There is a logic puzzle aiming on freeing same-color-eyed people from an island. The thing is that they must be certain of their own eye color so that they can leave. For that reason an external party must provide a [single] statement containing no piece of new information to the crowd.
This puzzle is described in the following link:
The answer provided is, basically (disregarding the eye color, which could be green or blue) : "At least one of you has blue eyes", or "I can see that at least one of you has blue eyes". It could even be "At least n-1 of you has blue eyes".;
However, when I introduced this puzzle to a few friends of mine, one of them mentioned that the information : "Your neighbor has blue eyes"
He argues this is no new information since it will require inferences. I argue that it means the following:
(1) For each one of you, there is a neighbor having blue eyes;
Which could mean the same thing as:
(2) To all of you, there is a neighbor having blue eyes;
And allows the following inference:
(3) For each one of you, there is someone with blue eyes;
I am not a big Math expert, so maybe I wouldn't be able to precisely represent that in symbols, but the discussion revolves on the following: if (1) has the same value as (2) & (3) and if 3) contains new information, then (1) & (2) also contain new information.
Since the information cannot be conveyed to a single person on the island, I am uncertain if that would be a valid solution regarding the novelty of information and regarding the apparent individual-based message.
What do you think:
1- Are the statements (1),(2) & (3) equivalent?
2- This I should maybe post on the philosophy forum: Is the statement (1) valid, since it seems to be directed to every single individual instead to the collection of individuals at once?