
The Trinity isn't a contradiction because "one" and "three" are counting different things
My thesis is that one God in three persons doesn't break the law of non-contradiction, because the "one" counts beings and the "three" counts persons. A contradiction needs the same thing to be affirmed and denied in the same sense and that's not what the doctrine says.
Think about it this way. One apple and three apples in the same bowl at the same time is a contradiction. One bowl and three apples isn't, because a bowl and an apple are different categories. So 1+1+1=3 only works when you're counting the same kind of thing.
Being is what something is. Person is who someone is, someone who can say "I" and talk to you as "you". A rock has being but no person, so it's one being and zero persons. You and i are one being and one person each. The Christian claim is that God is one being and three persons. Not three beings (that's tritheism and the church rejects it) and not one person wearing three masks (that's modalism and the church rejects that too).
In Scripture each person speaks as an "I". The Father: "then came there a voice from heaven, saying, I have both glorified it, and will glorify it again" (John 12:28). The Son: "Who art thou, Lord? And the Lord said, I am Jesus whom thou persecutest" (Acts 9:5). The Spirit: "the Holy Ghost said, Separate me Barnabas and Saul for the work whereunto I have called them" (Acts 13:2).
Someone might say then the persons are parts of God. But parts add up and you lose something if you take one away. The Son doesn't have less power or knowledge than the Father, so the three don't add up to more God. The Athanasian Creed says "none is greater or less than another". What makes them three is how they relate, the Father begets, the Son is begotten and the Spirit proceeds, not a piece each one owns.
To be clear, i'm not saying logic alone gets you to three persons. You can argue from "God is love" that God is more than one person, but that it's exactly three we know from Scripture. My point here is only that it isn't a contradiction.