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When a child is circumcised he is entered into G‑d's covenant with Abraham and his descendants, he joins a chain linking him to the very beginning of our nation. It is only fitting to give him his Jewish name after he has become a full-fledged member of the Jewish nation.
Perhaps another reason why we postpone naming a child until the circumcision is because the first "Jewish name" was given in conjunction with this mitzvah. As we read in Genesis (17:5, 10):
"And your name shall no longer be called Abram, but your name shall be Abraham. . . . This is My covenant, which you shall observe between Me and between you and between your seed after you, that every male among you be circumcised..."
Yours truly,
Rabbi Menachem Posner
Providence, RI
However, they were already G-d’s chosen nation. Right in the beginning of the account of Abraham’s life He is told that G-d will create a great nation from his seed (Gen 12:2).
So while they didn’t have the full status of Jews with all the accompanying bells and whistles, they were set apart as a special nation.
G-d bless,
Cincinnati, OH