Principle of unrepeatable existents. What exists cannot be repeated. E.g. I exist and another I is impossible, since whatever would be I is me. One can have many cats since “cat” as a unified whole or singular nature exists only in mind and not in reality. Again, we can’t have many things that are this particular white thing, but we can have many white things since “white” as opposed to “this particular white thing” has its singular unity only in mind and not in reality.
Though just explained using a conceptualist account of universals, it is also true on a platonic account. One of the motives for platonism was to explain, say, many cats as all partaking in the unrepeatable unity of felinity, the auto-cat or cat itself. Here again, the existent felinity cannot be multiplied.
Some corollaries:
1.) Existence is proportional to uniqueness and vice versa. Haecceitas is superfluous to explaining singularity since it is properly and formally explained by existence as such, even while existence is also seen as a common property of all existent things.
2.) Leibniz’s axiom has a simpler, less paradoxical formulation. It’s not that identical things are indiscernible but that the existent is unique. Things are identical, or even alike, only in virtue of something other than existence, whether this is an idea that exists only in mind (Conceptualism) or a relation to a thing-in-itself that cannot be multiplied (Platonism) or an instinctual association of the mind with no intelligible basis (Nominalism). We don’t need to puzzle over the problem of how things can be identical if they are indiscernible.
3.) Simpler account of the real distinction. Multiplicable things can be multiplied, but an existent as such cannot be, so multiplicable things are something other than their existence.
4.) Trinity is an instance of divine simplicity. If “God” is a divine person and “god” is divinity, then simplicity is the claim that god exists. Leaving aside Platonism, this is unique to divinity: “cat” does not exist but “god” does; and so it follows that there can be more than one cat but not more than one god. Simplicity demands that God (say, the Father) is god; but this is true irrespective whether the proper name “God” is used once or more than once. In each case, God is god – not a god but literally god.
Simplicity demands God is god. Trinity only specifies that “God” like “Bill” or “Joe” is a proper name that can mean more than one individual.