WherePeterIs has published another article which attempts to argue that all Papal teaching, even non-definitive Papal teaching (i.e. any exercise of the Magisterium of the Pope) is 100% infallibly protected from all error. You can read it here.
In fact, we have at least one historical scenario, completely ratified by Pope St. Leo II, where the opposite is proven. At the 6th Ecumenical Council held in Constantinople (681), the deposed Monothelite Macarius of Antioch had dug up the 2 letters of Honorius, Pope of Rome, which were written to the Patriarch of Constantinople, Sergius. The Council read aloud these 2 letters (they were obviously not ex-cathedra), and proceeded to anathematize Honorius as a heretic alongside Sergius, Cyrus, Pyrrhus, Pharan, Theodore, et al.
When the Council sent its proceedings to the Pope St. Leo II, a dogmatic reply was sent on May 7, 682 AD which states:
“And in like manner we anathematize the inventors of the new error, that is, Theodore, Bishop of Pharan, Sergius, Pyrrhus, Paul, and Peter, betrayers rather than leaders of the Church of Constantinople, and also Honorius, who did not attempt to sanctify this Apostolic Church with the teaching of Apostolic tradition, but by profane treachery permitted its purity to be polluted”
If the author of this article is willing to admit that a Pope might be able to pollute the purity of the Apostolic See through profane treachery, then he will at least be in light with an Ecumenical Council which was ratified by the then Pope of Rome.
Dom John Chapman, one of the greatest Patristic scholars of the 19th/20th centuries, has this to follow up with those who attempt to think that the condemnation given by Pope St. Leo II was merely to say that Honorius was merely inattentive to his duties:
“It has been sometimes said that St. Leo II in these words interprets the decision of the Council about Honorius in a mild sense, or that he modifies it. It is supposed that by ‘permitted to be polluted’ Leo II means no positive action, but a mere neglect of duty, grave enough in a Pope, but not amounting to the actual teaching of heresy. If Leo II had meant this, he would have been mistaken. Honorius did positively approve the letter of Sergius, as the Council pointed out. Further, the merely negative ruling of the typus had been condemned as heresy by the Lateran Council (649). As a fact, the words of Leo II are harsher than those of the Council. He declares that Honorius did not publish the Apostolic doctrine of his See, and he represents this as a disgrace to the Church of Rome itself, as a pollution of the unspotted. This no Eastern bishop had ventured to say” (The Condemnation of Pope Honorius, Pg. 115)
In his letter to the Bishops of Spain, Pope Leo II had written:
“With Honorius, who did not , as became the Apostolic authority, extinguish the flame of heretical teaching in its first beginning, but fostered it by his negligence” (ibid)
In his letter to King Erwig, Leo II writes:
“And with them Honorius of Rome, who allowed the immaculate rule of Apostolic tradition, which he received from his predecessors, to be tarnished” (ibid)
This last letter seals the deal. Leo says Honorius tarnished the Apostolic tradition. A mere withdrawal from duties does not do that. But one might say, “Ah, but he himself did not tarnish anything, but rather he allowed others to so tarnish”. That is manifestly false, since in his letter to Sergius, Honorius affirmed one will, and forbade the notion of two operations. Even if by “one will” he meant something quasi-orthodox (as some have argued), he still forbade the expression “two operations” and “two wills”, which the Council of Lateran 649 thought was heretical enough to condemn in the Ecthesis of Emperor Heraclius.
And if there were still some who thought Leo II only condemned Honorius was being inattentive to his duties (hardly a reason to dig up his memory and give anathema for), we have Pope Adrian II who, at the Council of Rome (869), whose decrees were read aloud at the Council of Constantinople IV (869), stated the following:
“Although we have read of the Roman pontiff having passed judgement on the bishops of all the churches, we have not read of anyone having passed judgement on him. For even though Honorius was anathematized after this death by the easterners, it should be known that he had been accused of heresy, which is the only offence where inferiors have the right to resist the initiatives of their superiors or are free to reject their false opinions” (from the Acts of Constantinople IV, ed. Leonardi, 238)
Thus, the Council of Constantinople (681), Pope St. Leo II , Pope Adrian II, and the Council of Constantinople IV (869) all together believe that a Papal teaching (non-definitive) was at odds with both the definitive Papal teaching and Conciliar teaching of Pope St. Agatho and the 6th Ecumenical Council, respectively.
In other words, one could appeal to the authority of the Church in order to deflect the abused authority of a particular Pope, even in the realm of doctrine and theology. In any case, the idea that Popes are protected from error in all of their Magisterium is simply untenable by the facts of history. Even if Honorius himself was technically not heretical by some stretch, his successor, Pope St. Leo II, by condemning Honorius for heresy, gave into that belief, as did two Ecumenical Councils. Time has kept me from produces more evidence from the life of Vigilius viz a viz the 5th Ecumenical Council of Constantinople II (553), where said Pope defended, with Magisterial authority, the orthodoxy of the letter of Ibas of Edessa, which later he admitted was wrong. Catholics are only obliged to believe that a Pope is infallible when he, in the exercise of his office as Pastor and teacher of all Christians, produces a decree on faith and morals to be held by all the faithful with catholic and divine faith, at whose dissent we forfeit our salvation. Otherwise known as an Ex-Cathedra teaching. There are some other corollary conditions as well (see this link for details)