Is it true that the Shafi'i Madhab permit men to marry their daughters who are concieved from zina? If so, why?

Can someone verify if this is true: > Ibn Al-Arabi said, “Lineage is an expression referring to the mixture > of fluids between a male and a female from a religious legal point of > view. However, if this union (between male and female) occurs through > disobedience (fornication) **then the resulting child is not considered** > **part of a person’s true lineage. That is why a daughter born from zina** > **is not mentioned in Allah’s saying, ‘Prohibited to you (for marriage)** > **are: your mothers and daughters’ (Surah 4:23) because she is not** > **considered a daughter according to the most authentic teaching of our** > **religion.. If there is no legal lineage then there is no legal** > **relationship; for zina does not prohibit (from marriage) the daughter** > of the mother (you committed zina with) nor the mother of a woman (you > committed zina with). […]I (Qurtubi) said: **The scholars have differed** > **on the permissibility of a man marrying his daughter who was the** > **result of an adulterous relationship; or for that matter marrying his** > **sister or granddaughter who was the result of zina.** Some prohibited > this type of relationship; among them was Ibn Al Qasim, which is also > the saying of Abu Hanifa and his companions. Others however al**lowed > this type of marriage such as Abdul Malik Al Maj’shun,** **which is also** > **the saying of Al-Shafi’i.** This was detailed in the explanation of > Surah 4 (Al Nisa’). > > Tafsir Al Qurtubi 25:54 Al Maududi seemed to have mentioned this and showed his disproval: > The prohibition about daughter also applies to the daughter of the son > and the daughter of the daughter. There is, however, a difference of > opinion in regard to a girl born of an illicit relationship. Imam Abu > Hanifah, Imam Malik and lmam Ahmad-bin-Hanbal (may Allah bless them > all) are of the opinion that she too is unlawful like the lawful > daughter, **but Imam Shafi 'i does not consider an illegitimate > daughter unlawful.** But the very idea is repugnant that one should > marry a girl about whom one has the knowledge that she is of one's own > seed. > > Tafsir Al Maududi 4:23 If this is true, why did Imam Al Shafi'i permit it. On what basis from the Qur'an and Sunnah? Or was it simply ijtihad?