1. Israel is the Church–the Church is Israel.
2. God commanded Israel to circumcise their households (believers, their children and anyone else of any status in the household).
3. Divine Commands remain in place if they are never rescinded.
4. Israel, by the plan of God, was a mixed multitude of believers whose circumcision signified the circumcision of their hearts (their faith and repentance), and unbelievers whose circumcision was a constant reminder to them of their need to circumcise their hearts (to repent and believe).
5. In Christ, the sign of circumcision–the sign and seal of faith (Romans 4:11)–was changed to baptism (see how the one is identified with the other in Colossians 2:11-12), and this sign continued to be given not only to individuals, but also to households in the New Testament.
6. The command to apply the sign and seal of faith to households is nowhere rescinded in the New Testament; therefore, the command remains in force.
7. For this reason, it is biblical to baptize the infant children of believers before they make a profession of faith.
8. In conclusion, the Bible teaches Christians to baptize their children with a view to raising them to repent and believe in the providential timing of the Lord.
The Misadventures of Captain Headknowledge 
How a Particular Baptist would answer these…
1). The church is the New Covenant ‘spiritual’ Israel.
2). God commanded all Hebrew people to circumcise their sons.
3). This is shaky ground
4). Circumcision was a type in the Old Covenant of which ‘circumcision of the heart’ is the anti-type in the New.
5). Rom 4:11 doesn’t speak to baptism
1. If believing Jews are also “spiritual Israel,” then is not “spiritual Israel” the true Israel, which would include the engrafted believing Gentiles? Does not “spiritual Israel” correspond to the invisible church, as over against the visible church, which includes the unbelieving members of the covenant community?
2. Which is the covenant of grace principle that the sign and seal of the righteousness which is by faith is to be given to the household (including infant children) of believers in any and every administration of the covenant of grace.
3. “To give a human example, brothers: even with a man-made covenant, no one annuls it or adds to it once it has been ratified” (Galatians 3:15)…”This is what I mean: the law, which came 430 years afterward, does not annul a covenant previously ratified by God, so as to make the promise void” (Galatians 3:17). Here is the principle I appealed to in action. It is the reason the Reformed see Scripture as retaining the practice of applying the sign and seal to children of believers in the New Testament, even after circumcision is replaced by baptism.
4. It was the visible sign of the circumcision of the heart in both the Old and New Covenants. God commands OT Israel to circumcise their hearts, and Paul writes “But a Jew is one inwardly, and circumcision is a matter of the heart, by the Spirit, not by the letter. His praise is not from man but from God” (Romans 2:29).
5. It applies to baptism as the New Covenant sign and seal of the righteousness that is by faith.
Abraham received circumcision as a sign and seal of his faith – faith in God’s promise to give him a land and a seed. That promise was carnally fulfilled in Abraham’s physical descendants under the Old Covenant. Abraham’s physical descendants (through Jacob) were born (really and truly) into the Covenant with God and were heirs of the promise. The sign was given in the reproductive organ as a sign that the promise was inherited through birth.
I agree with you that baptism is the sign/seal of being in the New Covenant. I do not believe there there is any genealogical principle in effect in the New Covenant. No one enters the new covenant through natural birth, but through a spiritual birth. The sign is given to believers.
1) Spiritual Israel is indeed the “invisible church” – the redeemed of God
2) I do not believe that the Old Covenant was the covenant of grace
3) Galatians provides a strong contrast (especially chapter 4) between the legal covenant and the grace covenant. I don’t think this helps your cause.
4 & 5) are answered above.
We must bear in mind that the New Covenant is not like the carnal/physical covenant made with the physical descendants of Abraham (physical land/seed). It is a new spiritual covenant that has spiritual blessings and a spiritual people. These blessings come through a spiritual birth, not a physical one.
On circumcision: “The Old is in the New explained.” The Land promise was a type of the actual promise, that of inheriting “the world” (Romans 4:13). The Old must be interpreted through the lens of Christ. The burden of proof is on you to show when “the genealogical (household) principle” was abrogated. The sign was explicitly abrogated (Acts 15), and the candidates were extended, not diminished. The New Testament evidence indicates continuation of the household principle, and the objective holiness of the members of the households of believers, and thus the “better” New administration of the covenant of grace extends the New Covenant sign of the covenant to not only the sons of believers, but also the daughters.
On the Old Covenant: Are you saying you do not believe the covenant mediated by Moses on behalf of Israel in the land was an administration of the covenant of grace?
Under the Old Covenant, Hebrew children were born into their inheritance in the Land of Canaan. Under the New Covenant, one must be spiritually born in order to receive the inheritance merited by Christ. The Old Covenant no longer exists. The New Covenant is entered by a spiritual birth. I think I need no further proof than that.
The Scriptures do not say that the Old and New Covenants are the same covenant. We are told the Old Covenant passed away. Was the Old Covenant a legal covenant? Yes, it’s chief character was “do this, and you will live”. Israel was punished because she transgressed the old covenant. A grace covenant could not be broken.
Was Abraham under the Old Covenant or New? If the former, how is he the father of those who have faith and are justified by grace through faith? Why is he given as an exemplar for New Covenant members?
The Old Covenant refers to the Mosaic administration of the covenant of grace, not an old legal covenant. Otherwise, David must’ve been saved by virtue of his keeping the law. In any case, circumcision was instituted some five centuries prior to the ratification of the Sinai (Old) covenant.