The fully developed theory of infallibility cannot be “proven” — it is a matter of faith. But the principles behind it can be. I deal with various facets of infallibility on my Papacy web page. As usual, it is an accumulation of many individual evidences and indications that make it compelling. All the evidence taken together make it exceedingly “probable” (from a standpoint of human reason alone. I think Newman’s Essay on the Development of Christian Doctrine remains the best treatment of the overall case from the historical perspective.
No pope ever issued a binding proclamation contrary to the received faith. Infallibility applies to what is dogmatically proclaimed, not everything a pope does, or perfection on a human level. Catholics know enough to not equate holders of an office with the dignity of the office itself (when they fall short). We’re not like Protestants, who, too often, slavishly follow leaders (Calvin, Luther) no matter how foolish their teachings.
According to the Old Testament, the prophet was to be judged by whether his prophecies came true or not. If they did not, he was regarded as a false prophet and stoned. That was the criterion of truthfulness, and a strong motivator for a person to be sure he was a prophet before claiming that! I would say the New Testament covenant was fundamentally different insofar as all were now filled with the Holy Spirit; thus had much more of a power of discernment than the masses under the old covenant.
Therefore, the standard then was simply whether a person spoke verifiable truth or not.
I explained the exact nature of this analogical argument, at the end of chapter three of my book, Bible Proofs for Catholic Truths. I made roughly the same arguments in another paper of mine, online: Bible on Papal & Church Infallibility.
In a nutshell, the argument is (as I stated in my book):
“If prophets spoke with inspiration, then popes can plausibly speak infallibly, since the latter is a far less extraordinary gift than the former.” Or, from a different angle: “if those with lesser gifts can do the great thing (inspired utterance), then those with greater gifts can certainly do the lesser thing (infallible utterance).”
Certain parts of the argument are indisputable. There were prophets in the Old Testament. These did speak the Word of the Lord, which was not only infallible, but in retrospect, inspired, too, insofar as they are now recorded in inspired Scripture.
It’s a simple analogy: there is such a thing as a de facto infallible person in the Old Testament; there also is such a thing in the New Testament. The presence of false prophets in both covenants does not nullify that. It simply means there are false prophets! The true ones do not cease to exist because there are fake ones; imitators.
I am arguing for the existence of infallible authority; not the non-existence of fallible authority or illegitimate authority. Both can clearly exist simultaneously. Therefore, the presence of such fakers (freely conceded) constitutes no disproof at all of my analogy.
Photo credit: Prophet Isaiah, by Antonio Balestra (1666-1740) [public domain / Wikimedia Commons]