I figured that I'd jot down some things I've been learning this summer; maybe someone else will be interested, maybe not, but it'll force me to think through it. The main thing I've been reading up on so far has been 13th century trinitarian theology; mainly Duns Scotus with some Henry of Ghent, but also a little Bonaventure and Aquinas (especially after being corrected on some errors in thinking on my last post). Much of the following is recapping a dissertation I've been reading by Russell Friedman, entitled, In Principio Erat Verbum: The Incorporation of Philosophical Psychology into Trinitarian Theology, 1250-1325.
All of the thinkers in this period look back to Augustine and Anselm in their formulations of the Trinity. Many of the strategies of these two are incorporated into later, more precise Scholastic thought, though in different ways. Aquinas and Bonaventure are similar in their thinking, yet develop differences which are further developed by succeeding thinkers.
One model of thinking about the Trinity involves looking at the relations between the persons. Augustine in particular champions this model (though I believe that similar steps were taken by at least one of the Cappadocians; I'll have to look this up). The key thing is that whatever distinguishes the persons in God can't be an essential property as they all have one essence. Neither can it be an accidental property, as God doesn't have any accidents in an Aristotelian sense (besides, we don't exactly want the persons to be accidental to God's nature, even if we are going with a contemporary, non-Aristotelian framework). The only other Aristotelian category which makes sense once you remove talk of accident and substance is relation. The divine persons are therefore distinguished from one another through their relational properties. In most beings, relations are a cheap manner of being, even below accidents; the fact that there are more books than DVDs in this apartment has nothing to do with the being of the books or DVDs. However, when these relations are within the divine essence, they have a manner of being as full as any substance.
Aquinas in particular takes up this line of approach. The persons are distinguished not merely by relations, but by opposed relations; the Father has the relation of paternity to the Son, and the Son has the relation of filiation to the Father. The Holy Spirit must have a relation to both Son and Father in order to be distinct from them, and so they both have the relation of active spiration to the Spirit and the Spirit has the relation of passive spiration from both the Son and Father (thus, the relation account demands the filioque).
Bonaventure, on the other hand, tends to take the lead of Richard of St. Victor and his emanationist model. In this model, the relations of origin become the most important; what makes the Son a distinct person is the origin of generation (procession in the mode of nature), and what makes the Spirit a distinct person is the origin of spiration (procession in the mode of will). The Father is who he is through being ungenerated. Generation and spiration are simply intrinsically different from each other and do not need opposing forces to distinguish them.
Now, here is an important thing to note: even though I have said that Aquinas and Bonaventure foreground the respective views, for them this is merely a conceptual distinction. Both speak of relations and both speak of emanations, but the ordering is the ordering of our concepts; these are really identified within God. Franciscans and others sympathetic to the Franciscan viewpoint on this issue, however, took the matter a step further. In particular, Henry of Ghent incorporated Augustine's psychological model of the Trinity along with Aquinas' philosophical psychology (with some tweaking). Augustine, in his De Trinitate, likened the Trinity to the memory, intellect, and will. Henry identifies the procession of generation with the procession of the mental Word (intellect/understanding) (thereby identifying procession in the mode of nature with procession in the mode of intellect), and the procession of the will (linked to love) with the spiration of the Spirit. Duns Scotus, Henry's pupil, follows in this, making this psychological emanation scheme his complete account of the Trinity by collapsing the relations into the emanations (this is, of course, leaving out many details on all sides).
I'm struggling with the psychological view of the Trinity myself. It's foreign to much of contemporary theology, and seems to be rather anthropomorphic. At the same time, I've been working through some reasonings for looking at it. Looking at the Son as God's understanding (as a simplification, at least; all of the above accounts separate God's understanding qua Son from God's understanding qua God), one can then make parallels to John's prologue, or Proverbs 8 with wisdom, or Genesis 1 with the ordering of the world from chaos. Similarly, identifying the Spirit with God's will makes sense considering who the Spirit is and how the Spirit acts in the Christian (at least as a broad characterization; I'm going for the big picture here). Similarly, it does seem like there is an outward movement from ourselves when we know something and when we will something, which nonetheless remains with ourselves. Finally, it does not seem (overly) anthropomorphic to say that God has a will or understanding; these seem to be pretty standard traits attributed to God and even within philosophy flow out of the cosmological argument. None of this is to imply that we have even the foggiest idea of God's phenomenology.