There is over on truthbecomeslies an effort to answer objections to paedobaptism by appealing to the nature of the church. If I follow the argument, the first two options represent possible credobaptist views and the third that of the paedobaptists. Within the credobaptist options, the first is what the church would look like if they were consistent with this doctrine. Only baptized believers would participate in the life of the church. The first option is desribed as "reasonable" and "consistent;" however, in a subsequent examination of the premises behind this option, it is revealed to be a reductio ad absurdam. From the premise that the great Commission does not explicitly mention covenants or the Kingdom of God, but just says, "Go. Train people," Scott concludes that Jesus wanted the "monasticizing of the Earth." Everyone needs to quit their job and become a preacher just like Jesus' disciples did. He has a valid point against a lot of credobaptist hermeneutics. The second option presents what credobaptist churches actually do look like. Unbaptized children are not members; however, they participate in all aspects of membership except for communion. Scott writes, "Not only is this completely unhinted at in Scripture, but it is an unfair practice to the children." Children are either in the covenant or they are not and they shouldn't be left wondering which it is.
I appreciate the idea of arguing the proper recipients of baptism from the nature of the church rather than from specific prooftexts, but I don't believe that option 2 succeeds. The implied concept of the covenant seems to me to be bit one dimensional, as though there were an exact correspondence between it and the church as a visible institution. But the covenant is deeper than that. Ultimately, the covenant is made with Christ. The elect are his eternal inheritance. While it is appropriate on one level to speak of baptized church membership as being in the covenant, this should not be confused with being in Christ. Baptism is a form of water ordeal that destroys all who pass through it. Only for those who pass through it in faith is it a means of grace. This faith unites them with Christ is his baptism on the cross and subsequent resurrection. All who are baptized are members of the covenant in the sense of being obligated to its terms and are promised blessings for keeping them and curses for breaking them. Still , there is a sense in which the visible church is not the covenant itself but reveals the covenant. The promises are made to Christ and the grace exihibited to us is that he has fulfilled all of the covenantal obligations.
If there are different dimensions within the covenant, may the church determine who is where? It can- this is what happens in the extreme end of church discipline. Someone within the church is judged to have been without faith and is excommunicated. But this does not mean that he has been expelled from the covenant. Instead, excommunication means that, barring repentance, he is going to be meeting the terms of his baptism all by himself. The set of those qualified for covenant membership in the visible church is larger than and contains the set of those qualified to take communion. The first requires baptism; the second requires faith. Which brings up the issue of paedocommunion. Scott also alluded to this in option 2. The question here is not whether faith is a requirement, but whether the church can assume faith in its immature members. Does it allow all to participate unless they fall away, or should it require sufficient maturity to make and understand a profession of faith? Historically, the Reformed position opts for making them wait (see WLC 177 -"only to such as are of years and ability to examine themselves"). This position does not deny that children are in the covenant, nor does it deny the possibility that they have true faith. It only claims that communion is reserved for those whom the church has judged, not assumed, to have true faith. It recognizes that there are different levels to the covenant.
Credobaptists argue that the physical covenant of OT Israel is fulfilled in those who have faith. Consequently, although the sign of circumcision was given after physical birth, the sign of baptism should only be given after spiritual birth. The argument is valid in that the conclusion follows from the premises. Nor does this view of the covenant contradict the value of the visible church as a place to train up one's unbaptized children in the Lord. Baptist theolgy does not recognize baptism as a sacrament. It is done in obedience to the command of Christ but is not a means of grace. [I'm speaking from having grown up as a dispensational baptist; please correct me if this is not the case for Reformed Baptists.] In holding to this view, Baptists are in error; however, they are not heterodox. This is not a matter of knowing the truth and rejecting it, but of being honestly mistaken in what scripture teaches. Baptism is a means of grace, whether anyone thinks so or not (unless paedobaptists would like to argue that infants have an opinion on the matter). And, to state the obvious, Baptists do get baptized. Furthermore, their children are a part of the covenant just as much as those of paedobaptists. This is not to deny the nature of baptism, not to say that it is a matter of indifference for those who know better. Baptism is a means of grace, but it is neither inseparable from grace nor is its efficacy "tied to that moment of time wherein it is administered" (see WCF XXVIII.5,6).
Baptists may have the wrong view of baptism and, for that matter, the covenant as a whole. Nevertheless, they are following scripture when they bring up their children in the Lord. This includes disciplining them toward righteous living, prayer, knowledge of the Scripture, and participation in worship. That the parents themselves may not rightly understand the doctrine of baptism is no justification otherwise to shirk their duty. Children need to be evangelized. Disciplining them in these various activities is how this is accomplished.