There is a great deal of misunderstanding about Papal Infallibility among non-Catholics. Even many practicing Catholics are not clear on the exact meaning of the term. So here is a quick primer on infallibility and the Pope.
By His very Nature, God is unchanging and unchangeable Truth and Goodness and Love. So He cannot err in any way. All that God says is certainly true. All that God does is certainly Good. God is infallible. All other infallibility comes from God.
Anyone can make a true assertion: two plus two equals four. Even when a statement is certainly true, it is not necessarily infallible. The term “infallible,” as it is used in Roman Catholic theology, refers only to teachings on matters of faith, morals, or salvation, that are found in Divine Revelation (Tradition and Scripture). All the teachings of Sacred Tradition and Sacred Scripture are infallible, since Tradition and Scripture are God speaking to us, and God is infallible. The infallible teachings of Tradition and of Scripture are called “material dogma”.
The Magisterium is the ability and authority, given by God as a gift to the whole Church, to teach the truths found in Tradition and Scripture on matters of faith, morals, and salvation. The Magisterium is exercised solely by the Pope and the Bishops. The rest of the Church on earth participates in the Magisterium in a general way, by solemn and prayerful consideration of the truths found in Tradition and Scripture, by expression of informal pious opinions and of formal theological opinions about those truth, in so far as they have not yet been defined by the Magisterium. The Magisterium draws from the understanding of the faithful in issuing new doctrinal definitions, and in teaching from Tradition and Scripture. This is particularly true in the case of the theological insights of the Saints, Doctors, and Fathers of the Church.
All of the teachings of the Magisterium are either:
1. infallible – no possibility of error, and therefore irreformable
2. non-infallible – limited possibility of error, and therefore non-irreformable
The infallible teachings of the Magisterium are called “formal dogma”. Every formal dogma is also a material dogma.
The infallible teachings require the full assent of faith, also called theological assent or divine and catholic faith. Infallible teachings do not allow for any obstinate denial, nor any obstinate doubt, without grave sin.
The non-infallible teachings require religious assent, also called the religious submission of will and intellect. Non-infallible teachings, because there is a limited possibility of error, allow for some limited licit theological dissent, without sin or without grave sin.
The Magisterium teaches infallibly in any of three ways:
1. Papal infallibility
2. Conciliar infallibility
3. the ordinary and universal Magisterium
All other magisterial teachings are non-infallible and non-irreformable.
The personal opinions, private theology, temporal judgments, rules and rulings of the Pope do not fall under the Magisterium at all. These are fallible. (Dogmatic facts are infallible, but that is another subject.) The teaching of the Pope as Bishop of Rome, directed to His diocese, and the teaching of the Pope as the Roman Pontiff, directed to all of the faithful in the world, falls under the Magisterium. But the Pope only teaches infallibly when his teachings meet certain conditions, as taught by the First Vatican Council and as reiterated, the same dogma in different wording, by the Second Vatican Council.
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 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 25]
The exercise of Papal Infallibility by the Roman Pontiffs, throughout the history of the Church, has been a rarity. Most Popes have not exercised Papal Infallibility at all. There are fewer infallible papal definitions than there have been Popes.
Typically, a Pope exercises Papal Infallibility by issuing a papal teaching document: a bull or an encyclical or an Apostolic Letter or an Apostolic Constitution. In theory, a Pope could teach infallibly apart from a formally-issued papal document. But he would have to be clearly teaching the whole Church, in a way that was binding on all the faithful. And his teaching would need to be a definitive act. That would be difficult, but not impossible, to accomplish without a formally-issued papal document.
The Pope cannot teach infallibly in a private conversation, nor in a private letter, because infallible teachings must be binding on the whole Church, and so they necessarily must be directed to the whole Church. An off-handed remark, whether verbal, or in a letter, or on the internet would not meet that standard.
A papal tweet on the Twitter website does not meet the criterion that the teaching must be a definition or a defining act: “by a definitive act, he proclaims”. A papal tweet would also not meet the criterion that the teaching must be expressed to the whole Church in a way that is binding on the whole Church: “must be held by the whole Church”, “in accordance with revelation itself, which all are obliged to abide by and be in conformity with”.
It is barely possible that a Pope could exercise the ordinary non-infallible papal Magisterium in the informal setting of a question and answer session on Twitter. If he is answering a question on a matter of faith, morals, or salvation, and doing so publicly, then he is able to teach non-infallibly. But it would be highly questionable whether the Pope could be considered to have exercised even the non-infallible Magisterium in that type of setting.
So I would expect that the answers to questions that the Pope gives on Twitter will fall into the realm of “private” theological opinion, just as his published books of theology also do.