- Jan 2019
there is no guarantee that the generality signified by a word will convey the same idea to all users of the language
This goes back to my point that there really is no such thing as common knowledge. It's impossible to expect everyone to recall the same things, just like you can't expect words to mean the same thing to everyone.
The term common knowledge has become a slippery slope, especially in situations where socioeconomic backgrounds and country of origin come in to play.
- Sep 2016
it’s easy to argue by induction
It may be easy to argue by induction, but what is actually going on?
Let's take a group of three children for a larger example: Abe, Ben and Cindy. Let's denote the situation "Abe has a clear forehead and Ben and Cindy have muddy foreheads" as [OXX], and so on.
Before the teacher tells them anything, Abe knows that either [XXX] or [OXX] (because he sees the mud on Ben's and Cindy's foreheads). If [OXX] were the case, Ben would know that either [OXX] or [OOX]. If [OOX] were the case, Cindy would know that either [OOX] or [OOO].
This picture demonstrates the situation.
The teacher's announcement makes it common knowledge that [OOO] is not possible. Then if Abe supposed [OXX] and provided that Ben supposed [OOX], Cindy would be sure that [OOX] (meaning she has mud on her forehead). After the first call, she doesn't stand up so we know that if Abe supposed [OXX], in that idea Ben could no longer suppose [OOX] and would be instead sure that [OXX] (meaning he has mud on his forehead).
This example suggests that the children don't need the general notion of common knowledge to efficiently reason whether they have muddy foreheads. It suffices to use any chain "Abe considers possible that Ben considers possible that Cindy considers possible etc.". that visits every child exactly once in an arbitrary order.
“would you like to come up for some intercourse?”
Incidentally, this straightforward approach is taken by Phoebe towards Chandler in the episode 5x14 titled "The One Where Everyone Finds Out" of the TV show Friends:
I'm really looking forward to you and me having sexual intercourse.
As pointed out by James Miller in his video introducing the concept common knowledge, this episode contains a nice informal demonstration of the concept.