According to Catholicism, does an individual bishop have the safeguard of Infallibility or Indefectibility?

According to Catholicism, do individual bishops have the safeguard of Infallibility or Indefectibility?

The short answer on both counts is no.

To start off with, I would like to quote Pope St. Paul VI in his Dogmatic Constitution on the Church, Lumen Gentium.

Although the individual bishops do not enjoy the prerogative of infallibility, they nevertheless proclaim Christ's doctrine infallibly whenever, even though dispersed through the world, but still maintaining the bond of communion among themselves and with the successor of Peter, and authentically teaching matters of faith and morals, they are in agreement on one position as definitively to be held. This is even more clearly verified when, gathered together in an ecumenical council, they are teachers and judges of faith and morals for the universal Church, whose definitions must be adhered to with the submission of faith.

First of all what is indefectibility?

Indefectibility

Definition

Imperishable duration of the Church and her immutability until the end of time. The First Vatican Council declared that the Church possesses "an unconquered stability" and that, "built on a rock, she will continue to stand until the end of time" (Denzinger 3013, 3056). The Church's indefectibility, therefore, means that she now is and will always remain the institution of salvation, founded by Christ. This affirms that the Church is essentially unchangeable in her teaching, her constitution, and her liturgy. It does not exclude modifications that do not affect her substance, nor does it exclude the decay of individual local churches or even whole dioceses.

By definition the local ordinary of a diocese does not enjoy the safeguard of indefectibility since it’s definition states clearly that it is accorded to the Church in its entirety. It is not the pope or a bishop that is indefectible, but the Holy Catholic Church itself. Whole dioceses can vanish and no longer exist. Historically, this has already happened many times over. However the Church will last until the consummation of the world.

Dioceses that no longer exist are generally referred to as titular sees.

A titular see in various churches is an episcopal see of a former diocese that no longer functions, sometimes called a "dead diocese".

The ordinary or hierarch of such a see may be styled a "titular metropolitan" (highest rank), "titular archbishop" (intermediary rank) or "titular bishop" (lowest rank), which normally goes by the status conferred on the titular see.

The term is used to signify a diocese that no longer functionally exists, often because the diocese once flourished but the territory was conquered by Muslims or no longer functions because of a schism. The Greek–Turkish population exchange of 1923 also contributed to titular bishoprics. The see of Maximianoupolis along with the town that shared its name was destroyed by the Bulgarians under Emperor Kaloyan in 1207; the town and the see were under the control of the Latin Empire, which took Constantinople during the Fourth Crusade in 1204. Parthenia, in north Africa, was abandoned and swallowed by desert sand.

Do individual bishops have the safeguard of infallibility?

This part of the question needs some clarification before going on. On their own, an individual bishop does not enjoy the privilege of infallibility. Only the pope and only when he speaks ex cathedra—that is, when in discharge of the office of pastor and teacher of all Christians, by virtue of his supreme apostolic authority, he defines a doctrine regarding faith or morals to be held by the universal Church, by the divine assistance promised to him in Blessed Peter, is possessed of that infallibility with which the divine Redeemer willed that his Church should be endowed in defining doctrine regarding faith or morals; and therefore such definitions are irreformable of themselves, and not in virtue of consent of the Church". Most bishops never teach erroneous doctrine and many are saints. But nevertheless they do not enjoy the privilege of infallibility in their episcopal office. It belongs to the Successor of St. Peter the Apostle.

Infallibility

Definition

Freedom from error in teaching the universal Church in matters of faith or morals. As defined by the First Vatican Council, "The Roman Pontiff, when he speaks ex cathedra—that is, when in discharge of the office of pastor and teacher of all Christians, by virtue of his supreme apostolic authority, he defines a doctrine regarding faith or morals to be held by the universal Church, by the divine assistance promised to him in Blessed Peter, is possessed of that infallibility with which the divine Redeemer willed that his Church should be endowed in defining doctrine regarding faith or morals; and therefore such definitions are irreformable of themselves, and not in virtue of consent of the Church" (Denzinger 3074).

The bearer of the infallibility is every lawful Pope as successor of Peter, the Prince of the Apostles. But the Pope alone is infallible, not others to whom he delegates a part of his teaching authority, for example, the Roman congregations.

The object of his infallibility is his teaching of faith and morals. This means especially revealed doctrine like the Incarnation. But it also includes any nonrevealed teaching that is in any way connected with revelation.

The condition of the infallibility is that the Pope speaks ex cathedra. For this is required that: 1. he have the intention of declaring something unchangeably true; and 2. he speak as shepherd and teacher of all the faithful with the full weight of his apostolic authority, and not merely as a private theologian or even merely for the people of Rome or some particular segment of the Church of God.

The source of the infallibility is the supernatural assistance of the Holy Spirit, who protects the supreme teacher of the Church from error and therefore from misleading the people of God.

As a result, the ex cathedra pronouncements of the Pope are unchangeable "of themselves," that is, not because others in the Church either first instructed the Pope or agree to what he says. (Etym. Latin in-, not + fallibilis; from fallere, to deceive: infallibilis, not able to deceive, or err.)

If individual bishops in union (at the same time) with the Supreme Pontiff were to make an infallible teaching within the Catholic Church, they would enjoy this privilege, uniquely by virtue of being united to and in union with the pope at that exact moment would they be infallible. Apart from this unique perspective, generally speaking, bishops are not infallible. They can make theological errors.

A bishop in full communion does not have the grace of infallibility. Bishops do not have the grace of infallibility, that is they do not have the safeguards of the assurance that they will always be infallible.

Thanks be to God, the vast majority do not.

Deo Gratias.

It should be noted that bishop who writes a theological work must still obtain an imprimatur. He can not grant an imprimatur to his own works! Papal encyclicals do not carry an imprimatur! There is no need to do so.