How many "degrees" of knowledge is conceivable?

Question

Not sure if this is the right place for this, but there's something I've been pondering about and can't wrap my head around.

Say person A knows something, that person B does not. Let's call that the 0th degree of knowledge.

Now imagine that person B knows that person A knows something, but person A does not know that person B knows that. Let's call that the 1st degree of knowledge.

Now imagine that person A knows that person B knows that person A knows something, but person B does not know that person A knows that. Let's call that the 2nd degree of knowledge.

Now extrapolate that loop to the nth "degree of knowledge".

My head can only wrap itself around the 2nd degree. For example:

0th degree: Adam knows who the killer is, but Brian does not.

1st degree: Brian knows that Adam knows who the killer is, but Adam doesn't know that Brian knows that Adam knows.

2nd degree: Adam knows that Brain knows that Adam knows who the killer is, but Brian doesn't know that Adam knows that Brian knows that Adam knows.

In all three scenarios, I can imagine how a conversation may play out. But past this, it is really straining to even conceive of what a 3rd degree or higher of "knowledge" would mean. Does it even have any meaning? Does the possible degrees of knowledge end at 2? or 3? Or can it theoretically be limitless? How does each additional degree impact how Adam and Brian may interact with each other?

I'm not sure if this question is too abstract or not but I hope what I'm asking makes sense. And is there a name for this type of "knowledge"? I've been using the term "degree" only because I don't know what else to call it.

Answer 1

In epistemic logic, a one-agent version of this problem arises per the idea that kS yields kkS, etc., i.e., "If it is known that A, then it is known that it is known that A." A simple countermove would be to draw an analogy between the (alleged) trivialization of iterated possibility/necessity operators in modal logic, and the epistemic iteration; the indefinite k-string can be safely collapsed to the unit case, without substantial logical losses being incurred.

Or so the dialectic might go. Recall that saying a given word "too many times" in a suitably circumscribed timeframe often generates jamais vu, and that at any rate, tracking a larger structure's content can be thereby generically more difficult than tracking the content of a smaller structure. So troubles with an intuitive handle on epistemic (or other such) iterations are more than forgivable; but also, encouragingly, can they be "overcome." There are, for example, non-useless logics involving infinitely long conjunctions and/or disjunctions and intelligible infinitely tall proof-theoretic graphs/trees.

Answer 2

I mean the 3rd would obviously be that "Brian knows, that Adam knows, that Brian knows, that Adam knows, who did it". Though the question is does Brian got to know something he didn't already know. And well yes if you picture Brian as a detective interested in the murder (amateur or professional doesn't matter).

Then in the case of 0th degree knowledge he'd have no clue. In terms of 1st degree knowledge he'd have a clue but hopes on being able to sneak up on Adam to learn more and on 3rd order knowledge he'd have certainty that it is futile as Adam is not as hoped unsuspecting that. Whereas Adam on 0th degree might have the suspicion of being surveilled by someone who knows. Has certainty about that on 2nd degree but might have hopes of fooling a watchman by putting on a play, while on 4th degree having certainty that this is futile.

Beyond that this either is complete knowledge of the situation. Adam knows what who did it and what Brian knows and Brian knows what Adam knows.

Or it could turn into a game of rock-paper-scissors where the futility means that they don't try but not trying would make it no longer futile and vice versa. So essentially Adam picks rock and Brian doesn't know. Brian knows and switches to Paper, Adam doesn't know that. Adam knows and switches to Scissors and so on.