No. If there are 2 or more blue eyed people on the island, the oracle stating “There is at least one blue-eyed person on this island.” provides no further information. In that scenario, everyone knew that there was at least one blue eyed person on the island.
Perhaps, the word “statement” is ambiguous. When I say “the statement itself” or the “statement, in and of itself”, I mean the statement regarded as a proposition. When I say “announcement”, I mean a publication of the statement that tells each person that the proposition is true and that everyone else has also received notice. Perhaps, we should use the terms “proposition” and “publication” in this specialized sense to be more precise.
Proposition: There is at least one blue-eyed person.
Publication: The oracle announces publically (in front of everyone at the same time and place so that they each know that everyone else hears the announcement), “There is at least one blue-eyed person.”
If there are 2 or more blue-eyed people, the proposition conveys no further information. However, the publication of the proposition does; see the comments about chains by me, ZenBean, and Lance Turbo.
I still don’t get it.
Which includes an assumption that B cannot see A, which is absurd.
I see six blue eyed people every day of my life.
The Oracle says at least one person has blue eyes.
“Yeah, I mumble, at least one. I already knew that. There are six.”
Each of the six has that same information, except they see five or six depending on whether I have blue eyes. But none of us has any reason to change their behavior on the difference between six or five. Neither is proof of my own blue eyes, or brown eyes.
And what possible difference is there between day six, and day five, for the six or seven of us? How is it different for a brown eyed person in observable behavior changes among the blue eyed? The case where there are seven, or six is not different. I might not have blue eyes.
I don’t assume that any individual one blue eyed person has assumed any other one individual blue eyed person must see my eyes and know they are blue, because he obviously sees the other five that we both see, and five is more than one. What difference in behavior arises out of the cases of different amounts greater than two?
I see three people with blue eyes. Either there are three, or four. Either the other two see three, or two. No one sees one. Unless someone tells us that there are more than two, there is no chain of logic that necessarily proves that there are three.
Still confused.
Tris
That’s the essence of the argument for a flaw. The Oracle has given the tribe no new information. Therefore, nothing happens.
The new information the Oracle gives is the setting of the clock. It sets a t=0 that induction can proceed from. Otherwise, you couldn’t have the induction start, as the “solution” depends on all islanders making similar deductions about their own eyes based on the number of islanders and days that have passed. But if there was no arbitrary time point to start at, they could not make these deductions.
I knew there were blue eyed people yesterday, I know today.
Six blue eyed people are greater than one.
I don’t know that I have blue eyes, and neither does anyone else. The sequence assumes that I have no perception of someone today that I percieved easily yesterday. That is nonsensical.
Tris
Yes, but with the Oracle’s public announcement, now everyone knows that everyone knows it, and that’s what’s important. Previous to that, if I saw blue-eyed bitwise (sorry to keep picking on you, but it’s a neat name), then I might think that the reason he doesn’t kill himself is because, even though I know he has blue eyes, and everyone else on the island knows he has blue eyes, maybe he doesn’t know it himself. With the Oracle’s announcement, there is now no longer any doubt. So the next day, when bitwise shows up still alive, I realize because he hasn’t killed himself even though he knows there’s someone with blue eyes, he therefore doesn’t know it’s him. And that means he must be able to see someone else with blue eyes. And since I can’t see anyone else with blue eyes, that means it’s me. bitwise makes the same conclusion, and we both kill ourselves.
If people are bothered by timing and backstory, how about the same problem in different clothes?
(from Scientific American)
A monastery is populated by monks who are expert logicians. One night, someone goes through the sleeping quarters and paints a blue dot on the foreheads of a random number of sleeping monks. The next morning, the abbot (who has his own locked room and could not be painted) looks out over the crowd and sees blue. He announces: "At least one of you has a blue spot on your forehead. When I ring this bell, raise your hand if you know you have a blue spot. I will continue ringing it every minute.
The logic then follows in the same way. If N monks have blue spots, they will all raise their hands on the Nth ring.
Ah!
Iterations are introduced by the Oracle.
On the first day, no one would die unless there was only one.
So, on the second day, we all know there are at least two.
etc.
Subtle savages, aren’t we?
Tris
I think the key element here is that “All the tribespeople are highly logical, highly devout, and they all know that each other is also highly logical and highly devout.” Let’s assume here that devout means that if any blue eyed person has any means of discovering that they have blue eyes, then unless they kill themselves rather than spend eternity in heaven, they will be damned forever to hell. This would mean they’d be eager to kill themselves if they discovered they had blue eyes. Immediately after death they find themselves in eternal paradise. Before the traveller announced “how unusual it is to see another blue-eyed person like myself in this region of the world”, because it was taboo to mention eye color they could not discover their eye color. Once the traveller announces this breaking the local taboo: “After nobody commits suicide on the n-1^th day, each of the blue eyed people then realizes that they themselves must have blue eyes, and will then commit suicide on the n^th day.” The motivation to want to commit suicide if they have blue eyes makes a difference.
I am not assuming that B sees 4 people with blue eyes. I know that this isn’t true.
I am assuming that A assumes that B sees 4 people with new eyes. This is a logical assumption, because I am assuming that I have brown eyes and that a assumes that A has brown eyes. So form my perspective it is logical to assume that A assumes that B sees 4 people with blue eyes. Even though I know this assumption to be false.
The new information is that everybody knows that there is at least one person with blue eyes. Or, to put it in a much longer way. When the oracle speaks, I know that A knows that B knows that C knows that D knows that E knows that F knows that at least one person has blue eyes.
This assumption:
I assume that A assumes that B assumes that C assumes that D assumes that E assumes that F sees 0 people with blue eyes.
Is a safe assumption before the oracle speaks, but after the first night it is proven false, and so begins the chain of logic.
For n = 3 or more, the oracle’s statement makes absolutely no difference. If I see 2 other people with blue eyes and they’re still alive after 2 days, then I know I must kill myself. Everyone already knew that there was at least one person with blue eyes. That knowledge alone should set the whole cycle in motion, assuming each person myst obey the rules of logic. The daily passage of time and their suicide law is completely independent from the oracle.
I think the only problem with the argument is this:
- we must first explicitly state that all people can see!
- murder must never occur. Consider the case where I observe only man with blue eyes. If I don’t want to find out that I also have blue eyes, I should kill him immediately before he would have killed himself (if he was the only one), and thus before it becomes necessary to kill myself.
oops…that should say:
“the case where I observe only one man with blue eyes”
I notice that you didn’t say “For n = 2 or more…”, even though, for n = 2, it’s also true that everyone already knew the truth of the Oracle’s proposition. Consider the logic you used to go from n = 1 to n = 2; the same logic will also take you from n = 2 to n = 3, and from 3 to 4, and so forth indefinitely.
I’m not so sure about that. For n =2, everyone knows that there is at least one other person with blue eyes-- yes, that is true. However, the tribesman themselves DO NOT know for certain that all the others know this. If you observe one person with blue eyes, it is still a possibility that they are the ONLY one with blue eyes, and thus they wouldn’t know that “at least one person” had blue eyes. Thus, I stand by my argument that the oracle’s statement is important only for n<3, as this possibility I just described disappears when n = 3 or more.
Let me explain it, from the point of view of someone who just figured it out.
We had no information about how many blue eyed people there were, until today. So, I had no way of knowing if I have blue eyes. Today the Oracle told us that at least one of us has blue eyes. No one is going to kill themselves today, cause you are supposed to do it first thing in the morning.
The Oracle has begun a process of iterative logical problems, that start with one bit of knowledge, and by the consequences of our decisions, adds new information each day. Today, I know that there are six people I can see who are blue eyed. So, I know that the number of blue eyed people is either six, or seven. But I also know, as does every single one of us, that if there were only one, He would see none, and he would know that he has blue eyes, and kill himself first thing tomorrow.
Tomorrow, no one has died. So, a new datum has been added to our knowledge. There were at least two blue eyed people yesterday, because if there had been one, he would have known it. So, today I look around and see the same old six blue eyed devils. But this time I know that everyone knows that there are at least two.
Morning of day three, and still no one dead. That means that no one saw two blue eyed people on day two, because if they had, they would have suicided this morning. So we know now that there logically must be at least three blue eyed people.
Each day, the absence of any suicides creates a new datum. The Oracle has given us a starting point for an inevitable concatenated induction based on new information each day. If there was no suicide this morning, the number of blue eyed people required by our logical examination increases by one. On day seven, I see only six blue eyed people (as do the other six blue eyed people) and know that I too must have blue eyes. Since I know that seven is the largest number possible, and all six of my blue eyed compatriots know this as well, we each get up a bit early, and throw ourselves into the volcano. Later that morning, everyone else finally finds out the color of their own eyes.
The new information is created first by the event of the oracular violation of our taboo. After that, new data is created every day, by our own observations, and actions. Without an initiated first iteration, and a periodic reiteration, the process is not logically necessary.
Tris
“I have always thought the actions of men the best interpreters of their thoughts.” ~ John Locke ~
Sorry! Thanks, Chronos, I now see that for n=3, we are equivalent to the situation where n=2, because a person with blue eyes will only see 2 other people with blue eyes, so it is a possibility (from his perspective) that there are only 2 total blue-eyed people.
Likewise, for n=4, a person with blue eyes sees three other blue-eyed people. So, we must consider the possibility that n =3, and so on.
So far, I’ve seen people repeating induction argument and the explanation of what new information the oracle adds.
However, I haven’t really seen anyone address whether there is a flaw in the induction argument. I was really hoping someone could point out an improper step, like the division by zero in the “proof” that 1 = 2.
I don’t know about the fallacy that you are looking for but I for one have a problem with the idea that blue eyes, a recessive trait is gonna show up at all on an island of brown eyed people without some intervention from another race.
There are no blue eyed people on the island so they throw the oracle into the volcano.
Otherwise everyone is obligated to commit suicide during the night.
This is a thought experiment on a hypothetical, fantasy island. You are making this more difficult than it is.