To clarify this issue, I'd like to post a section of David Engelsma's, The Covenant of God and the Children of Believers. This online version is slightly abridged, and they will send the full text (in pamphlet) to those who email them. The final section (not posted in this post) is titled, "The Call to Believers' Children to be Converted". I think it remarkable for its clarity and simplicity.
The Inclusion of the Children of Believers in the Covenant
The children of believers are included in the covenant as children, that is, already at conception and birth. They receive forgiveness of sins through the blood of Jesus the Holy Spirit of sanctification, and church membership -- as children. For they have God as their God, and are His people -- as children. Therefore, they have full right to baptism. Parents must present them for baptism. And the church that would maintain pure administration of the sacraments as instituted by Christ must see to it.
This is an important feature of the central doctrine of the covenant. It is important to children. Are they God's children or the devil's? It is important to the parents. We love our children and regard the rearing of our children as one of the most important tasks in our lives. May we regard them as children of God? Or are we compelled to regard them as Satan's "little vipers," as must all those who deny that children are included in the covenant and as certain Calvinistic theologians, e.g. Jonathan Edwards. Inclusion of the children in the covenant is important to the church. The church asks, "Are they members of the church or do they stand outside?" Does the church have a calling to them too, to feed and protect them as lambs of the flock of Christ, or are they nothing but heathens, little heathens to be sure, but heathens nevertheless, like all other ungodly people, whom the church at most should evangelize?
But above all, the place of the children in the covenant is important to God. He said at the beginning of the history of the covenant with Abraham, "I will establish My covenant between Me and thee and thy seed after thee in their generations... to be a God unto thee, and to thy seed after thee." (Genesis 17:7). He inspired the apostle, on the very day that the covenant became new, to proclaim as the gospel, "the promise is unto you, and to your children... even as many as the Lord our God shall call."(Acts 2:39). Rebuking His unfaithful wife in Judah, in Ezekiel 16:20,21, God exclaims like an aggrieved Husband and Father, "Is this of thy whoredoms a small matter, that thou hast slain My children..." In Malachi 2:15 God condemns the divorcing that was prevalent in Judah, because divorce jeopardizes the "godly seed." (And still today the unchangeable God hates divorce in the covenant community because it is destructive of the children who, as covenant children, are His children.)
How important our children's inclusion in the covenant is to God is shown in the New Testament (Covenant) by Christ's command, "Suffer little children (infants) to come unto Me...for of such (infants of believers) is the kingdom of God (made up)." (Luke 18:15ff.). "Children, obey your parents in the Lord...and, ye fathers, provoke not your children to wrath: but bring them up in the nurture and admonition of the Lord."
In light of our confession of the inclusion of the children of believers in the covenant (about which fact there is no dispute among Reformed people or churches), we must now answer the question, what exactly do Scripture and the Reformed confessions mean when they say that our children are included in the covenant?
The Reformed creeds are clear and emphatic about children's being included in the covenant of God. The Heidelberg Catechism teaches that infants must be baptized "since they, as well as the adult, are included in the covenant and church of God; and since redemption from sin by the blood of Christ, and the Holy Ghost, the author of faith, is promised to them no less than to the adult..." (Question & Answer 74).
Our question (what this means) is occasioned by the incontestable face that not all children of believers are saved. Both parents and church experience the hard, painful fact that some of our children grow up ungodly, unbelieving, and disobedient, and perish. God is not their God; and they are not His people. Scripture prepares us for this bitterest of all parental and ecclesiastical sorrows. Abraham had a grandson, Esau, who was a profane reprobate. (Cf. Genesis 25:19-34; Hebrews 12:16 and 17; Romans 9:6-13). Deuteronomy 21:18ff. prescribed the procedure by which the Israelite parents of gluttonous, drunken, rebellious, and stubborn sons were to bring these children to the elders to be excommunicated and stoned. Hebrews 10:29 speaks of the baptized son of believers in the time of the new covenant who treads under foot the Son of God, counts the blood of the covenant, with which he was sanctified, an unholy thing, and insults the Spirit of grace.
We cannot presume that all our children are regenerated and elect. To presume this is contrary to the Scripture and experience. Nor may we parents be bitter about this. For it is pure mercy that any of our children are saved.
But what then does the Reformed faith mean by the inclusion of the children of believers in the covenant of God?
Although all our children are in the sphere of the covenant and therefore receive the sign of the covenant and are reared as covenant members, the covenant of God, the relationship of friendship in Jesus Christ, is established with the elect children only. The promise of the covenant is for the elect children only. The promise does not depend upon the faith of the child, but the promise itself works the faith by which the child receives the grace of the covenant in every child to whom God makes the promise. It is the elect children among our physical offspring who constitute our true children, even as the seed of Abraham was not all his physical descendants, but only Christ and those who are Christ's according to election. (Cf. Galatians 3:7,16,29).
Our grounds for this explanation of the inclusion of children in the covenant are the following.
First, only this view harmonizes with the rule of faith in Scripture. God's saving, covenant mercy is particular, i.e., for the elect alone. (Romans 9:15). Predestination makes distinction not only between the visible church and the world but also within the visible church itself (Romans 9:10-13). God's salvation never depends upon the will or action of the sinner (Romans 9:16). Christ's death is efficacious (Romans 5:6 to 11). The promise of God is sure to all the seed. (Romans 4:16).
Second, Scripture itself gives exactly this explanation of the precise matter under discussion. It does this in Romans 9:1ff. The concern of Paul is that so many physical children of Abraham perish in light of God's promise to Abraham to establish His covenant with Abraham's seed (vss. 1-5). The chief difficulty of the apostle is not that dear relatives perish (although he could wish himself accursed for these brothers -- v. 3), but that it might seem that "the word of God hath taken none effect," that is, that the promise of God has failed to establish the covenant with many to whom the promise was given (v. 6). But it is not the case that the promise has proved to be a powerless failure in even one instance. Why not? Because the seed of Abraham, to whom the promise was given, never was all the physical children of Abraham. "For they are not all Israel, which are of Israel: Neither, because they are the seed of Abraham, are they all children: but, In Isaac shall thy seed be called. That is, they which are the children of the flesh, these are not the children of God: but the children of the promise are counted for the seed" (vss. 6 to 8). There is a distinction between two kinds of children of believing Abraham: children of the flesh and children of the promise. This distinction is determined by election and reprobation, illustrated plainly by the history of Esau and Jacob (vss. 9-23).
Paul's difficulty is exactly our problem. By promise, God includes our children in His covenant of salvation, but not all of our children are saved.
Scripture's solution of the apostle's difficulty solves our problem as well. The children of believers to whom God graciously promises membership in the covenant are not all the physical offspring of believers. They are rather the "children of God" among our offspring. And the children of God are those who are chosen in Christ. These are the ones whom God counts for the seed when He says, "I will be the God of your seed." These and these only are "the children of the promise." To them, and to them only, is the promise given. In every one of them is the promise effectual to work faith in Jesus Christ.
God realizes His covenant in the line of generations. He gathers His church from age to age from the children of believers. As the Puritans were fond of saying, "God casts the line of election in the loins of godly parents." For the sake of the elect children, all are baptized.
It is the covenantal election of God that determines the viewpoint believing parents and the church take toward their children that governs their approach in rearing them. We do not view them as unsaved heathens ("little vipers"), though there may well be vipers among them, any more than we view the congregation as a gathering of unbelievers because of the presence of unbelievers among the saints. But we view them as children of God.
Viewing their children as God's covenant children, believers must approach them as elect children in their teaching and discipline, even though there may indeed be reprobate and unregenerated children among them. Election determines the approach. All the children must receive the instruction that the regenerated must have and will profit from. By means of this rearing in the nurture and admonition of the Lord, the covenant promise will work the fruit of conversion in the elect children.