Objection 1. It would seem that there are more than three persons in God. For the plurality of persons in God arises from the plurality of the relative properties as stated above (Article 1). But there are four relations in God as stated above (Question 28, Article 4), paternity, filiation, common spiration, and procession. Therefore there are four persons in God.
Objection 2. The nature of God does not differ from His will more than from His intellect. But in God, one person proceeds from the will, as love; and another proceeds from His nature, as Son. Therefore another proceeds from His intellect, as Word, besides the one Who proceeds from His nature, as Son; thus again it follows that there are not only three persons in God.
Objection 3. Further, the more perfect a creature is, the more interior operations it has; as a man has understanding and will beyond other animals. But God infinitely excels every creature. Therefore in God not only is there a person proceeding from the will, and another from the intellect, but also in an infinite number of ways. Therefore there are an infinite number of persons in God.
Objection 4. Further, it is from the infinite goodness of the Father that He communicates Himself infinitely in the production of a divine person. But also in the Holy Ghost is infinite goodness. Therefore the Holy Ghost produces a divine person; and that person another; and so to infinity.
Objection 5. Further, everything within a determinate number is measured, for number is a measure. But the divine persons are immense, as we say in the Creed of Athanasius: "The Father is immense, the Son is immense, the Holy Ghost is immense." Therefore the persons are not contained within the number three.
On the contrary, It is said: "There are three who bear witness in heaven, the father, the Word, and the Holy Ghost" (1 John 5:7). To those who ask, "Three what?" we answer, with Augustine (De Trin. vii, 4), "Three persons." Therefore there are but three persons in God.
I answer that, As was explained above, there can be only three persons in God. For it was shown above that the several persons are the several subsisting relations really distinct from each other. But a real distinction between the divine relations can come only from relative opposition. Therefore two opposite relations must needs refer to two persons: and if any relations are not opposite they must needs belong to the same person. Since then paternity and filiation are opposite relations, they belong necessarily to two persons. Therefore the subsisting paternity is the person of the Father; and the subsisting filiation is the person of the Son. The other two relations are not opposed to either of these, but are opposed to each other; therefore these two cannot belong to one person: hence either one of them must belong to both of the aforesaid persons; or one must belong to one person, and the other to the other. Now, procession cannot belong to the Father and the Son, or to either of them; for thus it would follows that the procession of the intellect, which in God is generation, wherefrom paternity and filiation are derived, would issue from the procession of love, whence spiration and procession are derived, if the person generating and the person generated proceeded from the person spirating; and this is against what was laid down above (27, 3 and 4). We must frequently admit that spiration belongs to the person of the Father, and to the person of the Son, forasmuch as it has no relative opposition either to paternity or to filiation; and consequently that procession belongs to the other person who is called the person of the Holy Ghost, who proceeds by way of love, as above explained. Therefore only three persons exist in God, the Father, the Son, and the Holy Ghost.
Reply to Objection 1. Although there are four relations in God, one of them, spiration, is not separated from the person of the Father and of the Son, but belongs to both; thus, although it is a relation, it is not called a property, because it does not belong to only one person; nor is it a personal relation--i.e. constituting a person. The three relations--paternity, filiation, and procession--are called personal properties, constituting as it were the persons; for paternity is the person of the Father, filiation is the person of the Son, procession is the person of the Holy Ghost proceeding.
Reply to Objection 2. That which proceeds by way of intelligence, as word, proceeds according to similitude, as also that which proceeds by way of nature; thus, as above explained (27, 3), the procession of the divine Word is the very same as generation by way of nature. But love, as such, does not proceed as the similitude of that whence it proceeds; although in God love is co-essential as being divine; and therefore the procession of love is not called generation in God.
Reply to Objection 3. As man is more perfect than other animals, he has more intrinsic operations than other animals, because his perfection is something composite. Hence the angels, who are more perfect and more simple, have fewer intrinsic operations than man, for they have no imagination, or feeling, or the like. In God there exists only one real operation--that is, His essence. How there are in Him two processions was above explained (27, 1, 4).
Reply to Objection 4. This argument would prove if the Holy Ghost possessed another goodness apart from the goodness of the Father; for then if the Father produced a divine person by His goodness, the Holy Ghost also would do so. But the Father and the Holy Ghost have one and the same goodness. Nor is there any distinction between them except by the personal relations. So goodness belongs to the Holy Ghost, as derived from another; and it belongs to the Father, as the principle of its communication to another. The opposition of relation does not allow the relation of the Holy Ghost to be joined with the relation of principle of another divine person; because He Himself proceeds from the other persons who are in God.
Reply to Objection 5. A determinate number, if taken as a simple number, existing in the mind only, is measured by one. But when we speak of a number of things as applied to the persons in God, the notion of measure has no place, because the magnitude of the three persons is the same (42, 1,4), and the same is not measured by the same.
Source: Summa Theologica (New Advent)