Chapter 16: The Divine Persons
Person in general is a being which has intelligence and freedom. Its classic definition was given by Boethius: Person is an individual subject with an intellectual nature.  Hence person, generally, is a hypostasis or a suppositum, and, specifically, a substance endowed with intelligence.  Further, since person signifies substance in its most perfect form, it can be found in God, if it be stripped of the imperfect mode which it has in created persons. Thus made perfect, it can be used analogically of God, analogically, but still in its proper sense, in a mode that is transcendent and pre-eminent. Further, since revelation gives us two personal names, that is, the Father and the Son, the name of the third person, of the Holy Spirit, must also be a personal name. Besides, the New Testament, in many texts, represents the Holy Spirit as a person. .
Now, since there are three persons in God, they can be distinct one from the other only by the three relations which are mutually opposed (paternity, and filiation, and passive spiration): because, as has been said, all else in God is identical.
Comment: The persons are distinct, numerically distinct. And they are really distinct: distinct in reality, not merely relative to our thought. What makes the persons distinct given that each is God and there is only one God? What is the principium individuationis within the Godhead? The relations between them. Thus the Father is distinct from the Son because the Father stands in the paternity relation to the Son but not vice versa. It is difficult to see, however, how a relation between x and y can constitute the numerical difference between x and y. I should think that the numerical difference between x and y is a logically prior condition of their standing in any relation. So I am already having difficulty following the Thomist account.
These real relations, since they are subsistent (not accidental): and are, on the other hand, incommunicable (being opposed): can constitute the divine persons. In these subsistent relations we find the two characteristics of person: substantiality and incommunicability.
Comment: If the relations were accidental, i.e., accidents, then they would be dependent in their being on something else, and the objection I just made would hold. So they are said to be subsistent, i.e., substances in their own right. And since they are 'incommunicable,' they have two characteristics of persons. The problem, however, is to understand how the relata of the relations (of paternity, filiality, etc.) can be (identical to) the relations. Paternity and filiality are different relations. So if the Father = paternity, and the Son = filiality, then it is easy to see how the Father and the Son are distinct. But what is difficult if not impossible to understand is how the Father could be identical to paternity and the Son to filiality.
A divine person, then, according to St. Thomas and his school, is a divine relation as subsistent.  Elsewhere the saint gives the following definition:  A divine person is nothing else than a relationally distinct reality, subsistent in the divine essence.
These definitions explain why there are in God, speaking properly, not metaphorically, three persons, three intellectual and free subjects, though these three have the same identical nature, though they understand by one and the same intellective act, though they love one another by one and the same
essential act, and though they freely love creatures by one and the same free act of love.
Comment: So the Father loves the Son and the Son loves the Father by the same act of loving. But acts are individuated by their objects. So loving the Father is a different act than loving the Son. It cannot be the same act on pain of incoherence. But Aquinas says that they love by the same act. He has to say this because he cannot admit that there are three separate unities of consciousness in the Godhead. For this would entail that there are three Gods.
Hence, while we say: The Father is God, the Son is God, the Holy Spirit is God, we also say: The Father is not the Son, and the Holy Spirit is not the Father, and the Holy Spirit is not the Son. In this sentence the verb "is" expresses real identity between persons and nature, and the negation "is not" expresses the real distinction of the persons from each other.
Comment: This is contradictory as I have explained many times before, assuming that 'nature' refers to an individual existing nature. If the 'is' is taken to be the 'is' of identity, logical inconsistency is unavoidable. If F = G and S = G. then F = S, by the symmetry and transitivity of identity. You cannot consistently with that go on to say that it is not the case that F = S.
These three opposed relations, then, paternity, filiation, and passive spiration, belong to related and incommunicable personalities. Thus there cannot be in God many Fathers, but one only. Paternity makes the divine nature incommunicable as Father, though that divine nature can still be communicated to two other persons. To illustrate. When you are constructing a triangle, the first angle, as first, renders the entire surface incommunicable, though that same surface will still be communicated to the other two angles; and the first angle will communicate that surface to them without communicating itself, while none of the three is opposed to the surface which they have in common.
Comment: Garrigou-Lagrange is fudging now. He says that the opposed relations belong to related personalities. This is not what he said before. Before he said that the persons just are subsistent relations. Well, which is it? Are the relations identical to persons, or do the relations belong to persons? This fudge is to be expected since the doctrine attempts to articulate discursively a reality that lies beyond the discursive intellect, a reality that is mystical.
Here appears the profundity of Cajetan's  remark: the divine reality, as it is in itself, is not something purely absolute (signified by the word "nature") nor something purely relative (signified by the name "person"): but something transcending both, something which contains formally and eminently  that which corresponds to the concepts of absolute and relative, of absolute nature and relative person. Further, the distinction between nature and the persons is not a real distinction, but a mental distinction (virtual and minor): whereas the distinction between the persons is real, by reason of opposition. On this last point theologians generally agree with Thomists.
Comment: Cajetan's remark is profound. The divine reality must be absolute, not relative. But it must also in some sense be personal since the reality of persons surpasses that of every other category of entity. But persons are relative to each other. So the divine reality must in some sense be multi-personal and yet absolute. As I see it, theology issues in 'necessary makeshifts' that try to articulate in coherent discursive terms a trans-discursive reality. So it is no surprise that every doctrine of the Trinity issues in problems, questions, and outright inconsistencies. The doctrines point beyond themselves to a reality that cannot be grasped in discursive terms.
This is why doctrinal fights are absurd. Some doctrines are better than others, but in the end all are untenable. The divine reality is not a doctrine!