A student in the logic class I was teaching said that the Trinity doctrine was contradictory. He argued the following: Since the Father is God, the Son is God, and the Holy Spirit is God; and since the Father is not the Son, the Father is not the Holy Spirit, and the Son is not the Holy Spirit; then the result is that each person is simultaneously God and not God. This is, he reasoned, a violation of the law of noncontradiction (A cannot equal A and equal non-A).
This evaluation of the Trinitarian formulation is also a straw-man (misrepresentation), for again it fails to recognize the essence/subsistence distinction. The members of the Trinity all share equally the one divine nature and are thus the one God. However, the relational (personal) distinctions in the Godhead do not in any way subtract from each individual person’s possession of the divine nature. Thus the three persons are distinct from each other, but they nevertheless remain fully and equally God.
How one Being can simultaneously be three persons is an unfathomable mystery, but it is not a formal contradiction. It is not contradictory to predicate deity to all three members of the Trinity, while simultaneously asserting that they possess distinct personal identities: Father, Son, Holy Spirit.
The Christian view of God is different from other forms of monotheism in asserting that God is superpersonal (more than merely personal).
For more on the historic Christian doctrine of the Trinity, see “How Can God Be Three and One?” in my book Without a Doubt: Answering the 20 Toughest Faith Questions.