> So if partition logic had exponentiation it would just be Boolean logic.
I don't see why you say that. Are you saying that any poset with this chain of adjunctions must be a Boolean algebra? That's imaginable, but I don't know such a theorem.
Anyway, the poset of partitions is very far from a Boolean algebra: it's a lattice, but it's not even a distributive lattice! Distributivity of \\(\vee\\) over \\(\wedge\\) fails already here: