If a man had a son and a daughter but one of them predeceased him without producing progeny, according to Rav Huna he has still fulfilled the mitzva of procreation, as he maintains that the mitzva is fulfilled with their birth. Even if a child lives for only a short time, his life has value. His soul revealed something positive in the world and even brought redemption closer, for “The (messianic) son of David will not arrive until all the souls of the body have been finished.” In contrast, R. Yoḥanan maintains that the mitzva is only fulfilled if a man’s children live after his death, because the objective of the mitzva is to ensure the continued habitation of the world (Yevamot 62a-b). Sadly, R. Yoḥanan himself buried all of his children before they had children of their own, and so he referred to himself when he spoke of a person who was not privileged to fulfill the mitzva. We rule in accordance with R. Yoḥanan. However, if someone leaves behind a son and daughter when he dies, even if they remain unmarried and too old to have children, he has still fulfilled his mitzva (SA EH 1:5).
If someone was predeceased by his son and daughter, but had a grandchild from each, he has fulfilled his obligation through his grandchildren. What matters is for his descendants to continue after him through his son and daughter.
If someone had a son and daughter, and his son has many children but his daughter passes away childless during his lifetime, he has not fulfilled his obligation, since he has offspring not from both children, but only through his son. The same applies if his daughter bears many children while his son passes away childless during his lifetime (SA EH 1:6). If someone has a son and daughter, but one of them is sterile or infertile, he has not fulfilled his obligation, since he did not have a son and daughter capable of having children (y. Yevamot 6:6; SA EH 1:5). However, if his son and daughter were themselves able to have children, but one or both of them married people who were not able to, or they did not marry at all, he has still fulfilled his obligation, since the children themselves are not sterile (Ḥelkat Meḥokek ad loc. 6).
If someone has a child who is deaf-mute or mentally incompetent, he fulfills his obligation, since the child is physically capable of having children (Rema, EH 1:6). Accordingly, someone with an autistic child fulfills the mitzva; however, someone with a Down syndrome child might not, because many children with Down syndrome (especially males) are infertile.
If a non-Jew had children and later converted to Judaism, some say that he fulfills his obligation to procreate if his children convert as well (Rambam; SA EH 1:7; Yam Shel Shlomo). Others maintain that even if his children do not convert to Judaism, he has still fulfilled the obligation of procreation (Tosafot; Maharil; Ḥelkat Meḥokek; Beit Shmuel; Bi’ur Ha-Gra ad loc. 17).
A Jewish man who has children with a non-Jewish woman does not fulfill the mitzva to procreate, since his children are not Jewish and are not halakhically considered of his lineage at all.
. Most authorities maintain that if a man has an adulterous affair with a married woman which results in the birth of a mamzer, he has fulfilled the mitzva of procreation despite his sin. However, Responsa Radbaz 7:2 states that it is inconceivable that one fulfills a mitzva by committing a sin. (See 6:6 below for a discussion of this case.) It would seem based on this reasoning that if a couple did not observe the laws of family purity and conceived when the wife was a nidda, they have not fulfilled the mitzva of procreation. For this case, too, would be fulfilling a mitzva by committing a sin. However, many explain that even Radbaz would agree that the mitzva is fulfilled in the case of nidda. A child conceived when a woman was a nidda is not in the same category as a child conceived from an adulterous relationship. A mamzer could not possibly have been conceived without sin, since the union itself is forbidden, whereas a married couple is fundamentally permitted to one another, and the woman could have immersed in a mikveh, so the relationship is not inherently sinful.