As the child of a Jewish mother, a child who dies before brit milah, or before a naming, is still considered Jewish. As the Mishnah writes, “A one-day old infant, if he dies, is considered to his father and mother and all relatives like a full bridegroom”. Therefore, according to Jewish law, the child must be buried in a Jewish cemetery with proper honor and sensitivity. In fact, some classic commentators even had the custom of performing brit milah on the child on the day of burial. This is not the common custom but demonstrates how the history of Jewish law upholds the Jewishness of such a child in a very real and tangible way.
While there are multiple opinions as to the exact nature of the mourning and burial practices to be followed in such a case (because full mourning is not required by Jewish law for any child who died at less than thirty days old), within the Conservative movement, for those families who wish, there is a tradition of utilizing various traditional Jewish rituals in these situations. Further, there is little debate about the permissibility of giving the child a name. A child who dies before a brit milah or naming can certainly be given one. The parents can either choose the one they had intended to give the child or one that reflects the nature of the tragedy, like Nehamah or Menachem (both meaning “comfort”). The name can then be used in the burial ceremony and during future remembrances.
First, in case this is a question that comes out of practical experience, let me express my sadness at having to undergo such a painful experience. A baby's death is terribly sad, and we can only hope that God sends comfort to the bereaved very soon. As to the technical sides of the question, there is actually no obligation to circumcise a child who has passed away. The Torah's obligation to circumcise applies only to live children, and only after the 8th day. If the baby was ill on that day, we would delay the berit, the circumcision, until the baby was better. Should the baby pass away, the obligation to circumcise (like in any mitsvot) goes away.
There is, however, a longstanding custom to circumcise the baby anyway, connected to the wish that when the time of the Resurrection comes, this baby, too, will return to life circumcised like all Jews. That would seem all the more true if the failure to circumcise the child was a matter of neglect (if the child never was well enough for circumcision, there would be less opprobrium attached to his uncircumcised state).
I would certainly recommend following the custom in such sad circumstances, but that custom does not make the baby more or less Jewish-- a baby born Jewish is Jewish; circumcision brings the child to a new aspect of Jewishness, a new part of the covenant between God and Jews, but the uncircumcised child is Jewish as well. As to naming, incidentally, it does not have to happen at the circumcision (as we see in the case of girl babies), it is just customary to do it there.
So: if, sadly and God should protect us from such cases, a baby dies before being circumcised, custom decrees that we perform the ritual even after death. If, for some reason, this did not happen, that does not limit or affect the Jewishness of the baby. Similarly, I would recommend giving the child the name before burial, but that, too, is not essential.
To answer the first question, b'rit milah (circumcision) is not like baptism in Christianity. A child is Jewish at birth. Milah is a mitzvah upon the parents, then the community, and if not done by Bar or Bat Mitzvah, upon the child. An exception might be a child who is born to one Jewish parent. For the Reform movement, a child with only one Jewish parent only becomes Jewish as they participate in the rituals of Jewish life. I would assume that most Reform rabbis would consider this child Jewish and officiate at a Jewish funeral. As to name, there is a tradition not to name (or a superstition not to announce a name) before b'rit. Giving a child a Jewish name is not the province of the rabbi, who only provides advice. Parents give a Hebrew name, in the same way that they give an English name.
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