What is the apostolic office?
We look back using reason at the early church, we see books that will eventually become Scripture, the Church Fathers, councils and synods. The problem is disunity in the accounts here. How do we know Origen was a heretic, or that Tertullian was wrong on Baptism, or that Pelagius was wrong on anthropology/Original Sin.
Protestants take into account the fathers, the councils, and the books, and argue that by the very nature of Scripture (God-Breathed, Revelation, etc) it is superior to all other sources. This is an after-the-fact decision in an attempt to make up for difficulties and differences between Christian writers. Ultimate authority must be given to Scripture and it is thus assumed that Scripture is clear in its meaning. People cite inclarity between Protestant traditions as disproof, but one forgets that apostolically succeeding bishops disagree as well (Old Catholics, Roman Catholics, Eastern Orthodox, Coptics).
Catholics argue that Papal Supremacy is the apostolic office (or Petrine office as they might say). That St. Peter's authority was passed on to his successors and that they exercise his role as supreme among the apostles. This argument is ironically based off of scripture and the nature of the church (Spirit-led) and thus epistemologically it is equal to the after-the-fact system of sola scriptura, as the transfer of Petrine authority is not clearly taught (just like sola scriptura).
There are (paradoxically) three different Catholic answers:
1. Two-Source Method: Some argued that revelation is stored partly in Scripture, partly in oral traditions. In this view St. Paul could have taught the church at Ephesus the doctrine of the Immaculate Conception, or Indulgences. This to me seems a little ridiculous, but technically this is not a disproof of this view, it is merely my own incredulity. There are problems here in that many Catholic doctrines like indulgences, or certain Marian doctrines are not taught in the fathers. There might be 'seeds' but there is no fully defined doctrines as such.
2. Magesterium/Development Method: This was Cardinal Newman's solution, namely that as the church thought about these issues, over time, they managed to come up with new implications of each doctrine. I.E. Immaculate Conception from Genesis 3:15. Problematic to this opinion is that the Church declares revelation to be a finished process, this seems to add to the deposit of faith, which we have been told to guard (Jude 3) and presumably not to add to. Pelikan notes that the medievals stated that to add any doctrine was temerity, and so all doctrines must be proved to have been part of the deposit of faith.
3. Vatican II Method: Dei Verbum & Pope Benedict XVI seem to see Tradition as the bounds within which we are to read the bible. The Bible is the materially sufficient deposit of faith, and Tradition helps us interpret it.
Problematic is that each of these 3 views is on equal epistemological footing as Sola Scriptura. They're all after the fact ways of sorting out the problems of historical theology