> I believe "learning more" amounts to moving to a finer partition.
You're telling a very nice consistent story here, which everyone here should ponder!
But just to muddy the situation, the opposite attitude also makes sense. If you're trying to learn to distinguish things, i.e. "learn that initially similar-looking things are actually different", then learning more amounts to moving to a finer partition. But if you're trying to learn to relate things, i.e., "learn that initially different-looking things are actually similar", then learning more amounts to moving to a coarser partition.
In my lecture here I gave an example of the latter: a detective comes to an island and meets 5 seemingly unrelated people, but gradually discovers that some of them are relatives.
However, in the Fong-Spivak conventions we say \\(P \le Q\\) if the partition \\(P\\) is finer than the partition \\(Q\\). If we then think of partitions as propositions and decree that \\(P \le Q\\) means \\(P \implies Q\\), as I'd been doing with the logic of subsets, it makes sense to say \\(P\\) "knows more" than \\(Q\\) in this case.
So, after writing my lecture here, I decided I should switch to an example where "learning more" amounts to moving to a finer partition. I will do that.
There are a number of arbitrary conventions here, which provide ample scope for left-right dyslexia. For example, it seems at first glance odd to say \\(P \le Q\\) means \\(P\\) "knows more" than \\(Q\\). Less is more!
Of course, the cause of all the flip-flopping is that every poset has an "opposite", in which \\(\le\\) is redefined to mean \\(\ge\\). Both the poset of partitions and its opposite are useful!