I've been searching for the right kind of mathematical structure to think about the phenomenon of grounding or partial grounding with. The orthodoxy is that the right structure is a partial ordering. That the axioms of partial ordering are satisfied by partial grounding has been challenged and defended. But even the critics have tended to take partial grounding to be something like[note 1] a single relation, or perhaps a small collection of related relations, between pairs of propositions or between a proposition and a set of propositions. I've offered two suggestions (first and second) on how to model grounding using graphs. But I now think all of these approaches abstract away too much of the structure of grounding and/or are unable to capture all the prima facie possibilities that a theory of grounding should recognize.
For instance, the relational approach loses sight of the structural fact that one can sometimes have two different grounding relations between a pair of propositions. Let W be the proposition that Smith is drinking water and let H be the proposition that Smith is drinking H2O. Let D be the disjunction of W and H: the proposition that Smith is drinking water or drinking H2O. If we think of grounding as a relation, we can certainly say that H grounds D. But we want to be able to say that there are two groundings between H and D: H grounds D directly by being one of its disjuncts and indirectly by grounding W which is another disjunct. And this structure is not captured by the orthodoxy. The graph approach nicely captures this sort of thing, but it does not adequately capture the compositional structure which is that the indirect grounding that H provides for D is a composition between a grounding by H of W and a grounding by W of D. There are ways to make it capture this, say by identifying composition of grounding with sequences of arrows in the directed graph, but this won't work for infinite sequences of arrows, something that we should not rule out in the formalism. I realized this when trying to finish a paper on grounding and fundamentality.
I am now wondering if the right structure isn't that of a category. Maybe the objects are true propositions. The arrows are groundings, i.e., token grounding-like relations. Every arrow is at least a partial weak grounding (weak, because there are identity arrows). Some arrows may be full groundings. There is a nice associative compositional structure.
There will be further structure in the category. For instance, perhaps, every family of true propositions will have a coproduct, which is the conjunction of the propositions in that family. The canonical injections are the partial groundings that conjuncts give to the conjunction, and the universal property of the coproduct basically says that when a proposition is weakly partially grounded by each member of a family of propositions, then there is a coproduct weak partial grounding arrow from the conjunction of that family to the proposition. This is very nice.
We might also consider a move to a category where the objects are all propositions, but that creates the challenge that we need to keep track of which propositions are true and which groundings are actual. For that p partially grounds q is, in general, a contingent matter.[note 2] Truth and actuality of grounding respects the category structure. Actual groundings only obtain between truths, and compositions of actual groundings are actual.
The category structure is going to yield transitivity of weak partial grounding. There are apparent counterexamples to transitivity. But it is my hope that when we keep track of the additional structure, and think of the token groundings as central rather than the relation of there being a token grounding, the result is not going to be problematic.
It is now interesting to investigate what different category theoretic phenomena occur in the grounding category, and how they connect up with metaphysical phenomena. One thing I'd like to see is if there is a neat category theoretic characterization of full, as opposed to partial, grounding.
I have an intuitive worry about the above approaches. Intuition would suggest that if conjunctions are coproducts, then disjunctions would be products. But they're not. For in general there is no grounding from a disjunction to disjuncts. This could be related to another worry, that because categories include identity arrows, I had to take the arrows to be weak groundings—i.e., I had to allow each proposition to count as having a grounding between itself and itself.
I do not know if category theory will in the end provide a good mathematical home to grounding structures, but I am hopeful.