Luke, recording the words of Jesus to Peter, wrote:
But I have prayed for thee, that thy faith fail not: and when thou art converted, strengthen thy brethren. (Luke 22:32)
Many Catholic apologists appeal to this verse as evidence, not just of Petrine primacy, but the infallibility of Peter and his successors (i.e., the Papacy [e.g., Tim Staples in his 2000 debate against James White on Papal Infallibility]).
Responding to Ignaz von Döllinger, Luke Rivington, an Anglican convert to Roman Catholicism, wrote the following about this particular texts and its post-biblical application by Roman bishops:
Dr. Döllinger parodies the Church’s application of this text to the successor of St. Peter when he calls it ‘far from being a guarantee of infallibility for every single dictum on an article of ecclesiastical doctrine.’ No theologian ever laid down such a childish principle nor did the Church ever call on Dr. Döllinger to believe it. He insinuates the same absurdity when he says, ‘the exhortation that Peter should strengthen his brethren by no means involves a promise that he would really do so in every single instance.’ Our Lord promised the security arising from his own prayer; and that security need not be, and never was, stretched to include ‘every single instance,’ of whatsoever kind. (Luke Rivington, The Primitive Church and the See of Peter [London: Longmans, Green, and Co, 1894], 310 n. 1, italics in original, emphasis in bold added)
Some may retort that Rivington is refuting the naïve view that everything a pope does/says is free from error, an absurdity that Döllinger, who was very learned, would not have claimed (see his The Pope and the Council [1869]). Furthermore, it also avoids that a pope, when he exercises his authority on issues such as the form of sacraments and other issues, is free from error, even if he does not exercise the criteria explicated by Vatican I to meet an ex cathedra statement as such would fall under the category of “secondary objects of infallibility.” As John Salza and Robert Siscoe wrote:
[A]ccording to the teaching of the Church’s theologians, the Church also speaks infallibly on other matters, which fall into the category of secondary objects of infallibility. These include (a) theological conclusions (i.e., inferences deduced from two premises, one of which is immediately revealed, while the other is a truth known by natural reason), (b) dogmatic facts, (c) universal disciplines, and the (d) canonization of saints. These secondary objects of infallibility are not believed with Divine and Catholic Faith but with Ecclesiastical Faith, which is faith in the infallible Church teaching (but not in God revealing). (John Salza and Robert Siscoe, True or False Pope? Refuting Sedevacantism and Other Modern Errors [Winona, Minn.: St. Thomas Aquinas Seminary, 2015], 379, italics in original)
An example of a papal statement fitting this is discussed by the authors:
We have an example of this in Paul VI’s encyclical Humanae Vitae, which condemned contraception. Even though the teaching in the encyclical was not infallible by virtue of the Extraordinary Magisterium(since Paul VI did not issue a “definition”), nevertheless, the teaching was infallible by virtue of the Ordinary and Universal Magisterium, and therefore requires the assent of faith. Cardinal Felici explained this following the issuance of the encyclical. From L’Osseratore Romano: “On this problem we must remember that a truth may be sure and certain, and hence it may be obligatory even without the sanction of an ex cathedra definition. So it is with the encyclical Humanae Vitae, in which the pope, the supreme pontiff of the Church, utters a truth which has been constantly taught by the Church’s Magisterium and which accords with the precepts of Revelation” (L’Osservatore Romano, October 19, 1968, p. 3, emphasis added). (Ibid., 202 n. 40)