Objection 1. It would seem that the hypostases remain if the properties or relations are mentally abstracted from the persons. For that to which something is added, may be understood when the addition is taken away; as man is something added to animal which can be understood if rational be taken away. But person is something added to hypostasis; for person is "a hypostasis distinguished by a property of dignity." Therefore, if a personal property be taken away from a person, the hypostasis remains.
Objection 2. Further, that the Father is Father, and that He is someone, are not due to the same reason. For as He is the Father by paternity, supposing He is some one by paternity, it would follow that the Son, in Whom there is not paternity, would not be "someone." So when paternity is mentally abstracted from the Father, He still remains "someone"--that is, a hypostasis. Therefore, if property be removed from person, the hypostasis remains.
Objection 3. Further, Augustine says (De Trin. v, 6): "Unbegotten is not the same as Father; for if the Father had not begotten the Son, nothing would prevent Him being called unbegotten." But if He had not begotten the Son, there would be no paternity in Him. Therefore, if paternity be removed, there still remains the hypostasis of the Father as unbegotten.
On the contrary, Hilary says (De Trin. iv): "The Son has nothing else than birth." But He is Son by "birth." Therefore, if filiation be removed, the Son's hypostasis no more remains; and the same holds as regards the other persons.
I answer that, Abstraction by the intellect is twofold--when the universal is abstracted from the particular, as animal abstracted from man; and when the form is abstracted from the matter, as the form of a circle is abstracted by the intellect from any sensible matter. The difference between these two abstractions consists in the fact that in the abstraction of the universal from the particular, that from which the abstraction is made does not remain; for when the difference of rationality is removed from man, the man no longer remains in the intellect, but animal alone remains. But in the abstraction of the form from the matter, both the form and the matter remain in the intellect; as, for instance, if we abstract the form of a circle from brass, there remains in our intellect separately the understanding both of a circle, and of brass. Now, although there is no universal nor particular in God, nor form and matter, in reality; nevertheless, as regards the mode of signification there is a certain likeness of these things in God; and thus Damascene says (De Fide Orth. iii, 6) that "substance is common and hypostasis is particular." So, if we speak of the abstraction of the universal from the particular, the common universal essence remains in the intellect if the properties are removed; but not the hypostasis of the Father, which is, as it were, a particular.
But as regards the abstraction of the form from the matter, if the non-personal properties are removed, then the idea of the hypostases and persons remains; as, for instance, if the fact of the Father's being unbegotten or spirating be mentally abstracted from the Father, the Father's hypostasis or person remains.
If, however, the personal property be mentally abstracted, the idea of the hypostasis no longer remains. For the personal properties are not to be understood as added to the divine hypostases, as a form is added to a pre-existing subject: but they carry with them their own "supposita," inasmuch as they are themselves subsisting persons; thus paternity is the Father Himself. For hypostasis signifies something distinct in God, since hypostasis means an individual substance. So, as relation distinguishes and constitutes the hypostases, as above explained (2), it follows that if the personal relations are mentally abstracted, the hypostases no longer remain. Some, however, think, as above noted, that the divine hypostases are not distinguished by the relations, but only by origin; so that the Father is a hypostasis as not from another, and the Son is a hypostasis as from another by generation. And that the consequent relations which are to be regarded as properties of dignity, constitute the notion of a person, and are thus called "personal properties." Hence, if these relations are mentally abstracted, the hypostasis, but not the persons, remain.
But this is impossible, for two reasons: first, because the relations distinguish and constitute the hypostases, as shown above (Article 2); secondly, because every hypostasis of a rational nature is a person, as appears from the definition of Boethius (De Duab. Nat.) that, "person is the individual substance of a rational nature." Hence, to have hypostasis and not person, it would be necessary to abstract the rationality from the nature, but not the property from the person.
Reply to Objection 1. Person does not add to hypostasis a distinguishing property absolutely, but a distinguishing property of dignity, all of which must be taken as the difference. Now, this distinguishing property is one of dignity precisely because it is understood as subsisting in a rational nature. Hence, if the distinguishing property be removed from the person, the hypostasis no longer remains; whereas it would remain were the rationality of the nature removed; for both person and hypostasis are individual substances. Consequently, in God the distinguishing relation belongs essentially to both.
Reply to Objection 2. By paternity the Father is not only Father, but is a person, and is "someone," or a hypostasis. It does not follow, however, that the Son is not "someone" or a hypostasis; just as it does not follow that He is not a person.
Reply to Objection 3. Augustine does not mean to say that the hypostasis of the Father would remain as unbegotten, if His paternity were removed, as if innascibility constituted and distinguished the hypostasis of the Father; for this would be impossible, since "being unbegotten" says nothing positive and is only a negation, as he himself says. But he speaks in a general sense, forasmuch as not every unbegotten being is the Father. So, if paternity be removed, the hypostasis of the Father does not remain in God, as distinguished from the other persons, but only as distinguished from creatures; as the Jews understand it.
Source: Summa Theologica (New Advent)