Catholic Church
According to Catholic belief, Simon Peter was distinguished by Jesus to hold the first place of honor and authority. Also in Catholic belief, Peter was, as the first Bishop of Rome, the first Pope. Furthermore, they consider every Pope to be Peter's successor and the rightful superior of all other bishops. However, Peter never bore the title of "Pope" or "Vicar of Christ".
The Catholic Church's recognition of Peter as head of its church on earth (with Christ being its heavenly head) is based on its interpretation of passages from the canonical gospels of the New Testament, as well as sacred tradition.
The first passage is John 21:15–17 which is: "Feed my lambs... Tend my sheep... feed my sheep" (within the Greek it is Ποίμαινε i.e., to feed and rule [as a Shepherd] v. 16, while Βόσκε i.e., to feed for v.15 & v. 17) – which is seen by Catholics as Christ promising the spiritual supremacy to Peter. The Catholic Encyclopedia of 1913 sees in this passage Jesus "charging [Peter] with the superintendency of all his sheep, without exception; and consequently of his whole flock, that is, of his own church".
In this passage, the evangelist writes: "First, Simon called Peter..." The Greek word for "first" (protos), derived from the ancient Greek πρῶτος, can mean primacy in foundation, not just in a numerical sense.
Another passage is Matthew 16:18:
I tell you that you are Peter, and on this rock I will build my church, and the gates of Hell will not overcome it. I will give you the keys of the kingdom of heaven; whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven.
In the story of the calling of the disciples, Jesus addresses Simon Peter with the Greek term Κηφᾶς (Cephas), a Hellenised form of Aramaic ܟ݁ܺܐܦ݂ܳܐ (kepha), which means "rock", a term that before was not used as a proper name:
:ἐμβλέψας αὐτῷ ὁ Ἰησοῦς εἶπεν Σὺ εἶ Σίμων ὁ υἱὸς Ἰωάννου, σὺ κληθήσῃ Κηφᾶς ὃ ἑρμηνεύεται Πέτρος.
Having looked at him, Jesus said, "You are Simon the son of John; you will be called Cephas," which means Petros ("rock").
Jesus later alludes to this nickname after Peter declares Jesus to be the Messiah:
:κἀγὼ δέ σοι λέγω ὅτι σὺ εἶ Πέτρος [Petros] καὶ ἐπὶ ταύτῃ τῇ πέτρᾳ [petra] οἰκοδομήσω μου τὴν ἐκκλησίαν, καὶ πύλαι ᾅδου οὐ κατισχύσουσιν αὐτῆς.
I also say to you now that you are Peter, and on this rock I will build my Church, and the gates of Hades will not prevail against it.
The Peshitta Syriac version renders Jesus' words into Aramaic as follows:
:ܐܳܦ݂ ܐܶܢܳܐ ܐܳܡܰܪ ܐ݈ܢܳܐ ܠܳܟ݂ ܕ݁ܰܐܢ݈ܬ݁ ܗ݈ܽܘ ܟ݁ܺܐܦ݂ܳܐ ܘܥܰܠ ܗܳܕ݂ܶܐ ܟ݁ܺܐܦ݂ܳܐ ܐܶܒ݂ܢܶܝܗ ܠܥܺܕ݈݁ܬ݁ܝ ܘܬ݂ܰܪܥܶܐ ܕ݁ܰܫܝܽܘܠ ܠܳܐ ܢܶܚܣܢܽܘܢܳܗ܂
Also I say to you that you are Kepha, and on this kepha I will build my Church, and the gates of Sheol not will subdue it.
Paul of Tarsus later uses the appellation Cephas in reference to Peter.
To better understand what Christ meant, Basil of Caesarea elaborates:
Though Peter be a rock, yet he is not a rock as Christ is. For Christ is the true unmoveable rock of himself, Peter is unmoveable by Christ the rock. For Jesus doth communicate and impart his dignities, not voiding himself of them, but holding them to himself, bestoweth them also upon others. He is the light, and yet you are the light: he is the Priest, and yet he maketh Priests: he is the rock, and he made a rock.
In reference to Peter's occupation before becoming an apostle, the popes wear the Fisherman's Ring, which bears an image of Peter casting his nets from a fishing boat. The keys used as a symbol of the pope's authority refer to the "keys of the kingdom of Heaven" promised to Peter. The terminology of this "commission" of Peter is unmistakably parallel to the commissioning of Eliakim ben Hilkiah in Isaiah 22:15–23. Peter is often depicted in both Western and Eastern Christian art holding a key or a set of keys.
In the original Greek the word translated as "Peter" is Πέτρος (Petros) and that translated as "rock" is πέτρα (petra), two words that, while not identical, give an impression of one of many times when Jesus used a play on words. Furthermore, since Jesus presumably spoke to Peter in their native Aramaic language, he would have used kepha in both instances. The Peshitta Text and the Old Syriac texts use the word "kepha" for both "Peter" and "rock" in Matthew 16:18. John 1:42 says Jesus called Simon "Cephas", as Paul calls him in some letters. He was instructed by Christ to strengthen his brethren, i.e., the apostles. Peter also had a leadership role in the early Christian church at Jerusalem according to The Acts of the Apostles chapters 1–2, 10–11, and 15.
Early Catholic Latin and Greek writers (such as John Chrysostom) considered the "foundation rock" as applying to both Peter personally and his confession of faith (or the faith of his confession) symbolically, as well as seeing Christ's promise to apply more generally to his twelve apostles and the Church at large. This "double meaning" interpretation is present in the current Catechism of the Catholic Church.
Protestant arguments against the Catholic interpretation are largely based on the difference between the Greek words translated "Rock" in the Matthean passage. They often claim that in classical Attic Greek petros (masculine) generally meant "pebble", while petra (feminine) meant "boulder" or "cliff", and accordingly, taking Peter's name to mean "pebble", they argue that the "rock" in question cannot have been Peter, but something else, either Jesus himself or the faith in Jesus that Peter had just professed. These popular-level writings are disputed in similar popular-level Catholic writings.
The New Testament was written in Koiné Greek, not Attic Greek and some authorities say no significant difference existed between the meanings of petros and petra. So far from meaning a pebble was the word petros that Apollonius Rhodius, a writer of Koiné Greek of the third century BC, used it to refer to "a huge round boulder, a terrible quoit of Ares Enyalius; four stalwart youths could not have raised it from the ground even a little".
The feminine noun petra (πέτρα in Greek), translated as rock in the phrase "on this rock I will build my church", is also used in 1 Cor. 10:4 describing Jesus Christ, which reads: "They all ate the same spiritual food and drank the same spiritual drink; for they drank from the spiritual rock that accompanied them, and that rock was Christ."
Although Matthew 16 is used as a primary proof-text for the Catholic doctrine of Papal supremacy, some Protestant scholars say that prior to the Reformation of the 16th century, Matthew 16 was very rarely used to support papal claims, despite it being well documented as being used in the 3rd century by Stephen of Rome against Cyprian of Carriage in a "passionate disagreement" about baptism, and in the 4th century by Pope Damasus as a claim to primacy as a lesson of the Arian Controversy for stricter discipline and centralised control. Their position is that most of the early and medieval Church interpreted the "rock" as being a reference either to Christ or to Peter's faith, not Peter himself. They understand Jesus' remark to have been his affirmation of Peter's testimony that Jesus was the Son of God.
Despite this claim, many Fathers saw a connection between Matthew 16:18 and the primacy of Peter and his office, such as Tertullian, writing: "The Lord said to Peter, 'On this rock I will build my Church, I have given you the keys of the kingdom of heaven [and] whatever you shall have bound or loosed on earth will be bound or loosed in heaven' [Matt. 16:18–19]. ...Upon you, he says, I will build my Church; and I will give to you the keys, not to the Church."
Paul's Epistle to the Romans, written about AD 57. greets some fifty people in Rome by name, but not Peter whom he knew. There is also no mention of Peter in Rome later during Paul's two-year stay there in Acts 28, about AD 60–62. Some Church historians consider Peter and Paul to have been martyred under the reign of Nero, around AD 64 or 68.
Other theologically conservative Christians, including Confessional Lutherans, also rebut comments made by Karl Keating and D.A. Carson who claim that there is no distinction between the words petros and petra in Koine Greek. The Lutheran theologians state that the dictionaries of Koine/NT Greek, including the authoritative Bauer-Danker-Arndt-Gingrich Lexicon, indeed list both words and the passages that give different meanings for each. The Lutheran theologians further note that:
We honor Peter and in fact some of our churches are named after him, but he was not the first pope, nor was he Roman Catholic. If you read his first letter, you will see that he did not teach a Roman hierarchy, but that all Christians are royal priests. The same keys given to Peter in Matthew 16 are given to the whole church of believers in Matthew 18.
Oscar Cullmann, a Lutheran theologian and distinguished Church historian, disagrees with Luther and the Protestant reformers who held that by "rock" Christ did not mean Peter, but meant either himself or the faith of his followers. He believes the meaning of the original Aramaic is very clear: that "Kepha" was the Aramaic word for "rock", and that it was also the name by which Christ called Peter.
Yet, Cullmann sharply rejects the Catholic claim that Peter began the papal succession. He writes: "In the life of Peter there is no starting point for a chain of succession to the leadership of the church at large". While he believes the Matthew text is entirely valid and is in no way spurious, he says it cannot be used as "warrant of the papal succession". Cullmann concludes that while Peter was the original head of the apostles, Peter was not the founder of any visible church succession.
There are other Protestant scholars who also partially defend the historical Catholic position about "Rock". Taking a somewhat different approach from Cullman, they point out that the Gospel of Matthew was not written in the classical Attic form of Greek, but in the Hellenistic Koine dialect in which there is no distinction in meaning between petros and petra. Even in Attic Greek, in which the regular meaning of petros was a smallish "stone", there are instances of its use to refer to larger rocks, as in Sophocles, Oedipus at Colonus, v. 1595, where petros refers to a boulder used as a landmark, obviously something more than a pebble. In any case, a petros/petra distinction is irrelevant considering the Aramaic language in which the phrase might well have been spoken. In Greek, of any period, the feminine noun petra could not be used as the given name of a male, which may explain the use of Petros as the Greek word with which to translate Aramaic Kepha.
Yet, still other Protestant scholars believe that Jesus in fact did mean to single out Peter as the very rock which he will build upon, but that the passage does nothing to indicate a continued succession of Peter's implied position. They assert that Matthew uses the demonstrative pronoun taute, which allegedly means "this very" or "this same" when he refers to the rock on which Jesus' church will be built. He also uses the Greek word for "and", kai. It is alleged that when a demonstrative pronoun is used with kai, the pronoun refers back to the preceding noun. The second rock Jesus refers to must then be the same rock as the first one; and if Peter is the first rock, he must also be the second.
Unlike Oscar Cullmann, Confessional Lutherans and many other Protestant apologists agree that it's meaningless to elaborate the meaning of "Rock" by looking at the Aramaic language. While the Jews spoke mostly Aramaic at home, in public they usually spoke Greek. The few Aramaic words spoken by Jesus in public were unusual, which is why they are noted as such. And most importantly the New Testament was revealed in Koine Greek, not Aramaic.
Lutheran historians even report that the Catholic church itself did not, at least unanimously, regard Peter as the rock until the 1870s:
Rome's rule for explaining the Scriptures and determining doctrine is the Creed of Pius IV. This Creed binds Rome to explain the Scriptures only according to the unanimous consent of the Fathers. In the year 1870 when the Fathers gathered and the pope declared his infallibility, the cardinals were not in agreement on Matthew 16:18. They had five different interpretations. Seventeen insisted Peter is the rock. Sixteen held that Christ is the rock. Eight were emphatic that the whole apostolic college is the rock. Forty-four said Peter's faith is the rock, The remainder looked upon the whole body of believers as the rock. – And yet Rome taught and still teaches that Peter is the rock.