Some of the greatest minds of the Church, in what appears to be a sign of desperation to preserve what they perceive as infallibility, end up coming to firm conclusions that are frankly ridiculous and that should have been held tentatively. Most of that is because infallibility was poorly defined until relatively recently (even as late as Vatican II with respect to the obsequium religiosum), and a number of these theologians just didn't have a clear grasp of how infallibility was supposed to function. The most glaring example that I've found is in the brilliant historian Dom John Chapman's defense of infallibility, in which he ends up saying things (mostly in a footnote) that are just hard to read with a straight face. (Ironically, it was published by the Catholic Truth Society in 1907.) Specifically, Fr. Chapman says the following of Honorius's letters:
It was natural for the Byzantines, therefore, to treat it as giving the Roman view, natural that it should be followed by Sergius (whom in fact it bound since it was addressed to him), natural that it should remain a tower of strength to heretics until it had been authoritatively declared by Rome to be no embodiment of her tradition. Such a disavowal had become absolutely necessary as the complement of the Roman condemnation of the ecthesis and the typus, which had both been founded on Honorius, as we saw.
But once disowned by Rome, the words of Honorius were harmless against Rome. They were instantly reduced to their true value, as the expression of his own view.*
(* Infallibility is, as it were, the apex of a pyramid. The more solemn the utterances of the Apostolic See, the more we can be certain of their truth. When they reach the maximum of solemnity, that is, when they are strictly ex cathedra, the possibility of error is wholly eliminated. The authority of a Pope, even on those occasions when he is not actually infallible, is to be implicitly followed and reverenced. That it should be on the wrong side of a contingency is shown by faith and history to be possible, but by history as well as by faith to be so remote that it is not usually to be taken into consideration. There are three or four examples in history. Of these the condemnation of Galileo is the most famous, and the mistake of Honorius makes a good (or rather bad) second. But in this case the mistake was rectified within a few months, and after that, no one followed Honorius in good faith.)
The infallibility of the Pope is for the sake of the Church. Wherever his fall would necessarily involve the Church in the same error, he is infallible. Therefore he is infallible whenever he binds the Church by his authority to accept the ruling, and only then. It is a matter of history that no Pope has ever involved the whole Church in error. It is a matter of history that Pope after Pope has solemnly defined the truth and bound the Church to accept it. It is a matter of history that Pope after Pope has confirmed Councils which decided rightly and wrongly. It is a matter of history that Rome has always retained the true faith. If this was wonderful in the 7th century, it is more wonderful after thirteen more centuries have passed.
At this point, Fr. Chapman's view of infallibility is completely unprincipled, and it's because he is, at least in this case, a bad philosopher. Adherence to error is never natural; in fact, it's exactly the opposite. The idea that obedience to the Pope binds one to error is therefore a metaphysical impossibility. Fr. Chapman here is watering down infallibility to the point of triviality in order to preserve it, but infallibility needs to be far more robust than this from a philosophical standpoint to function effectively. For example, Fr. Chapman's assertion that one can correctly believe that a statement is binding only to find out a few months later that the statement has been reduced to its "true value" would be a manifest absurdity. It would require conditional assent, rather than the assent of faith, to papal teaching. The position that Fr. Chapman defends here would make papal authority senseless for purposes of divine revelation, for exactly the same reason that Scripture does not function as divine revelation without inerrancy.
The distinction that Fr. Chapman identifies, however, is a correct one: that the Pope is infallible "whenever he binds the Church by his authority to accept the ruling." But he makes the frankly ridiculous assertion that this is the case only "[w]hen they reach the maximum of solemnity, that is, when they are strictly ex cathedra." This is the same position taken by the liberal German bishops in the 1960s and all of the progressives that have since followed them, and it is absurd. Cardinal Franzelin pointed out that there are numerous instances where the Pope offers some kind of a definitive judgment from a disciplinary standpoint that does not amount to the kind of doctrinal definition required for an ex cathedra statement. Such commands are therefore infallibly safe to obey and can never lawfully be resisted. Thus, the pyramid Fr. Chapman describes is not a continuum; it has layers. At the apex are ex cathedra statements, which are infallibly true; at the next layer are those definitive disciplinary judgments that bind the entire Church in a human way, which are infallibly safe; at the next layer are private opinions that are to be "implicitly followed and reverenced," the deference to which is subject only to prudence.
One can rightly question the prudence of any judgment by the Pope, even of ex cathedra statements. One can plead for clarification or (where appropriate) reversal. What one cannot do is to question the authority of such statements, including the suggestion that the Pope himself has denied the faith or has led (or attempted to lead) the flock into heresy by such authoritative action. Fr. Chapman is correct that Pope Honorius was offering a private opinion along with his binding direction on how Sergius was to proceed (i.e., to adhere to the Chalcedonian faith), but Sergius was never bound (nor was anyone else) to Pope Honorius's private opinions. To say that there was binding but erroneous teaching in the letters is a contradiction in terms. What was condemned in the letters was never binding and never should have been followed.
