Talk:Common knowledge (logic)

message: Beware of the use first order logic which has a precise meaning in logic. As second order logic and higher order logic, see section Comparison with other logics in article first order logic  for a classification. Pierre de Lyon (talk) 21:54, 19 February 2008 (UTC)
thread_title:  First order logic 

message: For k>1, how would the outsider's statement add anything not already known? 75.118.170.35 (talk) 16:17, 12 August 2009 (UTC)
thread_title:  New knowledge 

message: This puzzle and its proof are completely bogus.
As no new knowledge was introduced, and everybody could easily infer what was said, it cannot possibly have any effects. Everybody could logically tell that everybody knows that some people have blue eyes, and everybody could already logically tell than everybody knows that everybody knows, and so ad infinitum. This fact is already common knowledge. Taw (talk) 02:38, 22 September 2009 (UTC)
replies: text: Wrong. If k=2 then not everyone knows that everyone knows that there is a blue eyed.Scineram (talk) 19:49, 13 November 2009 (UTC)
replies: text: Why not? Can everyone see color? Does everyone look at each other eyes? If people do these things then indeed everyone knows there are blue eyed people and everyone knows they know and so on. If not, then the k=2 case is bogus. When no one leaves, a blue eyed person simply assumes that the other blue eyed person must be colorblind or unobservant and therefore under the impression that there are a bunch of blue eyed people those people have not yet figured out they must go. Thus no one leaves on day 2, or any day for that matter.  — Preceding unsigned comment added by 69.242.116.124 (talk) 14:54, 28 December 2011 (UTC)
replies: text:  For purposes of the problem, it is already common knowledge that all of the islanders have correct color vision, pay attention, always count the right numbers of other colors, and always make perfect deductions. If we grant all of these, then it must work for any number. For example, suppose you agree that it works with two, but not three. Then we see that:
* If a (male) islander sees exactly two pairs of blue eyes, then he will know that the total number of blues is either two or three (a simple induction)
* If the total number is exactly two, then the two blues he sees will leave on Day 2 (it doesn't matter what color "our" islander's eyes are, so long as they are not blue; it also makes no difference that we are "in" the problem, since the same event should be observed the same way from all perspectives)
* If no one leaves on Day 2, then the total number must not be exactly two
* Somehow, not one of the blue-eyed people will deduce that the total is three (and thus, that he himself has blue eyes)
The last premise is, of course, flawed. 71.162.46.164 (talk) 17:48, 11 February 2012 (UTC)

text: See my analysis further up the page. The "solution" is invalid, and this "puzzle" should be removed. -- Jane Q. Public (talk) 07:30, 25 April 2016 (UTC)
thread_title:  This puzzle is bogus 

message: I think there is an important mistake in the Steven Pinker reference from Stuff of Thought. It currently states "...the notion of common knowledge (dubbing it mutual knowledge, as it is often done in the linguistics literature)". In fact, Pinker and linguists specifically differentiate mutual knowledge - that which we all know - from common knowledge - that which we all know that we all know (and we all know that we all know that we all know, ad infinitum). Unless I'm misunderstanding it, I'd like to change it. Dashing Leech (talk) 19:39, 18 December 2010 (UTC)
replies: text: No. The problem specifically distinguishes between what everyone knows, and what an individual knows. If that were not so, it would not be a puzzle.  -- Jane Q. Public (talk) 06:24, 20 January 2018 (UTC)
thread_title:  Common versus Mutual Knowledge 

message: Am I missing something? First, you say that there are n modal operators Ki (with i = 1, ..., n). Then you define a conjunction of the Ki with i in G (a possibly infinite set) and complain that this conjunction is not finitary. But you just finished saying that there are only finitely many Ki. Which is it?
In my opinion, G should always be countable (although that is not crucial as far as I can see) but the conjunction should always be of only finitely many of the Ki. (I think G should be countable in any real world situation, but I see no harm in allowing uncountable G. However, for common knowledge to be meaningful, I can't see how an infinite conjunction can be allowed.)
I'll leave it to someone else to edit this, using the correct symbols.
Dagme (talk) 20:02, 21 May 2011 (UTC)
thread_title:  Modal logic (syntactic characterization) --- finite or infinte? 

message: Firstly, let us assert that people don't *want* to leave the island, so they will work to avoid knowing it if they possibly can, and that this is common knowledge too.
Secondly, let us assert the the number of blue eyed people on the island is 'large' - let us assume 10.
What prevents the reasoning:
"What if the blue-eyed people I see don't leave the island, despite the taboo? Then I would leave, in the mistaken belief that I had blue eyes! I should not leave the island because I won't know my own eye colour. Since there are a large number of blue-eyed people on the island, they can come to the same conclusion, so they may choose not leave the island, despite of the taboo - because the taboo is not broken. No-one must leave."  — Preceding unsigned comment added by 2.26.62.147 (talk) 10:42, 29 June 2011 (UTC)
replies: text: All of the above is indeed correct. However, for purposes of the problem, the islanders simply cannot help but count the number of blue-eyed people and work out all the deductions. And they all know this about each other, and they all know that they all know this about each other, etc. So no "deliberate ignorance" is possible. 71.162.46.164 (talk) 17:37, 11 February 2012 (UTC)
thread_title:  Can someone please explain why this logic doesn't work? 

message: I feel like I'm being trolled on a mass scale.
Everyone uses examples of 1 and 2 people, where it works, and then jumps to it working for k > 2, when it just doesn't.
If k > 1, *everyone* already sees blue eyes, and the announcement doesn't introduce any new knowledge in any way.
We are to assume the islanders can count the number of blue eyes people to induce the number of days to wait as k,
but they have some inexplicable internal block that says despite everyone seeing at least one blue eye person (in k>2) they don't know it until the oracle states it explicitly?
I've spent the last few days trying to "understand" the solution, wracking my brain, and now I'm calling bullshit. I really just think people are being dishonest here and introducing a secretive vague absurdly ridiculous rule (everyone can see blue eyes, yet doesn't *technically* know that everyone else knows there are blue eyes) to justify a CLEARLY flawed example.
I also appreciate the correct answer, which is that if(k > 1) everyone blue-eyes person leaves on the kth day /from when they arrived/. But it doesn't seem to be anything more than a deceptive puzzle, and has little to do with the actual topic of common knowledge, and any attempt to amend the puzzle to account for this vague extra rule would seem to make it lose its appeal and just be convoluted.
"Oh yeah, everyone sees blue eyes but no one's really sure that everyone else sees them, oh but, wait - no, they do know that later when they are counting days, because someone announced it... dur de dur"
The premise of not being able to assume that the other agents are paying attention to their surroundings, really breaks down when they all pay attention to their surroundings to induce the number of days they have to wait to leave.
 --173.228.31.160 (talk) 11:42, 14 July 2012 (UTC)
replies: text: While this comment is several years later, you are correct. The others are wrong. See my explanation above. -- Jane Q. Public (talk) 11:50, 22 January 2018 (UTC)
thread_title:  The blue eyes puzzle is wrong 

message: I see a lot of formal logic chains confusing and overwhelming people. Here's a simpler explanation. Consider that as an islander, you can consider two possibilities: 1. You are a blue-eyed islander. 2. You are an observer to what happens with the blue-eyeds since you are not one.
* With 1 blue-eyed islander: As blue-eyed yourself, you only learn that blue eyes exist after announcement, you see none so it must be you, and you leave. As observer, you watch the one person leave.
* With 2 blue-eyed islanders: As blue-eyed yourself, you see only one person with blue eyes, so you wonder if you are just an observer of a 1-blue-eyed situation. But since you don't see the one person leave, you must be a participant in the blue-eyed group, so you leave the second day.
* With 3 blue-eyed islanders: As blue-eyed yourself, you see only 2 blue-eyed islanders. So, you wonder if you are an observer of a 2-blue-eyed situation. There is no way for a 2 blue-eyed situation not to work out as above, so you wait to just watch it play out. When it doesn't, you realize you have blue eyes too, and you leave.
* With 4 blue-eyed islanders: As blue-eyed yourself, you see only 3 blue-eyed islanders. So, you wonder if you are an observer of a 3-blue-eyed situation. There is no way for a 3 blue-eyed situation not to work out as above (because if it's really just 3, each of the participants will wonder if it is just 2 etc), so you wait to just watch it play out. When it doesn't, you realize you have blue eyes too, and you leave.
Going into layers upon layers of knowledge of what others know just makes this hard to follow. And treating it like a chain that actually plays out as an order of operations is confusing too. The approach above simply shows that if you are blue-eyed, you can discover this when you fail to see the logic play out without you.
Backfromquadrangle (talk) 19:43, 23 June 2016 (UTC)
replies: text:  Being logical they all know how this puzzle works and that it requires a starting date from which to start counting the days. Not being able to discuss blue eyes they can't agree on a starting date. The extra common knowledge brought by the visitor is the start date. 194.207.86.26 (talk) 22:39, 25 June 2018 (UTC)
thread_title:  Better puzzle explanation as observer vs partcipant 

message: I'm tempted to put this in External Links but will leave it for others to consider. 2602:24A:DE47:BB20:50DE:F402:42A6:A17D (talk) 00:09, 21 October 2020 (UTC)
thread_title:  Islander fiction