Objection 1. It would seem that the persons are not distinguished by the relations. For simple things are distinct by themselves. But the persons are supremely simple. Therefore they are distinguished by themselves, and not by the relation.
Objection 2. Further, a form is distinguished only in relation to its genus. For white is distinguished from black only by quality. But "hypostasis" signifies an individual in the genus of substance. Therefore the hypostases cannot be distinguished by relations.
Objection 3. Further, what is absolute comes before what is relative. But the distinction of the divine persons is the primary distinction. Therefore the divine persons are not distinguished by the relations.
Objection 4. Further, whatever presupposes distinction cannot be the first principle of distinction. But relation presupposes distinction, which comes into its definition; for a relation is essentially what is towards another. Therefore the first distinctive principle in God cannot be relation.
On the contrary, Boethius says (De Trin.): "Relation alone multiplies the Trinity of the divine persons."
I answer that, In whatever multitude of things is to be found something common to all, it is necessary to seek out the principle of distinction. So, as the three persons agree in the unity of essence, we must seek to know the principle of distinction whereby they are several. Now, there are two principles of difference between the divine persons, and these are "origin" and "relation." Although these do not really differ, yet they differ in the mode of signification; for "origin" is signified by way of act, as "generation"; and "relation" by way of the form, as "paternity."
Some, then, considering that relation follows upon act, have said that the divine hypostases are distinguished by origin, so that we may say that the Father is distinguished from the Son, inasmuch as the former begets and the latter is begotten. Further, that the relations, or the properties, make known the distinctions of the hypostases or persons as resulting therefrom; as also in creatures the properties manifest the distinctions of individuals, which distinctions are caused by the material principles.
This opinion, however, cannot stand--for two reasons.
Firstly, because, in order that two things be understood as distinct, their distinction must be understood as resulting from something intrinsic to both; thus in things created it results from their matter or their form. Now origin of a thing does not designate anything intrinsic, but means the way from something, or to something; as generation signifies the way to a thing generated, and as proceeding from the generator. Hence it is not possible that what is generated and the generator should be distinguished by generation alone; but in the generator and in the thing generated we must presuppose whatever makes them to be distinguished from each other. In a divine person there is nothing to presuppose but essence, and relation or property. Whence, since the persons agree in essence, it only remains to be said that the persons are distinguished from each other by the relations.
Secondly: because the distinction of the divine persons is not to be so understood as if what is common to them all is divided, because the common essence remains undivided; but the distinguishing principles themselves must constitute the things which are distinct. Now the relations or the properties distinguish or constitute the hypostases or persons, inasmuch as they are themselves the subsisting persons; as paternity is the Father, and filiation is the Son, because in God the abstract and the concrete do not differ. But it is against the nature of origin that it should constitute hypostasis or person. For origin taken in an active sense signifies proceeding from a subsisting person, so that it presupposes the latter; while in a passive sense origin, as "nativity," signifies the way to a subsisting person, and as not yet constituting the person.
It is therefore better to say that the persons or hypostases are distinguished rather by relations than by origin. For, although in both ways they are distinguished, nevertheless in our mode of understanding they are distinguished chiefly and firstly by relations; whence this name "Father" signifies not only a property, but also the hypostasis; whereas this term "Begetter" or "Begetting" signifies property only; forasmuch as this name "Father" signifies the relation which is distinctive and constitutive of the hypostasis; and this term "Begetter" or "Begotten" signifies the origin which is not distinctive and constitutive of the hypostasis.
Reply to Objection 1. The persons are the subsisting relations themselves. Hence it is not against the simplicity of the divine persons for them to be distinguished by the relations.
Reply to Objection 2. The divine persons are not distinguished as regards being, in which they subsist, nor in anything absolute, but only as regards something relative. Hence relation suffices for their distinction.
Reply to Objection 3. The more prior a distinction is, the nearer it approaches to unity; and so it must be the least possible distinction. So the distinction of the persons must be by that which distinguishes the least possible; and this is by relation.
Reply to Objection 4. Relation presupposes the distinction of the subjects, when it is an accident; but when the relation is subsistent, it does not presuppose, but brings about distinction. For when it is said that relation is by nature to be towards another, the word "another" signifies the correlative which is not prior, but simultaneous in the order of nature.
Source: Summa Theologica (New Advent)