The opinion of theologians is that a pope may be guilty of private heresy, but that the Holy Spirit will not permit him to impose it on the Faithful with the force of infallibility.
If a pope were to begin to teach heresy, what could the Church do about it?
Can a pope teach and believe heresy? This was always taken as a possibility. In the Traditional Roman Missal there are prayers that ‘neither the obedience of the flock to the shepherd should fail, nor the watchfulness of the Shepherd over the flock.’ S. Thomas Aquinas speaks very clearly on the scope and limits of obedience: there are times when compliance must be withheld. There was never a consensus reached on the matter of how to deal with a pope teaching heresy. It was actively discussed since the 11th century, but things quietened down (after the Catholic {Counter-} Reformation of the 17th century}, and the matter was held un-finalised. There are three possibilities:
(i) A Pope could be declared heretical and deposed by a Council of Bishops, after three public hearings in which he was appraised of his heretical statements and refused to withdraw them. This would prove ‘pertinacity’ and the Council could [on the hypothesis] declare that he had resigned – that he had relinquished the Papacy in favour of heresy;
(ii) If the Pope was 'manifestly' heretical (and that is a technical word: it means 'apparent to effectively *everybody*', not merely the well-informed – a mistake the Sedevacs make) he could be deposed by public acclamation;
(iii) No pope can be declared heretical because 'pertinacity' cannot be proven or declared without a lawful superior, of which there is none on Earth. The Faithful will simply have to keep the Faith and wait prayerfully for the pope to repent, or to die and give his successor a chance to revive the Church.
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It is possible that this issue will be tested and resolved in the present time – even during the present Papacy of Francis.