While it is true that 2+2=5 is irrational (and a great Orwell reference), it might be more appropriate to use the example of 3x1=1 since theists seem to readily accept that idea. Here is my attempt to dissect the logic of the Trinity doctrine:

Trinity doctrine states that Jesus, the Father and the Holy Spirit are each omniscient, omnipresent, omnipotent and omnibenevolent. It seems necessary to add omnitemporal (existing simultaneously throughout time) since God is said to exist outside of time. So here are a few questions to ponder:

Is there anything that one Person of the Trinity is capable of doing that the others are not? Is any member of the Trinity able to act independently of the other 2? If all 3 are omniscient and omnipresent, then wouldn't all 3 persons be participating in every action by definition?

Do the different persons have different personalities? If so, wouldn't they have to have different ideas? If not, then how can they be said to have different thoughts and thus be distinct?

How is it logically possible to "beget" an eternally existent son?

Person and being are synonymous. Saying 3 persons in one being really has no logical meaning. None of the analogies holds up. For instance you might logically argue that ants have multiple beings in one uber "nest" organism. But those ants are all parts of a whole with distinct limitations. The 3 Persons of the Trinity are not each 1/3 of God. They are all 3 stated to be completely God. That means 3 Gods. 3x1 cannot equal 1.

If the Trinity is impossible for Man to understand, why did God bother? If all 3 persons are God, why muddy the waters? He could have just said that he is God and left it at that. It would have been an honest statement and would have removed an unnecessary cause for doubting the truth of the teaching. Or he could have explained the Trinity concept in the Bible. Surely if the concept had any value to our understanding, God could have made the idea comprehensible. It seems to cause a lot of heartache to those trying to believe as this post shows. That poor guy seems to need a good dose of the Rational Response Squad.

The most logical reason I can see for the Trinity concept is that put forward in Doherty's book. God was thought to be too holy to be reachable by corrupted flesh. Thus the Son was required as a bridge between corrupt man and holy Father. 1 Timothy 2:5: "For there is one God, and one mediator between God and men, the man Christ Jesus". The Holy Spirit appears to be the miracle facilitator. In modern theology, we no longer put any real limitations on the 3 persons so the whole Trinity concept has become incomprehensible.

Here is the best article I have found from the Christian side but ultimately they fall back on the tried-and-true "our minds cannot grasp it" explanation.

Interestingly, Christianity did not originate the concept of the Trinity. It can be found in Hinduism at least 1000 years before Christ:

Hare Jesus: Christianity's Hindu Heritage