Logically, what does belief in the Trinity entail?
P1 The Father is God
P2 The Son is God
P3 The Holy Spirit is God
P4 The Father is not the Son
P5 The Father is not the Holy Spirit
P6 The Son is not the Holy Spirit
P7 There is exactly one God
These 7 premises form an inconsistent set—in other words, they cannot all be true.
However, Trinitarianism affirms all 7 premises.
What exactly is the problem? The basic problem is Tritheism.
If the Father, Son, and Holy Spirit are God, and not each other, it logically follows that there are 3 gods.
This then calls into question how the statements are to be analysed.
The key distinction is over the phrase "is God" in premises 1 to 3.
if "is God" is taken to be an "'is' of identity", then by classical identity,
the Father, Son, and Holy Spirit would be identical to each other, which would entail Modalism.
Modalism is a heresy. That's something Christians will want to avoid.
Maybe it's an 'is' of relative identity?
This would mean that 2 things (Father and Son)
could be identical (to God) and yet not identical (to each other).
This violates classical identity—and most people are reluctant to do so.
What about analysing "is God" in terms of predication?
Well, if each person is ascribed the quality of divinity & they are not each other—then we are back to 3 gods.
No matter how similar they are in terms of their attributes, will, actions, etc.
There may be another option.
Maybe each person "is God" in the sense that they are "parts" of God.
This is William Lane Craig's solution in an attempt to avoid Tritheism.
However, this is clearly unorthodox Partialism, as none of the persons is said to possess the divine nature.
What if all 7 premises are true, and we accept that true contradictions exist?
This has recently been proposed by JC Beall, but involves denying the Law of Non-Contradiction,
which most people will be reluctant to do—