Interesting question
This a toy question that one of my colleagues/friends told me of. You may have heard it, but I thought I'd share.
Suppose there are 100 perfectly logical people on an island, all of whom have either blue eyes or brown, and they all know each other. There are NO mirrors anywhere on this island, and none of the people EVER discusses the color of anyone's eyes. For some reason, if any person knows or can prove that he himself has blue eyes, he must kill himself before the end of the day. (Also, they are not trying to avoid killing themselves or get anyone else to kill himself erroneously.) They do not kill blue-eyed people who are not themselves, and they don't discuss eye color, so they never suggest to someone that he kill himself.
Now, suppose that every day, all 100 people gather in one place on the island to listen to an oracle which never lies. On Day 0, this oracle tells them, "at least one of you has blue eyes."
Here are the questions:
(1) If exactly one person has blue eyes, on what day does he kill himself?
(2) If exactly two people have blue eyes, when do they kill themselves?
(3) If exactly N people have blue eyes, when do they kill themselves?
Auxilliary question:
(A) Did the oracle introduce new information?
(B) Is question (A) mathematics?
I thought this cute.
1 Comments:
This is going to be too long. And there's always some blaringly obvious aspect to questions like these that I never fail to overlook. But what the hell, I'll give it a shot.
First off, let me assume the fact that the people know each other means that they know each other's eye color. That's not necessarily true (I don't think I could recall any of your eye colors, for instance), but it makes the question a lot less tedious. So, on we go.
(1) If exactly one person has blue eyes, he should kill himself before the end of Day 0. He knows that everyone else on the island has brown eyes; perforce, he is the one with blue eyes. So, off he goes to Davy Jones's Locker, or whatever. Not too hard. I think.
(2) This, unsurprisingly, is a bit trickier. My first temptation was to say that that neither of the blue-eyed folk - let's call them persons X and Y - would ever kill themselves. Reason being, the "at least" component of the oracle's statement is satisfied by only one blue-eyed person. For self-clarification here, all it takes is one blue-eyed dude, and the oracle's statement is logically sound. Since X can see Y, and vice versa, the oracle's statement thus holds true without either of them being able to logically infer that they themselves also have blue-eyes. So, no one dies. Except maybe of old age or if they get eaten by a shark or something. Seemed simple enough.
Then I realized that the above is a red herring though. And here's where the issue of intersubjectivity crops up. Assuming this isn't an island of annoying-as-shit radical Berkeleyans or anything, the individual perfectly logical islander must work under the notion that everyone else on the island is perfectly logical as well. Which means that question (2) actually should play out like this: X sees Y, and Y sees X. They both notice that the other is not killing himself immediately. In tandem, they should both think the following:
"Since we're all perfectly logical, if there were exactly one blue-eyed person, then [X or Y] should have inferred that he is that person and killed himself. But he hasn't yet, which means there is more than one blue-eyer here. And since I know everyone else's eye color, that other person must be...me!"
[sound of X and Y committing suicide simultaneously]
But of course this all hinges on when X and Y get around to figuring this stuff out. Throughout Day 0, both of them should be waiting for the other to kill himself; when Day 0 ends with no death, that should set the logical machinery in motion. This means that both X and Y should kill themselves no later than the end of Day 1.
Intersubjectivity, logic and mathematics. Lacan would be proud! I'm a dork.
(3) We've already covered situations where N = 1 or 2. Also, I can't do the "less than or equal to" sign, which pisses me off. Anyway, here's where I probably go wrong: I'm getting a similar answer as (2), but with an extra step. Let me try and clarify. X, Y, and Z are the blue-eyed people. Here's how the days run:
Day 0: Each runs through the scenario outlined in (2), so they all figure the other two will realize the situation tomorrow. No one dies.
Day 1: Each spends the day waiting for the other two to commit suicide. Since none of them yet suspect that there are more than two blue-eyers on the island, no one dies.
Day 2: Since no one has killed themselves yet, XYZ should all figure out that this means there are indeed more than two blue-eyers on the island. And since everyone knows everyone's eye color, XYZ should all infer (deduce? I don't care) that they are the three blue-eyed people on the island. All three commit suicide before the end of the day.
What I think this ultimately adds up to is that if exactly N people have blue eyes, then they kill themselves on Day (N - 1).
(A) That all depends. For N = 1, the oracle does introduce new information, but only for the blue-eyed guy: everyone else can see him of course, but I don't think he'd have any other way of knowing that blue-eyed people existed on the island. For all N greater than or equal to (grrrr) 2, the oracle obviously does not introduce any new empirical information - the islanders can already see everything with their own eyes. However, the oracle does introduce newlogical information, in the sense that it makes the islanders aware that everyone else must be going through the mental calculus outlined above. Let me try and clarify that. For all N = 2+ (you know what I mean), everyone already knew that "at least one of you has blue eyes." That's the empirical information. What they didn't know is that everyone else also knows this too. That's the logical information. And there's that intersubjectivity again.
So I guess, actually yes. The oracle does introduce new information, just of different types.
(B) Sure, why not? I used numbers and variables and stuff. Certainly the most mathy-type thinking I've done in awhile.
Anyway, I hope my needlessly verbose comment is reasonably close to the right answer. Whatever the case, good question Cameron! That was fun.
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