Consider 2+2=4.
As long as everyone accepts the logical definitions of 2, +, = and 4, they are necessarily going to agree with this equation.
If the scenario is that there are “2 here” and “2 there” and nothing besides, and if the sum of those things equals 4, then everyone on an island leaves the island on a ferry at midnight (otherwise nothing changes) then the only things that have any consequence are 2, +, = and 4, and the logic that 2+2=4.
Given that everyone involved is a perfect logician and thus accepts logical definitions, and everyone is aware of the limited number of things that have any consequence, you can know that everyone is ONLY going to be considering these things within these limits. In the above scenario, 2+2=4 is non-negotiable, and the ONLY thing of consequence, so you can KNOW everyone is going to be deducing this in this way, and nothing else. There is no worry of anyone thinking differently.
Now, in the actual puzzle, blue-eyed islanders are ONLY going to be considering the consequences of whether there are 100 blue-eyed islanders or 99. They see 99 and do not know whether they themselves have blue eyes - if they did there would be 100, if not there would be 99. So there are ONLY 2 possibilities to explore:
- The consequence of there being 100 is that each blue-eyed islander knows they have blue eyes, and leave on the ferry in the knowledge that they have blue eyes.
- The consequence of there being 99 is that 99 blue-eyed islanders would be ONLY considering the consequences of whether there are 99 blue-eyed islanders or 98. They would see 98 and not know whether they themselves have blue eyes - if they did there would be 99, if not there would be 98. So there would be ONLY 2 possibilities to explore:
…1) The consequence of there being 99 is that each blue-eyed islander knows they have blue eyes, and leave on the ferry in the knowledge that they
…have blue eyes.
…2) The consequence of there being 98 is that 98 blue-eyed islanders would be ONLY considering the consequences of whether there are 98 blue-eyed
…islanders or 97. They would see 97 and not know whether they themselves have blue eyes - if they did there would be 98, if not there would be 97. So
…there are ONLY 2 possibilities to explore:
…1) The consequences of there being 98…, and
…2) The consequences of there being 97… this is recursive…
Until: - The consequences of there being 1: that they would leave on the ferry in the knowledge that they have blue eyes, and
- The consequences of there being 0… BUT THERE ARE NOT 0! - the Guru revealed otherwise.
So it’s finally certain that if there were 1, they would leave at the next opportunity. They don’t so there aren’t one. We move back up to:
- The consequences of there being 2: that they would leave on the ferry at the NEXT opportunity in the knowledge that they have blue eyes, and
- The consequences of there being 1… BUT THERE ARE NOT 1! - nobody left at the FIRST opportunity.
This is also recursive. It leads back to there definitely being 100 on day 100, and thus all the blues finally know they have blue eyes… AND ALL FROM EXAMINING THE ONLY POSSIBILITIES. There were no other possibilities that were missed out here. And all possibilities were examined.
THIS is why there are no alternatives to the correct solution.
This is the best way in which I’ve explained it yet.