Below is the proof that the Christian Trinity belief (that God is three Persons in One) is illogical.
Question.—What is the meaning of the Trinity, of the Three Persons in One?
Answer.—The Divine Reality, which is purified and sanctified from the understanding of human beings and which can never be imagined by the people of wisdom and of intelligence, is exempt from all conception. That Lordly Reality admits of no division; for division and multiplicity are properties of creatures which are contingent existences, and not accidents which happen to the self-existent.
The Divine Reality is sanctified from singleness, then how much more from plurality. The descent of that Lordly Reality into conditions and degrees would be equivalent to imperfection and contrary to perfection, and is, therefore, absolutely impossible. It perpetually has been, and is, in the exaltation of holiness and sanctity. All that is mentioned of the Manifestations and Dawning-places of God signifies the divine reflection, and not a descent into the conditions of existence. 1
God is pure perfection, and creatures are but imperfections. For God to descend into the conditions of existence would be the greatest of imperfections. Some Answered Questions, pp. 113
There is a Trinity belief that is logical and you can read about it here:
27: THE TRINITY
To summarize that chapter:
God is One, not divided into parts.
The Holy Spirit is the Bounty of God, an emanation from God. God is like the sun and the Holy Spirit is like the rays of the sun. God remains in His own high place, and does not ever descend to earth.
The Holy Spirit is the Bounty of God which became visible and evident in Jesus Christ.
Jesus Christ was like a clear mirror, and God became visible in the mirror, but God did not descend into the mirror. This is why Christ said, “The Father is in the Son,” meaning that the God was visible and manifest in this mirror.