Objection 1. It would seem that the fire of Purgatory does not deliver from the debt of punishment. For every cleansing is in respect of some uncleanness. But punishment does not imply uncleanness. Therefore the fire of Purgatory does not deliver from punishment.
Objection 2. Further, a contrary is not cleansed save by its contrary. But punishment is not contrary to punishment. Therefore one is not cleansed from the debt of punishment by the punishment of Purgatory.
Objection 3. Further, a gloss on 1 Corinthians 3:15, "He shall be saved, yet so," etc. says: "This fire is the trial of tribulation of which it is written (Sirach 27:6): The furnace tries the potter's vessels," etc. Therefore man expiates every punishment by the pains of this world, at least by death, which is the greatest punishment of all, and not by the fire of Purgatory.
On the contrary, The pains of Purgatory are more grievous than all the pains of this world, as stated above (Article 3). Now the satisfactory punishment which one undergoes in this life atones for the debt of punishment. Much more therefore is this effected by the punishment of Purgatory.
I answer that, Whosoever is another's debtor, is freed from his indebtedness by paying the debt. And, since the obligation incurred by guilt is nothing else than the debt of punishment, a person is freed from that obligation by undergoing the punishment which he owed. Accordingly the punishment of Purgatory cleanses from the debt of punishment.
Reply to Objection 1. Although the debt of punishment does not in itself imply uncleanness, it bears a relation to uncleanness by reason of its cause.
Reply to Objection 2. Although punishment is not contrary to punishment, it is opposed to the debt of punishment, because the obligation to punishment remains from the fact that one has not undergone the punishment that was due.
Reply to Objection 3. Many meanings underlie the same words of Holy Writ. Hence this fire may denote both the present tribulation and the punishment to come, and venial sins can be cleansed from both of these. That natural death is not sufficient for this, has been stated above (Sent. iv, D, 20).
Source: Summa Theologica (New Advent)