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Matrilineal vs. Patrilineal Descent

(יב) כָּל מָקוֹם שֶׁיֵּשׁ קִדּוּשִׁין וְאֵין עֲבֵרָה, הַוָּלָד הוֹלֵךְ אַחַר הַזָּכָר. וְאֵיזֶה, זוֹ כֹהֶנֶת, לְוִיָּה וְיִשְׂרְאֵלִית שֶׁנִּשְּׂאוּ לְכֹהֵן וּלְלֵוִי וּלְיִשְׂרָאֵל.

וְכָל מָקוֹם שֶׁיֵּשׁ קִדּוּשִׁין וְיֵשׁ עֲבֵרָה, הַוָּלָד הוֹלֵךְ אַחַר הַפָּגוּם. וְאֵיזוֹ, זוֹ אַלְמָנָה לְכֹהֵן גָּדוֹל, גְּרוּשָׁה וַחֲלוּצָה לְכֹהֵן הֶדְיוֹט, מַמְזֶרֶת וּנְתִינָה לְיִשְׂרָאֵל, בַּת יִשְׂרָאֵל לְמַמְזֵר וּלְנָתִין.

וְכָל מִי שֶׁאֵין לָהּ עָלָיו קִדּוּשִׁין אֲבָל יֶשׁ לָהּ עַל אֲחֵרִים קִדּוּשִׁין, הַוָּלָד מַמְזֵר. וְאֵיזֶה, זֶה הַבָּא עַל אַחַת מִכָּל הָעֲרָיוֹת שֶׁבַּתּוֹרָה. וְכָל מִי שֶׁאֵין לָהּ לֹא עָלָיו וְלֹא עַל אֲחֵרִים קִדּוּשִׁין, הַוָּלָד כְּמוֹתָהּ. וְאֵיזֶה, זֶה וְלַד שִׁפְחָה וְנָכְרִית:

(12) Wherever there is betrothal and no sin, the child follows the male [with regard to familial status]. Which is this? This is a Kohenet, a Levite woman, or an Israelite woman who married a Kohen, a Levite or an Israelite. [ie- women adopt the status of the man that they marry]

Wherever there is [a valid] betrothal and there is a sin, the child follows the defective one. Which is this? This is a widow married to the Kohen Gadol [High Priest], a divorcee or a chalutzah [the widow of a childless man released from the obligation of levirate marriage by a ceremony performed by her brother-in-law] [married] to an ordinary Kohen, a mamzeret [the female offspring of certain prohibited relationships who may not marry into the general Jewish population] or a Netina [female Gibeonite] to an Israelite, a female Israelite to a mamzer or Netina.

[With] any [woman] for whom there is no betrothal to him, but there is betrothal to others, the child is a mamzer. Which is this? This is one who has sexual intercourse with any one of the forbidden sexual relations [mentioned] in the Torah. Any [woman] for whom there is no betrothal either to him or to others, the child is like her. Which is this? This is the child of a [non-Jewish] maidservant or a non-Jewish woman.

ולדה כמותה מנלן דאמר קרא (שמות כא, ד) האשה וילדיה תהיה לאדוניה
from where do we derive that her offspring is like her? The Gemara answers: As the verse states with regard to a Hebrew slave who marries a Canaanite maidservant: “The wife and her children shall be her master’s” (Exodus 21:4). This indicates that the offspring of a Canaanite maidservant and a Hebrew slave are slaves, as she is.
נכרית מנלן אמר קרא (דברים ז, ג) לא תתחתן בם אשכחנא דלא תפסי בה קידושי ולדה כמותה מנלן
§ The Gemara asks: From where do we derive that betrothal with a gentile woman is ineffective? The verse states: “Neither shall you make marriages with them” (Deuteronomy 7:3), which teaches that marrying gentile women is halakhically meaningless. The Gemara asks: We have found that betrothal is ineffective with her; from where do we derive that her offspring is like her?