More on grounding of universals

The standard First Order Logic translation of “All As are Bs” is:

1. ∀x(A(x)→B(x)).

Suppose we accept this translation and we further accept the principle:

2. Universal facts are always partially grounded in their instances.

Then we have the oddity that the fact that all ravens are black seems to be partially grounded in my garbage can being black. Let R(x) and B(x) say that x is a raven and black, respectively, and let g be my garbage can. Then an instance of ∀x(R(x)→B(x)) is R(g)→B(g), and the latter material conditional is definable as ¬R(g)∨B(g). But a disjunction is grounded in its true disjuncts, and hence this one will be grounded in B(g) (as well as in ¬R(g)).

There are three things to dispute here: the translation (1), the grounding principle (2), and the claim that a material conditional is grounded in its consequent whenever that consequent is true. Of these, I am most suspicious of the translation of the two-place universal quantifier and the grounding principle (2).

Grounding of universals and partial grounding

It is common to claim that:

1. The fact that everything is F is partially grounded in the fact that a₁ is F and in the fact that a₂ is F and so on, for all the objects a_i in the world.

But this can’t be right if partial grounds are parts of full grounds. For suppose you live in a world with only two objects, a and b, which are both sapient. Then everything is sapient, and by (1) it follows that:

2. The fact that everything is sapient is partially grounded in a being sapient and in b being sapient.

But suppose partial grounds are parts of full grounds. The facts that a is sapient and b is sapient are not a full ground of the fact that everything is sapient, because the full grounds of a fact entail that fact, and a being sapient and b being sapient doesn’t entail that everything is sapient (since it’s possible for a to be sapient and b to be sapient and yet for there to exist a c that is not).

So we need to be able to add something to the two particular sapience facts to get full grounds. The most obvious thing to add is:

3. Everything is a or b.

Clearly fact (3) together with the facts that a is sapient and b is sapient will entail that everything is sapient.

But applying (1) to (3), we get:

4. Fact (3) is partially grounded in the facts that a is a or b and that b is a or b.

But, once again, if partial grounds are parts of full grounds, then we need a fact to add to the two facts on the right hand side of the grounding relation in (4) such that together these facts will entail (3). But the obvious candidate to add is:

5. Everything is a or b.

And that yields circularity.

So it seems that either we should reject the particular-grounds-universal principle (1) or we should reject the principle that partial grounds are parts of full grounds.

Here is a reason for the latter move. Maybe we should say that God’s creating me is partially grounded in God. But that’s merely a partial grounding, since God’s existence doesn’t entail that God created me. And it seems that the only good candidate for a further fact to be added to the grounds so as to entail that God created me would be my existence. (One might try to add the fact that God willed that I exist. But by divine simplicity, that fact has to be partly constituted by my existence or the like.) But my existence is grounded in God’s creating me, so that would be viciously circular.