I think if we can see why papal infallibility is a logical necessity, it’s easier to see when the Pope is infallible.
It is a necessary consequence of the permanency of the primacy. The Church as established by Christ has one bishop and his particular Church (what we commonly think of as a diocese) that are head of the rest. The Church can never lose this element. By divine ordinance, this primacy is fixed in the particular church of Rome.
If this particular church were to separate itself from the rest of the Catholic Church through heresy or schism, the primacy would be lost to the universal Church, and that is impossible. Therefore, it is impossible for the Church of Rome to embrace any error that would rupture its communion with the universal Church. It is likewise impossible, by Christ’s promise, for the whole Church to defect into error.
Given all this, the Pope must be infallible in those doctrinal judgments that define the faith of this church or universal church, that if in error, would either rupture the Church of Rome’s communion with the rest of the Church or cause the entire Church to defect from the faith, since both are impossible.
Unfortunately, the unusual ceremonial attached to the two famous instances involving Marian dogmas (and their unusual circumstances, where rather than settling a controversy, they were a kind of capstone on a settled point) have also minimized the other instances (probably also a false irencism that wants to minimize this truth for ecumenical purposes). The fact is, Popes have intervened to provide definitive/irreformable judgments binding the whole Church many, many times.
At the First Vatican Council, when some bishops wanted to condition papal infallibility on the Pope following some procedure or using some verbal formula, the relator for the Commissio de fide (charged with providing official doctrinal explanations of Council documents to the Council Fathers) said this could not be done, because there were already so many instances with various procedures or even none at all:
But, most eminent and reverend fathers, this proposal simply cannot be accepted because we are not dealing with something new here. Already thousands and thousands of dogmatic judgments have gone forth from the Apostolic See; where is the law which prescribed the form to be observed in such judgments?
“Thousands and thousands” might be a bit hyperbolic, but the point remains–it has been a lot. I assume where a document contains a long list of definitively judged propositions, each one is considered a single judgment. The First Vatican Council also notes the variety of ways such definitions have come about:
The Roman pontiffs, too, as the circumstances of the time or the state of affairs suggested, sometimes by summoning ecumenical councils or consulting the opinion of the Churches scattered throughout the world, sometimes by special synods, sometimes by taking advantage of other useful means afforded by divine providence, defined as doctrines to be held those things which, by God’s help, they knew to be in keeping with Sacred Scripture and the apostolic traditions.
