So you're talking about identity being transitive?
You're basically saying that if both (for instance) The Son and The Holy Spirit are both "God" then The Son and The Holy Spirit are therefore the same thing? Because they are both God? Is that what you're saying?
Son = God
Holy Spirit = God
Therefore: Son = Holy Spirit
Is that right?
But what if in this instance identity is not transitive?
Not all things that can be transitive are transitive:
I love my wife
My wife loves the postman
Therefore I love the postman
Under Trinitarian doctrine, Jesus is wholly God and synonymous with God. Jehovah is also wholly God and synonymous with God. The Holy Spirit is also wholly God and synonymous with God.
If Jesus is wholly God and synonymous with God, then that makes the statement "Jesus is God" transitive, by definition. They are completely equivalent under Trinitarian doctrine. Jesus is equivalent to God in the same way 2+2 is equivalent to 4. This means that we can substitute the word "Jesus" any time we see the word God. This is the Trinity under Trinitarianism:
Jesus is God.
Jehovah is God.
The Holy Spirit is God.
Jesus is not Jehovah.
Jesus is not the Holy Spirit.
Jehovah is not the Holy Spirit.
So let's do the replacement:
Jesus is Jesus.
Jehovah is Jesus.
The Holy Spirit is Jesus.
Jesus is not Jehovah.
Jesus is not the Holy Spirit.
Jehovah is not the Holy Spirit.
There's a contradiction here. The Holy Spirit is both Jesus and not Jesus. Jehovah is both Jesus and not Jesus. We can even go further than that, though. Since each of the terms simplify to "God is God" in the same way 2+2=4 simplifies to 4=4 we actually get:
God is God
God is God
God is God
God is not God
God is not God
God is not God
So it's not only a contradiction between "God is God" and "God is not God" but "God is not God" is itself a contradiction. This is the direct consequence of how Trinitarianism defines itself.
Under modal logic, claims that violate the laws of logic like the Law of Identity and the Law of Noncontradiction are considered "metaphysically impossible," which means that we can formally say that Trinitarianism is necessarily false. Since Trinitarianism violates logic, that makes it literally illogical to believe in.
And yet Trinitarianism is a mainstay of mainstream Christianity. I think it bolsters the point that Christianity didn't become popular because it's true, but because they were so brutally efficient at coercing people into accepting their doctrines that we now have people literally affirming a doctrine which is completely impossible and self-refuting.