Thomas Aquinas, Compendium of Theology

We must realize that, although the relations subsisting in the Godhead are the divine persons themselves, as was stated above, we are not to conclude that there are five or four persons corresponding to the number of relations. For number follows distinction of some sort. Just as unity is indivisible or undivided, so plurality is divisible or divided. For a plurality of persons requires that relations have power to distinguish by reason of opposition, since formal distinction necessarily entails opposition. If, then, the relations in question are closely examined, paternity and filiation will be seen to have relative opposition to each other; hence they are incompatible in the same suppositum. Consequently paternity and filiation in God must be two subsistent persons. Innascibility, on the other hand, although opposed to filiation, is not opposed to paternity. Hence paternity and innascibility can pertain to one and the same person. Similarly, common spiration is not opposed either to paternity or to filiation, nor to innascibility. Thus nothing prevents common spiration from being in both the person of the Father and the person of the Son. Accordingly common spiration is not a subsisting person distinct from the persons of the Father and the Son. But procession has a relation of opposition to common spiration. Therefore, since common spiration pertains to the Father and the Son, procession must be a person distinct from the persons of the Father and the Son.

Accordingly the reason is clear why God is not called "quiune," quinus, on account of the notions, which are five in number, but is called triune, on account of the Trinity of persons. The five notions are not five subsisting things, but the three persons are three subsisting things. Although several notions or properties may pertain to a single person, only one of them constitutes the person. For a divine person is constituted by the properties, not in the sense that He is constituted by several of them, but in the sense that the relative, subsisting property itself is a person. If several properties were understood as subsisting in themselves apart, they would be several persons, and not one person. Hence we must understand that, of the several properties or notions belonging to a single person, the one that precedes according to the order of nature constitutes the person; the others are understood as inhering in the person already constituted.

Thus it is evident that innascibility cannot be the first notion of the Father, constituting His person, because nothing is constituted by a negation, and also because affirmation naturally precedes negation. Further, common spiration presupposes paternity and filiation in the order of nature, just as the procession of love presupposes the procession of the Word. Hence common spiration cannot be the first notion of the Father, or of the Son either. The first notion of the Father is paternity and the first notion of the Son is filiation, whereas procession alone is the notion of the Holy Ghost.

Accordingly the notions constituting persons are three in number: paternity, filiation, and procession. And these notions must be strict properties. For that which constitutes a person must pertain to that person alone; individuating principles cannot belong to several individuals. For this reason the three notions in question are called personal properties, in the sense that they constitute the three persons in the manner described. The remaining notions are called properties or notions of the persons, but not personal properties or notions, because they do not constitute a person.