"...one point must be considered absolutely incontrovertible and placed firmly above any doubt whatever: the adhesion of the universal Church will be always, in itself, an infallible sign of the legitimacy of a determined Pontiff, and therefore also of the existence of all the conditions required for legitimacy itself.
...the aforementioned adhesion of the Church heals in the root all fault in the election and proves infallibly the existence of all the required conditions."
-Cardinal Billot
"It is of no importance that in past centuries some Pontiff was illegitimately elected or took possession of the Pontificate by fraud; it is enough that he was accepted afterwards by the whole Church as Pope, since by such acceptance he would have become the true Pontiff."
-Saint Alphonsus de Ligouri
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Thank you for posting these quotes, NIFH. As much as this idea is maligned, the logic behind it is very simple, is explained by probably both the eminent authors you quote, and has never been refuted.
The argument is simple: We must accept
with certainty any universal papal teaching on faith or morals. But we cannot accept the teaching of any pope with certainty unless we likewise know
for certain that such a man is pope. Therefore there must be some observable,
certain criterion to know that some particular man is pope. But the only criterion that fits that description is the universal acceptance of the whole Church of a particular man as pope. Why? Because
if the whole Church could be wrong about who the pope is, then the whole Church could be led into error, which is contrary to the promises of Christ.
People who reject this teaching are unable to provide any other universally observable, objective criterion by which the faithful could know for certain that some particular man is pope. I know this because I have asked this question before: "If a man can be universally accepted as pope by the whole Church and still not be pope, then what criterion can give infallible certainty to the whole Church that someone is pope?" I have never gotten an answer to this.
If someone asserts that a man can be accepted peacefully by the universal Church as the pope, and somehow not be the pope, then every papacy is called into doubt, and therefore every defined dogma and every canonized saint and every papal teaching is likewise called into doubt. There is no way to know whether someone is pope or not, or to know whether there was some legal problem in the way his election took place, or anything else.
If you deny Universal Peaceful Acceptance, then either there must be some other universally-observable criterion that can give everyone certainty that someone is pope, or no man can be certainly known to be the pope.
If there is a third possibility, I would love to hear what it is. This is why both St. Alphonsus and Cardinal Billot taught this idea, along with many others.