Formulating the Conclusion
When I was studying mathematics in university, a professor once told us that half the proof of a mathematical theorem consists of formulating the given premises and the conclusion correctly. In discussions about papal primacy, or infallibility, it is often the case that the conclusion, or what is supposed to be proved, is not clearly formulated. However when it is, the deductive gap between the passages of Matthew 16:17-19, and the claims of the papacy, becomes self evident at once.
So what is the actual conclusion of the claim? What is it that is supposed to be proved? It is simply this:
For no one can be in doubt, indeed it was known in every age that the holy and most blessed Peter, prince and head of the apostles, the pillar of faith and the foundation of the catholic church, received the keys of the kingdom from our lord Jesus Christ, the saviour and redeemer of the human race, and that to this day and for ever he lives and presides and exercises judgment in his successors the bishops of the holy Roman see, which he founded and consecrated with his blood (Council of Vatican I)
Thus the conclusion in bold is that Peter "somehow" lives on in the bishop of Rome. And not only that, but that he lives on in the Bishop of Rome alone. As such we should not confuse this very specific and unique contention of Rome with other issues to do with apostolic succession or the preeminence of Rome or whatever. The contention here is a very specific one, the key upon which the whole claims rest: that Peter somehow lives on in the Bishop of Rome. What needs to be demonstrated is this transmigration or transfer of Peter's soul/spirit/life into the Bishop of Rome.
On St Peter Alone the Church is Built?
In various Roman Catholic theological literature there is often a very heavy emphasis upon demonstrating the primacy of St Peter the Apostle and his significance over the rest and that it is upon him alone that the Church is built. Proving the premise that St Peter is the foundation of the Church as per Matthew 16:17-19 is the first step towards demonstrating the truth claims of the papacy.
There is however a logical problem with the premise, a logical problem whereby even if it were proved does not lead to its desired conclusion and may ironically even undercut it.
For all the arguments about how St Peter is the Rock upon which Christ built his Church, the complete identification of Peter as the "Rock" would not lead to the desired conclusion because if St Peter himself alone was the foundation of the Church, then that would be so to the exclusion of the bishops of Rome!
If truly St Peter alone was so vital and significant to the foundation of the Church, then only he alone, and no one else, not even the Bishop of Rome, can possibly be the foundation of the Church. Who knows, maybe as an alternative scenario the glorified St Peter in heaven still rules the Church as its "rock" there, coordinating the efforts of angels, wielding the Keys, ruling and judging our prayers and causes in his glorified state. (After all, Roman Catholics believe in the living communion with past saints don't they? Why can't St Peter himself still be in active communion and leadership with the Church?) This would be the logical implication that the person of St Peter himself was alone the Rock upon which the Church was built.
So it cannot be Peter himself, strictly speaking, that is the foundation of the Church, but something about him, not him directly or personally, which would be passed on and continued beyond himself, which must be the rock of the Church.
Once however this modified premise is admitted, all the Roman Catholics efforts to demonstrate the primacy and importance of St Peter and him as the foundation of the Church vanishes into irrelevance. For the premise of the importance of St Peter would play no part at all in proving the claims of papacy if it logically closes in upon itself and cannot move beyond St Peter to the Bishop of Rome. It doesn't matter how important, or how foundational, St Peter is if this importance or foundation cannot move beyond his own person.
So there must be something about St Peter, something which can be separated from his person, and continued beyond him. What is this something?
Peter Himself or Petrine Faith?
Unfortunately it is by no means self-evident from the text of Matthew 16:17-19 that this something, which goes beyond St Peter, is a petrine spirit or liferather than St Peter's faith.
Now historically the Matthew 16 passages have been interpreted both ways to refer to Peter himself or a petrine faith. Exegetically I would say that the text does not decide one way or another. Even granting that St Peter, the person, was the Rock upon which the Church is built, the text does not tell us how this foundation or rock continues to persist after his death or get transferred around, whether in the form of a shared faith or in his own person. The issue has to be decided by theological rather than exegetical arguments.
Transmigration of Petrine Spirit to the Bishop of Rome?
But let us suppose, for the sake of argument, that Peter himself, his spirit or soul or whatever, after the death of the apostle, can be transferred around, and it is this transmigrated petrine spirit which would serve as the "foundation" or "rock" of the Church.
Let us recall what our conclusion is. The conclusion is that Peter himself or his Spirit/Soul/Life has been committed to the (1) Bishop of Rome (2) alone. The (2) is important as some of the Eastern Orthodox maintains that Peter's ministry has been committed to all bishops, or maybe the key patriarchies. As such, there are two claims which needs proving, that "Peter" has been in fact committed to the Bishop of Rome and that the Bishop of Rome alone possesses it and no one else.
It is here where the weakest link in the proof lies. The main problem is simply this: no theological mechanism has ever been explicitly proposed to explain how Peter, or his spirit or life or whatever, got transferred to the Bishop of Rome.
There are however two vague narratives which attempts to show how the Bishop of Rome possessed the Petrine office:
(1) The Episcopate of Rome possesses the Spirit of Peter by virtue of it being occupied by St Peter himself.
(2) The Episcopate of Rome possesses the Spirit of Peter by virtue of it being founded by St Peter himself.
It is important to note the difference between the two claims, the first narrative would require that St Peter be the first Bishop of Rome himself and then his successor literally took his place. The second narrative on the other hand doesn't require St Peter to occupy the Episcopate of Rome himself, Peter merely needs to create the Episcopate of Rome but not be the Bishop of Rome himself.
The first narrative suffers from two problems (1) historical and (2) theological. First, there is no evidence that St Peter was ever the first Bishop of Rome. As Hermann Sasse puts it:
The Roman list which Irenaeus brings up to his time has nothing but authentic names. Its age and authenticity are evidenced by the fact that, in contrast with the current official list of popes, Peter does not appear as the first Roman bishop. Peter and Paul are presented as the founders of the church in Rome. They are said to have committed the episcopate to Linus. the whole following list with its numbering of the third, sixth, and ninth bishop is constructed on the presupposition that Linus was the first bishop, and that Peter and Paul put him in this office.
-Apostolic Succession (Letters to Lutheran Pastors No.14 April 1956)
The blessed apostles [Peter and Paul], then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate. Of this Linus, Paul makes mention in the Epistles to Timothy. To him succeeded Anacletus; and after him, in the third place from the apostles, Clement was allotted the bishopric... To this Clement there succeeded Evaristus. Alexander followed Evaristus; then, sixth from the apostles, Sixtus was appointed; after him, Telephorus, who was gloriously martyred; then Hyginus; after him, Pius; then after him, Anicetus. Sorer having succeeded Anicetus, Eleutherius does now, in the twelfth place from the apostles, hold the inheritance of the episcopate.
-Irenaeus: Against Heresies Book III; Chapter 3
Thus, the first Bishop of Rome was Linus, not St Peter. The succession list shifted over time to reflect St Peter as the first Bishop of Rome and thereby legitimise the idea that the Bishop of Rome itself directly possessed the Petrine office by virtue of being occupied by St Peter himself. But the historical evidence for such a thing is scant, if not virtually non-existent. While Irenaeus does speak of Peter and Paul founding the Church at Rome, there is no evidence that Peter was ever the Bishop of Rome. If founding a church makes one a bishop then Paul would also be a bishop of Rome. It seems however that this idea that Peter was the first bishop of Rome simply arose from a confusion of him having ordained the first bishop of Rome. However it should be obvious that to ordain a bishop of a locality is not to be the bishop of that locality.
Theologically even if we accept that St Peter was the first Bishop of Rome, it still does not follow that somehow St Peter serving as a bishop of a locality causes his spirit to be transferred to subsequent successors. Peter no doubt served as bishops in many other places, do all their successors possess the Petrine see as well? No theological mechanism has been proposed to explain how does being a bishop of a locality somehow cause the rock to magically transfer over to its successors.
Thus we are left with the other narrative to pass down the Petrine Spirit to the Bishop of Rome, that is, being founded and ordained by St Peter himself. However the theological problems are the same as the previous one, Peter also ordained and handed the episcopate of many other places to other people, and founded many other churches. Do all those other places and their episcopal sees possess the petrine spirit by virtue of being so founded and headed by St Peter? Again, no theological mechanism has been proposed to explain how does Peter founding a church or ordaining someone as bishop somehow transfer the rock or spirit over.
It is particularly significant that in Irenaeus's account he does not describe the episcopate of Rome as the "Chair of Peter" as later theologians would. He never says that the spirit/rock of Peter is somehow transferred over to the Roman see. Later theologians and "Church Fathers" would simply assert that the episcopate of Rome just is the Chair of Peter. Somehow it "just happened" without anyone explaining how it did.
The fact of the matter is that there is simply no ecclesiological, liturgical or theological principle which can connect the dots between the Petrine office and the Bishop of Rome alone. How the Petrine See gets passed on to the Bishop of Rome and him alone is the missing gap in the proof and I don't see anyway of making it work. Much more vitally, no Father, before Cyrian at least, has ever testified or witnessed to this transfer of the Petrine Spirit or the transmigration of this soul or the transfer of the Rock.
Conclusion: The Scandal of the Weakness of the Historical Connection
A few centuries ago the Jesuits debated the proposition: "It is not de fide that a particular person, e.g., Clement VIII, is the successor of St. Peter." The argument was that since we have no "de fide" certainty as to the historical fact as to whether a Pope validly baptised, canonically ordained, or elected without simony, therefore we can only have "moral certainty" and not "de fide" faith that a particular person now is Pope. The inquisition promptly intervened and, here's the interesting part, they judged that the proposition was "scandalous", but not heretical.
Do we not have sufficient grounds to draw a like inference, that it is not a "de fide" matter that the Petrine Spirit has been passed to the Bishop of Rome and him alone? This of course also does not include the other numerous problems which attends the doctrine of the papacy even if we could somehow miraculously demonstrate that the petrine spirit has been committed to the Bishop of Rome and him alone. What proof, historical or theological, could there be for the completely arbitrary restriction of the exercise of that Spirit only when he speaks of matter of faith and morals and only ex cathedra, which meaning and criteria no Roman Catholic could agree on?
It seems clear that in the end, for all the importance of St Peter, the ability of this claim to legitimise the transfer of the Petrine Spirit to the episcopate of Rome cannot even get off the ground. Perhaps we would be better off believing that St Peter really is ruling the Church from heaven.