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In Catholicism, a dogmatic definition is an extraordinary infallible statement published by a pope or an ecumenical council concerning a matter of faith or morals, the belief in which the Catholic Church requires of all Christians (although Christians who are not Catholic do not recognize the Catholic Church's authority in such matters).
It most often refers to the infallible teaching of a truth believed bona fina de fide credenda, meaning one defined as explicitly revealed in the deposit of faith and demanding supernatural faith in itself on the authority of the Word of God: Public Revelation in Scripture and Tradition. Popes and ecumenical councils may also extraordinarily infallibly define truths to be held "de fide tenenda," that is to say discerned as logically implied or intrinsically connected to the deposit of faith and guaranteed as infallible by the Holy Spirit, but not necessarily specifically and explicitly Revealed and so demanding supernatural faith not in themselves specifically, but as part of the faith in the Holy Spirit's guarantee of the infallibility and authority of the Church in such matters. The difference is technical; such teachings upon further discernment may be advanced to the status of "de fide credenda," and denial of either makes one a heretic as they are both infallible.
Contrary to the stereotype that Catholics think that everything that the pope says is infallible (or that the pope is impeccable, unable to sin), dogmatic definitions by popes are in fact very rare. Throughout the entire history of the Church, Papal Infallibility has only been invoked, at least unambiguously, twice: Ineffabilis Deus, the definition by Pope Pius IX in 1854 concerning the Immaculate Conception of Mary, and Munificentissimus Deus, the definition by Pope Pius XII in 1950 concerning the Assumption of Mary. Beyond these two, theologians disagree on which, if any, previous papal declarations may be considered as possible examples of extraordinary infallible dogmatic definitions on the part of a pope.
Dogmatic definitions by ecumenical councils, on the other hand, are significantly more common throughout history, including, for example, the Trinitarian and Christological dogmas of the early councils, the teachings on Grace, Justification, and the Sacraments by the Council of Trent, and most recently the definition of Papal Infallibility itself by Vatican I.
The Church holds her teachings to be infallible beyond merely those doctrines that have been dogmatically defined extraordinarily by a pope or ecumenical council. The so-called ordinary and universal magisterium of the Church is held to be infallible too, and in fact the ordinary manifestation of infallibility. The common consensus of the bishops throughout the world and over time is held to be infallible regardless of whether a Pope or Council has extraordinarily defined the teaching. This is to avoid a positivistic notion of Truth and belief. Like extraordinary definitions, the ordinary magisterium may propose teachings both de fide credenda and de fide tenenda, and the latter may advance to the status of the former, and both are held to be infallible. John Paul II, for example, clarified that the reservation of ordination to males only is infallible by the infallibility of the ordinary magisterium, without going so far as issuing a separate extraordinary dogmatic definition. Far from meaning that Catholics are thus free to question this teaching, it means that it is infallible and unchangeable, demands supernatural faith, and one who denies it is a heretic. It has been suggested that the reason that John Paul did not simply define it extraordinarily was in order to not weaken the understanding that the ordinary magisterium is also infallible, to remind Catholics that it is not merely the extraordinary definitions that are infallible and irreversible as if doctrine were positivistic — that it holds "only if the pope says so with a particular formula".