Mutual appreciation
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Today's Zits:
Mutual knowledge — at least the kind of mutual knowledge involved in theories of convention, theories of reference, and so on — is generally taken to require the logical equivalent of an infinite conceptual recursion.
Thus Herbert Clark and Catherine Marshall ("Definite Reference and Mutual Knowledge") spend several pages exploring a hypothetical situation where Ann uses the phrase "the movie showing at the Roxy tonight", in conversation with Bob, to refer to Monkey Business.
For this usage to be felicitous, it's obvious that Ann must know that "the movie showing at the Roxy tonight" is Monkey Business — which C&M abbreviate as "Ann knows that t is R". But C&M then create a series of scenarios demonstrating that it must also be true that "Ann knows that Bob knows that t is R", and also that "Ann knows that Bob knows that Ann knows that t is R", and furthermore that "Ann knows that Bob knows that Ann knows that Bob knows that t is R", and moreover that "Ann knows that Bob knows that Ann knows that Bob knows that Ann knows that t is R", and so on ad infinitum.
But does the relationship between felicitous reference and mutual knowledge have an analogy in the relationship between felicitous favor-solicitation and mutual appreciation? I don't think so, though maybe there are cases where mutual knowledge of one- or two-sided feelings is critical to some sort of social relationship or event.
tk said,
August 23, 2014 @ 6:04 pm
I knew that.
Ken said,
August 23, 2014 @ 8:16 pm
@tk: I knew that you knew that.
(Just because you were obviously fishing. But I stop here.)
rosie said,
August 24, 2014 @ 2:14 am
Three people A, B, C are seated so that each can see the others. An umpire tells them all that five hats are available — three red and two black — then tells them to shut their eyes while the umpire puts a hat on each. We know, but they don't, that the umpire puts a red hat on each. The umpire tells them to open their eyes, then asks A, B and C in turn:
"Do you know the colour of your hat?"
A says no. B says no. C says yes.
Clearly, if the umpire had started by asking B, then B would say no, and C (being in the position B was in, in the first scenario) would say no. This means that C knows his hat is red only if he had previously heard A answer no.
Now the only way A could say yes is if he saw two black hats. But everyone sees two red hats. So everyone knows that everyone else sees at least one red hat, and therefore knew — even before A spoke — that A would say no. So how come A's answering no tells C anything?
Greg said,
August 24, 2014 @ 3:44 am
@rosie, A's answering no doesn't tell C anything directly, but it provides extra information that, combined with B's answer, tells C enough. Suppose that C were wearing a black hat. Then A would still answer no, but B could now see one black hat and would know that A couldn't see two, so he'd know that his hat was black. He would then say yes and C would use that fact, together with A's no, to infer that he must be wearing a red hat and so would say yes as well.
Incidentally, if A were taken away when the hats were placed on heads, so that when B and C open their eyes they find that there's only two of them now, being told "You are not both wearing black hats" (A's contribution, essentially) is sufficient for C to be able to answer "yes" again.
I'm not sure that I'm convinced these infinite recursions actually happen in practice, as my feeling is that one I've understood that "Anne knows that Bob knows that Anne knows that t is R" we then assume that this is now common knowledge (at least in the context of the participants).
MattF said,
August 24, 2014 @ 9:38 am
How about 'a state of mutual knowledge that is representable as a limit of sequences of states of mutual knowledge'? That's a meaningful way to define of real numbers as a superset of 'constructible' numbers– perhaps something like that would work in this context. You'd like to show that the limit is unique…
Jules said,
August 24, 2014 @ 1:20 pm
That reminds me of this sketch: http://youtu.be/viLdm60aJtw: "You know I feel you feel I feel what you feel."
Ø said,
August 24, 2014 @ 9:12 pm
Rosie, although the fact that A says no tells C nothing, the fact that the fact that A says no tells B nothing tells C something.
Adam Roberts said,
August 25, 2014 @ 2:38 am
I think the key is the criterion 'knowledge'. 'I know that you know' adds information to the scenario; 'I know that you know that I know' perhaps adds a little more; but by this point the general case of mutual knowledge has been established, and further recursion adds no more information.
I'm reminded of an old Martin Gardner piece from Mathematical Games, which I can't lay my hand on at the moment and so will imperfectly recollect now. It concerned a scientific paper by Professor White, translated into Greek by a colleague, Dr Rastopopolos. White adds a footnote to the paper's title for the Greek edition, saying 'I am grateful to Dr Rastopopolos for translating this paper.' He then adds a second footnote to the first footnote, saying 'I am grateful to Dr Dr Rastopopolos for translating the preceding footnote.' He adds a third footnote to the second footnote saying the same thing. But then he stops. Asked how he avoided the need for an infinite recursion of footnotes, he replied: 'I may speak no Greek, but that doesn't prevent me from copying the second footnote without need for help from Dr R,'
carla said,
August 25, 2014 @ 10:08 am
"I know. You know I know. I know you know I know, and Henry knows we know it. We're a knowledgeable family." — A Lion in Winter
Lars said,
August 25, 2014 @ 10:17 am
In Rosie's problem as stated, telling us that B and C both know that A will say no is an interesting type of misdirection. The problem's apparent paradox is resolved (as explained in other words by Greg and Ø) when you realize that, unlike us, C didn't know that B knew that A would say no — but making us aware, as the people thinking about the problem, that B and C's knowledge is identical: that seems to force us to use an extra level of abstraction when thinking about what C actually knew.
Or shorter: Thinking about identical but non-mutual knowledge is hard.
KeithB said,
August 25, 2014 @ 1:24 pm
This reminds me of Knots by RD Laing.
DougW said,
August 25, 2014 @ 7:45 pm
I'm reminded of this Spy vs. Spy scenario described by Malcolm Gladwell in "Pandora's Briefcase":
Viseguy said,
August 25, 2014 @ 8:01 pm
Or something out of Yes, Minister.
pjharvey said,
August 26, 2014 @ 2:18 am
It's less that 'Ann knows that Bob knows that t is R', and more that 'Ann assumes that Bob knows that t is R', at which point the recursion can be halted. Further contextual clues within continued dialogue will show whether the assumption holds or not.
If the assumption holds, the dialogue will flow naturally, which will make the initial assumption look like shared knowledge that requires the infinite recursion to explain. If it does not hold, there will be moments of confusion, as snippets of information don't match up, until a point is reached where a direct question about the assumption will be asked.
The felicitous cases where the assumption holds look like magical powers of shared knowledge, but the occasional cases where people aren't on the same page show that it really is just good guessing based on context.
Alex Leibowitz said,
August 26, 2014 @ 4:26 am
I have a hard enough time following these examples without the letters. Once the letters are added in, I'm completely lost.
KeithB said,
August 26, 2014 @ 2:33 pm
And of course, the exchange that ends with this:
"You only think I guessed wrong! That's what's so funny! I switched glasses when your back was turned! Ha ha! You fool! You fell victim to one of the classic blunders – The most famous of which is "never get involved in a land war in Asia" – but only slightly less well-known is this: "Never go in against a Sicilian when death is on the line"! Ha ha ha ha ha ha ha! Ha ha ha ha ha ha ha! Ha ha ha… "