Status of the son of an unmarried Kohen

Question

If a Kohen had a relationship outside marriage with a Jewish-born, non-divorced woman, is his son a Kohen? I mean that the woman is not currently married (either single or a widow) and the child is the product of this union.

Answer 1

100% Kohen Kasher

Because this woman is not psula Likh'huna. The only problem is if she was well known in relationship with a non-Jew man or other man who make her Chalala.

To make the answer more sourced, we will read a little in Gemara Yebamot (59b):

ת''ר אנוסת עצמו ומפותת עצמו לא ישא ואם נשא נשוי אנוסת חבירו ומפותת חבירו לא ישא ואם נשא ר' אליעזר בן יעקב אומר הולד חלל וחכמים אומרים הולד כשר

The Gemara (Yebamot 59b) explained that if a Kohen Gadol (special status which is not existent in galut times) want to marry a celibate woman who was previously raped or seduced by him, we tell him not to make this;

But in Gemara above (59b), concerning the rules for a simple Kohen:

ר' אלעזר אומר פנוי הבא על הפנויה שלא לשם אישות עשאה זונה אמר רב עמרם אין הלכה כרבי אלעזר: ‏

Here two views are expounded regarding the status of a celibate woman who was in sexual relationship with a celibate man. One opinion (of Rabbi El'azar) is that she is not allowed for a Kohen and the second doesn't agree. The second one (which Rav Amram stated for Halacha) is adopted by the Gemara. According to Rabbi El"azar the fruit of the union of non marital relationship between Kohen and celibate Kasher woman is Challal. But we ruled as the second opinion.

Rambam Kedusha, Isure Bia, 18, 2 ruled explicitly that the woman remains allowed to Kehuna. So the Kohen doesn't make this Woman Chalala, and following the rule that if a woman is not non-compatible for a Kohen, her son will not be non-compatible to Kehuna (even if the relationship she has was matter of prostitution {against the view of Rabbi Akiva}, but only with men who arn't Chalal or men prohibited to marry with her or non-Jewish). The SA EH 6, 8 is an exact reproduction of the words of Rambam.

The baby can even be Kohen Gadol later.

Answer 2

is his son a Kohen?

YES

As it cleary says in the end of Shulchan Aruch E"H 3.9

a Cohen who has carnal relations with a single girl and admits that the child is his son, the child is a Cohen for all things and we are not concerned that perhaps she has made herself loose for others
כהן הבא על הפנויה ומודה שהוא בנו הבן כהן לכל דבר ולא חיישינן שמא הפקירה נפשה לאחרים 

(see Aruch Hashulchan 3.20 )

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The general rule is whenever the kohen can marry the woman legally the child is a Kohen (from Shulchan Aruch E"H 8.1)

but this is after making some assumptions

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the assumptions are:

1. halachicly it is considered his son (that we are not afraid that another man might be the father),

2. the woman was legal (even if only believed (ex post facto)) for him to marry (from Aruch Hashulchan E"H 6):

   1. she was never divorced

   2. she is not a "zona" (halachicly it is considered that 1. she was never a non-jew and 2. she was never penetrated by one of the men listed bellow (from Aruch Hashulchan E"H 6.19-20 and 15:)

      1. while she was married, any man who is not her husband
      2. a non-jew
      3. her son
      4. her husband's son (even after her husband died)
      5. her husband's brother (even after her husband died)
      6. her brother
      7. her grandfather
      8. her father
      9. her mother's husband
      10. her grandmother's husband
      11. her daughter's husband
      12. her grand daughter's husband
      13. her brother's son.
      14. her sister's son
      15. her husband's paternal brother's son (even after her husband died)
      16. her husbands father (even after her husband died)
      17. her sister's husband
      18. a slave (According to some)
      19. a Halal (According to some)
      20. a mamzer (According to some)
      21. a Ptzua Daka, and clus sfca (vasectomy, eunuch, castration...) (According to some)

   3. she is not a "Halala" (she is not an outcome of a Kohen paternal ancestor's relations with a woman forbidden to him, due to him being a Kohen (a Halala, divorcee, or a zona))

   4. she is not a "Mamzer" (this point needs more research Can a kohen be a mamzer? )

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even ezer 6.8 (his mother might even not be a zona, and a koen (even his father can marry her))

וְכֵן הַבָּא עַל הַפְּנוּיָה, אֲפִלּוּ הָיְתָה קְדֵשָׁה שֶׁהִפְקִירָה עַצְמָהּ שֶׁהִיא בְּמַלְקוֹת, לֹא נַעֲשֵׂית זוֹנָה וְלֹא נִפְסְלָה מֵהַכְּהֻנָּה, שֶׁהֲרֵי אֵינָהּ אֲסוּרָה לִנָּשֵׂא לוֹ.

And similarly a man that has relations with a single woman, even though she is a lewd woman that made herself freely available, which is punishable with lashes, she does not become a zonah and she does not become invalid to a kohen, because she is not forbidden to him.

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On one hand, his father has a mitzvah to marry his mother Shulchan Aruch E"H 177.5

But on the other hand as @DrM writes
it might be forbidden for him to marry her (Shulchan Aruch E"H 6.17-18) since by being pregnant she might have lost her presumption of being kosher for kohen.

Answer 3

It is not so simple.

Indeed, as others have noted, the shulchan aruch Even HaEzer 8:1, writes that the child of relations between Jews, follows after the father's lineage.

The Beis Shmuel expands on these last words citing a mishna in Kedushin 66b, that it is a general principal that the child follows after the father wherever there is or can be kedushin (marriage) and there is not aveirah (sin), i.e. it would be a kosher marriage.

But to simply cite this halacha and stop there, misses an important part of the question; the presumption of kashrus, of the man, and also of the women, and under what circumstances do they have or lose that presumption.

Similary, the answer expounding on a gemara in Yabomot where the question is whether she is permitted to him after premarital relations, misses this same point by neglecting the gemara in Ketubot 13a-b where the women is pregnant and not married. We might also remind ourselves that we do not poskin from gemaras.

The case concerning where a woman becomes pregnant outside of marriage, is addressed in the Shulchan Aruch, explicitly, in Even HaEzer 6:17-18

There we see (a) whether a pregnant women is believed regarding the father, (b) what circumstances can establish the presumption that the father is kosher and (c) whether the women would then be allowed to marry a Cohen "lechatchela". This derives in part from a well known gemara, Ketubot 13a-b.

The problem is that if we cannot assume the father of the child is kosher, she herself is regarded as possul. Surprisingly, up to here, the gezara can apply to her but not the child. But, those same circumstances might also make her possul from before they had relations. In that case, the baby is regarded as a Challal. So where there is a problem, there can be a problem for both the mother and the child.

Consequently, it would be irresponsible to give a blanket answer, or any answer without looking into the specifics of the case or speaking with the people involved.

It seems more appropriate to ask a Rav, who will consider the case in its particulars, and whose determination will be accepted in the larger Orthodox community.