Do the truths of Christianity have to make logical sense? The law of noncontradiction (the foundational principle for all logical thinking) asserts that two contradictory statements cannot both be true at the same time and in the same respect (A cannot equal A and also equal non-A). But how does logic apply to the Trinity doctrine?
Skeptics often claim the Trinity is a logical contradiction. For example, some assert that it violates the law of noncontradiction on the ground that the doctrine claims that God is one and not one, and that God is three and not three. This criticism is a straw-man argument, however, for orthodox trinitarianism does not assert that God is one and not one, three and not three. Rather the Trinity doctrine asserts that the way in which God is one (essence or being), he is not three, and that the way in which God is three (subsistence or personhood), he is not one.
Trinitarians assert that a person must distinguish God’s essence on one hand and God’s subsistence on the other. God is one in a different respect from the way in which he is three, and three in a different respect from the way in which he is one. Thus the Trinity, as one What and three Whos, is not a formal logical contradiction.
The Trinity doctrine is mysterious and incomprehensible to the finite mind of man, but it can be formulated in a way that is reasonable and avoids contradiction.
For more on the historic Christian doctrine of the Trinity, see “How Can God Be Three and One?” in my book Without a Doubt: Answering the 20 Toughest Faith Questions.