Since I’m the pastor of a Reformation church where there are very few Reformed churches, one question I get is, “Why do you baptize children? Are you Roman Catholic?” The short answer is no, we’re not at all Roman Catholic, we’re Reformed. We baptize children because we’re a Reformed church. All Reformed/Presbyterian churches baptize infants, not just ours. It’s written very clearly in our confessions!
I like how puritan Thomas Vincent (d. 1678) explained this in his commentary on the Westminster Shorter Catechism - specifically on Q/A 95. (Side note: this commentary on the WSC received a churchly stamp of approval by John Owen, Thomas Manton, Thomas Brooks, and Thomas Watson, among other puritans).
“How do you prove that the infants of such as are visible church members may and ought to be baptized?”
“That the infants of such as are visible church members may and ought to be baptized may be proved, because they are in the covenant [of grace]; and the promise of the covenant belonging unto them, this seal of covenant doth belong to them also. ‘The promise is to you and to your children” (Acts 2.39). It is upon account of the promise of the covenant that any have the seal; hence it was that not only Abraham, but all his seed, whilst in their infancy, received the seal of circumcision, because the promise of the covenant was made to both; and by the same reason, not only believing parents, but also their infants, are to receive the seal of baptism, the promise being made to both. See Gen. 17:7, 10.”
The next question basically asks this: How can you prove that we should baptize our children today based on circumcision in the OT?
A: “That the infants of Christians have the promise of the covenant of grace made with Abraham is evident, because that covenant was an everlasting covenant, Gen. 17:7, which covenant Christ is the Mediator of, and is renewed in the New Testament with all believers, and that as full as under the law: and, therefore, if the infants under the law were included, the infants under the gospel are included too. That the privilege of infants (being made church members) under the law, doth belong to the infants of Christians under the gospel, besides the parity [similarity] of reason for it, and equality of right unto it, is evident, because this privilege was never repealed and taken away under the gospel.”
I appreciate that point. The NT doesn’t exclude children from God’s covenant people; in fact, it speaks of them as part of God’s people much like the OT does (i.e. Acts 21.5, 1 Cor 7, Eph. 6.1-4, Col. 3.20-21, 2 Tim 3.15, etc.). The next question Vincent asks is: how do you prove that the privilege of infants being made visible church members under the gospel was never taken away?
“1) Because, if this privilege were repealed, we would have some notice of its repeal in the Scripture; but we have no notice or signification of God’s will to repeal this privilege throughout the whole book of God. 2) Because Christ did not come to take away or straiten [narrow] the privileges of the Church, but to enlarge them; and who can, upon Scripture grounds, imagine that it was the will of Christ that the infants of the Jewish church should be church members, but the infants of the Christian church should be shut out like heathens and infidels? 3) Because the Scripture is express [clear], that the infants of Christians are holy (1 Cor. 7.14). As the Jews are called in Scripture an holy nation, because by circumcision they were made visible church members; so the infants of Christians, as well as themselves, are called holy; that is, federally holy, as they are by baptism made visible church members.”
I summarized this very slightly; you can find the entire Q/A discussion in Thomas Vincent’s The Shorter Catechism Explained from Scripture (Edinburgh: Banner of Truth Trust, 2004).