You agreed that if there were only 99, they wouldn’t leave after 3 days.
If there were 100, they also wouldn’t leave after 3 days.
It doesn’t take much brain power, then, to realize, “Hey, them not leaving after 3 days is completely unrelated to the quantity of blue-eyeds there are, and therefore, I can’t use that information to deduce how many there are.”
You can’t use a fact that would be true regardless of the number of blue eyes, to determine the number of blue eyes.
You can’t use a fact that would be true regardless of your eye color to determine your own eye color.
Only if “starting a count from X” means “know it to be common knowledge that there are X blues”. So, for 100, if the blues know there are 100 blues from the get-go (e.g. the guru tells them “there are 100 blues”), they will all leave. But for 99 blues, it’s not enough that they know that there are 99 blues, they need to know that (the 99 blues know that there are 99 blues). And they don’t know that: they can see 99, they don’t know their eye color, so they don’t know if the 99 see 98 or 99. They can only leave on the 2nd day if they “start counting from 99”, i.e. “know it to be common knowledge that there are 99 blues.”
You are using a vague phrase, “start counting from”, which you’ve refused to define precisely. But the way that you’re reasoning from it is to treat it as them having common knowledge, i.e. for any hypothetical set of islanders, they know that there are X blue eyes. If they get the knowledge by looking at the other islanders, it doesn’t translate the same way: even if we see 99 blues, it’s clear that we can say that if there were only 3 blues, they couldn’t know that there are at least 99 blues.
I am the only person on this island. I don’t see anyone.
If there were 1 person on the island with blue eyes, it would be me.
1
Therefore, I have blue eyes.
In the canonized solution, it is proposed that a blue (you) is thinking that another blue is thinking that he might be brown, “if I am brown, then…”. Why would a blue be thinking that he is brown? He has no way to know that he is brown, especially since he really isn’t.
First, we need to be precise in our language, since this has gotten us into trouble before. It’s not what he “is thinking”, but what he logically could think.
Second, while he could assume he is brown, the relevant assumption to complete the modus tollens is that he is ‘not blue’. That is different from being brown. Being brown is one way to be ‘not blue’, but it is reasoning on the possibility of being ‘not blue’ that ultimately allows him to conclude that he’s blue.
Finally, he’s able to contemplate and reason on these assumptions because they are logically permitted. There are a million logical deductions he could make, most of which aren’t relevant to the solution of the problem, but which the givens state that every person on the island knows immediately; everything that is logically deducible. He doesn’t know his eye color, so he knows that he could be blue, not-blue, brown, not-brown, red, etc. Making a deduction that includes his being any of these does not contradict anything he knows. And he makes them all. One of those deductions is "if I am not blue, then those blues see 98 blues, and they might think, "If I am not blue, then those blues see 97 blues and they might think, "If I am not blue then those blues see 96 blues and they might think, “…” " " "
Anybody ready to try a different one?
Don’t worry, it’s far less ambiguous (I think so anyway…).
Again, no tricks/loopholes:
You’re talking to a woman, she says she has 2 children, you ask her, “Is at least one of your children a boy?” She says yes. What is the probability that the other child is a boy?
Just to be perfectly clear about no tricks/loopholes:
Assume she’s telling the truth.
Assume there’s no ambiguity about sex, everyone is either a boy or a girl in this universe.
Assume that each time a woman gives birth, it’s 50/50 boy/girl.
Assume that each child’s sex is independent of the other (no statistical anomalies caused by twins).
If you must cheat and look up the answer, at least put it in a [ tab ] thingy.
[tab].25 meters?
My reasoning is this: It will always be between .5 and 0 meters long, and the average of all the lengths between .5 and 0 should be .25[/tab]
You decide to play a game with your friend where your friend places a coin under one of three cups. Your friend would then switch the positions of two of the cups several times so that the coin under one of the cups moves with the cup it is under. You would then select the cup that you think the coin is under. If you won, you would receive the coin, but if you lost, you would have to pay.
As the game starts, you realize that you are really tired, and you don’t focus very well on the moving of the cups. When your friend stops moving the cups and asks you where the coin is, you only remember a few things:
He put the coin in the rightmost cup at the start.
He switched two of the cups 3 times, as follows:
The first time he switched two of the cups, the rightmost one was switched with another.
The second time he switched two of the cups, the rightmost one was not touched.
The third time he switched two of the cups, the rightmost one was switched with another. This was the last switch
Yes. It is what he COULD think.
And he COULD think, “if everyone were to simply start with the same number then…”
And just as he wasn’t actually brown when he thought of what COULD be the case, to see where it would lead, he wasn’t thinking of an actual number when he thought, “what if everyone simply started with the same number…” so as to see where that would lead. Just as he COULD possibly be brown, he COULD possibly think of a common number.
And very importantly, he MUST eliminate such a possibility before he can ASSUME that everyone isn’t thinking such a thing rather than thinking of whether they are brown (or “not blue”).
Perfect logicians don’t ASSUME, they ELIMINATE alternatives by EXAMINING them (not simply jumping to the first possible assumption of what 200 others might be thinking).