[[John Baez]] wrote:
> I couldn't make up my mind whether to think of "learning more" as moving from a coarser partition to a finer one - that is, learning more distinctions between things - or as moving from a finer partition to a coarser one - that is, learning more connections between things. I picked the latter, but then I realized this doesn't match a story I'm trying to tell later.

I believe "learning more" amounts to moving to a finer partition.

I base my opinion off of the model of knowledge proposed by the Nobel laureate Robert Aumann. Aumann proposed to model knowledge in game theory as partitions of information states. This is presented in his [Agreeing to Disagree (1976)](https://projecteuclid.org/euclid.aos/1176343654). Leonard Savage also proposed using partitions to model decisions under uncertainty in his book [The Foundations Of Statistics (1972)](https://books.google.com/books/about/The_Foundations_of_Statistics.html?id=zSv6dBWneMEC). The philosopher Jaakko Hintikka independently suggested using [*S5* Modalities](https://en.wikipedia.org/wiki/S5_(modal_logic)), which reflect partitions on information space. in his text [*Knowledge and Belief* (1962)](https://philpapers.org/rec/HINKAB). Hintikka's *epistemic logic* has been embraced by various philosophers since his initial proposal. For instance, the logicians van Ditmarsch, van der Hoek and Kooi have a logic with a relation *knows more than*. This relation expresses that one agent's partition partition on information states is *finer* than another. This research is presented in their paper [*Knowing More* (2009)](http://www.ijcai.org/Proceedings/09/Papers/162.pdf).

I will take a little artistic license with Aumann's presentation. If your partition on information states in a game is \$$JB\$$ and mine is \$$MD\$$, then \$$JB \wedge MD\$$ would be the information state if we "put our heads together" and colluded. So *learning* can be modeled in this logic as moving to a finer partition.

On the other hand, \$$JB \vee MD\$$ is our *common knowledge*. This reflects our *consensus* on groupings of possible information states. And, at the risk of being modest, I think it's safe to say that \$$JB \leq MD\$$. So the things we both know are rather restricted by the contents of my near-empty head :(

I can attempt to make a concrete example if you like. Jan van Ditmarsch has some papers where he adapts some John Conway puzzles to reasoning over partitions of information states.