village?", there is no logical contradiction between the liar's necessary responsive lie to the main explicit question, and the liar's necessary responsive lie to the implicit, "Does the left road lead to the village?" if the liar were to be asked that implicit question directly.

If a "similar argument" to the one I used to expose the internal logical contradictions of your proposed solution can be easily constructed and used against the published solution then I respectfully request that you construct such an argument, present it, and edify me.

All of my objections to your proposed solution question would be eliminated if the question were rephrased to read, "Would you say yes if asked if you are standing on the road to the village?"

"I hate it when I am forced to answer yes or no to a question that is actually a question within a question. "Would you say yes if asked if the left road leads to the village?" contains another question, 'Am I a liar?'.

...........................

That the liar is a liar is part of the defined structure of the Logician at the Crossroads problem, and therefore "Am I a liar?" is not a viable question. Analogously, in a game of chess, the board, the pieces, and how the pieces move, are all part of the defined structure of the game, and are not open to question.

The Logician at the Crossroads problem is comprised of two roads, one village, one liar, one truthteller, one randomizer, and one traveller, whereby one road leads to the village. That is how the problem is defined. Therefore the existence of any of these elements cannot be questioned. They are part of the structure of the problem. It cannot be argued, for example, that, "Is there a road?" or "Is there a village?" are implied questions within the published solution question, "Would you say yes if asked if the left road leads to the village?"

You have presented no argument that shows a logical contradiction within the published solution. You should therefore acknowledge that there is no logical contradiction within the published solution.

But your proposed solution question, "Does the man on the road to the village tell the truth?" certainly does contain an implicit question, "Are you the man on the road to the village?" that can viably be asked within the structure of the problem. The liar's necessary lie to the main, explicit question would contain an implicit truthful answer to the implicit question about which road he is on. Since if the liar were to be asked the implicit question directly and permitted to vocalize his answer, he would lie, it is therefore a reasonable argument that the liar cannot answer your proposed question without implicitly violating his tribal rules. That remains my position.

OK: change the question to "Am I the liar?".

The implicit question, "Am I the liar?" exists in any and all questions that could be asked of the liar, therefore the entire structure of the Logician at the Crossroads problem collapses if you argue that the liar, in saying, "Yes, I am the liar", in his truthful internal dialogue, embeds an implicit truth in his vocalized answer, resulting in a logical contradiction, and an implicit violation of his tribal rules. Since this would occur in every answer to every question posed to the liar, the integrity of the Logicians at the Crossroads problem depends on acceptance and agreement by problem enthusiasts that the liar's affirmative internal answer to that implicit question has no logical consequences.

Further, in case you were planning to introduce the following as a "similar argument" against the published solution question, I will preemptively write that the described implicit logical contradiction would be as universal as the above example and therefore must also be accepted and ignored by problem enthusiasts:

The liar, before replying to the published solution question, must think. Therefore any yes or no answer by the liar to that question contains the implicit truth that the liar exists ("I think, therefore I am." - Descartes). The implicit truth of the liar's existence is in conflict with his necessary vocalized lie to the traveller's question, which results in a logical contradiction, and is also a direct contravention of his tribal rules. The truth of the liar's existence cannot be doubted or denied. It is established, proven, and broadcast by the yes and no sounds emitting from his cake hole.

First of all I do not accept your implied proposal in #70 that the liar believes he only exists as part of a logic problem: so I am returning to the use of "Am I a liar?" as the implicit question that snookers nearly all possible answers.

The implicit question, "Am I a liar?" exists in any and all questions that could be asked of the liar, therefore the entire structure of the Logician at the Crossroads problem collapses if you argue that the liar, in saying, "Yes, I am a liar", in his truthful internal dialogue, embeds an implicit truth in his vocalized answer, resulting in a logical contradiction, and an implicit violation of his tribal rules. Since this would occur in every answer to every question posed to the liar, the integrity of the Logicians at the Crossroads problem depends on acceptance and agreement by problem enthusiasts that the liar's affirmative internal answer to that implicit question has no logical consequences.

That's my point exactly, except that you can't pick and choose which implicit questions you treat one way and which another way.

My argument is that the "Does the man on the road to the village tell the truth?" question involves a lot less introspection by the liar than the "double inversion" question.

However I still believe that "If I asked him if the left road leads to the village: would he say 'YES'?" (from now on just called "the question" in this post), does get us out of the bind.

First notice that it doesn't matter which road leads to the village, evaluation of the solution is the same. So without loss of generality let's assume that the left road does lead to the village.

Clearly if you ask the liar "Does the left road lead to the village?": he will say "NO", and, if you ask the truthteller the question, he will say "NO".

Again clearly if you ask the truthteller "Does the left road lead to the village?": he will say "YES". This is an indisputable fact, and therefore, if you ask the liar the question, he will say "NO".

If, as you write, 'Am I a liar?' is an implicit question that snookers nearly all possible answers, that is surely sufficient reason for a consensus among problem enthusiasts to exclude that implicit question from discussion. I can't see any difficulty with the liar answering that implicit question, in a truthful internal dialogue, with a 'yes'. He is a liar. That is part of the structure of the problem. It is understood that if he were asked directly, 'Are you a liar?', his vocalized answer would be 'no'. However, if he is allowed to answer 'no' in his internal dialogue to the question, 'Am I a liar?', and if 'Am I a liar?' can be considered a ubiquitous implicit question, then the entire structure of the Logician at the Crossroads problem would collapse into internal logical contradictions and paradoxes.

The two implicit questions I have put forward are not ubiquitous, and do not threaten the structure of the Logician at the Crossroads problem. I will call them first distillation implicit questions specific to the published solution and specific to your proposed solution. I call them first distillation implicit questions to indicate that I did not "pick and choose" from among many. One is the obvious first distillation implicit question contained within the published solution question. The other is the obvious first distillation implicit question contained within your proposed solution question.

The first distillation implicit (intrinsic?) question in the published solution is, 'Does the left road lead to the village?". If the liar were asked to vocalize an answer to this question, he would answer 'no' (assuming the left road leads to the village). So the liar's vocalized answer to the implicit question is logically consistent with his vocalized answer to the main question, 'Would you say yes if asked if the left road leads to the village?"

The first distillation implicit question in your proposed solution is, 'Are you the man on the road to the village?'. If the liar were asked to vocalize an answer to this question, he would lie, and this necessary lie would be logically inconsistent with the answer you assert he would vocalize to the main question, 'Does the man on the road to the village tell the truth?'. To get the result you want, you have to keep a lid on what would be his vocalized lie to the implicit question. To get the result you want, you need his answer to the main question to be consistent with the truth of where he is standing, rather than be consistent with what would be his vocalized lie if he were allowed to vocalize his answer to the implicit question, 'Are you the man on the road to the village?'. As illustrated in the previous paragraph, the published solution question does not need to keep a lid on what would be the liar's vocalized lie to the given implicit question, because it would be logically consistent with his vocalized answer to the main question. This is the difference between the published solution and your proposed solution, and in my view, exposes the flaw in your proposed solution.

I'm sorry, but I have read and re-read the last part of your post and I still don't know what you are trying to establish or what point you are trying to make with your introduction of the question, 'If I asked him if the left road leads to the village, would he say yes?' It's very murky. You don't explain from what "bind" it is supposed to be extracting us, or how the explained working of the introduced question accomplishes the extraction from the unspecified bind. Are you saying that the natives are positioned according to your set-up, one on one road and one on the other, and the traveller points to one of them (the "him") in your question), and asks the question? And after the traveller receives the answer, how does that relate to your proposed solution question? None of this is explained. I acknowledge that the question works, but it gives the traveller the answer he needs independent of your native positioning set-up, and independent of your proposed solution question. It is a separate solution, just as the published solution is a separate solution. You don't need either native to be standing anywhere. Your entire native positioning set-up is irrelevant. Just point to one of them and ask the question, and it would work. The point you say you are making completely eludes me. You will need to explain it more clearly.

Addendum: Hold off replying to my previous (#74) post. I have maybe found a way out of this. It is based on your proposed solution, and is in keeping with your goal of minimal introspection. I need a bit more time to think about it, and I am also busy tomorrow, so I probably won't be able to post again until tomorrow night (Canadian eastern standard time). If it works we can claim joint authorship.

"If, as you write, 'Am I a liar?' is an implicit question that snookers nearly all possible answers, that is surely sufficient reason for a consensus among problem enthusiasts to exclude that implicit question from discussion. I can't see any difficulty with the liar answering that implicit question, in a truthful internal dialogue, with a 'yes'. He is a liar. That is part of the structure of the problem. It is understood that if he were asked directly, 'Are you a liar?', his vocalized answer would be 'no'. However, if he is allowed to answer 'no' in his internal dialogue to the question, 'Am I a liar?', and if 'Am I a liar?' can be considered a ubiquitous implicit question, then the entire structure of the Logician at the Crossroads problem would collapse into internal logical contradictions and paradoxes."

I repeat: you can't pick and choose, and, your description just confirms the confusion that arises.

"The first distillation implicit (intrinsic?) question in the published solution is, 'Does the left road lead to the village?". If the liar were asked to vocalize an answer to this question, he would answer 'no' (assuming the left road leads to the village). So the liar's vocalized answer to the implicit question is logically consistent with his vocalized answer to the main question, 'Would you say yes if asked if the left road leads to the village?" "

But the implicit "Am I a liar?" question raises its ugly head.

"The first distillation implicit question in your proposed solution is, 'Are you the man on the road to the village?'"

No it is not: the subject of the question is "the man on the road to the village": not "you".

"You don't explain from what "bind" it is supposed to be extracting us"

The "bind" would be: "an implicit question that snookers all possible answers"; but, I wrote "nearly all". My proposed question provides an exception. You don't need to position the two people; simply indicate the other when asking "If I asked him if the left road leads to the village, would he say yes?"

_{Incidentally this is just recalling the question in post #59.}

I asked you in my #75 post to hold off responding to my previous post because I had an insight that would free us from this impasse, but you ignored my request.

It has occurred to me that your proposed solution question, "Does the man on the road to the village tell the truth?" can be rephrased to, "Is the truthtelling man on the road to the village?", or simply:

1) "Is the truthteller on the road to the village?"

or its alternative:

2) "Is the liar on the road to the village?"

Position one native on one road, and the other native on the other road, then ask either native either question (1) or question (2).

Here are the replies to question (1) "Is the truthteller on the road to the village?"

If the truthteller is on the village road and asked. Answer: Yes

If the truthteller is not on the village road and asked. Answer: No

If the liar is on the village road and asked. Answer: Yes

If the liar is not on the village road and asked. Answer: No

Therefore, if 'Yes' is the answer, the traveller takes the road on which the asked native is standing. If 'No' is the answer, the traveller does not take the road on which the asked native is standing.

Here are the replies to question (2) "Is the liar on the road to the village?"

If the liar is on the village road and asked. Answer: No

If the liar is not on the village road and asked. Answer: Yes

If the truthteller is on the village road and asked. Answer: No

If the truthteller is not on the village road and asked. Answer: Yes

Therefore, if 'No' is the answer, the traveller takes the road on which the asked native is standing. If 'Yes' is the answer, the traveller does not take the road on which the asked native is standing.

As can be seen, either of the two questions will provide the traveller with the needed directional information.

In re-framing your original proposed solution question, the same question is being asked, but so re-framed it eliminates all disallowable implicit truths and consequent logical contradictions formerly contained within the liar's vocalized necessary answers. The only implicit truths that remain embedded in the new questions, and implicitly contained within the liar's vocalized necessary answers, are allowable truths. That is: more than one road exists, a village exists, a road to the village exists, the liar exists. These are allowable implicit truths in the liar's vocalized necessary answers because they are all elements that are defined as part of the structure of the Logician at the Crossroads problem.

Also, unlike your original proposed solution question, the liar's vocalized necessary lie to the main explicit question expressed in either of the new re-framed questions, is never in logical contradiction with what would be his vocalized necessary lie to the first distillation implicit question, if he were to be asked to vocalize an answer to it. That first distillation implicit question is, "Are you on the road to the village?" I went through all eight native questioning possibilities above to confirm this logical consistency. You can check this on your own if you wish.

So there it is. Based on your proposed solution, and in front of us all this time. The choice of two, simple, non-introspective questions, either of which will provide the traveller with the needed directional information, and free of any implicit internal logical contradictions.

Good Answer. I think I can see why you think your question is better than mine, but, just sticking with question number 1.): the implicit question which breaks it in the same way as most other questions is "Am I the truthteller?".

Thankyou for your gracious GA.

I hold to my position that there are certain implicit questions that threaten the defined structure of the Logician at the Crossroads problem, and therefore cannot be treated as open questions. Examples: Is there a road? Is there a village? Is there a road that leads to the village? Am I the truthteller? Am I the liar? The truthteller and the liar are not people you could actually meet on any road. They are chess pieces on a board. They each have a defined way of moving, and how they move cannot be questioned, explicitly or implicitly. Implicit questions that threaten the structure of the problem are easy to recognize, and there are not many of them. So, while accepting a need to bar them from consideration, it does not follow that all implicit questions should be barred from consideration. They can sometimes provide useful insights.

OK: you pick and choose your set of allowable implicit questions and I'll pick and choose mine.

I now see that the question I believed circumvented all problems: "If I asked him if the left road leads to the village, would he say yes?" also fails because of the implicit question "Is he a truthteller?".

So although I think we have both come a lot further than the traveller, neither of us has a perfect answer.

Perhaps it would be easier to approach the problem if there was only one native (although the solution would still apply to two you just ignore one of them).

How about: "If I asked a member of the other tribe if the left road leads to the village, would he say yes?"

As before If the answer is: "NO", you take the left road; "YES", the right.

If you happen to be speaking to a truthteller there is no problem.

If you're speaking to a liar the question already contains the information that the person the question is about is from the other tribe therefore, he, the liar does not need to ponder any implicit questions.

You might need to add a preamble to the question so that the meaning of "the other tribe" is clear.

Just thinking aloud really.

I don't consider the recognition that there are certain implicit questions that threaten the very structure of the problem and there are those that don't, to be "picking and choosing". No solution to the problem is possible if implicit questions about the structure of the problem are allowed as open questions. To use the chess analogy again, imagine trying to play a game in which how all the pieces move is open to question. The game would no longer exist. Your new proposed solution falls apart immediately if the implicit question, "Is there a road?" is considered an open question. If asked, the liar would then reply, "No, there is no road.", and that is the end of, not just your proposed solution, but the entire Logician at the Crossroads problem.

The implicit question I used to test for a possible flaw in your proposed, "Does the man on the road to the village tell the truth?' was, "Are you the man on the road to the village?" That is not an implicit question that threatens the structure of the problem. Your earlier posted accusation that I had deliberately "picked and chosen" that specific implicit question to attack your solution can now be seen to be a false accusation, since I use the same implicit question to test for flaws in my re-framing of your proposed solution question ("Is the liar on the road to the village?").

Let's compare. Assume the liar is standing on the road to the village. If he were to be asked to vocalize his answer to the implicit question, "Are you the man on the road to the village?" , he would answer "No". This is perfectly logically consistent with his "No" answer to the explicit question, "Is the liar on the road to the village?" (the re-framing of your original proposed question). Now ask the liar your explicit original proposed question, "Does the man on the road to the village tell the truth?" How is the liar to answer this, and be logically consistent with his vocalized "No" answer to the implicit, "Are you the man on the road to the village?". He can't. He is caught in a logical contradiction. Same question, but framed differently, gets us out of the catch-22.

I appreciate the GA you gave me for my #77 post, but then you write the weasel words, "I think I understand why you think yours is the better solution". This is not an acknowledgement that it is a better solution. It is like someone who says, "I'm sorry you feel so upset that I swindled you.", and considers that an apology. Hey, I based the re-framed question on your proposed solution! I could not have done it without the foundation of your work.

I would be happy to claim co-authorship with you. Forget your meaningless speculations about structure threatening implicit questions. I think the, "Is the liar on the road to the village?" question is a significant and important new solution to the two natives problem, and also as a new final question to the three natives problem. All of the tricky introspection of, "Would you say yes if asked if the left road leads to the village?", is gone. What could be simpler? Put the natives on the roads and ask one of them, "Is the liar on the road to the village?" It is a breakthrough solution. I'm not saying I did it. I'm saying we did it!

"The implicit question I used to test for a possible flaw in your proposed, "Does the man on the road to the village tell the truth?' was, "Are you the man on the road to the village?" That is not an implicit question that threatens the structure of the problem. Your earlier posted accusation that I had deliberately "picked and chosen" that specific implicit question to attack your solution can now be seen to be a false accusation, since I use the same implicit question to test for flaws in my re-framing of your proposed solution question ("Is the liar on the road to the village?")."

I repeat: my objection to that implicit question is that it is not implicit, because the subject of the question is "the man on the road to the village" not "you".

Well, if the traveller is asking the liar, and if he is standing on one of the roads, the implicit question certainly is, "Are you the man on the road to the village?". It is critical information that will inform his reply to the main question - that is why I use the same implicit question to test both your proposed solution, and my solution. I don't call my solution "proposed", because unlike yours, mine contains no internal logical contradiction, and therefore actually works.

I assume from the vehemence of your reply that you have rejected my offer that we present ourselves as co-authors of the solution question, "Is the liar on the road to the village?" or its mirror, "Is the truthteller on the road to the village?". If that is the case, and that is your decision, then I will claim it exclusively as my own solution, and will be presenting it to the Solutions Examination Committee of the International Logicians at the Crossroads Society. After it passes examination (as it surely will), it will then go on to be ratified by the august members of their Board of Governors. I will be the recipient of many accolades, which unfortunately, in rejecting my offer of co-authorship, you will not share. I will not immediately submit this breakthrough solution to the Solutions Examination Committee. I will allow a couple of days for you to consider the consequences of your rejection of my offer. If it stands, all of the credit for the new breakthrough solution will go to me, and none to you. It is your choice. Consider carefully.

I am very sorry if my reply came across as vehement, that was certainly not my intension: I just wanted to point out that I had already made the point.

In the same way: I have also already made the point that the liar's position is irrelevant if the truthteller happens to be on the road to the village.

"the liar's position is irrelevant if the truthteller happens to be on the road to the village" (your quote)

But your set-up is one native on one road and one on the other. The traveller asks your question to a native who is standing on a road. If the question happens to be posed to the liar and the truthteller is on the road to the village, then the liar (by your set-up) is not on the moon. He is on the other road. Where he is standing determines where the truthteller is standing. Their positions, by your set-up, are interdependent. So the implicit question, "Are you on the road to the village?" is entirely relevant. In this case he would lie and answer 'Yes'. To be consistent with that lie he would also have to answer 'Yes' to your main question. This would be (as I have already pointed out many times) in logical conflict with the actuality of the truthteller being on the village road, and the 'No' answer you need for your proposed solution to work. The liar's only option is to remain silent.

If the main question is simply re-framed to, "Is the liar on the road to the village?", the logical conflict disappears. It is simple and elegant, and there can be no doubt whatsoever how the liar is going to answer. Ask the implicit question, "Are you on the road to the village?" and it is always logically consistent with the main question, "Is the liar on the road to the village?"

Since you have not even acknowledged my olive branch offer to present ourselves as co-authors of the solution question, "Is the liar on the road to the village?" (based on your set-up), I withdraw that offer. If you hadn't been so enamoured of your proposed solution, and so stubbornly unwilling to admit that it could be flawed, you might have more critically evaluated it - and as a result have had the insight to re-frame the main question yourself. But you didn't. So now you are stuck with defending something that works only if the truthteller happens to be the one asked. It was your choice.

"Where he is standing determines where the truthteller is standing."

Only from the point of view of the traveller, or, for example the solver of the puzzle.

Put yourself in the position of the liar: if the truthteller is on the road to the village when he is asked the question, then his own position is irrelevant.

Okay, I'll put myself in the position of the traveller. If the truthteller happens to be on the road to the village, the liar will always be on the other road. By the terms of your defined set-up, one native on one road and one native on the other, the truthteller is on the road to the village because the liar is on the other road. Their positions are interdependent and therefore always relevant. The liar can imagine himself anywhere he wishes, but by the terms of your set-up, he will always be on one of roads and therefore his answer will always be informed by that reality.

But let us suppose, as you suggest, that the liar, although he is standing on the road that doesn't lead to the village, he is in his mind, standing on the moon. If he were to vocalize an answer to the implicit question, "Are you on the road to the village?" he would still answer 'Yes', and therefore to be logically consistent with that lie, and consistent with the defined terms of your set-up, he must answer your main question as if he were, in his mind, on the road to the village (not on the moon), and the truthteller , to be consistent with the lie, and the defined terms of your set-up, is on the other road. The liar, in his mind, and in reality, is back in your defined framework, and once again, this throws the liar into a catch-22 situation, in which it is impossible for him to maintain logical consistence with what would be his vocalized lie with respect to his position, if he were to be asked, and the lie with respect to the truthteller's actual position on the road to the village. Silence is his only option.

I know you love your proposed solution question with an irrational love, and irrational love is blind. Nevertheless, it is always germane to point out that a re-framing of your main question, "Does the man on the road to the village tell the truth?" to, "Is the liar on the road to the village?" removes all possibility of the described internal logical contradiction.

It does not matter how or why the two natives got to their positions. The liar was not party to the decision making process; he has no reason to think that the "setup" was done to solve any kind of problem, and, does not need to know what the travellers motives are.

Let me propose a new situation: just after the traveller has finished positioning the two men, he sees a friend approaching from one of the two roads; he does not want to talk to him so he runs off in the direction he came from. As it happens the friend has just come from the village, and he knows that all liars wear one coloured shirt and all truthtellers wear a different coloured shirt, but he doesn't know which wear which; he approaches the man on the other road and asks "Does the man on the road to the village tell the truth?" When the man answers "NO" he realises that he has not learned anything.

The question "Are you on the road to the village?", is not implicit in the question "Does the man on the road to the village tell the truth?"

Solutions to the Logician at the Crossroads problem have never been about identifying who is the liar and who the truthteller - information the friend needs if he is to learn their shirt colour preference. Solutions are about identifying the village road without knowing who lies and who is truthful. Interestingly, by my reasoning, your question does give the friend the information he needs about the natives' choice of shirt colour, since by my reasoning, your question does not identify the village road, but does identify who is the truthteller and who is the liar. The native who answers 'No' would be the truthteller. The native who answers with silence would be the liar. The friend would then know their preference in shirt colour. Regardless of the questioner's motive, each native has to ask himself, "Am I standing on the village road?" in answer to the implicit question, "Are you on the village road?" before answering the traveller's or the "friend's" question. This is because of the interdependence of the natives' positions. Where one native is standing determines where the other native is standing. It determines who is "the man" on the village road. If either were asked to vocalize an answer as to whether he were standing on the village road, the truthteller would tell the truth, and the liar would lie, and with respect to the main question, that implicit lie puts the liar into a logical contradiction, a catch-22, which I have explained ad nauseam. That logical contradiction forces him to remain silent.

I'm not doing this anymore. I am sick of this discussion/argument. We are going around in circles. If you wish to hold to the position that your native set-up, followed by your proposed question, is a legitimate solution, that's fine, but in my reasoned opinion, it is flawed.

Your proposed solution does have merit. It is a solid step towards the goal of a non-introspective solution to the Logician at the Crossroads problem. With some insight, and a re-framing of your proposed question so that it reads, "Is the liar on the road to the village?" or to its mirror, "Is the truthteller on the road to the village?" (as I set forth in post #77), a sound, non-introspective new solution to the 'Crossroads' problem is achieved.

Months ago, in post #12, I submitted to CR4 a new first question that can be used in the three natives problem. With that question the traveller could separate the Randomizer from the non-Randomizer natives. Then the traditional two native question (posted by Rixter in #2) could be applied.

Now, with post #77, there is a new second question that can be used in the three natives problem, or used as the sole question in the two native problem. Further, if the solution in post #12 is combined with the solution in post #77, there is now an entirely new two question solution to the three natives problem. In conclusion, although our discussion has been onerous, I am very pleased with the outcome.

Don't give up. If you are as certain that you are right as I am, then we should continue 'til we resolve this. But I feel that you need to focus on the point in question at any one time.

You just typed several hundred words without addressing my main point in the previous post:

Why do you think it matters how or why the two natives got to their positions?

Do you honestly believe that the liar would be struck dumb in the situation I described above, where he has just seen the "friend" approach from the village. He knows that the friend knows where the village road is. All the nonsense about the shirt colours was just to give the friend a reason to ask exactly the same question: "Does the man on the road to the village tell the truth?"

Of course it makes absolutely no difference to the liar why anyone asks a question (he is not aware that he is part of a logic problem).

If you cannot refute this then: it is clear that the question "Are you on the road to the village?", is not implicit in the question "Does the man on the road to the village tell the truth?"

I am sick of this topic, and am prepared to abandon it, even if not resolved, but since you obviously think you have pulled a "gotcha" with the "friend" scenario, I will respond.

As a preamble: I have never given any importance to how or why the natives got to their positions. What is important is that they are in those positions, and their positions as defined in your set-up, are interdependent - that the position of one determines the position of the other. It follows from that, that if the liar lies about which road he is on, that lie does not just take him (in the lie) off the road he is on, it puts him on the other road, and the other native (in the lie) is placed on the road the liar (in the lie) just vacated. This is key to understanding the logical contradiction into which the implicit question, "Are you on the road to the village?" throws the liar.

Also,as a preamble: The introduction of a "friend" who knows which road leads to the village, and therefore knows which native is standing on the road to the village, and then asks one of the natives a question purportedly designed to determine which road leads to the village, is so twisted, that it's connection to the actual problem eludes me. Through the logical fog I think you are trying to say that since the friend knows the answer to the implicit question, "Are you on the road to the village?", that the implicit question doesn't exist.

Here is an analogy to demonstrate the fallacy of your argument (if that is your argument):

In a sleight of hand demonstration, an observer asks, "Is the red ball under the left cup?". The implicit question to the observer's question is, "Is there a ball under the left cup?". The observer doesn't know the answer to either question.

Now suppose there is a second observer who, unfooled, knows the red ball is under the left cup. Therefore he would also know the answer to the implicit question, "Is there a ball under the left cup?". I submit that it does not follow that because the second observer knows the answer to the implicit question, and does not need to ask it, the implicit question ceases to exist. It is a question implicit to the main question regardless of what an observer does or doesn't know. If we were to accept that the implicit question ceases to exist because an observer can be found who knows the answer to it, then we would also be required to accept the absurdity that the main question ceases to exist because an observer can be found who knows the answer to it.

Your proposed question, "Does the man on the road to the village tell the truth?", is asked face to face to "a man", and the implicit question, "Are you the man on the road to the village?" relates directly to who is that man. Both the logician traveller, and the "friend", will understand that the liar would not be truthful in answering that implicit question, if asked directly. The logician traveller will see the logical contradiction into which this will put the liar, so even if he could ask the extra question, the implicit question, he doesn't - but this does not mean the implicit question doesn't exist. As for the friend, he already knows whether the man before him is on the village road, so he doesn't ask the implicit question either - but in this case too, not asking the implicit question does not render it non-existent. Of course, if the friend really did want to know who wears what coloured shirt (which you now say he never did want to know), he would ask the implicit, "Are you on the road to the village?" and by the answer he would know which native had answered, and therefore know both their shirt colour preferences.

When you first posted your proposed question, way back when, I sensed there was a flaw in it. There was an expectation in the question that the liar would answer the question with a necessary lie, and the predicted lie would have as its foundation, and be dependent on, the truth of which road he was standing on - something he would never be truthful about if asked directly. That remains my position. Re-framing the question to, "Is the liar on the road to the village?" allows the liar to lie wholeheartedly. There is no sneaky truth the traveller is trying to get past the liar. The same implicit question, "Are you on the road to the village?" can be asked of the re-framed question, and if the liar were to vocalize an answer, no logical contradiction would be exposed in the re-framed question. I'm not afraid of that implicit question. You are. That is why you want it to disappear. That is why you introduce a "friend" that you hope will dynamite the entire problem (as you previously tried to do with structure-threatening implicit questions) and in the ensuing destruction you will somehow manage to de-legitimize all implicit questions and not have the logical contradiction buried in your proposed question exposed.

OK so we've got the truthteller on the road to the village; the liar on the other road, and, we are about to ask the liar a question.

"Re-framing the question to, "Is the liar on the road to the village?" allows the liar to lie wholeheartedly. There is no sneaky truth the traveller is trying to get past the liar. The same implicit question, "Are you on the road to the village?" can be asked of the re-framed question, and if the liar were to vocalize an answer, ":

He would answer "YES". So then in answer to your question, he would reply "NO".

So I think you'd be better sticking to your question "Is the truthteller on the road to the village?"

But put all that aside for now I'm going to try to put us both out of our misery by asking the other participants on the thread to vote. I realise it will take a little effort but we will need to agree on the actual wording of the post we ask them to vote on. After we have made the post I will PM all the other participants and ask them to read just that one post and vote if they agree with me or you.

IF ANYONE ELSE IS STILL FOLLOWING PLEASE DON'T VOTE UNTIL WE HAVE AGREED ON THE FINAL FORM OF THE QUESTION.

Because the main disagreement seems to have distilled to the situation where the truthteller is on the road to the village and we ask the liar my original question I propose that we just stick to that scenario.

Here is my first attempt at a post that people can vote on.

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canadianslidewinder and Randall have been arguing about a proposed solution to this problem for nearly 50 posts since the end of November, and, we would like to ask other participants to resolve the disagreement by voting. I am just going to use our initials C and R in the following description of the point of disagreement. Also I am going to assume that all people present are men, it doesn't make any difference to the logic, but makes it easier to explain.

We believe that you only need to read this post come to a decision and vote, but feel free to torture yourself by reading more of the thread.

The difference of opinion applies to the much simpler problem where there are only two natives: one truthteller and one liar.

We were both unhappy with the self-reflective question posed by the logician in the published solution. So R proposed the following solution.

First the traveller asks one man to stand on one of the roads and the other on the other. He then asks the question "Does the man on the road to the village tell the truth?"

If the answer is "YES" the traveller takes the road that the man he asked is standing on; "NO" the other.

C believed that there was an error in this solution, and, after a lot of discussion the point of difference could be distilled to a single situation: when the truthteller is on the road to the village and the traveller asks the question to the liar.

C believes that the question includes the implicit question "Are you on the road to the village?", because the positions of the two men are interdependent. The liar will ask himself this question before attempting to answer the travellers question. And by lying about the answer to both the implicit question and the question will defeat the solution.

R believes that their interdependence is only something which the traveller or an onlooker would perceive. As far as the liar is concerned it does not matter how or why the two men ended up where they are, and, when he is asked the question about the man on the road to the village his own position is irrelevant. The simple fact is that the man on the road to the village does tell the truth and the liar must answer the question "NO".

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Please vote by responding either C if you agree with him or R if you agree with him.

Stop the press!

My conclusion (as of half an hour ago) is that your solution and my solution are both flawed for almost identical reasons.

My proposed question, "Is the liar on the road to the village?" contains an embedded truth, just as does your proposed question, "Does the man on the road to the village tell the truth?". So if the liar were to be asked either question, regardless of the road on which he was standing, he is thrown into a logical contradiction, a catch-22, that he can only avoid by remaining silent.

My proposed solution first: My expectation that the liar would answer 'No' to my question, "Is the liar on the road to the village?" (assuming the liar is on the village road), contains the assumption that the liar would only lie about which road he is on, but not about the fact that he is the liar. But if he were to be asked directly, "Are you the liar?", his vocalized answer would be 'No'. Therefore, consistent with this lie, and consistent with the positional interdependence of the liar and the truthteller, it is the truthteller who is then on the village road (within the lie), and consistent with that, the liar would answer 'Yes" to the main question. But this 'Yes' turns out to be a truism with respect to the liar's actual position on the village road. He is caught in a catch-22. Silence is his only option.

Your proposed solution: Your expectation that the liar would answer 'Yes' to your question, "Does the man on the road to the village tell the truth?" (assuming the liar is on the village road), contains the assumption that the liar would only lie about whether the man on the village road tells the truth, but not about the fact that he is "the man on the village road". But if he were asked directly, "Are you the man on the village road?", his vocalized answer would be 'No'. Therefore, consistent with this lie, and consistent with the positional interdependence of the liar and the truthteller, it is the truthteller who is then on the village road (within the lie), and consistent with that, the liar would answer 'No' to the main question. But this 'No' turns out to be a truism with respect to the liar's actual position on the village road. He is caught in a catch-22. Silence is his only option.

Neither solution works!

PS: Both your solution and mine are doubly defeated. Firstly, by the embedded truth in each. Secondly, by the positional interdependence of the natives.

The logical contradiction of the embedded truth in each, taken alone, is sufficient to defeat them both.

The positional interdependence of the natives results in a chain of logical consequences, leading to the final catch-22 contradiction, of which the exposed embedded truth is the root cause, but even if you don't accept the existence of positional interdependence (a non-acceptance which I am unable to comprehend), in my view, analysis need go no further than exposure of the embedded truth. Both solutions collapse right there.

"(a non-acceptance which I am unable to comprehend)"

As I am unable to understand your rejection of my counterargument viz.

In the mind of the liar there is no interdependence: he was not party to the decision process which led to their positioning.

Please try to respond specifically to that point. And if we cannot agree: then why not submit to a vote?

Re: Positional interdependence

To accept the necessity of interdependence, there first must be the acceptance that any question put to the liar that is dependent on implicit acknowledgement by the liar that the answer specifically and truthfully relates to the identity of either the truthteller or the liar (an acknowledgement the liar would never give if asked directly), contains an internal logical contradiction, and fails. You have never accepted this test, nor any implicit question used as part of the test. But let us assume for the moment that you do.

If the liar were asked to vocalize an answer to the implicit, "Are you the man on the road to the village?" (assuming he is on the village road), he would say, 'No'. So, that puts him off the village road (in the lie). That could be the end of it, but the liar still hasn't answered the main question, "Does the man on the road to the village tell the truth?". He is being asked to answer that question, but he doesn't want his answer to contain the implicit truth that he is the one standing on the village road. Well, if he is to answer as if he not the one standing on the village road, the question demands to be answered as if someone is standing on the village road, and the only other someone, is the truthteller. So he prepares to answer as if the truthteller is on the village road, then he stops, realizing that he is in a logical contradiction. I concede that it is not necessary, within the liar's lie, for the liar and the truthteller to completely trade positions, but it is necessary, within the lie, for the liar to move off the village road, and the truthteller to move onto the village road. Otherwise there is no one (in the lie) standing on the village road, and therefore no one to whom to relate the "does he tell the truth" part of the question. Note that how or why the liar and the truthteller came to be standing each on one of the roads is not relevant to the liar's reasoning. He knows how - the traveller put them there. He doesn't know why. But neither the how nor the why has any relevance to his reasoning, as set forth here.

Re: Submitting to a vote

I am not interested in any vote count by CR4 forum participants. If someone has a contributing argument to this tempest in a teapot, that's fine. But a vote? Before Copernicus and Galileo, the majority vote would have been 'Yes' to the question, "Does the sun circle the earth?". And how about that anonymous poster who has been continuously giving "Off Topic" votes to my posts, and "Good Answer" votes to your posts - both for no good reason that I can comprehend. His activities have the appearance of malice, not reason. I can confidently predict how he would vote in your proposed ballot.

I suggest we let the question rest. I have concluded, by my reasoning, that neither your nor my proposed solution works. You assert that, by your reasoning, your proposed solution does work, and I don't think that you would deny that also by your reasoning, my proposed solution works. So your vote is thumbs up on both our proposed solutions, and my vote is thumbs down on both our proposed solutions. I am satisfied with that poll.

I think you dealt with most of these problems in post #60

"If the Logician at the Crossroads problem isn't going to descend into a hopeless tangle of logical conflicts and paradoxes, we have to assume that in the liar's internal dialogue he would be truthful with himself"

I still hold to the point made in the italicized quote. But in #92 I am not referring to the liar's internal dialogue. That is why I wrote that his vocalized answer to the question, "Are you the liar?" would be 'No'.

But he has no reason to vocalise those answers, and, would not. So those questions and answers remain in his internal dialogue.

True, he does not need to vocalize those answers, but he needs to consider what would be his vocalized answers, were he to answer the implicit questions. This is maybe our main point of difference in our analysis of the validity of the proposed solutions (both yours and mine). I assert that the liar does not want to give an answer to any main question, if that answer contains the implicit truth about the identity and position of either the liar or the truthteller. Therefore, to avoid such, in his truthful internal dialogue he would ask, "What would be my vocalized answer, were I to be asked, to any question about the identity of who is on the road to the village, and how can I make my answer to the main question logically consistent with it?" If he determines that such logical consistency is not achievable, it forces him to silence, and the proposed solution question fails.

Sorry this was a reply to the wrong post.

I think you are taking, and have taken, this discussion far, far too seriously. The Logician at the Crossroads is a recreational logic problem. It is supposed to be fun and entertaining. I have attempted here and there to inject some humour - suggesting we refer your proposed solution to the Native Elder's Council for their deliberation and judgement; asserting that Descartes' axiomatic "I think, therefore I am." disallows the liar from denying his own existence; expressing excitement about the possible accolades we could receive from the International Logician at the Crossroads Society (awards ceremony at their headquarters in The Hague) - but all these efforts at levity resulted in no responsive witticism. It was like throwing a stone into a bottomless well. Your unvarying tone has been deadly, deadly serious.

And now you announce that you consider the matter of such importance that you want us to draft a document summarizing our arguments, then convene a virtual meeting of CR4 forum members, and put the document before them for study, sombre reflection, and vote casting. Pulleeeze! I suggest you open your mind to the real possibility that few here give a fig about our inability to agree on what constitutes a valid Crossroads solution, and fewer still would want to devote any time dissecting the nuances of the arguments. It is such an inconsequential tiff, and here you are expressing a wish to get out the megaphone and call Hear Ye! Hear Ye! All the king's horses and all the king's men. Come forth. Gather 'round my learned colleagues and reflect upon this weighty matter! I am utterly sick of the topic, and I want no part of your grand spectacle. Where is the door?

The door is right in front of you. Please go through it, and slam it behind yourself.

Thankyou Tornado. If I don't get any witticisms from Randall, at least I got one from you.