The question used both rhetorically and also legally, is “Who is a Jew?”.
According to Rabbinic Judaism, which in its orthodox manifestation controls many Batei Din around the world, and dominates the Israeli scene, this is something which is passed on by the mother and not the father. A Jewish woman who intermarried with person of non Jewish heritage still gives birth to “Jewish” children, whereas the favour is not returned to the Jewish man who takes a wife of non Jewish heritage.
This is the well known story. Apart from conversion, which is the only other option, there is not much of an alternative. The Karaites do not have a clear position on this, although some will make a claim that Patrilineal descent is valid, whereas others insist on both parents being Jewish.
To try to understand this issue, it might help to define what “Jewish” actually means. In fact, there is something that preceeds “Jewishness”, and that is being an Israelite. Israel was a given name to Jacob. His descendants are Israelites. Israelites are the people to whom the Torah was given. Before we can discuss Jews we need to understand who the Israelites are.
Genesis Chapter 29
16 Now Laban had two daughters: the name of the elder was Leah, and the name of the younger was Rachel. 17 And Leah's eyes were weak; but Rachel was of beautiful form and fair to look upon. 18 And Jacob loved Rachel; and he said: 'I will serve thee seven years for Rachel thy younger daughter.' 19 And Laban said: 'It is better that I give her to thee, than that I should give her to another man; abide with me.' 20 And Jacob served seven years for Rachel; and they seemed unto him but a few days, for the love he had to her. 21 And Jacob said unto Laban: 'Give me my wife, for my days are filled, that I may go in unto her.' 22 And Laban gathered together all the men of the place, and made a feast. 23 And it came to pass in the evening, that he took Leah his daughter, and brought her to him; and he went in unto her. 24 And Laban gave Zilpah his handmaid unto his daughter Leah for a handmaid. 25 And it came to pass in the morning that, behold, it was Leah; and he said to Laban: 'What is this thou hast done unto me? did not I serve with thee for Rachel? wherefore then hast thou beguiled me?' 26 And Laban said: 'It is not so done in our place, to give the younger before the first-born. 27 Fulfil the week of this one, and we will give thee the other also for the service which thou shalt serve with me yet seven other years.' 28 And Jacob did so, and fulfilled her week; and he gave him Rachel his daughter to wife. 29 And Laban gave to Rachel his daughter Bilhah his handmaid to be her handmaid. 30 And he went in also unto Rachel, and he loved Rachel more than Leah, and served with him yet seven other years. 31 And the LORD saw that Leah was hated, and he opened her womb; but Rachel was barren. 32 And Leah conceived, and bore a son, and she called his name Reuben; for she said: 'Because the LORD hath looked upon my affliction; for now my husband will love me.' 33 And she conceived again, and bore a son; and said: 'Because the LORD hath heard that I am hated, He hath therefore given me this son also.' And she called his name Simeon. 34 And she conceived again, and bore a son; and said: 'Now this time will my husband be joined unto me, because I have borne him three sons.' Therefore was his name called Levi. 35 And she conceived again, and bore a son; and she said: 'This time will I praise the LORD.' Therefore she called his name Judah; and she left off bearing.
Jacob's first 4 sons were:
These were born from Leah. His 2nd wife, Rachel, was infertile at this stage, so she offered him her handmaid, Bilhah, to bear children for him. Bilhah bore:
Then Leah gave Jacob her handmaid called Zilpah, and she bore 2 sons:
Then Leah bore him 2 further sons:
And a daughter called Dinah
Finally, Rachel was successful in bearing a son named Joseph. And again later she bore Benjamin.
These 12 sons of Jacob, through 4 wives, become the 12 tribes of Israel. They are referred to throughout the Torah as Bnei Israel - “sons of Israel”.
If they are sons of Israel, from 4 different wives, 2 of whom are handmaids, then the Torah attaches more importance to their patrilineal lineage than their matrilineal lineage. That doesn't denigrate the matriarchs in any way, but they are descended from Laban, who was not the most “pure” or honest of people.
It is important to note that one of the tribes of Israel is called Judah, after the son of Leah and Israel.
So far, the only determining factor of identifying an Israelite is that he is one of the sons of Israel = Bnei Israel. Israel has 4 wives, 2 of whom are sisters, the other 2 are unknown handmaids. Hence the identity of the sons are by definition, after father. If it were otherwise, they would not be Israelites, but Leahites and Rachelites. The promise was made to Abraham, and not Laban. Laban was the grandson of Nahor, who was Abraham's brother. Again, it was Abraham, and not his brother Nahor who was chosen as the progenitor of Israel (nation) who would inherit the land of Israel.
Any suggestion that Israelite identity is matrilineal is both logically absurd, and contradictory tot he Torah. We have no idea who Laban's magical wife was, and why she should determine who is a Jew! BY what merits should she determine the future of Israel, when this was already promised to be the speciality of Abraham, Isaac and Jacob? So far, matrilinealism is a parallel universe, or alternate narrative , divorced from reality, history or the text of the Torah.
So what of Judah and the Jews?
As far as the Torah is concerned, Judah is only one of the 12 tribes of Israel. There is nothing more Israelitish about Judah than there is about Gad, Reuven, Dan, Naphtali, Benjamin etc. Technically speaking the other tribes are non-Jews. Because Jews are Yehudim, who are descended from Yehudah (Judah). Indeed, Jacob blesses all his sons, who will inherit the land of Israel and each get a portion of Israel.
Yehudah's family is also interesting. In Genesis 38, we see that Judah marries the daughter of a Canaanite. They have 3 sons: Er, Onan, and Shelah. Er marries an unknown woman named Tamar.
Er dies, and so Onan must perform a levirate marriage to continue his brother's seed. The levirate marriage is predicated on a patrilineal inheritance. By definition, only a male from the deceased's family, (usually brother) can continue his name. Onan gets buyer’s remorse, and is punished for refusing to bring up his brother's name. He dies as a result of this sin. So to continue with the Levirate marriage, which is Torah commandment, the youngest son of Yehudah – Shelah must marry Tamar. Yehudah is scared that the bad luck will go on to kill his 3rd and last son. So he puts this on hold. Even if Shelah does sire a son for Tamar, it is only to continue Er's name, not Tamar's.
Er, Onan, and Shelah were the first sons of Yehudah, and were Yehudim – the very first Jews in history. Their Jewishness was not from his Canaanite wife, but from himself. His wife's father was called Shua, a Canaanite. According to Rabbinic thought, since Judaism is matrilineal, then the inheritance was from Mr or Mrs Shua. In other words, Jews are not descended from Abraham, Isaac, or Jacob, but from Shua, and Canaan. Therefore, the new religion is in fact Shuism, and the descendants should be called Shoes not Jews.
Yehudah then ends up with what he thinks is a prostitute, but turns out to be Tamar. By this, she is continuing the name of Er, and not Yehudah per se. Yehudah becomes an unwitting Levirate for his son Er. The sons born to Yehudah and Tamar are called Perez and Zerah. These are now , along with Shelah, the 3 living Jews, because their father was Yehudah. If we follow the rabbinic argument, they are not Jews at all, but Tamarites. In fact, their religion would be Tamarism and not Judaism, but they are actually legally descended from Er.
The Torah, and the law of Levirate marriage therein offer strong proof that Israelite identity in general and Yehudish (Jewish) identity in particular is patrilineal. It cannot be otherwise. There is no Shuism, or Tamarism or Canaanite Torah. The promise of Israel was made to the forefathers, and not to Canaan or Tamar. The rabbinical question of “who is a Jew” and its answer, in fact is claiming that a Shuite (Shoe) is a Jew. This is absurd and meaningless.
Whilst is true that we later see in the book of Ruth that Perez is the ancestor of King David, this is patrilineal. It is from his line via Er and Yehudah that David becomes King and not from Tamar.
As far as the Torah is concerned , Israelite and Jewish identity is patrilineal, even where there is intermarriage with Canaanites, as was the case with Yehudah himself.