IX. (5) Because the children of believers are holy; therefore they ought to be baptized. For since they have the thing signified, they cannot and ought not to be deprived of the sign (Acts 10:47). “The unbelieving wife,” says the apostle, “is sanctified by the husband: else were your children unclean; but now are they holy” (1 Cor. 7:14). They are called holy, however, not because they are legitimate (as some suppose), because he was not speaking of that and nothing would thus be ascribed to them above the children of unbelievers, who can also be legitimate. But they are said to be holy by a federal holiness (i.e., Christians and belonging to the church). Therefore ἀκάθαρτον and ἅγιον here designate heathen and Christian (in which sense the word קדושה is often used by the Talmudists). “Offspring” is said to be “begotten in holiness” or “out of holiness” (i.e., when the parents were made proselytes or while they were still heathen). The children (τέκνα) of the Gentiles were unclean (ἀκίθαρτα) to them (from Acts 10:28) and of the Jews holy (ἅγιά), to which meaning (well known to the nation) allusion is made.
Francis Turretin, Institutes of Elenctic Theology, ed. James T. Dennison Jr., trans. George Musgrave Giger, vol. 3 (Phillipsburg, NJ: P & R Publishing, 1992–1997), 19.20.9 (p. 417–18). [modified according to the 1688 Latin text by the substitution of Greek and Hebrew words for the transliteration in the Dennison edition]