Something which keeps coming up, and being misunderstood, is the interesting interplay of πετρα and πετρος in Mt 16:18. Now, I am not a Catholic, but I would nonetheless say that that verse does support Petrine primacy, and I would like to explain how.
In Greek, Mt 16:18 reads καγω δε σοι λεγω (“and I-myself say to you”) οτι συ ει πετρος (“that you are Petros”) και επι ταυτη τη πετρα (“and upon this Petra”) οικοδομησω μου την εκκλησιαν (“I will build my church”) και πυλαι 'αδου ου κατισχυσουσιν αυτης (“and the gates of Hades will not stand against her”).
What some anti-Catholics claim is that πετρος and πετρα are wholly-dissimilar, and that Jesus is not saying that he will build his church on Peter; what some Catholics claim is that πετρος and πετρα are utterly identical, and that Jesus is saying that he will build his church only on Peter. Neither of these readings does justice to the text.
The myth of dissimilarity: “You are Rock, but upon this other Stone”
The two terms πετρα and πετρος are closely related. Greek had more than half a dozen other terms for stone (λιθος being the one most often used in the Bible), but the Gospel writer used a pair which share an obvious linguistic root in πετρ- rather than for the distinct difference which would have been achieved by using such terms as “πετρος … λιθωι”.
The “pebble” myth
One of the more common claims of anti-Catholic readings is that πετρα is a mass of stone, while πετρος is a small pebble. Although that description of πετρα is not a bad one, the description of πετρος just does not match the usage in Greek. Homer uses πετρος in Iliad 20.288 for a boulder too large for two mortals to lift; it is subsequently used for boundary stones, and for the stones dropped to kill besiegers ( 7.142Jewish Antiquities). Those are no pebbles. The consistent factor in the usage of πετρος is individuality: it refers to a distinct, separate chunk of stone.
The myth of identity: “You are Rock, and upon you”
The two terms πετρα and πετρος are not identical. The Gospel writer could have used the same term twice (πετρος … πετρωι), or a simple personal pronoun the second time (πετρος … σοι), but did not. This point can be confusing for people whose first language is English, and who thus think that the “natural sense” is that the two are the same because an English translation says “Rock … rock” (but see also Jesus’ “brothers”).
The neologism myth
Πετρος did exist as a common noun before the Gospel, and even, as seen in Josephus, after the Gospel. The claim that it was a neologism might have come from Petrus, the Latin form used in the first part of Mt 16:18, which does not seem to have predated the Latin translation of the Gospel.
The gender myth
Πετρος is not just the masculine form of πετρα, and Jesus did not “have to” use something other than πετρα because he was talking to Simon, a man. First, Greek gender was primarily grammatical, and so Greek could identify usually-masculine people with feminine nouns if one so wished, thus calling Simon “Petra” just like Jesus is “Sophia” (1 Co 1:24, 30). Second, Greek could also switch the grammatical genders of nouns, and we have examples of that with πετρος itself being feminine in Anthologia Palatina 7.274, 479.
“It does not matter, because it was not originally Greek”
The claim that Matthew’s Gospel was originally in “the Hebrew dialect”, and so all of this Greek does not matter, is rather complex. It is based upon Irenaeus’ Adversus Haereses 3.1.1 and Eusebius’ later Historia Ecclesiastica 3.39, both referring back to Papias’ lost work. There is doubt amongst scholars as to whether this does indicate the existence of a Hebrew original to Matthew, or whether Papias was referring to Matthew’s having compiled something like the uncanonized Gospel of the Hebrews. Even if Papias was referring to a Hebrew or Aramaic original for Matthew’s Gospel, we just do not have that text, which makes it very difficult to claim that it “obviously” said, “You are Kephas; upon this Kephas”, especially when Biblical Hebrew, which was closely related to Aramaic, had about ten different words for “rock”, and so another interplay could easily have taken place there. (Also, anyone who believes strongly in the canon is unlikely to be persuaded by an appeal to a text other than the canonized Greek.)
The resulting interplay: “You are Rock, and upon such Rockness”
In Greek, then, we have in Mt 16:18 πετρα, the undifferentiated mass, and πετρος, the distinct chunk of that mass. The different-yet-similar interplay places the two terms in a relationship with one another, and so, (as you can see if you look on p.1079 of Lampe’s Patristic Greek Lexicon,) the early, Greek-reading Church Fathers read it in a variety of interconnected ways, all of which centre around “Simon, you are a-shining-example-of-X and upon X will I build my Church”. Peter is thus identified as the first and foremost of what was to come, and denying his significance there would be as unfaithful to the text as would claiming that he is the only foundation stone in view. The text presents Peter (including his faith and his profession of that faith) as the archetype of Christianity, which makes it an excellent argument for Petrine primacy, if not quite for Petrine supremacy.