Okay, show me. Using the language of logic, explain to me how the concept of the Trinity is logical. I will pay close attention.
First, on one point you are right and that is I accepted on faith the explanation that the nature of God is best explained in using a three person description. That was true in the beginning. This is true in any educational endeavor in the whole of human education. We accept what teachers say on faith. On most matters it remains a faith. On some we move on into testing this information ourselvees, for some who do think about what they think at all.
But with the years, I have come to understand God from experience. Some of this I can share but there is a fundamental problem in your thinking that you will need to adjust. You are not alone. A lot of Christians tend to think this way as well and it is this. God is not a concept or a theory an explanation. God is a living being. So we are talking explaining or coming to understand a living being. This is different than a theory that needs to conform to the rules of logic. You might as well be asking me to explain the body my husband has in terms of logical constructs. While there is logic as to why his body works as it does, explaining the nature of a living being is not the same as a concept.
A living being needs to have a nature that enables it to perform all the functions it needs to perform. A theory does not. A theory can logically explain the functions living cells or beings perfom in that we read the theory, observe the living form, and see that what the theory proposes matches real life. The logic consists in the matching of the words to what is actually observed to be true and if all goes well, fits perfectly and the living subject performs its function prefectly. I assume you are following this. In not, just ask me and I will give you examples. I studied and work in science so I have a lot of examples from that field.
So if we move onto the nature of God, we are talking trying to understand how a living being operates. This I do with the personal permission of said living Being, by the way. His nature (different than His character) is how he is and interacts with man whom He made to interact with Him. In this goal, He is able to be with us as He is of a spiritual nature. That is, He is not limited to being within a physical body as we are. This enables Him to be with us and with many, or in fact all, men should those men want Him to be with them. While He is with men, He is also in a dimension that requires His presence in a kingly way. That is, He is ruling and judging being the Judge of all the Earth and so it performing that office at the same time. This gives us a short description of two elements or persons of his being defined by function.
There is the third which is the more troubling for most. In order to satisfy justice and mercy, He also submitted to being born as a man in order to save the other men (includes women) from the destiny they had deserved which would mean no relationship with Him at all. All the while He was in the body as a man, he was still the Judge of all the Earth and still moving by His spirit on said earth. So you see the three functions of God which we call triunity....unity in diversity.
This is the closest one can come in using logic to explain the nature of a living being as far as I can see. Again, if I were to explain to you the logical nature of the human body (studied medicine) I can do so in terms of physiology and its logical functions that allow us to perform all the functions necessary for life ideally (meaning not ill) but it will not be the logical constructs of a theoretical argument that has no counterpart nor way to measure its validity in real life.
I suspect that you will not accept the above as I have never met an atheist yet who was able to see God as a living Being and not merely an idea they want to discuss. There are a number of theologians, if not most of them, who suffer from the same limitation. And as I said, a number of Christians also feel this way about God except when it comes to Him answering their prayers. They discuss Him as though He were an idea about whom they can make up anything that suits them. Trying to find out what suits said living Being does not occur to them.