I haven't read every reply here, but I would say that it is most obviously illogical, and that's not a bad thing. It just means it does not fit within the human-created categories by which we can prove or support our suppositions. Being 3-in-1 and 1-in-3 is not logical. Obviously 3 and 1 are different numbers, and it's difficult to comprehend how anything can be both at the same time.
But I would say that we can know what the Holy Spirit has enlightened us to know. Anything further than that is speculation, which may or may not venture further than it is wise to venture in talking about matters of God. This is why in the traditional churches we stick to the Creed of Nicaea-Constantinople as what we can affirm about God without a doubt. Everything else may be enlightening in whatever way, but we only need to positively affirm that. In Arabic we call it "the law of faith" (qanun el iman), which I think sums it up pretty well. You can debate a law and call it illogical all you want, but at the end of the day, it's still going to be a law. Like gravity or the conservation of motion or something like that (not like "no drinking on the sidewalk" or whatever). It just is. I think that was one of the points HH St. Athanasius the Apostolic made to his letter to Serapion on the Holy Trinity (which I can't find online right now, or else I'd link it here), where HH starts off by saying something like "Foolish men pry too much into the workings of God; you say you believe in the Holy Trinity, so is that not enough to know and believe?" (very liberally paraphrased; the point being -- and this is usually the point in Egyptian Christianity, I have found -- that to believe is in itself enough, whether or not we are intellectually satisfied in the details; as St. Anthony the Great, the Father of Monks, said regarding a brother's correct answer to a question about the scriptures, "Abba Joseph (or whichever Abba it was; my books are in storage, so I can't look it up) has answered correctly, because he said 'I do so, not understand.'")
We worship the Father and the Son and the Holy Spirit + not because it is most logical to do so, but because this is the revelation of God to us, and so this is how we must worship Him in order to worship Him in spirit and in truth, as our Lord and Savior Jesus Christ said that the Father is seeking such worship. So to do otherwise is to betray God.