I think that those who deny infallibility (as properly defined) deny the truth of the Scriptures.
Speaking to the Apostles at the last supper, Jesus said (John 14:26), “But the Counselor, the Holy Spirit, whom the Father will send in my name, he will teach you all things, and bring to your remembrance all that I have said to you.”
He promises to send the Apostles the Holy Spirit to teach them ALL things.
Later on in the same discourse (John 16:23), Jesus continues, saying, “When the Spirit of truth comes, he will guide you into all the truth; for he will not speak on his own authority, but whatever he hears he will speak, and he will declare to you the things that are to come.”
Again, he promises to send the Spirit of Truth to guide the Apostles into ALL the truth.
So if He is telling the truth here (and we have no reason to believe that Jesus ever did anything other than proclaim the truth), He promises to guide the Apostles into the Truth. Any denial of this is a denial of Scripture (a pretty funny thing for a “sola scriptura” person to do, if you think about it — they may as well deny John 3:16).
The Apostles, operating with this charism of truth, appointed and empowered bishops as their successors. And St. Paul, operating with this charism of truth said that this would continue until the return of Christ (See Ephesians 4:11-13). Deny this and deny Scripture (you may as well deny John 3:16 if you deny the Scriptural support for apostolic succession).
The method for confirming doctrine or for making dogmatic decisions was originally by Church Councils, consisting of: bishops — the apostles and their successors. The first example of this is shown in Acts 15 (Gee, imagine, councils are scriptural, as well). Again, deny it — and you may as well deny John 3:16.
And guess who “ratified” Church Councils? Yup, the Pope. And guess what codified the doctrine of Papal Infallibility? A council.
(To a Catholic, this is confirmed also by the history of the popes...there have been some very “interesting” characters who have been named pope in history. Despite how “interesting” they were — the Catholic Encyclopedia has used the term “disappointing,” in fact — no official doctrine, much less dogma, was ever issued by those “interesting/disappointing” characters)
**He promises to send the Apostles the Holy Spirit to teach them ALL things.**
So true. Right there in Scripture!