I clarified my discussion of "finer versus coarser" in the logic of partitions. It will e convenient to say that having a finer partition means you know _more_. So:
Imagine you're a detective trying to solve a case on a small island with 5 people on it. At first you don't know any of them are related, so they're all in separate families, as far as you know:
But then you start digging into their history. Each time you learn that two people are related, you change your partition by putting them into the same part:
You keep doing this as secret family relationships are revealed:
In this example, as you learn more you move to _coarser_ partitions, because your goal is to find relationships between people, and lump them in as big bunches as possible. But often people think about partition logic a bit differently, where as you learn more you move to _finer_ partitions.
Suppose you are an amateur wine taster learning to distinguish different kinds of wine by their taste. At first all wines taste alike, so they're all lumped together:
When you learn to distinguish red and white wines, you move to a finer partition:
This sort of example will fit our story a bit better. We'll generally say that you know more if your partition is _finer_.