So if I am reading wikipedia right, if we take \$$p(X) = \\{ s | s \subset X \land s \text{ is finite}\\} \$$ then monotone functions to \$$\mathbf{Bool}\$$ from \$$p(X)\$$ are isomorphic to elements of the free distributive lattice.

Now, there's an identity on objects functor whose actions on the Hom objects is the inclusion of monotonic functions in the set of functions. So these are still preorders indexed by \$$p(X)\$$, but ones that respect inclusion, that is each link exists if and only if a expression built From elements of \$$X\$$, and the operators or and and is true, given that the variables in the indexing power set are true.

Now this is a interesting logic to work in, because it doesn't distinguish Boolean and intuitonistic logic. You need to be able to use "not" just to talk about q being false or unknown, let alone distinguish the two.

You can distinguish the partition logic though, because accumulation isn't true for dit sets on more then 2 elements.