Let's do a little theology.
Consider the following scenario. You and I are walking along a sandy beach one beautiful day, when we come across a strange black box.
"What is it?" you breathe, puzzled by its strangeness.
"It's a strange black box," I say blandly.
"No," you reply, unable to keep at least a touch of sarcasm out of your voice. "What is it?"
"I don't know," I reply.
In this crude example is found the point of negative theology. We often talk as if there were only one "What" question, with only one answer. There are, however, layers of "What" questions, and layers of answers to them. This helps us understand a famous saying by Aquinas: We cannot know what God is, only that He is. I have dealt with people (unfortunately, if you ever do history of philosophy, you will deal with many such people) who rather sarcastically respond to this with something like the words: "Well, you have to know what a thing is before you can know that it is." And this is, after a fashion, true. But there are layers to what a thing is.
Let's return a moment to our black box. I see that it is black, I see that it is box. I know what it is: a black box. In another sense, I also don't know what it is at all. In a sense, all I know is how the box is related to me: it is visibly black, it is visibly box-like, and that's all I know. I don't know what the box is in itself. I don't, however, need to know what the box is in itself (the deeper answer to a "what" question) in order to show that it exists. All I need is to have some knowledge of the box under some description. Mutatis mutandis, this is what Aquinas is noting about God. We do not know (in this deeper sense) what God is; we only know that He is (under some set of descriptions). The descriptions under which we know God do not capture Him in Himself, but only relative to us in some way (e.g., as Cause, or as Revealer, or whatever).
Here's another saying that belongs to Aquinas: We do not know what God is, but only what He is not. Again, the "what" in question is the deeper "what"; we cannot ever say what God is in Himself - to do so would be an impossible task for any finite creature - but only what He is not (in this deeper sense). The reason we can do this is that we know what God is (under some set of descriptions) and from these some things follow about what God is not (in the deeper sense). This is what is called negative theology. It has traditionally been the key to all good philosophical thought about God. It's hard to find these days.
Consider the doctrine of the Trinity. With negative theology as a restraint on irresponsible speculation, one would always be reminded that we do not, and can never, know what it is like to be God. It is not possible to 'give an account' of the Trinity. We cannot say what the Trinity is, only that it is. We cannot know what the Trinity is, only what it is not. The Church Fathers noted that there are very distant sorts of analogies to the Trinity. The doctrine of the Trinity is this: "The Father is not of anyone, the Son is of the Father, the Spirit is of the Father with the Son; the Father is God, the Son is God, the Holy Spirit is God; the Father is not the Son, the Son is not the Spirit, the Spirit is not the Father or the Son; yet there are not three Gods, but only one God." Basil recognized that we already recognize something like this in other things. For instance, take Paul, Silvanus, and Timothy, three human beings. Paul is human, Silvanus is human, and Timothy is human; Paul is not Silvanus, Silvanus is not Timonthy, Timothy is not Paul or Silvanus; yet there are not three human natures but one human nature in virtue of which they are all called human. Basil does not pretend this is an exact parallel; indeed, if anyone did claim to find such an exact parallel it would be in clear violation of the reasonable restraints of negative theology. The point is that, while the unity-without-confusion of the Three exceeds any other sort of unity-without-confusion that we know, nonetheless we are familiar with this sort of pattern. Some took thoughts like Basil's a bit too literally, and had to be corrected by Gregory of Nyssa, who clarified the difference between Paul, Silvanus, and Timothy on the one hand, and the Father, the Son, and the Spirit on the other. Each person of the Trinity is distinctly a person, but they share their attributes in common according to their distinction as persons. Take willing. What the Father wills, the Son and the Spirit will: their willing is one willing, but as Gregory says in To Ablabius on Not Three Gods, "every operation which extends from God to the Creation, and is named according to our variable conceptions of it, has its origin from the Father, and proceeds through the Son, and is perfected in the Holy Spirit" such that the power of willing is "issuing from the Father as from a spring, brought into operation by the Son, and perfecting its grace by the power of the Spirit." In other words, they will one thing with one will; but the Father wills it as the Origin or Principle of Godhead, the Son wills it as being immediately from the Father, and the Spirit wills it as being from the Father with the Son. This is itself little more than a restatement of the doctrine of the Trinity in terms of willing: having granted the doctrine of the Trinity, we had already implicitly granted this, and our restatement has just clarified one aspect of what we had granted in an obscure and general way.
Now, it has been argued by Richard Cartwright that the following seven claims are, when taken together, inconsistent:
l) The Father is God
2) The Son is God
3) The Holy Spirit is God
4) The Father is not identical with the Son
5) The Father is not identical with the Holy Spirit
6) The Son is not identical with the Holy Spirit.
7) There is one God.
But this is hasty. Where is the inconsistency? To take (1) through (6) all together is consistent with (7); we can see that from the Nyssan's point about willing. (7) can only be inconsistent with (1) through (6) if we assume that the Father, the Son, and the Spirit are not identical with each other because they are different Gods. But this is not the doctrine of the Trinity.
This will be found to be a common problem with claims of the doctrine's inconsistency: the set of (1) through (7) can only be proved inconsistent on the assumption that it is. I have no strict proof of this; but it seems to be the case. Cartwright holds that (1) through (6) are a consistent set, and that the set only becomes inconsistent when (7) is added. (4), (5), and (6) share no common term with (7), so cannot conflict with it. (1), (2) and (3) are not, taken together, inconsistent with (7); they lead to the conclusion that the Father, the Son, and the Spirit are one God. Let's call this conclusion (C). Put (C) with (4), (5), and (6); the resulting conclusion is that the Father, the Son, and the Spirit are one God but that the Father, the Son, and the Spirit are not identical to each other. This can only be an inconsistent if non-identity of each to the others means non-unity of divinity. But we (Cartwright and anyone else) have no basis on which to argue such a claim from deeper knowledge of the divine nature, since we don't have special insight into the divine 'inner nature', as we well know through negative theology. And Gregory has given us a statement of the divine unity that seems to allow for both non-identity and unity in a consistent way: the substance is not divided (it is explicitly one) and the persons are not confused. Indeed, if Gregory is right, it seems to follow that, given their relations, the distinctness of the Persons implies that they have one being, one willing, etc.; and that the nature they have in common is had in common in such a way that the persons are necessarily distinct. One might argue that there is some hidden inconsistency here, but again, it couldn't be from any special insight into the divine nature, and it is pointless to complain about the difficult of understanding what it would be like for something to be like this, because we aren't being asked to understand what it would be like for something to be like this - that's the deeper "what" question again (it just keeps sneaking back, doesn't it). Another saying, this time attributed to Augustine: If you understand it, it is not the Trinity. Negative theology again. If you understand it (deeper "what" question), it can't be attributed to God without pretending you know what it is like to be God. The whole point of negative theology is to keep us from such absurdity.
I've put a "Part I" in the title because I will probably discuss this issue more at some point in the future; this is all rather rough.