Earlier this year, in the comments on another post, there was an exchange concerning what I said about baptism in the body of the post and my use of Charles Hodge to back up my perspective.
In particular, I had appealed to Hodge's insistence that, in the believing
reception of baptism, the Holy Spirit reconveys and faith reappropriates the forgiveness of sins, ingrafting into Christ, the Spirit himself, and, in general, the benefits of Christ's mediation. I don't think my interlocutor disagreed with Hodge on this general perspective.
The real sticking point in the exchange seemed to be "regeneration" and its relationship to baptism, particularly in light of Hodge's assertion that "baptism can not be the only or even the ordinary means of conveying the grace of regeneration." Rather, according to Hodge, it is the preaching of the word by which such regeneration is wrought.
Hodge goes on to give no less than five reasons why baptism cannot be an ordinary means of regeneration, which would seem to render Hodge irreconcilable with the teaching of the church catholic, from the earliest Fathers onward, up through and including a number of reformational Protestant divines who thought that, at least in the case of elect infants, baptism was an occasion and means of the Spirit's regenerating work.
Nevertheless, I'm not sure Hodge was so at odds with wider Christian traditions as one might think. At that point in the exchange, my attempt to say something constructive proceeded along the following lines.
First, let's stipulate that by "regeneration" we mean the result of effectual calling, a sovereign work of the Spirit, renewing and renovating the heart, so that a person is enabled to put actual faith in the Gospel so to receive and rest upon Christ for salvation.
Second, I want to be absolutely clear that I believe that this regeneration can be enjoyed quite apart from the actual administration of baptism and that there are many who are baptized who are never regenerated. (By the way, the wider Christian tradition - from Chrysostom and Augustine to Lombard and Aquinas - would agree with this in general, allowing, for instance, that unbaptized catechumens and martyrs might still be saved and that others, though baptized, might fail to receive the grace of baptism due to their own heresy or unbelief.)
Third, I would whole-heartedly affirm that this regeneration is something ordinarily wrought by the Spirit through the preaching of the Word.
None of these points are in dispute in what I am saying. Neither are any of these points, however, in the least conflict with the following affirmations:
 While infants are incapable of experiencing actual regeneration and the exercise of faith in particular acts, infants are nonetheless capable of having the seed and root of regeneration and faith, that is to say, the Holy Spirit at work in their lives.
 In infant baptism, faithful parents should have the hopeful expectation that the Holy Spirit is present and active, so that, whatever prior operations of the Spirit may have been present in the infant, the Spirit ordinarily (re)conveys himself to our children in baptism as the seed and root of their regeneration and faith.
 We should, therefore, also expect that in the ordinary process of Christian nurture of children by faithful parents, the Spirit will use the preaching and teaching of Word to bring the seed and root of regeneration and faith to fruition in actual regeneration and the exercise of faith. That's to say, even our baptized children need to hear the Gospel and be called to repentance and faith, as do all God's people.
 Thus, following from the previous points, it is perfectly natural to say, with respect to our children, that "we baptize in order that the one who is baptized be made regenerate." That is to say, baptism is among the ordinary means at God's disposal by which he works in the lives of our children along the way to regeneration and faith, which are properly and ordinarily wrought by the Word.
 In the case of adult converts, they are presumably already believers before they come to baptism and thus are already regenerate.
 Nonetheless, as Hodge says, "the benefits of redemption, the remission of sin, the gift of the Spirit, and the merits of the Redeemer, are not conveyed to the soul once for all. They are reconveyed and reappropriated on every new act of faith, and on every new believing reception of the sacraments."
 That's to say, in baptism, faith is strengthened and increased so that we more and more die to sin and walk in newness of life, which is the progress and increase of regenerating grace (WCF 31.1; WLC 167).
 Thus, following from these points, it is perfectly natural to say, with respect to an adult convert, that "we baptize in order that the one who is baptized...might grow in his regeneration." That is to say, baptism is among the ordinary means at God's disposal by which he works in the lives of converts to strengthen and increase their faith unto newness of life.
 There are other complicating cases, of course, such as the adult who is baptized in unbelief but subsequently comes to faith or the child who dies in infancy prior to being able to be called through the preaching of the Word. But I'll set those aside for present purposes.
 Since we cannot look upon the heart, we extend the judgment of charity to all the baptized who profess faith and who are not living scandalously. That is to say, we have a hopeful expectation that what God has signified and sealed sacramentally is actually true in fact. Thus we speak to and about such baptized professors as "regenerate." Moreover, this judgment of charity is grounded in what Reformed theology has typically termed "regeneration" in a "merely external, sacramental, and conditional" sense.
I hope that is all relatively clear. Now, how does all of that intersect with what we find in Hodge and Miller and other 19th century Princeton theologians?
First, I certainly do not at all affirm that "baptism regenerates" in the sense that Hodge denies that "baptism regenerates." We are coming at the question and terminology from different angles and with different meanings.
Second, I don't know the details of Hodge well enough to say, but I suspect he probably would not agree with what I believe about baptized infants of faithful parents. The way he speaks of baptized children as needing to "ratify that covenant by faith" suggests this perhaps, though I certainly could agree with that statement in the sense that children need to grow up into repentance and faith through the preaching of the Gospel.
Third, I'm convinced that my probable disagreement with Hodge here is part of a historic and ongoing difference of opinion within the Reformed and Presbyterian tradition, going back to the Westminster Assembly itself and before. There were conservative 19th century American Presbyterians who essentially agreed with what I've said above and there were those who didn't. All of our views are well within the bounds of the Westminster Standards and this is historically demonstrable.
Fourth, I fail to see how what I have outlined above bears any direct relationship to the kind of Anglo-Catholic sacerdotalism that Hodge describes and opposes. I'm not an Anglo-Catholic and my views come largely out of the Puritans and 17th century Reformed scholastics - not Pusey or Newman.
I expect Presbyterians will disagree on some of these details, and that's okay. I think our confessional tradition is broad enough to include a healthy spectrum of opinion.
I should also further clarify that I am only speaking of what infants are capable and of what faithful parents, relying upon the promises of God, can ordinarily expect with joyful hope.
On the specific question of just what
precise operations the Spirit performs when and under what circumstances, I would commend caution against being overly dogmatic.
I'm fairly convinced in my own mind - based on the teaching of Scripture and how our tradition has reflected upon Scripture - that we can say (with a high degree of probable certainty) that the Spirit is the seed and root of regeneration in baptism for all infants who are elect
to eternal salvation.
Moreover, we may add that all infants who are baptized - at least who are baptized in the context of church communities where Christ is believed upon and the Gospel is preached - enjoy the privileges and benefits the Spirit grants to the church visible, which is no small matter.
Further than that, I think we are in the realm of speculation.
There certainly have been prominent figures in the 16th and 17th century Reformed tradition who believed (or at least hoped or thought probable) that all
baptized infants enjoy the remission of original sin and that this is what we mean by infant baptismal regeneration, adoption, sanctification, etc. In this case, those terms are used in a manner that is not univocal
with their application to those of riper years who can exercise faith in particular acts and who enjoy subjective grace in a way that infants do not and in connection with the preaching of the Word.
This was once the position of much of evangelical Anglicanism, as well as some Presbyterians and figures within the Reformed churches on the continent. But it is admittedly a position very much in the minority. Moreover, it was criticized and rejected by other Reformed figures such as Mastricht and Witsius, though they nonetheless saw the proponents as orthodox Reformed teachers. I myself am fairly persuaded of the view, though only as a matter of hope and probability, not dogmatic certainty.
On the whole, I think this is the best I can manage as a brief and non-technical summary of how I appropriate Reformed teaching on regeneration and baptism.