Some people have trouble with paradox because it seems illogical. However, this is largely because our thoughts are limited by our own experience.
No human being is able to understand even the whole universe simply because they can't observe all of it, or experience all of it.
All our "knowledge" of the universe is based on the tenuous assumption that the whole universe is exactly as we observe our little corner of it to be. It is entirely possible that other portions of the universe could be vastly different and we would never know it.
Taking a step further down the rabbit hole... Our knowledge of our little corner of the universe is also very limited, in reality, and our understanding very limited, because of the reality that our ability to observe and experience is very limited.
How many people have ever seen an electron? or a quark, or quanta. According to what we currently think the universe and its mechanics are based on particles only a few of which we can actually observe in any concrete way.
The irony here is that religious people have trouble with paradox, when all of our modern theoretical physics is based on a huge paradox that no one, as yet, has been able to resolve.
Wave - particle duality, according to physics is observed reality, and it is foundational to our understanding of the universe right now.. yet it is an apparent contradiction that boggles the mind.
The point being limited viewpoint, limited understanding, limited knowledge all can make the truth appear to be contradictory. Not only can it, we know that it does even in the physical world around us.
So then... we have a God who created everything. This logically requires that God existed before everything else and his mind gave rise to everything else (assuming that the creation was intentional as the bible describes it).
This means that God's mind is more complex and more vast than the whole universe and everything that it contains.
Now, we already know that we are currently incapable of understanding even one small corner of the unvierse... How then could you possibly, logically, come to the conlusion that God must be within the limits of your understanding?
In fact, as far as we know both from science and from the bible, Time itself is tied to the physical universe and it came into being simultaneously with space. Thus God existed "before" time. Which in itself is such a contradiction that it can't even be expressed in language without creating an oxymoron.
Try some time to answer the question in your mind "Where did God come from?".
We all know, of course, that God didn't come from anywhere.. he has no beginning, he simply is.
But its one thing to accept that, its another to wrap your mind around it. If you really try, you will find that you can't. The human mind is not capable of really understanding the idea of something that has no beginning, something that has no cause.
We can accept it, but we can't understand it.
Think, for a moment about the smartest human being you know.. or the smartest you have ever heard of. Think of the stuff that they understand that you don't. Perhaps try reading a thesis paper in advanced quantum theory or particle physics.
Now realize that God is not 10 times smarter than that, he isn't 1000 times smarter than that, he isn't a million times smarter than that.. he is infinitely smarter.
What that means is if you were the smartest human being who ever lived, and you did nothing but study and learn for all of eternity, you would never become as smart, or as wise, as God is right now.
If you can't understand even an advanced physics paper, or philosophy, or math, written by a human.. how could you possibly hope to understand God?
The thought that a mere man, even the smartest man who ever lived, could comprehend God is laughable.
Really, we should all be very happy about paradox, because without paradox, there would be no possability of relationship with God.
First, consider this...
We know from scripture that God is eternal, he existed before his creation.
We know from scripture that God is unchanging. His nature never changes, he is the same yesterday, today, and forever.
We also know from scripture that God is perfect and does not need anything.
So.. if God is simply one being, one person, who has always existed alone, how could relationship be part of his nature?
The answer is, it couldn't be. If God is unchanging, and he has existed always alone, and he is perfect in himself, needing nothing... then he must by nature be solitary and alone. The concept of relating to other beings would be completely foreign to him.
Another question.. how could this God who exists alone by nature, be a God of love? Love requires an object. It is by nature relational. If God is love.. what did he love before creation?
Who did he relate to before creation?
I have heard some people claim that God created so that he could have creatures to relate to. This suggests that God is relational and loving... but it also implies that he is imperfect and incomplete. In other words what they are saying is that God needed creation to complete himself. In other words, he depends upon his creation for self fulfillment.
Such an idea is repugnant to scripture of course.
If we know that God is love, and that he is relational by nature, and we know that he is unchanging (ie he has always been this way, he didn't become love etc), and we know that God is perfect and complete in himself... then we must necessarily conclude that God exists as a love relationship within himself.
It is the only logical possibility. given the facts we know from the bible.
But how is it possible that anyone could be a relationship within themselves, especially a love relationship. Keeping in mind that the biblical definition of love is selflessness. Selfish love is not a possibility for God.
The only viable conclusion is that God is a multiple person being. He is one being which consists of multiple persons who are in relationship with each other and selflessly love each other.
The only objection here is simply that we have never seen any other multuple person being. However this is not a logical objection. The fact that we have never seen a given type of being does not logically require that it does not exist. Especially given the fact that we are dealing with a being which logic has already shown to be beyond everything else we have ever experienced.
So... not only is the Trinity logical.. it is a logical necessity if you wish to have a complete biblical view of God without contradictions.
More than that, the Trinity is the only monotheistic explanation of God that allows for a God of relationship and love. (ever wonder why muslims don't believe in a God of love and relationship?)