The purpose of this post is to sketch a argument for the incoherence of the Christian doctrine of the Trinity.
The Doctrine of the Trinity Formulated
Let us begin by defining the doctrine of the Trinity as three theses:
(T1) There is exactly one God.
(T2) Father, Son, and Holy Spirit are not identical to one another.
(T3) Father, Son, and Holy Spirit are consubstantial.
The Argument for the Incoherence of the Trinity (also known as the "Logical Problem of the Trinity")
(1) If the doctrine of the Trinity is true, there is exactly one God, the Father Almighty. [from T1]
(2) The Father is a god. [From (1)]
(3) If the doctrine of the Trinity is true, the Son is consubstantial with the Father but not identical to the Father. [From T2 and T3]
(4) If there are x and y such that x is a god, x is not identical to y, and y is consubstantial with x, then it is not the case that there is exactly one God. [Premise]
(5) Therefore, it is not the case that there is exactly one God. [From (2), (3), and (4)]
(6) Therefore, it is impossible that the doctrine of the Trinity is true. [From (1) and (5)]
Because I am aware that Christian scholars have written numerous clarifications and defenses of the doctrine of the Trinity against arguments such as this, I shall make no attempt to argue that the argument is sound; rather, I only claim that the argument is valid.
Source: with the exception of premise (6), this material was taken from Michael Cannon Rea, Oxford Readings in Philosophical Theology: Trinity, Incarnation, and Atonement (New York: Oxford University Press, 2009), 3-4.