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—
          most people will not be likely to reject the Law of Non-Contradiction or the Law of Identity.

Maybe the 7 premises are all true and they are only apparently contradictory, but not actually contradictory.
Even if we cannot tell you how or why they aren't contradictory.
This is a form of Mysterianism, which has been proposed by James Anderson.

This isn't technically a "solution" to the Logical Problem of the Trinity from the Classical paradigm.
Rather, it is questioning whether or not a solution is required from an epistemological standpoint—
               "are mysteries acceptable in theology, and if so, when?"—
How do we determine when something is actually contradictory vs. merely apparently contradictory?

Questions to consider.

So, given the brief explanation of the Logical Problem of the Trinity and possible responses, we are left with:

1) Modalism
2) Denying Classical Identity with respect to this problem ('is' of identity option)
3) Tritheism or Partialism ('is' of predication option)
4) Denying the Law of Non-Contradiction
5) Affirming a form of Mysterianism
     and arguing that the Logical Problem of the Trinity may be unsolvable,
          but Christians are still justified in affirming the doctrine of the Trinity.

Admittedly, this is a brief sketch of the complexities of the problem
     and its various solutions proposed, but I hope you find this somewhat beneficial.

I, myself, am not satisfied with any of the above for reasons already stated.
This problem has been around since the 4th century CE,
     when the doctrine was authoritatively established.

Take, for example, the following sets of premises according to the classical paradigm:

					1) Adam is Human
					2) The Author is Human
					3) The Reader is Human
					4) Adam is not the Author
					5) Adam is not the Reader
					6) The Author is not the Reader
					7) There are exactly 3 Humans

Regarding what to believe for the Trinity, either:
	   I)		we can know what the true belief is
	   ii)		we can't know what the true belief is 			     		       (the belief is rationally unknowable)

	if I, either:
	   I.I)		we can know the true belief is not a contradiction
	   I.ii)	we can know the true belief is a true contradiction 		      (denying Laws of Non-Contradiction & Identity)

	if I.I, either:
	   I.I.I)	we know the belief is a statement of Predication
	   I.I.II)	we know the belief is a statement of Identity

	if I.I.I, either:
	   I.I.I.I)	the belief is classical Predication
	   I.I.I.ii)	the belief is non-classical Predication   (special-pleading numerical identity — Social-Trinitarianism	  [1+1+1=1])

	if I.I.I.I, either:
	   I.I.I.I.i)	the predication subsists within some other essence		   (the belief is Partialism		  [⅓+⅓+⅓=1])
	   I.I.I.I.ii)	the predication does not subsist within some other essence	   (the belief is Tritheism		  [1+1+1=3])

	if I.I.II, either:
	   I.I.II.i)	the belief is modes of identity, mutually equative Predication		(the belief is Modalism		  [1|2|3=1])
	   I.I.II.ii)	the belief special-pleads/denies the Law of Identity				  (Relative Identity Trinitarianism)

So, there are three types:
	1) (       ii, 			     I.ii) 	— Some type of Mysterianism
	2) ( I.I.I.ii,			I.I.II.ii) 	— What is not a solution classically
	3) (I.I.I.I.i,	  I.I.I.I.ii,	 I.I.II.i)	— Heresies

I don't think new solutions are coming.

May Allah guide us all.