FATHER <> SON <> HOLY SPIRIT
The Trinity.
I am not a theologian, but I recognize a logical construction when I see one. The trinity of the Father, Son, and Holy Spirit is a logical construction.
The Father is the Father throughout the discussion, and so is the Son and the Holy Spirit. Thus both the principle of identity and that of non-contradiction are fully respected. And since the Father is equivalent to the Son and equivalent to the Holy Sprit, the construction of the Trinity respects also the principle of equivalence.
Perhaps, a more complete understanding is derived when the three concepts are observed in an analytic format. Thus we clearly see that the concept of the Father is reflexive. The Father is always identical to the Father. And so is the Son and the Holy Spirit. The concept of the Father is symmetrical: the Father is a mirror image of the Son and the Holy Spirit. The concept of the Father is transitive: one can easily pass from the concept of the Father to that of the Son and the Holy Spirit.
The concept of the Son and that of the Holy Spirit are equally reflexive, symmetric, and transitive.
Not only is the construction of the Trinity logically consistent. It is also prominent in all major religions: it exists in Catholicism and most forms of Protestantism, of course. And it exists also in Judaism, in Islam, and in Hinduism. In Hinduism, the construction of Brahma, Shiva, and Vishnu is clearly a trinitarian construction. To see the structure of the trinity operating in Judaism and Islam one must see the Son, not a Jesus, but as the Messiah and as Mohammed.
To accept the logical necessity for the structure of the Trinity is not the same as believing in the Trinity. Do I believe in the Trinity? As a devout Catholic, I do. I deeply believe in the Trinity. Would I be able to convince others of its existence beyond its being as a logical construction? No, I would not. And therefore I would not even try.
The mind cannot comprehend the Triune God, because if it could it would be as comprehensive as God. It would be equal to God. Therefore there cannot be any rational explanation for the existence of God. But innumerable people experience God. Indeed, speaking for myself, I experience God whenever I succeed in emptying my mind to make room for God. At that point there is a union with God and a sudden realization that it is not God who is a mystery. It is life without God that is a mystery. God is love. God is not an affair of the mind; God is an affair of the heart.