it might be — flannel jesus
It is.
it exists in a sea of equally valid and arbitrary premises. Suppose they replace 2 with committing to leave on X + 5 days. Or even X - 10 days. — flannel jesus
The premises aren't arbitrary.
is the number of people seen with
eyes. Adding or subtracting some arbitrary number to or from
would be arbitrary though, which is why perfect logicians wouldn't do it.
Next we consider Argument B:
B1. There are 99 people with blue eyes and 101 people with brown eyes
B2. Every person commits to the rule: "if the
people I see with
eyes don't leave on day
then I will leave on day
and declare that I have
eyes"
B4. Therefore, from (B1) and (B2), every person with blues eyes commits to the rule: "if the 98 people I see with blue eyes don't leave on day 98 then I will leave on day 99 and declare that I have blue eyes"
B5. Therefore, from (B1) and (B2), every person with blues eyes commits to the rule: "if the 101 people I see with brown eyes don't leave on day 101 then I will leave on day 102 and declare that I have blue eyes"
B6. Therefore, from (B1) and (B2), every person with brown eyes commits to the rule: "if the 99 people I see with blue eyes don't leave on day 99 then I will leave on day 100 and declare that I have blue eyes"
B7. Therefore, from (B1) and (B2), every person with brown eyes commits to the rule: "if the 100 people I see with brown eyes don't leave on day 100 then I will leave on day 101 and declare that I have brown eyes"
B8. Therefore, from (B4), every person with blues eyes leaves on day 99 and declares that they have brown eyes
B9. Therefore, from (B7), every person with brown eyes leaves on day 101 and declares that they have brown eyes
This argument is also valid.
Now we get to the more interesting part. If
we know that these arguments are valid then
so too do our islanders. They might not yet know if any of the premises are true, but they do know that the arguments are valid.
For the next step let's start by considering a simplified version of the argument in the OP. The islanders arrive on the island together and are told that everyone has either blue or brown eyes — which is
not the same as being told that there is at least one person with blue eyes and one person with brown eyes (it could be that everyone has blue eyes or everyone has brown eyes); it is only meant to dismiss the possibility that one's own eyes are green or red or pink or whatever.
I am an islander.
I know that Arguments A and B are valid.
I see 99 people with blue eyes and 100 people with brown eyes. Therefore I know that either A1 or B1 is true.
Therefore, I know that if every person commits to the rule: "if the
people I see with
eyes don't leave on day
then I will leave on day
and declare that I have
eyes" then everyone will leave the island having correctly declared their eye colour.