It's "one nature, three persons." Consider the analogous case of human nature:
Mark is human. (A is B)
Christ is human. (C is B)
Therefore Mark is Christ. (A is C)
This is obviously false. Leaving out that all predication vis-รก-vis God is analogical, you would still need to assume a properly metaphysical premise like:
"More than one person cannot subsist in the same nature."
Traditionally, in "the Holy Spirit is God," "is God" refers to the Divine Nature. I suppose another premise that would work is: "'is God' must refer to univocal, numerical identity." However, this is exactly what is denied. As noted earlier, numerical identity is taken to be posterior to (dependent on) God, the transcendental property of unity, and measure. Numerical identity is a creaturely concept. — Count Timothy von Icarus
It's very hard to see what you are trying to do here, or how it might help your case. I had presumed you would be seeking to defend trinitarian dogma, but your example serves more to highlight the problem.
Presuming we read "Mark is human" and "Christ is human" as that Mark and Christ participate in a common nature, then we are not here talking about identity. That is, you have moved from identity to predication. If we were to follow that, you would end up with Christ and The Holy Spirit merely participating in godhood in the way that Mark, Christ and Tim participate in being human. You would have three gods, not one. Your conclusion would be polytheistic.
You seem to think that you can avoid this by claiming "numerical identity is a creaturely concept". But that is exactly the issue; classical logic does not permit us to just drop transitivity without contradiction. Your suggestion amounts to saying that logic does not apply to god - to claiming mysticisms.
That idea, "participates in a common nature", is a presumably Aristotelian or Thomist? It seems to be a way to render a relation as a predication. The idea is that individual substances (like Mark, Christ) "participate in" or "instantiate" universal natures or essences (like humanity). Instead of saying:
Mark stands in the relation of instantiation to human nature
The scholastic tradition reformulates this as predication:
Mark is human
Now traditional trinitarianism
requires identity, not participation - it requires that the Holy Spirit literally
is God.
Frankly you seem to be using the ambiguity of "is" to make an invalid logical move seem plausible. Either "is God" means identity and contradiction ensues, or it means predication and polytheism.
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Your comments on analogical reasoning are similarly puzzling. It doesn't address the issue - even if granted, it leaves aside whether analogical reasoning can actually do the logical work required. You would have it that God is so transcendent that normal logical categories don't apply, and yet claim analogical reasoning somehow captures this transcendence.
How?. Your aim may be to preserve the Trinity and avoid contradiction, but how you do this remains unexplained. How could "Analogically, the Holy Spirit is God" be represented logically? - and if it can't be, then it is illogical. Ok, so god is not just another creature - he is special; and again, what this amounts to is the claim that logic does not apply to God!
What you have set out does not help Bob derive the Trinity form first principles, nor show me how to understand the special kind of identity in "Christ is god, The holy spirit is god, but Christ is not the holy Spirit".
It just changes the topic.
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But moreover, adding more and more assumptions and explanations to an already
ad hoc account is not helpful. You seem to be simply digging a deeper hole.