A two-sorted logic and the Trinity

For a while this spring I’ve been thinking about ways of avoiding quaternity views of God.

The problem is that, first,

1. Father ≠ Son, Son ≠ Spirit and Father ≠ Spirit

and, second, we seem to have to choose between the following two options:

A. Father = God, Son = God and Spirit = God, or

B. Father ≠ God, Son ≠ God and Spirit ≠ God.

Add:

2. Father is divine, Son is divine, Spirit is divine, and God is divine.

If we accept (1) and (A), then we have a logical contradiction given classical identity.

If on, other hand, we accept (1), (B) and (2), then there are four that are divine, a quaternity!

A now-standard solution to problems like this is to go for a version of a relative identity theory rather than classical identity. Then the (negated) = in (1) and the = in (A)/(B) are different identity relations (e.g., sameness of person versus sameness of essence).

In this post, I want to consider a somewhat different take on non-classical identity. Suppose we take a variant on two-sorted logic. On a many-sorted logic, bound variables and names come with sorts, and predicates have grammatical restrictions on the sorts of terms that can be their arguments. Usually, the restrictions say for each argument place what sort of term can go in that place. But we can have a more complicated kind of sort restriction.

Suppose, then, we have two sorts: essence and hypostasis, and that = has the classical rules of inference, but has the sort restriction that only terms of the same sort can go on the two sides of =. Suppose that “Father”, “Son” and “Spirit” are of the hypostasis sort, and “God” is of the essence sort. Then we can have (1), but neither (A) nor (B) will express a truth, as both (A) and (B) will be ungrammatical. Even if = has the classical rules of inference, I think there will be no way for us to derive that there are four that are divine. Indeed, in the two-sorted logic, the only way to say “there are four that are divine” is:

3. There are four essences that are divine or there are four hypostases that are divine.

For we cannot mix the essence-variables and hypostasis-variables in an “=” formula.

Interestingly, I think we can have an even stronger non-classical logic of identity while avoiding incoherence and quaternity, though I don’t know if this will appeal to anyone. Make the sort restriction on = be that u = v is grammatical if and only if either u and v are both of the same sort or u is a hypostasis term and v is an essence term. Next, specify that the =-elimination says that from the sentence a = b and a formula ϕ(a), we are allowed to infer ϕ(b), but only if “ϕ(b)” is grammatically correct.

In this stronger logic, we can have all three of (1), (A) and (2), apparently without contradiction. The crucial thing is that from Father=God it is impossible to conclude God=Father, because the latter is ungrammatical. It seems to me to be the Holy Grail of Trinitarian logic to be able to affirm all of (1), (A) and (2).

That said, I don’t like the asymmetric sort restriction on =.