Eleanor,
This is from an old post of Pilgrm's that I had saved. I couldn't find it here on the search, but here it is.
Quote
Pilgrim wrote:

Acts 2:39?

"For the promise is unto you, and to your children, and to all that are afar off, [even] as many as the Lord our God shall call.
The way I read this passage, the bold phrase is the key to its understanding. For we know from O.T. history, Paul's quotes which I have included above, etc., that this "promise" is NOT all inclusive and indiscriminate. When God makes a promise, it is an infallible one. Thus, if this "promise" of salvation is given to ALL... then ALL shall receive it. We know that not all who hear the gospel, not all covenant children and/or adults are saved. Thus is it not correct, just on a logical basis that this text is not saying that there is an immutable salvation given to all covenant children? But exegetically, I have come to understand that this "promise" is given specifically to "as many as the Lord shall call", and to them only. These are the elect, the chosen in Christ who will come to repentance and faith, just as those who heard this message from Peter's very lips, did, including their children who did likewise. "By their fruit you shall know them."

...Here is my summary view: children of believers should receive the sign of baptism as the legitimate sign of the covenant of grace. Covenant children are to presumed unregenerate until proven otherwise, since they are born in sin and under the wrath of God, no differently than all of mankind. Covenant children are "holy" (1Cor 7:14), i.e, they are truly blessed in that they have been given the privilege of being born into a home where there is at least one believing parent through and by whom the gospel will be heard and used to regenerate them, if it so be God's electing choice to do so in His perfect time. And they are subject to the teaching of righteous living, according to the Scriptures. They also have the privilege of being present in the house of God where believers offer worship, hear the Word of God preached, witness the sacraments, hear the testimonies of godly men, women and children, etc. which are the means of grace given by God to convert sinners to Christ. Covenant children should be expected to make a valid profession of faith before the church when THEY are moved to do so; having been convinced of their own conversion and the necessity of doing so, thereby becoming full members, fellow heirs of Christ, with the Body of Christ.