We recognize the logical possibility of the Trinity not in spite of the fact that there is only one divinity, but because of it.
Either it is impossible for there to be more than one divinity or monotheism just happens to be true for the moment, but monotheism is not the sort of thing that could just happen to be true for the moment (you might as well say that a mathematical theorem is true in this way. The theorem might be true or false, but not true for the moment.)*
At the same time, divinity is essentially communicable to many, formal, and abstract. The impossibility of polytheism or the necessity of monotheism has to be defended in a way that preserves the indefinite or purely formal character of divinity.
God or divinity, like all intelligible things, is communicable because he is purely formal, in the same way that a mathematical theorem both the one I know and you know and the one common to both of us and to anyone who might know or have known it. In God however, this intelligible existence is real existence and not just logical abstraction. Divinity or God is not just an idea but a self.
But this unity of God arises precisely from his intelligible or abstract existence and so must be communicable to many individuals.
So, in comparing divinity to other sorts of intelligible or abstract existence, we get
1.) Unlike other sorts of intelligible or abstract (IA) existence, divinity/God exists really and not just logically. Divinity/ God is truly a self and not just an idea. The capitalization of ‘god’ means only that the abstraction ‘a god’ or ‘divinity’ differs from other abstractions by being a true self.
2.) Like all other sorts of IA existence, divinity/God is formal and therefore can exist in many individuals.
3.) Like all other IA existence, the nature is not a thing in addition to what it is communicated to. The DNA one learns about in biology class, which is capable of making us know any relevantly similar real strand of DNA, is not another instance of DNA. You can count up the supposed dozen people who knew the theory of Relativity in 1920, and thereby get twelve instances of the theory, but the theory itself would not make for a baker’s dozen. Though divinity/ God is a true self, (cf. #1) he is not, say, a fourth self in a trinity of divine individuals.
4.) Unlike all other IA existence, when divinity is communicated to another so as to make it fully divine it is logically impossible for that individual to exist apart from all other individual of the same description. Any existent frog can continue to be if another frog dies, but this is a logical impossibility for any divine individual. As a corollary to this:
4a.) Like all other IA existence, divine individuals are relations. What exists abstractly or intelligibly is a formal sign, which is not a sign imposed on some already existing substance but a pure relationship making an object known. But because it is a logical impossibility for one divine individual to exist without another, their existence as individuals is relative.
*This claim even serves as common ground for atheists and theists, especially when reflecting on the Ontological Argument.