According to apostate Dale Tuggy, the Trinity is false because it violates the "law of identity" or Leibniz's law. That's what his objection to the Trinity always comes back to.
He defines numerical identity by reference to Leibniz's law (which he reformulates). And he considers numerical identity to be a precondition of personal identity.
Is personal identity consonant with organ transplantation? Suppose I have a heart transplant. Am I still the same individual? It's no longer my heart. Someone else's heart is keeping me alive.
Is personal identity consonant with prosthetics? Suppose I have knee-replacement surgery. I now have an artificial joint. Am I still the same individual?
Suppose you kept replacing my organs and body parts with donated organs or prosthetics. Would I still be the same individual? Is there a threshold beyond which I'd cease to be the same individual?
Suppose you transplanted my brain into the body of an android. Would still I be the same individual?
Suppose you transplanted my (male) brain into a female body. Would I still be the same individual?
Suppose you transplanted my brain into the body of a dolphin. Would I still be the same individual?
Is my body incidental to my core identity? What's the core of my core identity? That's especially dicey if you're a physicalist.
One way to finesse the conundrum is to distinguish between intrinsic and extrinsic properties. There are, however, some problems with that distinction:
i) It's not a distinction that derives from Leibniz's law. Rather, it modifies Leibniz's law.
ii) But if we grant that distinction, why can't we apply it to the Trinity?
Why is having a human brain inside an android body or dolphin body consonant with numerical identity, but having three "persons" or "modes of consciousness" disconsonant with numerical identity?
What makes a body an extrinsic property that's compatible with the same individual, but distinct personhood an intrinsic property that's incompatible with the same individual (i.e. one and only God)?
iii) This also goes to ambiguities regarding the nature of intrinsically or extrinsically. If, according to David Lewis, duplicates never differ with respect to their intrinsic properties, then the distinction between Father, Son, and Spirit is an extrinsic property.
Suppose we modeled the Trinity on enantiomerism (or enantiomorphism). Suppose we regard the members of the Trinity as enantiomers. They mirror each other in every respect, yet they are nonsuperimposable.
Is that in tension with numerical identity? But why not consider superimposability an extrinsic property? What's the criterion?