Assume the word “God” is the name for a nature* that, while remaining a true nature, is nevertheless not a logical abstraction but a true reality. Three things follow:
1.) Like any nature, “God” can be said of many individuals.
2.) Unlike any nature, “God” is not a mere logical abstraction, but is as equally real as what it is said of.
3.) Like any nature said of many, it is categorically different from any individual or number of individuals it is said of. To think that “canine” is a fourth member that can be added to Lassie, Fido, and Checkers is a category mistake; just as thinking one could add “even number” as a fourth member to 2,4, and 6.
From (1), God can be said of many – Assume it is said of three. But, from (2) such a predication of the nature “God” does not require that there are three Gods as opposed to one, for “God” subsists of itself and not merely in the individuals who it is said of. From (3) this God subsisting in itself is not a fourth being in addition to the three; nor are the three individuals the second, third, and fourth instances of the nature which subsists.
If by the word “self” one means an existent, intelligent being, then “God” is a self from (2) and the persons are each selves from (1). But this does not give us four selves from (3). Nor, from the juction of (1) and (3) do we have to say that each of the persons is the other.
Taken in this way, when Trinitarians deny that there are three gods, what they mean is that the nature does not have a merely logical existence, but is, if you like, a self. This self-as-nature has all the properties that can be attributed to divine nature, while preserving what is necessary for something to count as a nature, i.e. ability to be predicated of many.
But doesn’t this bring us to the point of saying that the divine nature is absolutely unique as divine but communicable as nature? And Isn’t this to speak of an incommunicable communicable thing? But I think all this comes to is an opposition between nature and individual for us, which is precisely what (2) is trying to deny.
The mystery is that uniqueness and communicability reduce to a common source in existence, which is manifest in the fact that existence is simultaneously the most general and most formal of predicates; it is simultaneously what makes a thing differ from the non-existent, fictional, abstract, and other while also being something that does not allow for any difference among existents.
If this is right, it explain why Thomists deny haecceity: it would be superfluous since existence is at once the most abstract and most concrete of forms. This, however, seems to make “existence” a mere pointer word, indicating a convergence of the abstract and concrete that is not presently intelligible to us.
For the purposes of this post, the following words are perfect synonyms of “nature”: essence, abstract form, form, divinity, idea, sort of thing etc.