The Trinity and classical identity, again

Let me re-phrase an argument from an earlier post.

Suppose “=” is governed by the classical rules of identity, and G, F, S, and H are names for God, the Father, the Son and the Holy Spirit. Let x ≠ y abbreviate Not(x = y). Let D(x) say that x is divine. There then are three predicates A, B and C such that:

1. A(S) and not A(F)

2. B(H) and not B(S)

3. C(S) and not C(F).

For instance, we can let A(x) say that x is begotten, B(x) say that x is not begotten, and C(x) say that x proceeds.

Then add:

4. Either (a) F = G and S = G and H = G or (b) F ≠ G and S ≠ G and H ≠ G.

5. D(F)

6. D(S)

7. D(H)

8. D(G)

Premise 4 says that the Father, Son and Holy Spirit are on par with respect to being God. Either each is classically identical with God, or classical identity does not apply to any person and God (in which case we explain “The Father is God” as something other than classical identity).

It follows from (1)-(3):

9. F ≠ S and S ≠ H and H ≠ S.

It then follows that (4)(a) is false, so we must have:

10. F ≠ G and S ≠ G and H ≠ G.

It then follows from (9) and (10) and classical identity being an equivalence relation that:

11. ∃w∃x∃y∃z(D(w)∧D(x)∧D(y)∧D(z)∧w≠x∧w≠y∧w≠z∧x≠y∧x≠z∧y≠z).

But (11) is the standard classical logic translation of:

12. There are at least four that are divine.

Heresy!

What are the ways out? We can’t reject (1)-(3). Even if we have some quibbles about the specific examples I chose for A, B and C, every orthodox Trinitarian agrees that for each pair of persons there is something that is truly predicated of one that isn’t truly predicated of the other.

Rejecting any of (5)-(7) is a non-starter: one isn’t a trinitarian if one does not say that the Father is divine, the Son is divine and the Holy Spirit is divine.

That leaves (4) and (8). Start with (8). That seems as uncontroversial as anything about God can be. God is divine!

But here is one way of rejecting (8): reject the presupposition that there is a name, “G”, for God. If we do that, we also end up rejecting (4), of course, but not in a way that threatens the parity of the three persons of the Trinity with respect to being God. I think there are two ways of doing this:

1. Reject the claim that there is a proper name for God as such.

2. Reject the very existence of classical identity.

It is tempting to say that instead of rejecting the very existence of classical identity to God, one can reject its applicability to God. But that can’t be done. It is part of the very concept of classical identity that it applies to everything, that for any name N it is axiomatic that N = N and that it is a theorem that ∀x(x=x).

What about (I)? Surely this is a non-starter. Doesn’t the Christian tradition constantly talk about names of God? Well, yes, but there are names and proper names. What if we say this? There are proper names for the Father, the Son and the Holy Spirit. But there is no proper name for God. Instead, what we have is something like a definite description like “the divine one”.

We still haven’t solved the problem. The normal way to understand definite descriptions is the Russellian way. When you say “The divine one created the world”, you are saying:

13. ∃x(D(x)∧∀y(D(y)→y=x)∧C(x,W)).

That won’t do, however. For (13) leads to a contradiction when it is combined with (1) together with the non-negotiable Trinitarian claim

14. D(F) and D(S)

that the Father is divine and the Son is divine, as well as classical inference rules for identity.

So, what do we do? Here is a suggestion. Let R be an equivalence relation. Then we can have an R-based article “the_R”, and sentences with “the_R” are translated in the Russellian way except with R in place of =.

(Compare how Aquinas makes the distinction between talking in the neuter and talking in the masculine of God, and where when one applies substantives in the neuter, one is talking of the divine essence. One can think of “the_E” as a neuter article, which English doesn’t distinguish from the personal—masculine or feminine—articles, and which Latin lacks altogether, since it lacks articles.)

Thus, “The divine one created the world” translates to:

15. ∃x(D(x)∧∀y(D(y)→yEx)∧C(x,W)).

No contradiction results from (1)-(3) and (14)-(15).

We can call “the divine one” an E-definite description, while “the begetter” is an =-definite description. Aquinas at times in his discussion of the Trinity makes a distinction between substantives used in the neuter and substantives used in the masculine—the masculine is personal in a way that the neuter is impersonal and more suited to when we talk of the divine essence.

Thus, on our present theory, God as such has no proper name, but he does have E-definite descriptions.

Now, what are we to make of the truth value of the following?

16. The Son is identical with the divine one.

If “The Son” is just a proper name and “the divine one” is “the_E divine one”, then (16) is:

17. ∃x(D(x)∧∀y(D(y)→yEx)∧x=S).

And this is false, because it contradicts (5), (6) and (9). So, on the theory under consideration, we have to deny (16). That sounds kind of bad. But perhaps it’s not bad if we realize that “identical” here is classical identity, and we think that classical identity comes to “is the same hypostasis as”, since in the divine case, same hypostasis means same person, and it sounds wrong to say that the Son is the same person as God.

So, I think there is a way of holding on to classical identity while defending the Trinity, but it is costly: we need to say that there is no proper name for God as such and that definite descriptions for God are E-definite descriptions. But denying classical identity is also costly.