Well, I guess that since we can't expect Salamander to regale us with his extensive knowledge of the historical background of the Bible, it's up to me to explain why if he knew anything about a) Calvinism or b) first-century Roman culture, he wouldn't have asked the question at the top of this thread.
The Biblical doctrine of adoption does not in any sense contradict the Calvinist doctrine of election. In fact, they complement one another.
Paul writes:
I mean that the heir, as long as he is a child, is no different from a slave, though he is the owner of everything, but he is under guardians and managers until the date set by his father. In the same way we also, when we were children, were enslaved to the elementary principles of the world. But when the fullness of time had come, God sent forth his Son, born of woman, born under the law, to redeem those who were under the law, so that we might receive adoption as sons. And because you are sons, God has sent the Spirit of his Son into our hearts, crying, "Abba! Father!" So you are no longer a slave, but a son, and if a son, then an heir through God. (Gal. 4:1-7)
In Roman society, extended families lived together under a male head, the
paterfamilias. He was the absolute ruler of the family, and his authority even extended to life or death over disobedient family members. Effectively, the
paterfamilias owned his family.
This is why Paul writes that the son was no different from a slave. Biologically he was the son, but he had no rights. He was unable to tend to his own business affairs. He was not allowed to leave the house except in the company of the
paidagogos (what Galatians calls the "schoolmaster"), an educated slave that was responsible for his upbringing. Nonetheless, Paul writes, he is "the owner of everything." Not at present, but in spite of his
current low status, he was his father's legal heir.
This legal standing would not last forever, however. There was a time in a Roman boy's life, typically between the ages of 14-17, when he came of age. He received the right to marry, to manage his own business affairs, and to participate in the civic life of Rome. He was entitled to put aside the clothing of his childhood and don the
toga virilis, the pure white outfit that was the symbol of Roman citizenship.
The Romans believed in filial piety: that their ancestors watched over them and their property. Therefore, it was a disgrace for a man to die without an heir. If he had none, he might try to get himself adopted by another family, in which case his property would be passed to them, and in this way his ancestors would be satisfied. Better yet, he would adopt an heir himself, often the son of a lesser family, or a slave. There was a ceremony in which the adopted son was formally separated from his own family and legally bound to his adoptive father.
Adoption was not merely the bringing of an orphaned child into a welcoming family. It was a change of legal status, in which the son was formally recognized as the father's heir.
Even if the son as yet had no legal standing, the father knew that he was his son and that he would inherit his birthright. Similarly, as Paul writes to the Ephesians, God "predestined us for adoption through Jesus Christ, according to the purpose of his will" (Eph. 1:5); "In him we have obtained an inheritance, having been predestined according to the purpose of him who works all things according to the counsel of his will (1:11). At the appointed time we, as adopted sons of God, receive the inheritance for which we were adopted, and of which we receive a down payment now, in the person of the Holy Spirit, "who is the guarantee of our inheritance until we acquire possession of it, to the praise of his glory" (1:14).
Adoption does not contradict election, or vice versa. Adoption is but one way of
explaining election.