Popes can err if they are not making proclamations in the limited circumstances in which we believe papal infallibility applies to them, per the 1870 dogma. The pope is not some kind of inspired oracle, like he is a walking Bible.

If, for example, the pope said over lunch, “the earth is flat,” that would not be in line with the special conditions of what we define as “infallible” because it was merely a private remark and has nothing to do with faith or morals.

If he wrote an encyclical, however, and stated, “I define and declare ex cathedra, as pope, speaking for all Catholics, that henceforth, all the Catholic faithful must regard the earth as flat,” then this would be a crisis for Catholics, since it would be a blatant example of a falsehood promulgated under the proper conditions: thus overthrowing the dogma.

We say that this has never happened. If we look at the very “best” case our critics come up with Pope Honorius, but in his case, we find that he was making a statement in a private letter. Right off the bat, then, it had nothing to do with infallibility. Their best case is literally a non sequitur. It doesn't disprove infallibility in the slightest.

Proper definitions are crucial in any debate. On this issue, that seems to be about 75% of the battle. Folks don’t want to accept Catholic explanations, and want to almost re-write the dogma according to their own whims.

I understand that. I played that game myself in 1990 when I detested infallibility and thought it was the most ridiculous thing in the world (or at least in theology). But it is ultimately an intellectually dishonest endeavor.           

Protestants seem often unable to distinguish infallibility from inspiration. This is the main problem of comprehension, I suspect. It's understandable, because it’s a new thing. The Protestant outlook is diametrically opposed to infallibility because it is so contrary to sola Scriptura: one of the two pillars of the so-called “Reformation.” That's why I fought it so ferociously myself: because I was a huge sola Scriptura guy and Luther devotee to boot, and I had already been doing apologetics for nine years, so I loved to argue and dispute.

It comes back to prior premises. You can’t look at infallibility from Protestant eyes. You have to try to understand it from Catholic eyes, even though not yet Catholic. We always see things through our own premises, so to change them provisionally, to grasp another, is difficult.

It's like debate, too. To debate well, one should understand the opposing position at least as well as the opponent does himself. And that takes a lot of work, and sort of a “mental discipline.”           

It seems over-dramatic (in ex cathedra statements like those concerning the Immaculate Conception and Assumption of Mary), to say that whoever disbelieves has fallen away from the faith, etc., but again, one has to understand that Catholics believe that we have to accept all that the Church teaches; therefore, if a pronouncement at this high level is made, it has to be accepted, lest we have destroyed the principle of Catholic authority, and in that sense have rejected the Catholic faith.

The opposite of this is the “cafeteria Catholic” game that liberal dissidents play. It would be like a Protestant rejecting sola Scriptura. He or she isn’t much of a Protestant anymore, then.

To use the analogy of government, it's the difference between having an idea for a bill and a bill drawn from that thought passing the House and Senate and being signed into law by the president. The second is binding as law; the first is not at all. The president or Senate leader or House speaker might have talked about a proposed bill, but it has no binding force when it is simply being talked about. When it is made law, it does have binding power.

So what we're saying is that when the pope binds the entire faithful to a dogma, and declares that it is to be held by all Catholics, we believe in faith in such cases that he is given the gift of infallibility by God: protection from error.

If the objection is that it is fideism with no objective proof, then we appeal to history and show how alleged counterfactuals like Honorius don't succeed, and that no such scenarios have succeeded in disproving papal infallibility.

Most Protestants (especially if they're like I used to be) simply assume that such a thing is impossible. They lack faith in God's capacity to enact his supernatural gifts in men.

I reply that it takes much more faith to believe in inspiration of Scripture (so that the huge amount of words in the Bible are not self-contradictory and all “God-breathed”) than infallibility in limited circumstances, so that if one is believed, the other should not be all that difficult to accept in faith.

God had to use lousy sinners like Moses and David and Paul and Peter and Matthew to write the Bible. What else did He have: except for a few recorded words from sinless Mary? Thus, sin and the foibles and follies of men are no barrier to him if he has some purpose. (He even used a donkey once.) Protestants forget that these inspired words in Scripture were written by very human and sinful men like we all are. They came from men, just as papal decrees come from men.

All those objections make no sense, analogically. If one thing is believed, the other should not be impossible to accept. The real root of the error, I believe, is in the sola Scriptura mentality, that holds (with no biblical support) that only the Bible is infallible, and that no human institutions (Church, popes, councils) can be, because only the Bible is, because the Bible says so (but in fact it never does, and states the contrary). That is where the root error and circular reasoning lies.