When things were simpler there were simpler answers. A child who is born to a Jewish mother and a non-Jewish father is without doubt Jewish (Yevamot 45b). Judaism is transmitted through the mother. Yet the child is considered as one who has no father (Yevamot 98a). Rabbi Moses Isserles says therefore that the son of a Jewish mother and a non-Jewish father is called to the Torah as “…son of Abraham” and not by his father’s name. (Shulchan Aruch, Orach Chayyim 139:3) There are some who say that there is no obligation to respect one’s non-Jewish father (or mother in the case of a convert).
However, this cannot truly be the case. There is another principle that is operative in Talmud and Jewish law: It cannot be the case that halakhah is more lax than general ethical principles (or more literally it cannot be the case that Jewish law permits something that non-Jews are forbidden to do). Further, in regards to respecting one’s parents, the Talmud (Kiddushin 31a) lauds the behavior of one Dama ben Netina, a non-Jew as exemplary. If we derive the extent to which we should respect our parents from the behavior of a non-Jew, are we not to be obligated in respecting a non-Jewish parent?
The naming of a child is very important. The name given to a child is the one that s/he will use to be called to the Torah and the name that will be written on his or her ketubah (and God forbid, the get). The naming signifies the tradition that the child carries forward. Here is where the rub is.
I would say that in a situation where the father is not actively part of another faith tradition, there is no problem naming the child as the son of his/her mother and father. This is especially so if to not do so would shame the father. If the father is an active believer of another religion then the child should be called son of the mother (and perhaps of Abraham).
Based on the wording in the Torah and as delineated in the Talmud, Jewish law has always recognized matrilineal descent in defining one’s Jewishness. Thus, in the above scenario, the child is clearly a Jew, but it is not so clear how s/he would be named. Jewish custom has generally used the person’s mother’s name (i.e. Rachel bat Sarah) when we pray for such person’s healing, and the father’s name when calling a man for an aliyah to the Torah or in identifying people in documents such as the ketubah (i.e. Yitzchak ben Avraham or Rachel bat Avraham). These names identified people before the advent of family names, and we continue to maintain this custom today.
Due to the tragic increase of assimilation, the above scenario is common, but as the child is Jewish, it is important that s/he be given a Jewish name. The Midrash (homiletical and legal teachings on the Torah) stresses that by giving their children Jewish names, the assimilated Jews in Egypt maintained their identity. Based on discussions with colleagues, there would be two options in our case. S/he could be called ben or bat Avraham, since Avraham is considered to be the father of all Jews, or ben or bat of the mother, without using the father’s name. It would clearly be inappropriate to use the non-Jewish father's English name in giving a Jewish/Hebrew name to the child.
Here, you ask not about substituting one name for another, but using a non-Jewish name, the name of the non-Jewish father. Without doing any research, I can tell you what I have often seen and done in a situation such as that you describe.
The child is Jewish, and when named it is Ploni ben Floni in Hebrew. In other words, it is Child’s Hebrew name son of (or daughter of) Mother’s Hebrew name. In the Hebrew no mention is made of the non-Jewish father’s name.
In any English usage, such as on a certificate, I do include the father’s name in the English – the father’s name is omitted only in the Hebrew.
I know that I have encountered some documents where the person filling in the names transliterated (wrote in Hebrew letters the English name of the non-Jewish parent). I am not in favor of that practice generally, because it leads to confusion down the road. Since these documents are sometimes used to confirm Jewish identity or status, transliteration of a non-Jewish name can lead to confusion and mistaken assumptions at some later time.
The answer I would offer you is that the child would be named Ploni bar (bat) Floni (the mother’s Hebrew name), with no reference to the father in the Hebrew.
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