The Logic of the Trinity
There are principles of knowledge, one of which is the law of non-contradiction and it is the fundamental law of all rational thought. The law of non-contradiction informs us that something cannot be true and false at the same time and in the same sense. The doctrine of the trinity does not violate this law. To show this, I will state what the trinity is not. Taken from the Baker Encyclopedia of Christian Apologetics:
"The Trinity is not the belief that God is three persons and only one person at the same time and in the same sense. That would be a contradiction. Rather, it is the belief that there are three persons in one nature...Further, the Trinity is not the belief that there are three natures in one nature or three essences in one essence. That would be a contradiction. Rather, Christians affirm that there are three persons in one essence. This is not contradictory because it makes a distinction between person and essence. Or, to put it in terms of the law of non-contradiction, while God is one and many at the same time, he is not one and many in the same sense. He is one in the sense of his essence, but many in the sense of his persons. So there is no violation of the law of non-contradiction in the doctrine of the Trinity."
So, one could say that God has one "What" and three "Whos." The three persons (Whos) share the same essence (What). Norman Geisler stated, "So God is a unity of essence with a plurality of persons. Each person is different, yet they share a common nature." God is one in his substance. The unity is in his essence, while the plurality is in God's persons, or how he relates within himself. Descriptions of this relationship are within the Bible, showing how the Son and the Father relate, and how the Father sends the Spirit as a Messenger, and the Spirit is a Witness to the Son (John 14:26). Reading descriptions like that help us to understand the functions within the unity of the Godhead. "Each is fully God, and each has his own work and interrelational theme with the other two. But it is vital to remember that the three share the same essence, so that they unify as one Being," (Geisler).
I hope this helps to see the logic and model of the Trinity. Think of this post as one to wet your appetite for more study (there will be more). If I have time, I'll probably continue this in tomorrow's post with some illustrations and scriptural defense. It's important to know that the Trinity is not at all a contradiction, so that's why I laid the foundation as I did.
Norman Geisler, "Baker Encyclopedia of Christian Apologetics."