Hello Arch-
The idea that Papal Supremacy in the form that the RC Church teaches papal supremacy requires leaps of logic, to the extent that I don't perceive those leaps in the Bible itself.
Jesus does not explicitly state something like: "Peter, you are the administrative head of all Christians, and this status is permanent, and therefore all your successors in your See will rule the Church, and this only applies to the See where you serve when you are killed, and Papal Supremacy entails you being practically all bishops' bishop with total universal jurisdiction, and it entails you having Papal Ex Cathedra Infallibility."
Therefore the only way to see this idea as Biblical is to assert that the Bible's statements indirectly entail this kind of line of logic, using the leaps in italics. The way that one would try to assert this is by looking at statements like Jesus' one to Peter: You are Peter/Petros, and on this rock/petros I will build my Church.
However, Jesus' statement is ambiguous as regards to Peter's position. First, it's amibiguous what "this rock" refers to - either Peter himself, or to Peter's statement in the preceding passage that Jesus was the Son of God.
"And I say unto thee, Thou art Peter, and upon this rock will I build my Church;
that is, on the faith of his confession. Hereby He signifies that many were now on the point of believing, and raises his spirit, and makes him a shepherd."
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Chrysostom, Homily LIV,
"And wherefore said He not, But I did not suffer it, rather than, I have prayed? He speaks from this time lowly things, on His way to His passion, that He may show His humanity.
For He that has built His church upon Peter's confession, and has so fortified it, that ten thousand dangers and deaths are not to prevail over it;"
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Chrysostom, Homily LXXXIV
"In a passage in this book, I said about the Apostle Peter: 'On him as on a rock the Church was built', But I know that very frequently at a later time, I so explained what the Lord said: 'Thou art Peter, and upon this rock I will build my Church,' that it be understood as built upon Him whom Peter confessed saying: 'Thou art the Christ, the Son of the living God,' and so Peter, called after this rock, represented the person of the Church which is built upon this rock, and has received 'the keys of the kingdom of heaven.' For, 'Thou art Peter' and not 'Thou art the rock' was said to him.
But 'the rock was Christ,' in confessing whom, as also the whole Church confesses, Simon was called Peter. But let the reader decide which of these two opinions is the more probable."
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Augustine, "Retractions"
"To Peter the Father revealed that he should say, 'Thou art the Son of the living God.'
Therefore the building of the Church is upon this rock of confession; this faith is the foundation of the Church."
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Hilary
"Faith,
then, is the foundation of the Church, for it was not said of Peter's flesh, but of his faith, that 'the gates of hell shall not prevail against it.' But his confession of faith conquered hell."
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Ambrose of Milan
"When [Peter] wisely and blamelessly confessed his faith to Jesus saying, 'You are Christ, Son of the living God,' Jesus said to divine Peter: 'You are Peter and upon this rock I will build my Church.'
Now by the word 'rock', Jesus indicated, I think, the immovable faith of the disciple.
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Cyril of Alexandria, Commentary on Isaiah 4.2
"For all bear the surname rock who are the imitators of Christ, that is, of the spiritual rock which followed those who are being saved, that they may drink from it the spiritual draught. But these bear the surname of rock just as Christ does. But also as members of Christ deriving their surname from Him they are called Christians, and from the rock, Peters.
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Origen, Commentary on the Gospel of Matthew (Book XII), sect. 10,11
"
Christ, you see, built his Church not on a man but on Peter's confession. What is Peter's confession? 'You are the Christ, the Son of the living God.' There's the rock for you, there's the foundation, there's where the Church has been built, which the gates of the underworld cannot conquer."
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Augustine, John Rotelle, O.S.A., Ed., The Works of Saint Augustine, © 1993 New City Press, Sermons, Vol III/6, Sermon 229P.1, p. 327
"You are Peter and on this rock from which you have taken your name, that is, on myself,
I will build my Church, upon that perfection of faith which you confessed I will build my Church by whose society of confession should anyone deviate although in himself he seems to do great things he does not belong to the building of my Church...Metaphorically it is said to him on this rock, that is, the Saviour which you confessed, the Church is to be built, who granted participation to the faithful confessor of his name."
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Bede, Homily 23, M.P.L., Vol. 94, Col. 260. Cited by Karlfried Froehlich, Formen, Footnote #204, p. 156.
"It will not be moved is said about the Church to which alone that promise has been given: 'You are Peter and upon this rock I shall build my Church and the gates of Hell shall not prevail against it.' For
the Church cannot be moved because it is known to have been founded on that most solid rock, namely, Christ the Lord."
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Cassiodorus, Expositions in the Psalms, Psalm 45.5, M.P.L., Vol. 70, Col. 330)