It is philosophical. Coming from a Baptist tradition, I think I can fairly say that Baptists make very little use of philosophy at all. The unfortunate part of that is that they insist on teaching and accepting doctrines that are by the nature necessarily philosophical, but failing to understand that, they think theological language is the same as an explanation. They do this (again, in my experience, which is extensive in Baptist circles) certainly with the Trinity, but also with the hypostatic union and even to some extent creation
ex nihilo and pretty much anything that say with reference to God's omnipotence, immutability, and aseity. The Calvinist argument that has been raging for the last two decades in the SBC is completely and totally rooted in a failure to distinguish correctly between philosophical and theological problems.
OK, so much for that rant. Bringing it back to this thread, it's important to note that the Trinity is not a doctrine to be explained. It
is the explanation of the biblical data. So asking for an explanation of the Trinity is just asking for an explanation of the explanation. And that's fine, but then we have to ask what kind of explanation the Trinity is. And the answer is that it is a philosophical analysis of the biblical data. Therefore, you
can't explain the Trinity using only (or even primarily) biblical language, because that's not what it is. Again, the Trinity is, by definition, a philosophical explanation for the biblical data, and it is that data that is the theological language we are trying to understand.
So theological data is actually pretty simple. You need to affirm these statements:
1. There is only one God
2. There is a Person called the Father is who God
3. There is a Person called the Son who is God
4. There is a Person called the Holy Spirit who is God
5. These Three Persons are distinct from one another (that is, the Father is not the Son or the Holy Spirit, and the Son is not the Holy Spirit).
If you affirm those five statements, you are orthodox. The problem becomes answering the question, "How can we hold those five statements to be true together?" The
answer to that question is the doctrine of the Trinity. Therefore, to be clear, you could--in theory--reject the Trinity (insofar as you reject
the explanation) even as you accept all five statements and still be orthodox. But practical problem is that in rejecting the explanation of the Trinity, you end up denying at least one of the five. And sadly, many people--especially Baptists--don't understand the explanation because they don't understand philosphy, such that they do hold all five statements to be true, but their own attempts to explain it theologically rather than philosophically fall short such that they do not accept the Trinity as actually stated, and worse, their own explanations (unintentionally) entail a rejection of one of the five statements. This usually takes the form of either a latent Tritheism (i.e., a social gospel) or else a latent Modalism (i.e., comparing the Father, Son, and Spirit to the three states of water or to the three sides of a triangle).
If, then, I've sold you on the importance of the philosophical language, I'll just address the notion of procession. Aquinas discusses it in some detail here:
http://www.newadvent.org/summa/1027.htm
So strictly, a procession is just something that proceeds or comes from something else. So in any cause/effect relationship, the effect is a procession of the cause. Taken one way, the effect proceeds from the cause; taken another way, the cause proceeds to the effect. My wife and I producing children might be an example of the former: our children proceed from (are processions of) us. The second is more common, where this causes that: a baseball shatters a glass, in which the shattering is a procession of the ball hitting it in a certain way at a certain time. Now the Bible clearly says that the Son proceeds from the Father (John 8:42), so we know there is procession in God. The problem is that in these processions, that which processes from is different that which processes to. The effect and cause are different, and that obviously. But taken seriously, this should imply that the Father and Son are different things entirely, where the Father is the cause of the Son. But that would create very seriously problems, as you can imagine.
The solution is to recognize that the types of processions discussed above are called "outward processions." This proceeds externally to that. But at least in the case rational creatures, for every outward procession there is a corresponding inward procession. In the case of the will, we call this inward procession an intention. In the case of the intellect, we call this knowing. So if I knowingly throw a baseball through the window, then there is an outward procession of the second type discussed above, but there are also two inward corresponding processions--there is my knowledge that if I throw the ball through the window it will shatter (I had to think about that to know it), and then there is the related intention to actually throw the ball through the window. In both of these cases, there is an internal action that
precedes the outward action and, in fact, gives rise to it. Now, in any inward procession, that which proceeds from and proceeds to are identical. It is I myself who knew that the ball would shatter the glass (you see the inward procession from me to what I know), and it is I myself who intended to so throw the ball (again, you see the inward procession). In both cases, I am the same--I was that which proceeded in myself, from this, to that. There are more technical reasons this is true when you get into a deeper philosophical understanding of the the intellect and the will actually are, but let that pass for now.
Now if there is inward procession in us, then how much--how much higher--ought any procession in God be inward as well. And to quote Aquinas here, as I think he is rather clear on this point: "Procession, therefore, is not to be understood from what it is in bodies, either according to local movement or by way of a cause proceeding forth to its exterior effect, as, for instance, like heat from the agent to the thing made hot. Rather it is to be understood by way of an intelligible emanation, for example, of the intelligible word which proceeds from the speaker, yet remains in him. In that sense the Catholic Faith understands procession as existing in God."
So you will find out that there are two processions in God. There is the procession of the Word (which will for still other reasons come to be called generation) and there is a procession of love (which for still other reasons is not generation, but simply a procession absolutely). And these processions will form the basis of the relations between them, and those relations are identical with the three Persons.
Somewhat technical, I know. But thoughts so far?