John Lightfoot on Matthew 16:18

[Thou art Peter, &c.] I. There is nothing, either in the dialect of the nation, or in reason, forbids us to think that our Saviour used this very same Greek word, since such Graecizings were not unusual in that nation. But be it granted (which is asserted more without controversy) that he used the Syriac word; yet I deny that he used that very word Cepha, which he did presently after: but he pronounced it Cephas, after the Greek manner; or he spoke it Cephai, in the adjective sense, according to the Syriac formation. For how, I pray, could he be understood by the disciples, or by Peter himself, if in both places he had retained the same word Thou art a rock, and upon this rock I will build my church? It is readily answered by the Papists, that "Peter was the rock." But let them tell me why Matthew used not the same word in Greek, if our Saviour used the same word in Syriac. If he had intimated that the church should be built upon Peter, it had been plainer and more agreeable to be the vulgar idiom to have said, "Thou art Peter, and upon thee I will build my church."

II. The words concerning the rock upon which the church was to be built are evidently taken out of Isaiah, chapter 28:16; which, the New Testament being interpreter, in very many places do most plainly speak Christ. When therefore Peter, the first of all the disciples (from the very first beginning of the preaching of the gospel), had pronounced most clearly of the person of Christ, and had declared the mystery of the incarnation, and confessed the deity of Christ, the minds of the disciples are, with good reason, called back to those words of Isaiah, that they might learn to acknowledge who that stone was that was set in Sion for a foundation never to be shaken, and whence it came to pass that that foundation remained so unshaken; namely, thence, that he was not a creature, but God himself, the Son of God.

III. Thence, therefore, Peter took his surname; not that he should be argued to be that rock, but because he was so much to be employed in building a church upon a rock: whether it were that church that was to be gathered out of the Jews, of which he was the chief minister, or that of the Gentiles (concerning which the discourse here is principally of), unto which he made the first entrance by the gospel.