This is a hard topic on which to write. I anticipate some negative reaction. However, it is a question I have been asked repeatedly by anti-Calvinists.
First, let us look at the confessional standard: "Elect infants, dying in infancy, are regenerated and saved by Christ
through the Spirit, who worketh when, and where, and how He pleaseth. So
also are all other elect persons who are incapable of being outwardly
called by the ministry of the Word" (Westminster Confession of Faith X:3). The divines described what happens to elect infants who die. Of course, since they are elect, that would be God's plan for them. What about non-elect infants? On that the divines were silent. Charles Hodge and his son, A. A. Hodge, took that to mean that all infants who die are elect. I think that is presumptuous, taking an argument from silence where it does not lead.
For the Christian, there is extensive biblical justification to believe that his dead child is in Heaven. First, God claims the children of believers for Himself in Ezekiel 16:20-21: "You took your sons and daughters, whom you had borne to Me, and these you sacrificed to them [i. e., idols] to be devoured. Were your whorings so small a matter that you slaughtered My children and delivered them up as an offering by fire to them?" When the Israelites had become so given over to idolatry that they even practiced human sacrifice, God's anger was directed at the theft of what belonged to Him by covenant.
And second, what does God promise to these children who are His? "Your children shall be taught by the LORD, and great shall be the peace of your children" (Isaiah 54:13). Also in Acts 2:39: "The promise is for you and for your children and for all who are far off, everyone whom the Lord our God calls to Himself." So, in both Testaments, we have a promise from God to be covenantally faithful to the children of believers. Is this a promise that every child of believers will be saved? No, it isn't, as we know both from personal experience and from the biblical examples of Esau and Ishmael.
However, we also have a biblical example of the comfort that covenantal promise is to the believer. When David's first son with Bathsheba died (II Samuel 12:15-23), David took comfort in his assurance that his son would be waiting for him is Heaven: "I shall go to him, but he will not return to me" (verse 23).
I think that the most-important verse on this issue is I Corinthians 7:14: "The unbelieving husband is made holy because of his wife, and the
unbelieving wife is made holy because of her husband. Otherwise your
children would be unclean, but as it is, they are holy." Paul tells us that the children of at least one Christian parent are holy. He doesn't say saved. Rather, he speaks of the covenantal connection between the believing parent and the child, such that the child is federally holy on the basis of the parent's faith.
However, we should notice his exact words. Paul speaks negatively. He doesn't just say, "The child with a believing parent is holy." Instead, he adds, that the child would otherwise be "unclean." And this is logical because we know there is no neutral moral state. But what are the consequences to the unbelieving parents regarding their own children?
"Nothing unclean will ever enter it [i. e., the New Jerusalem, v. 10], nor anyone who does what is
detestable or false, but only those who are written in the Lamb’s book
of life" (Revelation 21:27). Here is where the unbeliever must consider his standing with God, because it affects not just himself but also his children. If he decides that the pleasures of this life outweigh the eternal consequences, can he also say that they outweigh the eternal consequences for his child? Of course, this doesn't mean that every child of an unbeliever will himself be an unbeliever. We know from experience that the Holy Spirit often breaks into the families of unbelievers to bring one to Himself. I myself was such a convert. But the generality can be predicted, just as above with the children of believers.
This is my personal interpretation. Though I consider it a rational conclusion from the relevant Scriptures, I am aware that it goes beyond the confession. Therefore, other Calvinists should not be blamed for my personal opinion. I am especially conscious that I am going against some theological giants when I disagree with the Hodges. All I can say is that the Scriptures compel me.
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