Short answer is, “identity.” Each Divine Person is perfectly identical to the Divine Essence.
Perhaps the best exposition of this idea is given by St. Thomas Aquinas in the Summa theologiae [S.Th.].
Keep in mind that the Persons, fundamentally, are relations of origin: the Father is Fatherhood; the Son is Sonship or Generation, and the Spirit is Procession. (There is a fourth relation, active spiration, that is not a Person, because it is proper to two of the Persons: Father and Son. Regarding the Divine Relations, see S.Th., Ia q. 28, a. 4, corpus.)
However, in God there are no “accidents;” rather, whatever perfections creatures derive from their accidents, God possesses in their fullness in His very Essence (or Substance).
The same holds true for the Divine Relations. As St. Thomas puts it,
And thus it is evident [after a long demonstration, of course!] that a relation really existing in God is the very same Essence in reality; it does not differ [from the Essence] except according to the notion by which it is known. (S.Th. Ia q. 28, a. 1, corpus; my translation. The original is “Et sic manifestum est quod relatio realiter existens in Deo, est idem essentiae secundum rem; et non differt nisi secundum intelligentiae rationem.”)
In other words, although we creatures need to use two different notions to signify the Divine Essence and each one of its intrinsic relations (including the Persons), in reality, the notions refer to the very same reality.
To use Frege’s terminology, we could say that Divine Essence and the Person have the same Bedeutung (reference) but differ in Sinn (sense).
When speaking of theology, there is the usual caveat that the concept of being or existence (“is”) changes according to the subject, in a way that perhaps Frege did not consider in his logic.
For example, the following statements, although they formally appear very similar, actually have very different meanings:
- The greenness of the tree is (exists).
- The tree is (exists).
- The cow is (exists).
- John (a man) is (exists).
- God is (exist).
(A similar observation could be made for all four of Frege’s meanings of to be. Probably the best exposition of this idea is found in Aristotle’s Metaphysics, especially book 4, in which he talks about the manifold meaning of being, all in reference to one; and in book 5, 1017a10-1017b10, in which he outlines his fourfold division of “being.”)
These statements are arranged in a hierarchy, from the least noble subject to the most noble. This is simply to observe that God can be (exist) in a way that creatures cannot. Strictly speaking, in God, all four of Frege’s senses of being would coincide. For example,
- “God is good”; this is not merely a predicate of God, but is identity. God is His goodness. It is also His very being—He is Goodness Itself.
- “The Son is God”; this is not a “subset,” but rather identity. The Son is God, and He is (so to speak) “all” of God, the “entire” Divine Essence. (And so is the Father and the Spirit.)
Answer to addendum: why this does not violate transitivity.
Remember, “is” changes in meaning depending on the subject to which it is predicated.
The being of God is special: God does not simply “exist,” the way His creatures exist; He is best characterized as Being Itself (Ipsum Esse, in St. Thomas Aquinas’ lingo). What that implies is that the being (existence)* of every creature is a participation in God’s Being.
The Persons, however, don’t merely participate in God’s Being, they possess it completely. And they possess the very same Being. And when we predicate the verb “to be” to the Divine Essence, we mean it in this very special sense: that He is Being Itself.
In this very precise sense of the verb “to be”—which means “to possess the Divine Being in all its fullness” (and note that possessing Being in its fullness means perfect identity with that Being)—there is no problem in saying:
- The Father is God
- The Son is God
- The Holy Spirit is God
However, when we use the verb “to be” to distinguish Persons, the meaning changes, because the only way to compare two relations is by analyzing he terms of those relations.
- The Father is not the Son, because the Father is Unbegotten (a term of the relation of Fatherhood) and the Son is Begotten (the other term). And vice versa, regarding the Sonship of the Son.
- The Father and the Son are not the Holy Spirit, because the Father and the Son are one term of the relation of spiration, and the Holy Spirit is the other (and vice versa, regarding procession).
Hence, when we say
Father = God
Son = God
Holy Spirit = God
we mean “the Father possesses all the Divine Being, which is His Essence” (and so forth with all the Persons).
However, such an affirmation would be false if we tried to apply it between two Persons, because the very meaning of “to be” actually changes:
Father ≠ Son
Father ≠ Holy Spirit
Son ≠ Holy Spirit
If you will, the use of the same symbol (=, or the verb “to be”) fools us into thinking the meaning is the same.
In reality, in proper theological terminology, we would say, “The Father is really distinct from the Son,” etc. The Persons are really distinct, because we are comparing the relations to one another, and, in fact, the terms of the relations are different (unbegottenness vs. begottenness, spiration vs. procession).
* It should be kept in mind that Aquinas distinguished very carefully between exsistentia (existence) and esse (being), a distinction that was largely lost in St. Thomas’ commentators (Cajetan, Francesco Silvestri, etc.), and certainly by Duns Scotus and Suarez. For Aquinas, Esse means the perfection or—more precisely, the act—of being, which is more than the mere “fact” of being. Exsistentia, on the other hand, means the presence or emergence of being to some kind of observer. A thing “is” whether or not we perceive it. It “exists,” strictly speaking, when we are aware that it is. In English, too, the distinction is largely lost.