See Part One. Patrick's words will be in green. He is a philosopher.
I'm not sure I see how Elena's post helps. The alleged problem with the Trinity isn't that we can't really understand it or that it's nonsense. If this were the supposed problem, then analogies could help. But, the problem is different. The alleged problem is that we understand the doctrine quite well enough to see that it is logically contradictory. The foe of the Trinity says, in effect, you can use all the Lewisian "flatland" analogies or musical analogies or geometrical analogies you like: none of them will change the fact that there's a self-contradiction in the heart of the doctrine.
To elaborate. The doctrine, very roughly, tells us that the Son is God, the Father is God, and the Holy Spirit is God; but the Son is not the Father or the Holy Soirit, the Father is not the Son or the Holy Spirit, and the Holy Spirit is not the Father or the Son. In other words, it looks like the doctrine commits us to the following contradictory views. First, the Son is God: S=G. Second, the Father is God: F=G. Third, it's not the case that the Father is the Son: ~(F=S).
If S and F are both identical with G, then S must be identical with F. Think of it mathematically: 4+4=8, and 6+2=8. Therefore 4+4=6+2. But in the case of the Trinity, this necessary and obvious truth about identity is denied. So, the charge goes, the doctrine is contradictory.
One might try to reply to the charge by cashing out the "is" in "The Son is God" as indicating something other other than identity. But what? Obviously, it's heretical to think of it as a parthood relation. (The Son is not part of God: God is not composed of the the three Divine Persons, for God is absolutely simple and incomposite.) So what other sense of "is" can you substitute if not identity?
One might try invoke the doctrine of "relative identity." This is a rather technical move, which some very competent philosophers have endorsed in recent years (most notably van Inwagen). But it's tricky, not least because it involves the rejection of a real identity relation and may wind up heretical after all. And the logically revisionary character of the move detracts from its attractiveness. I do believe, though, that Suarez endorsed relative identity in the case of the Trinity, so it surely has precedent in the philosophical tradition. Of course, St. Thomas had no sympathy for this sort of move: but again, it's awfully tough to make St. Thomas's view work out.
I myself believe with the certainty of Faith in the revealed truth of that God is Three Persons in one Substance, but I still recognize the seriousness of the logical problem here. And I don't think thought experiments like Lewis's or Elena's really help with that.
Interesting. Isn't the logical problem overcome, though, by the prior recognition of the possibility that one Being can subsist in three persons? For us, one Being is one person, but how can it be ruled out logically (or axiomatically) that Being and person may not always be in a one-to-one relationship? I think, then, that the flatland analogy is quite relevant, precisely because it hits upon this difference of perception and defined realities which is the prior axiomatic consideration before we even get to logic. The flatlander says that there are only two dimensions, so that talk of a third dimension is meaningless and incomprehensible to him.
So Lewis was maintaining that this is how we are with regard to the Holy Trinity. We can't imagine one Being (even God) existing with more than one Person. But who's to say that we understand all of reality and that there isn't something more?
We know that there is a third dimension and that a cube has a oneness in a "greater" sense than a square possesses oneness. It's more complex, yet it remains a single entity. We can't comprehend with our thinking abilities alone, how God could subsist in three Persons, but it is not logically impossible or intrinsically self-contradictory, in my opinion. I think it's just very difficult to grasp, and must be accepted primarily by faith.
Revelation claims that it is a communication from this greater world. We arrive at the Trinity from revelation, not natural reason. But it's not inherently contradictory if we allow the possibility of three persons in one God, and don't rule it out beforehand.
That's how we can say the Father is not the Son, etc., and not be contradictory, because it isn't polytheism we're talking about, but rather, distinction of Person only.
You ask: "Isn't the logical problem overcome, though, by the prior recognition of the possibility that one Being can subsist in three persons? For us, one Being is one person, but how can it be ruled out logically (or axiomatically) that Being and person may not always be in a one-to-one relationship?"
I don't see how the logical problem is overcome by saying (something that I, of course, believe wholeheartedly) that person and substance need not always stand in a one-to-one relationship. Indeed, there are philosophers today who would gladly grant that there can be more than one person in relation to only one substance. Imagine you believe that being a person is strictly a matter of having the capacity for (conceptual) thought. So if an object can think, it's a person. But now imagine a human being who has suffered some kind of serious brain surgery (a commisurotomy, which severs the brain hemispheres, say), and can, as a result, think in two disconnected "centers" of thought. In a case like this, many people would be inclined to say that there are two persons that somehow inhabit that one human body. So it's not inconceivable, by any means, that there be more than one person in relation to just one substance.
The real question is—what's the nature of those person/substance relations? In the imaginary case above, we wouldn't be at all tempted to say it's an identity relation. Neither of the two persons is identical with the human body. But in the case of the Trinity, the claim is exactly that: the persons are all identical with the Divine substance, but they are not identical with one another.
I think it is easy to gloss over the difficulty here. Even the great Dominican Thomist Fr. Garrigou-Lagrange treats the issue far too cavalierly. In his Reality, he wrote: "The objection runs thus: Things which are really identified with one and the same third thing are identified with one another. But the divine relations and the divine persons are really identified with the divine essence. Hence the divine relations and the divine persons are identified with one another. The solution runs thus: Things which are really identified with one and the same third thing are identified with one another; yes, unless their mutual opposition is greater than their sameness with this third thing. Otherwise I say no. To illustrate. Look at the three angles of a triangle. Are they really distinct one from the other? Most certainly. Yet, each of them is identified with one and the same surface."
Now I myself find this very perplexing. The angles surely are really distinct. But nobody would try to suggest that angle A=surface Y, where surface Y is the whole triangle! Angle A=surface Y if and only if surface Y just is that particular part of the triangle that the angle, so to speak, occupies. But then there's no problem of identifying angle A and angle B with some third thing (i.e. surface Y). For it would be silly to say angle A=surface Y, and angle B=surface Y. On the contrary, angle B=surface Z. Indeed, as I pointed out in my initial post, it's quite obviously heretical to cash out the "is" in "the Son is God" as implying a parthood relation between the Son and God. So there's no comparison between Fr. Garrigou-Lagrange's "illustration" and the actual case at hand.
I'm inclined to think the matter cannot simply be brushed off as a matter of our failure to grasp the mystery. Of course, we can't adequately grasp the mystery, and the full truth of the matter will not be revealed to me until (God willing) I behold the Beatific Vision. But we can deal with the logical issues. I think they have to be solved, and, as I say, I'm skeptical that flatland-type illustrations help.
Changing keys: I'm sure illustrations like that do help people who aren't deeply troubled by the (slightly technical) logical worry I'm posing. St. Patrick's lesson with the shamrock, Frank Sheed's brilliant discussion of the Trinity, Lewis's analogy or Elena's might very well be of great help in leading people to a better grasp of this central truth of the Christian Faith, and so I don't mean at all to deny their value. I only want to deny their value in defusing the real logical worry.
What I'm not grasping is your objection! You wrote:
"the persons are all identical with the Divine substance, but they are not identical with one another."
I still don't see how this is a logical contradiction, once you grant the possibility of the trinitarian premise: God can subsist in three persons. To make a VERY imperfect analogy, I am a father, a son and a husband. All those things are me (though not all of me; as I said, it's imperfect):
A. Dave is a son.
B. Dave is a father.
C. Dave is a husband.
Once you grant what you did ("Indeed, there are philosophers today who would gladly grant that there can be more than one person in relation to only one substance"), then why do you still maintain that a logical contradiction is entailed? I don't think it is. So in the above analogy, I can be all three things, but that doesn't mean they are identical with each other:
a son qua son is not a father qua father, (etc.)
The concept of "Dave" (the totality of my being, self, or whatever term you wanna use) includes all these things (and many more) as an overarching concept, without contradiction:
1A) Dave is a son of Graham and Lois Armstrong.
1B) This son of Graham and Lois Armstrong is Dave
Armstrong. So "son" = "Dave".
2A) Dave is a father of four children: three boys and a girl.
2B) This father of four children: three boys and a girl is Dave. So "father" = "Dave."
3A) Dave is the husband of Judy.
3B) This husband of Judy is Dave. So "husband" = "Dave."
But no contradiction is entailed. This father and this son and this husband are all me, but it doesn't follow that a father equals a son: that they are the same thing in relation to each other; no, they are different in that relational sense.
Therefore, if I can be all three things simultaneously and yet remain the same person, yet son and father and husband remain distinct categories, why is it contradictory for God to contain three persons (analogous to my three relational attributes) and remain one God, and also for the three persons to be distinct in relation to each other, yet each being God?
Perhaps I'm missing something here, and I am quite aware that you are trained in philosophy and I am much less so, but that's what was in my mind (for better or worse), so I thought I'd throw it out to keep the dialogue going. I really want to learn more about this. I think it's very stimulating. This is the kind of "bottom line" philosophy and mind-exercise that I love.
I think the trouble with your analogy about your being a husband, a father and a son is that at least the first two are what might be called "phase sortals," as opposed to what might be called "substance sortals." Your being a husband is an entirely contingent matter: you could have existed without being a husband. (I demonstrate this truth by pointing out that you did exist, for quite a few years, without being a husband, and it's always valid to infer possibility from actuality!) Similarly, your being a father is entirely contingent.
Your being a son is a trickier matter, since one might think that sonhood is an essential property—something you couldn't possibly fail to instantiate if you exist. This is hard to make a decisive ruling on: is it possible that God have created you—not someone otherwise exactly like you, but actually you—without your having been the son of your parents? I'm inclined to say "no," but I admit to being in over my head here. So sonhood is a tough one to figure, as far as its essentiality goes. But we'll say more on this in a moment.
Go back to fatherhood and husbandhood. Given what we've said about the contingency of your exemplifying these properties, it follows straightaway that you, Dave, are not identical with some object (or essence) we might name "Father Dave." If you, Dave, were identical with that thing, then you could not possibly exist without that thing existing. But you did exist without that thing existing. So fatherhood is just a property you exemplify contingently, and thus it is not going to help with the Trinity, where, again, the central logical worry is the identification of the persons with the Divine Essence.
You might think sonhood is different than fatherhood, if you agree with my tentative suggestion that your being a son is an essential property. Since (for the sake of argument) you could not fail to exist without being a son, then that property is, at any rate, relevant to the discussion. I still don't think so. To see why, consider the following. It might make sense to say that you, Dave, are (identical with) a son. It might make equal sense to say you are identical with an animal; to say that you're identical with a carbon-based life form; to say that you're identical with a thing with an immortal soul; to say that you're identical with etc. All of these identity statements might be accurate, but that's just because they're all different, partial ways of describing what it is to be a human being. None of the descriptions really purports to be identifying you, Dave, with that partial description. If they did, they'd be false. (I mean, it's true that you're identical with a son, but that's not all that's true of you: it's only a very partial, and pretty uninformative, description.) In the case of the Divine Persons, this isn't going to work, or at least, I don't see clearly how it will work.
It certainly will not work if it makes Sonship a mere essential property of Godhood, in the way that sonship is an essential property of yours. For then "sonhood" becomes simply a partial way of describing God, and that's no good. (I don't know if this reduces more clearly to Modalism, or to a rejection of the incompositeness of God, but I think it reduces to one or the other.) But if this isn't what it's doing, then I don't see what it is doing.
A further point is that all the properties that you identified as part of the concept of Davehood (so to speak) are quite compatible. There is no conflict—nothing at all worrisome—about your being both a husband and a father. It is, however, precisely the point of the Trinity that the persons have incompatible relations. The Holy Spirit, for example, proceeds from the Father and the Son. The Father, however, does not proceed from anyone. These properties are contradictory. So the issue is a lot more complicated than your having a number of quite compatible properties.
I'd sure be happy if I could wind up endorsing something like St. Thomas's solution to the problem. But as I say, it's trickier than it might seem. The central issue is identity. I think if you're going to solve the puzzle, you've got to clearly show how the real identification of two things with a third thing doesn't require you to identify the two things. I don't think the analogies help with this.
Incidentally, there are some recent papers on these problems available online. They are, in some instances, pretty technical. Michael Rea has written a couple of papers [one / two ].
A nice paper by Trenton Merricks is available. [link]
A key paper to see in terms of defending Relative Identity is van Inwagen's "And Yet They are Not Three Gods, But One God," but I don't believe that one is online. If you have access to an academic library, you can find it in his collection God, Knowledge, and Mystery.
I should also point out that I have never really thought philosophically about the Trinity. I believe the doctrine, and so I believe there is no contradiction to be found. But I've never even tried to work out my own solution. Perhaps, once I get to it, I'll find St. Thomas's compelling. Perhaps I'll lean more towards Suarez. I don't know. Anyway, I just wanted to point out that I'm groping here as much as (or, perhaps, significantly more than) anyone. My perplexity over the details is very real.