The assumption behind Paul’s reasoning is that the child will bear the image of the father. Since Abraham was characterised by faith, then his children will be also.
The Abrahamic covenant had two parts. One part was made with Abraham and his physical descendants and included promises to possess a land, to be protected from enemies, to be given plentiful harvests, and to prosper as a nation. This part was made with Abraham’s descendants whether or not they had faith. These however did not benefit from the spiritual promises and all that God gave them was in the physical realm, since they were not capable of understanding or receiving the spiritual promises. The other part was made with those who are Abraham’s spiritual descendants and included only those who possessed the faith that Abraham had. This part of the covenant gave spiritual promises and an inheritance in heaven. Only those who were in this spiritual part of the Abrahamic covenant could have salvation.
These two parts of the covenant were related to each other as a type is related to its antitype, but in themselves they were quite distinct. A person could be in the physical covenant and not in the spiritual covenant – these were unbelieving, unconverted Jews. A person could be in both parts – these were believing converted Jews. Paul tells us here that there are also those who could be in the spiritual side of the covenant and not in the physical side – these are believing Gentiles. For our instruction God so arranged things that in the early generations that the physical side of the covenant and the spiritual side of the covenant unfolded in parallel. Those descendants of Abraham who were chosen to inherit the land, were also the ones who were converted and exercised true faith. After the generation of Jacob, the two sides of the covenant began to diverge, and many among the physical descendants of Abraham never had personal faith and were only outward members of the covenant.
The case of Ismael is particularly instructive, for although he was circumcised, yet he did not receive the blessings of the covenant, not even the typical ones, for he was given no inheritance in Canaan. He did not enter the promised land which was a type of heaven, let alone heaven itself. God wished it to be seen that, though he was circumcised, he was excluded from the promise. Why did God cause him to be circumcised even though he was shut out of the promise? So that he would be a type of those who are members of the covenant outwardly, but who never receive the spiritual blessings.