Divine simplicity is often summarized in the axiom, "All that's in God is God." If this is taken at face value, though, it conflicts with Trinitarian orthodoxy.
1) By this axiom, the Persons are identical with the essence of God.
2) But the Persons have (or are) identifying particularities or relative properties or relations of origin that distinguish them from one another.
3) There are various ways to name these particularities: ungenerate, generate, proceeding; paternity, filiation, spiration; etc. Whatever the name, the particularity is unique to the Person. Only One, the Father, proceeds from none. There are not two Sons, because the Second Person proceeds by generation and the Third by spiration.
4) We cannot apply the "all that's in God is God" to these particularities. Though paternity is "in God," it's not the case that paternity "is God," for then it would have to be a particularity of both Son and Spirit as well as Father. Sonship is "in God," but neither the Father nor the Spirit are sons. We can say "God is Father," but we cannot say "the Father is the Son."
Of course, Trinitarian defenders of simplicity recognize this issue, and offer solutions. Thomas serves as a representative example. For Thomas, the Persons are distinct from one another, but identical to the nature, attributes, and acts of God. They can be really distinct because the Persons are nothing but opposed relations.
Thomas depends on Aristotle's distinction between relations and absolute being, using Aristotle's illustration of the road from Athens to Thebes. The road is one thing, but relationally it is two, the road to Athens and the road to Thebes. These are identical to the one road, but they aren't identical to one another. The properties of the road can be attributed to the relational realities (made of gravel, curvy) but the relational properties can't be attributed to one another (uphill, southward). The relational reality doesn't add anything to the road, but only adds reference to another.
So too it is coherent to say the Persons are identical to the divine essence while being distinct from one another.
That's a pretty elegant solution, but I'm not sure it answers the objection I raised above. The relations of the road depend on something other than the road - on the reality of Athens and Thebes. If we consider the destinations of the road as properties "in" the road, we don't have a simple road. It's not the case that "all that is in the road is the road."
Of course, Thomas didn't offer the road as a perfect analogy to God. Still, even on Thomas's account, there's something in God that isn't identical to God. Unless this is acknowledged, divine simplicity will continue to labor under the suspicion that it muffles the Triunity of God.
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