Grounding overdetermination

Consider:

1. The sky is blue or snow is white.

This is grounded by:

2. The sky is blue.

But's it's also grounded by:

3. Snow is white.

This is a case of grounding overdetermination. I was tempted to characterize this as follows:

4. p is grounding overdetermined iff p is grounded by q and grounded by r, and q≠r, and neither q grounds r nor r grounds q.

The proviso at the end ensures that grounding chains do not count as cases of overdetermination. But (4) isn't good enough. Consider this:

5. The sky is blue or the sky has a color.

This seems to be grounding overdetermined by:

6. The sky is blue.
7. The sky has a color.

But (6) grounds (7), so the proviso in (4) kicks in. Oops!

This odd phenomenon is related to a deficiency of the causal analogue of (4):

8. E is causally overdetermined iff there are C₁ and C₂, with C₁≠C₂, and each of them is a full cause of E and neither of which is a full cause of the other.

But counterexamples akin to (5) are easy to manufacture. Boff is a trainee exterminator and Biff is his boss. Biff tells Boff to imitate everyting he does. Biff pours a lethal dose of crocodile poison into the customer's pond. Boff imitates him and pours another lethal dose. While Biff poured first, Boff poured closer to the crocodile, and as a result both doses arrived at the crocodile simultaneously, each sufficient to kill it. The alligator's death was overdetermined by the pouring of the two doses, even though one pouring caused the other.

We could try this:

9. p is directly grounding overdetermined iff p is directly grounded by q and directly grounded by r and q≠r.

And then we could say that:

10. p is grounding overdetermined iff it is either directly grounding overdetermined or it is grounded in something directly grounding overdetermined.

(And there is an obvious analogue for causation.) But this is only plausible if grounding is transitive. Maybe what we need to add to (10) something like "in a way that doesn't generate a relevant failure of the transitivity of grounding"?

Another difficulty with (9) is that the notion of direct grounding is not so easy to define. The tempting definition of direct grounding is, after all:

11. p is directly grounded by q iff p is grounded by q and there is no r such that p is grounded by r and r is grounded by q.

But cases of overdetermination like (5) are counterexamples to (11), since (5) is directly grounded by its first disjunct, even though the first disjunct grounds the second and the second grounds all of (5). So one wants to say:

11. p is directly grounded by q iff p is grounded by q and there is no r such that p is grounded by r and r is grounded by q and this is not a case of grounding overdetermination.

But of course then we have circularity.

All this suggests to me that we need a notion not reducible to grounding to make the above distinctions. We might, for instance, take direct grounding to be primitive. Or overdetermination. Or maybe we could use some version of my grounding graph approach.