What does it mean for two distinct people to know a fact?

Question

Let there be two people A and B. They both encounter a strange object O. What do they know about each others knowledge about O?

More strictly, we are speaking of a situation where, say, you and me encounter object O. I know that you know about existence of O, and you know that I know about existence of O; I know that you know that I know about existence of O, and you know that I know that you know about existence of O; and so on...

The issue here is that, traditionally, when we both see O, we simultaneously acknowledge in such a way that if we were to ask either of us any of the question (from above) of the form, "do you know that I know that you know that I know that you know .... about existence of O", response will be, correctly, "Yes". Let's call this a stable condition.

Now consider an alternative situation. Situation: A encounters O. So A knows O. At:

• t=0: A knows 0.
• t=1: B is privately told, "A knows O".
• t=2: A is privately told, "B knows that A knows O"
• t=3: B is privately told, "A knows that B knows that A knows 0" and so on...

Do A & B, at some time t, reach stable condition? Of course, we are setting these values to be true at time t, t+1, t+2, ... but I am unable to figure out: after many alternating knowledge of each others knowledge bases, why is either of them unable to confidently assert that both know everything (or reach a stable condition) -it makes sense for lower levels, but shouldn't higher level degrees simply "collapse" -all I want is each to know that both know that both know O, so is there some redundancy after a level? Or why is this situation logically different from first situation?

Answer 1

With an actual example, the difference is easier to grasp.

Suppose Bob and Alice are spies, and stole some information (object O) from the Pentagon. They hide it somewhere, but leave a trace that allows special agent Oscar to find and capture Bob.

Alice knows that Bob knows the location, and Bob knows that she knows that he knows, and she knows he knows and so on, like in your first case.

Oscar knows some information was leaked, but not what it is nor where Alice and Bob hid it. He interrogated Bob. He knows that Bob knows the location, and Bob knows Oscar knows he knows (otherwise, he wouldn't interrogate him). Oscar knows Bob knows he knows... And so on, yet, Oscar does not know what Bob knows, just that he knows it, which corresponds to the 2nd situation.

You can see that, assuming Bob never talks, the Oscar and Bob pair and Alice and Bob pair have the same structure of knowledge that the other has knowledge, yet Oscar will never know what Bob knows.