Why are children who have a Jewish Father and gentile mother not considered Jewish?
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The principle derives from the Mishna, Kiddushin 3:12. There it gives four different examples of possible sexual unions and relates the status of the child in each example:
This is the earliest formulation of what has come to be known as the matrilineal principle: in the event that a woman cannot contract an halakhic marriage with the man whom she has had sexual intercourse with, or any other man, the child follows her in status. Given that one of the two examples of such a woman is a gentile, the practical upshot is that the offspring of a non-Jewish woman is always non-Jewish, irrespective of who her partner is.
(There have always been dissenting voices to this principle, an opinion to the contrary even being recorded in the post-Mishnaic Avot deRebi Natan, but the halakha is as I have outlined it. For further information on the development of this principle and its potential social origins, I recommend Shaye J.D. Cohen, The Beginnings of Jewishness: Boundaries, Varieties, Uncertainties (Berkeley: University of California Press, 1999).)
Well basically, the Torah says Jewishness is determined matrilinearly, so that's all there is to it. In fact, the product of a Jewish mother and non-Jewish father is considered by Jewish law to technically have no father. (They are identified in a ketubah, for instance, as "the son of [mother's name]", or "the son of [maternal grandfather's name]." A kid whose father is a Jewish product of adultery is a lot worse-off in Jewish law than a kid whose father wasn't Jewish (in which case dad's parentage is totally moot).
I don't know where you got this "sins to the soul" business, but it doesn't fit with the laws described in the above paragraph, nor sound like anything I've heard in classical Jewish sources.
If you wanted to propose some basic suggestion as to understanding why (not "oh the reason is x and therefore x doesn't apply so I don't have to keep it"), it may be that mom tends to be more of an influence on kids than dad. Note that the Biblical source for matrilineal descent occurs in a warning on pagan influences -- "don't marry your son or daughter to a pagan, as the male pagan will influence your grandchildren" -- implying that the female pagan's children aren't even "yours", Jewish grandpa.