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The Trinity and Two Natures of Christ are "clear" in the Bible with the convenience of historical hindsight that we have, yet they weren't fully defined till 325 and 451, respectively.
Protestants claim that the canon of the Bible was so "clear" from the outset, yet no one listed all 27 NT books till St. Athanasius in 367.
Protestants claim sola fide is so clear in the Bible, yet even such a one as Norman Geisler states that no one taught imputed, forensic justification between the times of Paul and Luther (!!!).
Why should the papacy be any different? Why would anyone expect it to be if they were familiar at all with how other doctrines necessarily developed and became better understood over time?
Yes, the papacy (like the Trinity) is very clear in the Bible, once one does some serious exegesis (an exegesis which -- in this case -- Protestant laymen are most reluctant to discuss), but that doesn't mean that everyone would have perfectly understood it from the outset. As men reflected upon it, and as it was attacked, it became more clear, as all doctrines do. Cardinal Newman wrote about it:
Let us see how, on the principles which I have been laying down and defending, the evidence lies for the Pope's supremacy.
As to this doctrine the question is this, whether there was not from the first a certain element at work, or in existence, divinely sanctioned, which, for certain reasons, did not at once show itself upon the surface of ecclesiastical affairs, and of which events in the fourth century are the development; and whether the evidence of its existence and operation, which does occur in the earlier centuries, be it much or little, is not just such as ought to occur upon such an hypothesis.
. . . While Apostles were on earth, there was the display neither of Bishop nor Pope; their power had no prominence, as being exercised by Apostles. In course of time, first the power of the Bishop displayed itself, and then the power of the Pope . . .
. . . St. Peter's prerogative would remain a mere letter, till the complication of ecclesiastical matters became the cause of ascertaining it. While Christians were "of one heart and soul," it would be suspended; love dispenses with laws . . .
When the Church, then, was thrown upon her own resources, first local disturbances gave exercise to Bishops,and next ecumenical disturbances gave exercise to Popes; and whether communion with the Pope was necessary for Catholicity would not and could not be debated till a suspension of that communion had actually occurred. it is not a greater difficulty that St. Ignatius does not write to the Asian Greeks about Popes, than that St. Paul does not write to the Corinthians about Bishops. And it is a less difficulty that the Papal supremacy was not formally acknowledged in the second century, than that there was no formal acknowledgment on the part of the Church of the doctrine of the Holy Trinity till the fourth. No doctrine is defined till it is violated . . .
Moreover, an international bond and a common authority could not be consolidated, were it ever so certainly provided, while persecutions lasted. If the Imperial Power checked the development of Councils, it availed also for keeping back the power of the Papacy. The Creed, the Canon, in like manner, both remained undefined. The Creed, the Canon, the Papacy, Ecumenical Councils, all began to form, as soon as the Empire relaxed its tyrannous oppression of the Church. And as it was natural that her monarchical power should display itself when the Empire became Christian, so was it natural also that further developments of that power should take place when that Empire fell. Moreover, when the power of the Holy See began to exert itself, disturbance and collision would be the necessary consequence . . . as St. Paul had to plead, nay, to strive for his apostolic authority, and enjoined St. Timothy, as Bishop of Ephesus, to let no man despise him:so Popes too have not therefore been ambitious because they did not establish their authority without a struggle. It was natural that Polycrates should oppose St. Victor; and natural too that St. Cyprian should both extol the See of St. Peter, yet resist it when he thought it went beyond its province . . .
On the whole, supposing the power to be divinely bestowed, yet in the first instance more or less dormant, a history could not be traced out more probable, more suitable to that hypothesis, than the actual course of the controversy which took place age after age upon the Papal supremacy.
It will be said that all this is a theory. Certainly it is: it is a theory to account for facts as they lie in the history, to account for so much being told us about the Papal authority in early times, and not more; a theory to reconcile what is and what is not recorded about it; and, which is the principal point, a theory to connect the words and acts of the Ante-nicene Church with that antecedent probability of a monarchical principle in the Divine Scheme, and that actual exemplification of it in the fourth century, which forms their presumptive interpretation. All depends on the strength of that presumption. Supposing there be otherwise good reason for saying that the Papal Supremacy is part of Christianity, there is nothing in the early history of the Church to contradict it . . .
Moreover, all this must be viewed in the light of the general probability, so much insisted on above, that doctrine cannot but develop as time proceeds and need arises, and that its developments are parts of the Divine system, and that therefore it is lawful, or rather necessary, to interpret the words and deeds of the earlier Church by the determinate teaching of the later.
(Essay on the Development of Christian Doctrine, 1878 ed., Univ. of Notre Dame Press, 1989, pp. 148-155; Part 1, Chapter 4, Section 3)
If one reads the above carefully, they'll see that Newman is reasoning as follows (using mostly his words, but clarifying and expanding upon them a bit):
1. Suppose there is a good reason for saying that the Papal Supremacy is part of Christianity (there certainly is: the biblical Petrine data and much indication in the Fathers).
2. Let us construct a provisional, falsifiable theory of development of doctrine to account for facts as they lie in the history that we can objectively observe and verify.
3. Let us also assume the general probability and likelihood (indeed, dare we say, inevitability?) that doctrine develops as time proceeds and need arises, and that these developments are parts of the Divine system.
4. In order to illustrate #3, we will show several analogies to other doctrines which are held in common by Protestants and Catholics.
5. Let us pursue a theory which connects the words and acts of the Ante-nicene Church with that antecedent probability of a monarchical principle in the Divine Scheme, and that actual exemplification of it in the fourth century.
6. Assuming the premises we have established, let us see if there is anything in the history of the early Church to contradict it.
7. By checking those facts and applying this provisional analysis, we see that nothing in the early history of the Church contradicts it
8. Therefore, we accept it as a plausible explanation of the course of developmental history as God ordained it from the outset, and as indicated in kernel form in the apostolic deposit itself; particularly the New Testament.Far from being some sort of "historical dogmatism" or a Quasi-Platonic forcing of actual history into what Catholics merely wish it to be; this is, to the contrary, a very scientific way of approaching the issue: almost Popperian, rather than Platonic. Newman tries, in effect, to falsify his own theory by applying it provisionally to history, and he sees that it succeeds; therefore, he accepts it as a reasonable, epistemologically justified explanation till something better is offered. Thus Newman writes in section 21 of his Introduction to his classic work:
. . . the increase and expansion of the Christian Creed and Ritual, and the variations . . . are the necessary attendants on any philosophy or polity which takes possession of the intellect and heart, and has had any wide or extended dominion; that, from the nature of the human mind, time is necessary for the full comprehension and perfection of great ideas; and that the highest and most wonderful truths, though communicated to the world once for all by inspired teachers, could not be comprehended all at once by the recipients, but, . . . have required only the longer time and deeper thought for their full elucidation. This may be called the Theory of Development of Doctrine; . . .Note how Newman views his theory very scientifically (rather than dogmatically in the sense that it is proclaimed as true and expected to be believed):
It is undoubtedly an hypothesis to account for a difficulty; but such too are the various explanations given by astronomers from Ptolemy to Newton of the apparent motions of the heavenly bodies, and it is as unphilosophical on that account to object to the one as to object to the other. Nor is it more reasonable to express surprise, that at this time of day a theory is necessary, granting for argument's sake that the theory is novel, than to have directed a similar wonder in disparagement of the theory of gravitation, or the Plutonian theory in geology.There was one apostolic deposit. Ecclesiology was part of that. The fact that no brand of ecclesiology works or has worked perfectly throughout history is irrelevant to the question of which brand is true. Christians are not philosophical pragmatists or utilitarians.
The Trinity is biblical and true, yet it has always been the case that people calling themselves Christians have denied it.
The doctrine of eternal hellfire has always been biblical and true, but there have always been people calling themselves Christians who denied it (e.g., 7th-Day Adventists or universalists).
Even baptism is not practiced by Quakers and the Salvation Army.
Now, that being the case, does this prove that the Trinity, hell, and the necessity of baptism (however one interprets its significance) are disproved, because not all agreed with them? No, of course not.
Likewise, the fact that not all through history have accepted the papacy as construed by the Catholic Church is not any sort of argument against it. It does not prove that it is not the 1) biblical form of ecclesiology, or that 2) it is not the best form in practice, in terms of creating a unified structure of belief.
Of course, the question of whether Catholic theology is the best one is a separate issue, but it is certainly true that we have preserved our doctrines intact. To the extent that one likes what they see in Catholic theology, the papacy must be seen as the primary unifying factor in Catholicism. That proves nothing as to its truthfulness, but if the argument is over what system has worked the best in practice, then ours has a prime claim for "workability," wholly apart from the question of which ecclesiology is the apostolic, biblical, and patristic one.
And the Great Western Schism does not disprove this. As C. S. Lewis said, "the rules of chess create chess problems." The fact that a problem arises in chess does not mean there are no rules of chess which can be ascertained. How does the fact that three people claimed to be pope for a while prove that the Bible teaches no such thing as a papacy? It does not.
By the same token, Bible difficulties and problems to be worked out do not disprove biblical inerrancy or infallibility or inspiration. Protestants believe in that despite all these "difficulties" that whole books are written about (e.g., Gleason Archer's Encyclopedia of Bible Difficulties).
Protestants also believe that inspired Scripture was written by many great sinners (people like David and Paul and Peter). Yet they find it difficult to believe that sinful men can be infallible (a much less extraordinary gift than inspiration) in limited circumstances. This is to be marveled at, but Protestant bias runs very deep and looks for any and every opportunity to manifest itself. Hence the constant double standards that are not consistently applied to its own systems of thought.