“everyone knows that everyone is a perfect logician”
“everyone knows that everyone can see everyone”
“everyone knows that everyone is thinking the same scheme”
proving that there is no alternative to them. A perfect logician cannot deduce anything until he can prove to himself that there is no alternative, “nothing is possible until something is impossible”.
The first three we have been taking as a given and I suspect that if we don’t, the puzzle isn’t worth addressing. But the fourth is the one I still see as an issue. There is nothing in the canonized version that says that there is no possibility of any other person on the island thinking anything else, such as merely a higher number to begin the count down.
The incentive issue really isn’t an issue because perfect logicians with instant thought, would instantly know what number every perfect logician would be using and thus instantly be using it himself trapping himself into destiny and his fate.
The rules exist after the fact by logical extension, some people find it hard to work them out, some people think there are none, some more ignorant people thing there are not just no rules but they have no need of them, but there are, and that is the point of the problem, it’s why I liked it, and hated it at the same tiem.
“Standing before the islanders, she says the following: “I can see someone who has blue eyes.”” - though nothing is said about their attention/auditory/comprehension capacity. An assumption to which it would be valid to object, given the information explicitly given.
“They are all perfect logicians… Everyone on the island knows all the rules in this paragraph.” - known.
“Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves).” - known.
“if a conclusion can be logically deduced, they will do it instantly.” - this means that everything that can be deduced certainly from certainty will be known instantly. I have explained what these things are - no assumptions, just fact and deduction from fact. Apart from the unspecified hearing and understanding of the Guru’s words, this is no issue, and it gives rise to the one and only solution, which I have explained and that you have ignored.
As just shown, there is no alternative to any of the above assumptions, other than the first one, where they may merely not be paying attention, perhaps distracted by the bewitching green eyes of the only female on the island (known by paying attention to the pronouns). They may be deaf or they may not be - this is not specified. It appears, though, that they are within hearing range, assuming they are both able and willing to listen. There is also nothing to say they did not hear and understand the Guru’s words. I would prefer it you came up with one of your theories based on this ambiguity, rather than from missing certainty and miscontruing from there, as you have done so far.
Everyone has only shown that there is at least one conclusion series that can be known (and everyone here has known that for a week). “At least 1” =/= “no possible alternatives”.
And until there are no possible alternatives, the perfect logicians cannot use that one.
…and until everyone here knows that everyone knows that everyone here knows that there is no alternative, the arguing continues and no one “leaves this island” (assuming that any two care).
This is another example of your mistaking what we from the outside know with what they from the inside know.
The correct statement would be, “And until there are no possible alternatives, we don’t know that the perfect logicians will use that one.”
What we do and don’t know doesn’t affect what they can and can’t use.
I wasn’t talking about what WE know, but what THEY know.
THEY have to know that there is no alternative concerning any method they are using to predict what others on the island are doing, else they cannot depend on that scheme.
Imagine that you find 200 logicians who have never failed a logic test of any kind. You stick them on an island with those rules. There happens to be 100 blue-eyed. But on the 4th day, one of them leaves. The others are thinking, “what the hell?? How did he do that? Now what? Is someone else going to leave tomorrow?”
James, isn’t it clear that given a syllogism,
A - > B, ~B |- ~A
we don’t have to separately prove that A isn’t also an alternative? The syllogism itself proves that A is not an alternative
So, if the canonical solution relies only on knowns and syllogisms built from them, the proof proves that there are no other alternatives. The logic makes the conclusion necessary; any other alternative would produce a contradiction, which must be inherent in the problem statement (thus my previous question; I didn’t understand your answer).
Carleas, anyone on the island or off can easily see that they need only start counting days from a common number. There are two easy to see common numbers; 1 and 200. If by 1, it takes 100 days to know. If by 200, it takes 51 days to know. But using either method, it must be known as to which method everyone else is using. Both work if chosen by everyone. Neither work if not. Why would they necessarily all choose to use 1 rather than the 200?
This is a problem only for your solutions that rely on everyone somehow choosing the same assumptions without explicit communication.
This really isn’t a problem for the correct solution. As I have pointed out, all they are doing is deductively expanding on what they definitely know for sure, and this just so happens to solve the puzzle. Nothing more is needed for it, it just works from logic and certainty. Nothing is left out or gone to waste, nothing is missed, everything that comes from logic and certainty goes towards the one solution and no other solutions - as shown.
Sure, maybe a breach of the conditions of the puzzle might allow for the otherwise illegal assumptions necessary for your solutions, and at least the pier one would work if the problem was altered in order to allow it.
I think I’ve even said already: if you would only realise/admit that your solutions were only appropriate for an altered version of the puzzle, then that would be fine. It’s the fact that you seem to insist that they’re appropriate for the puzzle as it is that’s incurring the criticism of myself and others. You’ve only been exploring alternative solutions to the puzzle if it were altered - and if you do alter it, there ARE alternative solutions.
We’re waiting for a proof of that. If you feel you’ve provided one, link to it. Or, correct my version of it (which I truly meant to be a good faith restatement of my understanding of your proposed solution).
Carleas, before I go make some more formal proof for a hypothetical, I need to ensure that you actually understand the hypothetical. If you understand it, it seems blatantly obvious that it would work, so either you do not understand it (which I only give any credit to at all because FJ couldn’t grasp it), or you are doing your political rhetoric thing again where you divert the conversion upon seeing that you might have a flaw in your reasoning. If it is the former, the proof isn’t going to help because you wouldn’t understand what is being proven. If it is the later, it wouldn’t do any good to present a proof anyway because you would just divert from it again.
So before I go put together some kind of unnecessary proof, how about you show me that you actually understand what it is that I am saying with “IF THEY ACTUALLY START WITH THE SAME NUMBER (between 0-99), IT WILL ALWAYS WORK.” What do you think that statement means?
This whole thing is about clarity and verification.
I agree, and I must ask that you be much more clear, as I’ve asked before: when you say they “start with” or “pick” the same number, what do you mean. Just that they have number in their head? We’ve agreed the number must be meaningful, what meaning are you ascribing to it? What meaning are the people who “start with” or “pick” that number ascribing to it?
It may seem obvious to you, but you know what you’re talking about before you even start talking. We can only discern what you’re talking about to extent you tell us. It is frankly poor form to allege that it is the fault of those you’re trying to convince that you have failed to convince them. As I said before, communication is a two way street. I don’t understand, neither it seems does FJ, but you can’t conclude from that that we can’t understand. It’s at least as true that you can’t explain yourself. To be clear, I’m not saying that you can’t, but that in the same way that you can but haven’t explained yourself adequately, we can but haven’t understood what you’re saying.
So, as you said, let’s focus on clarity so that we can verify the canonical solution (or verify your challenge to it). The problem for me is that your words are capable of many meanings. I’ve offered my interpretation of them, which you said was wrong but about which you said nothing more. Say more.
Granted. So what? What does that tell us about what “start with” or “pick” means? What does that tell me about how my restatement of your case was wrong? Clarity, James.
Ok, James, now we have “start with”, “pick”, and “use”, all of which you refer to how their holding some number in their heads leads to their conclusion that they have a certain eye color. You haven’t provided any additional insight into what “start with”, “pick”, or “use” mean in this context, other than to suggest that it doesn’t mean “X islanders with blue eyes know that there are X islanders with blue eyes,” unless that’s not why my earlier restatement of your solution is incorrect (which you’ve still yet to clarify).
I asked for clarification:
Perhaps you would be able to help me get it by actually explaining what you and James are talking about? Make a syllogism, or tell me specifically why mine is wrong.
EDIT: And by “mine”, I mean my restatement of what I think your intended syllogism to be.
