On the one hand, we have the abstract, textbook definitions of papal infallibility when applied to papal assertions about, for example, the ordination of men only:
Does this statement meet all five criteria of Papal Infallibility, as defined by the First and Second Vatican Councils?
1. “the Roman Pontiff”
2. “speaks ex cathedra” (“that is, when in the discharge of his office as shepherd and teacher of all Christians, and by virtue of his supreme apostolic authority….”)
3. “he defines”
4. “that a doctrine concerning faith or morals”
5. “must be held by the whole Church” [Pastor Aeternus, chap. 4.]
1. “the Roman Pontiff”
2. “in virtue of his office, when as the supreme shepherd and teacher of all the faithful, who confirms his brethren in their faith (cf. Lk 22:32),”
3. “by a definitive act, he proclaims”
4. “a doctrine of faith or morals” (“And this infallibility…in defining doctrine of faith and morals, extends as far as the deposit of revelation extends”)
5. “in accordance with revelation itself, which all are obliged to abide by and be in conformity with” [Lumen Gentium, n. 25, paragraph 3.] . . .
All five criteria for Papal Infallibility are met by the declaration on priestly ordination found in the Apostolic Letter Ordinatio Sacerdotalis. Therefore, the declaration falls under Papal Infallibility and is, without doubt, the Infallible Teaching of Christ. This teaching on priestly ordination is an example of the use of the first charism of the Sacred Magisterium: Papal Infallibility.
Moreover, at this point in time, the same teaching is also infallible under the ordinary and universal Magisterium. So the infallibility of the teaching should not be a matter of dispute among the faithful.
Whosoever obstinately denies or obstinately doubts this infallible teaching commits the sin of heresy.
This is good because we know where we stand.
On the other hand, there is the politics of the Roman Catholic Church:
Debate over the reach of infallibility has swirled ever since the First Vatican Council in the 19th century, and has become steadily more intense since the early 1980s.
Vatican I formally defined papal infallibility in 1870, and most experts say it has been clearly invoked only with two dogmas, both about Mary: the Immaculate Conception in 1854, and the Assumption in 1950. In that light, some theologians and rank-and-file believers argue that on other contentious matters that have never been formally proclaimed as infallible, such as the ordination of women, contraception and homosexuality, dissent remains legitimate.
Other voices in the church, however, insist that a tight focus on rare public proclamations downplays the role of the church’s “ordinary and universal magisterium,” meaning things that have been taught consistently across time. Such teachings are effectively infallible, according to this understanding, even if no pope has ever formally declared them as such, and thus Catholics are bound to accept them.
Throughout the 1980s and ’90s, a leading advocate of this more expansive view of infallibility was Cardinal Ratzinger, today Pope Benedict XVI.
In the 1980s, these clashing views were at the heart of an exchange between Ratzinger and Fr. Charles Curran, an American moral theologian fired in 1987 by The Catholic University of America in Washington after a lengthy investigation by Ratzinger’s office. In back-and-forth correspondence with Ratzinger, Curran defended a right of dissent from what he called “authoritative non-infallible hierarchical teaching.”
Ratzinger responded that such a restricted view of the church’s teaching authority derives from the Protestant Reformation, and it leads to the conclusion that Catholics are obligated only to accept a few core dogmatic principles — the Trinity, for example, or the resurrection of the body — while everything else is debatable. In fact, Ratzinger said, the Second Vatican Council (1962-65) used the phrase the “secondary object of infallibility” to refer to a wide range of teachings on faith and morals that are intrinsically connected to divine revelation, and therefore infallible. . . .
American Jesuit Fr. John Coleman called it a form of “papal fundamentalism.” The Catholic Theological Society of America endorsed a 5,000-word study that concluded “there are serious doubts” about whether the teaching is infallible, and called for “further study, discussion and prayer.” The Canon Law Society of Great Britain and Ireland likewise concluded in 1996 that the teaching on women priests was not infallible.
In December 1996, the then-secretary of the Congregation for the Doctrine of the Faith, Bishop Tarcisio Bertone, published an article in L’Osservatore Romano in which he asserted that certain papal teachings should be considered infallible, even in the absence of a formal statement. Bertone mentioned three such documents: Veritatis Splendor, Ordinatio Sacerdotalis and Evangelium Vitae.
Bertone is today a cardinal and the Vatican’s secretary of state.
In January 1997, the doctrinal congregation published a collection of documents supporting its reasoning on women’s ordination. In a press conference, Ratzinger addressed the question of whether Catholics who believe that women should be priests are heretics. Technically, he said, the term “heresy” refers to denial of a revealed truth such as the Incarnation or the Resurrection. The ban on women priests, he said, is a doctrinal conclusion derived from revelation, and as such those who deny it are not literally heretics. They do, however, “support erroneous doctrine that is incompatible with the faith” and exclude themselves from communion with the church.
In his 1998 commentary on Ad Tuendam Fidem, Ratzinger argued that a host of teachings are infallible because they’re joined to the revealed truths of the faith, either by a historical relationship or by a logical connection.
Examples of doctrines connected by historical necessity, according to the Ratzinger commentary, include: the legitimacy of the election of a given pope; the acts of an ecumenical council; the canonizations of saints; the declaration of Pope Leo XIII in the papal bull Apostolicae Curae on the invalidity of Anglican ordinations. Examples of doctrines connected by logical necessity include: the doctrine that priestly ordination is reserved only to men; the doctrine on the illicitness of euthanasia; the teaching on the illicitness of prostitution; the teaching on the illicitness of fornication.
(Notice how little teaching about the work of Jesus Christ is considered to be infallible.)
What are Protestants to think? Heck, what are Roman Catholics to believe? But for a doctrine, nay, a reality, that is supposed to produce such certainty, it sure looks like Roman Catholics stumble over it the way that Protestants fail to agree on what their Bibles teach.