Commentary for Kiddushin 136:12

Exodus 21 says that if a master gives a slave woman to his Hebrew slave the offspring belong to the master. This shows that the offspring of a Jew and a slave woman follow the status of the slave.

The Talmud now explains how we know that kiddushin is invalid with a non-Jewish woman and that her offspring is not Jewish. First of all, the Torah says “Do not marry them.”
As to the second question, the verse warns that if a non-Jewish man marries a Jewish woman, the son will be turned away from God. Thus the child of a Jewish woman is called “your son” meaning an Israelite. By deduction, we can conclude that the child of a Gentile woman is not your son.
Note that all of these explanations are clearly “derashot”—interpretations of the verse meant to accord with known halakhah. As I said in my intro to the mishnah, in the Torah there is little doubt that genealogy goes through the father. None of these verses actually means that it goes through the mother. This is the rabbis reading their own halakhah into these verses. Indeed, later in this passage, the Talmud will basically admit that.

The Talmud now explains how we know that kiddushin is invalid with a non-Jewish woman and that her offspring is not Jewish. First of all, the Torah says “Do not marry them.”
As to the second question, the verse warns that if a non-Jewish man marries a Jewish woman, the son will be turned away from God. Thus the child of a Jewish woman is called “your son” meaning an Israelite. By deduction, we can conclude that the child of a Gentile woman is not your son.
Note that all of these explanations are clearly “derashot”—interpretations of the verse meant to accord with known halakhah. As I said in my intro to the mishnah, in the Torah there is little doubt that genealogy goes through the father. None of these verses actually means that it goes through the mother. This is the rabbis reading their own halakhah into these verses. Indeed, later in this passage, the Talmud will basically admit that.

Since the Jewish woman cannot have kiddushin with this man, because he is not Jewish, but she can have it with other men, if they are Jewish, this case should fit into the category of the mishnah that says the child of such a union is a mamzer. This would mean that the child of a non-Jewish man and a Jewish woman is a mamzer, and would not be allowed to marry an Israelite.
The Talmud rejects this—this child is not “fit” which means she cannot marry a Kohen. This is because there was a transgression in the union. But she/he is not considered a mamzer. She/he is Jewish and can marry an Israelite.

Since the Jewish woman cannot have kiddushin with this man, because he is not Jewish, but she can have it with other men, if they are Jewish, this case should fit into the category of the mishnah that says the child of such a union is a mamzer. This would mean that the child of a non-Jewish man and a Jewish woman is a mamzer, and would not be allowed to marry an Israelite.
The Talmud rejects this—this child is not “fit” which means she cannot marry a Kohen. This is because there was a transgression in the union. But she/he is not considered a mamzer. She/he is Jewish and can marry an Israelite.

Canaan (listed in verse 7:1). Kiddushin are invalid with members of those seven nations. But what about members of other nations? The answer is found in the next verse—kiddushin are invalid with any non-Jew because they may turn your son away.
[The assumption here seems to be that the problem with intermarriage is not purely genealogical. It is social/educational/religious. Intermarriage will lead to foreign worship.]

There is a debate in the Talmud over whether one searches for the reason behind mitzvoth. According to R. Shimon, one does. The prohibition of intermarriage is lest the non-Jewish partner turn the offspring to idol worship. Thus it applies to any non-Jew, not just the seven nations.
But the other rabbis do not interpret the reasons of Scripture. And therefore the verse “Do not marry them” applies only to the seven nations.
They, instead, find another verse to derive the rule that kiddushin with non-Jews is invalid. According to rabbinic interpretation, the Torah allows one to marry a captive woman only after she has converted. From here they derive that kiddushin with her before she converts is invalid.

There is a debate in the Talmud over whether one searches for the reason behind mitzvoth. According to R. Shimon, one does. The prohibition of intermarriage is lest the non-Jewish partner turn the offspring to idol worship. Thus it applies to any non-Jew, not just the seven nations.
But the other rabbis do not interpret the reasons of Scripture. And therefore the verse “Do not marry them” applies only to the seven nations.
They, instead, find another verse to derive the rule that kiddushin with non-Jews is invalid. According to rabbinic interpretation, the Torah allows one to marry a captive woman only after she has converted. From here they derive that kiddushin with her before she converts is invalid.

According to the reading of this verse, if the marriage is valid then the children follow his status—the woman bears the children to him. But if the kiddushin are not valid, the children do not follow his status, they follow hers.

If we have a verse that teaches that in all cases where the woman cannot be betrothed to an Israelite, the children follow the mother, then what do we need the verse from Exodus for? Why have a specific midrash when we have a general one?
The answer is that the verse from Exodus can be used for another midrash. A master tries to free a female slave but tells her that her future children will be his. According to R. Yose the Galilean, this does not work. Once she goes free, her kids are hers. However, the sages say the offspring belong to the master even after the mother is freed.
The verse quoted seems like it should explain the position of the rabbis. And in some ways perhaps it does—the children belong to the master even if the wife does not.
But Rava does not read it that way. He reads it as referring back to R. Yose the Galilean. The children belong to the mater only if the mother does as well. If the mother does not, then her children born to her when free are also free. Yeah!