Turretin’s third argument for infant baptism is by circumcision. He considers the similarities of circumcision and baptism and uses them as a support to the argument that baptism should be administered to children as circumcision was. The two sacraments are the same in their essence and purpose therefore the administration of both should also be the same.
In Genesis 17 God commands that circumcision was to be administered to infants. The command carries forward to the New Testament as baptism is now the covenant sign signifying what circumcision did. This is proven in the New Testament where baptism and circumcision are shown to meet at the Cross as Paul writes in Col 2:11-12
Col. 2:11 In him also you were circumcised with a circumcision made without hands, by putting off the body of the flesh, by the circumcision of Christ, 12 having been buried with him in baptism, in which you were also raised with him through faith in the powerful working of God, who raised him from the dead.
Paul shows that circumcision was pointing to the Cross. And now that covenant sign of circumcision, being fulfilled perfectly by Jesus, is administered as baptism. The sign has changed from the bloody cutting of the foreskin to the non-bloody sign of water. The similarity shown here is that circumcision was a rite of initiation so is baptism.
The other similarity is in the thing signified by the two sacraments. Both signify the same essence, some of those similarities are the internal circumcision of the heart, the righteousness of God that is by faith, and the grace of regeneration.
The grace of God is not diminished or restricted in the New Testament but extended and enlarged. Therefore we should not add restrictions to the administration of the sign. The ability of the child to mentally or rationally comprehend the terms of the covenant was not a prerequisite for circumcision. Therefore we should not place those qualifications as a barrier to baptism for the children of believers. No one would say God waisted blessings on the children of the Jews in the Old Testament then neither would the blessing of God be wasted in the New.
To point out that it is not explicitly stated to baptize infants as it has been explicitly stated to circumcise infants is not a proper argument either. Some of God’s commands are more explicit than others. Some commands are implicit. But both are commands of God. See the post “Turretin: Infants of Believers Should be Baptized” where this is flushed out citing Matt 28:19. Jesus says to go and baptize all nations and children are part of that group. Turretin states “…for a precept concerning a genus includes all its species.” (3.415)
The differences between circumcision and baptism do not keep children from receiving the sign of baptism. Because the emphasis is on what the sign communicates. That they are confirmations of the covenant because they are signs of the covenant. They are both symbols of invitation and badges of profession that are used by the Jews and Christians to distinguish themselves from the non-believers in the world. The differences between circumcision and baptism don’t change the essence of the sacraments or their purposes nor the requirements for administration.