James Durham puts it this way
I think what is hanging me up on this discussion is trying to distinguish between Jesus died due to/for the sins of humanity, and/but that is just for sake of the Elect of God, as I hold to both points as of right now. I see Jesus died in some sense for all, but not in the sense that Christian Universalists have seen it meaning!
James Durham explains a connection between Christ's death and common grace:
"Doctrine Two. We may consider Christ’s sufferings and death in the fruits of it, either as they respect common favors, and mercies, common gifts, and means of grace, which are not peculiar and saving, but common to believers with others, being bestowed upon professors in the visible Church; or as they are peculiar and saving, such as faith, justification, adoption, etc. Now when we say that Christ’s sufferings and death are a price for the sins of his people, we exclude not the reprobate simply from temporal and common favors and mercies that come by his death; they may have, and actually have, common gifts and works of the Spirit, the means of grace, which are some way effects and fruits of the same covenant. But we say, that the reprobate partake not of saving mercy and that Christ’s death is a satisfaction only for the elect, and that none others get pardon of sin, faith, repentance, etc. by it, but they only; it was intended for none others. And this we clear and confirm from, and by, these following grounds and arguments, which we will shortly hint at."
And Bavinck:
"Although vicarious atonement as the acquisition of salvation in its totality cannot therefore be expanded to include all persons individually, this is not to say that it has no significance for those who are lost. Between the church and the world there is, at this point, not just separation and contrast. It is not the case that Christ has acquired everything for the former and nothing for the latter. In rejecting universalism one may not forget that Christ’s merit has its limits even for the church and its value and meaning for the world. In the first place, it must be remembered, after all, that though Christ as such is indeed the Re-creator, he is not the Creator of all things. Just as the Son follows the Father, so re-creation presupposes creation, grace presupposes nature, and regeneration presupposes birth. Not included in Christ’s merits, strictly speaking, is the fact that the elect are born and live, that they receive food, shelter, clothing, and an assortment of natural benefits. One can say that God would no longer have allowed the world and humankind to exist had he not had another and higher purpose for it. Common grace is indeed subservient to special grace, and along with salvation God also grants the elect many other, natural, blessings (Matt. 6:33; Rom. 8:28, 32; 1 Tim. 4:8; 2 Pet. 1:3)."
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Concerning offering Christ in the gospel, it should be noted that we do not offer election or effectual calling. The Scriptures tell us to offer Christ indiscriminately and promising salvation on the
condition of faith and repentance. Since the general call is distinguished by definition from effectual calling, the basis of such a call to individuals who are either elect or reprobate that conditionally promises Christ to both on the same conditions, can only be a product of a general design in Christ's suffering, though this is not to be confused with a salvific purpose of God in relation to His decree.
Some Scriptures for consideration: John 3:14-18 as it relates to the offer. Col. 1:20 in relation to common grace benefits. Even though Calvin doesn't apply the sufficient/efficient distinction to 1 John 2:2 as does Hodge, John's particular use of the phrase "whole world" in 1 John only applies to the unbelieving world who is "under the sway of the wicked one" (5:19) if we look at how he actually uses the phrase. (Also consider the usage in Revelation: 3:10, 12:9, 16:14.) For us to consider the "whole world" in 1 John 2:2 as the elect world would be a dramatic departure from John's usage in these later books.
Further, we confess that God is a God of mercy and justice. His mercy that extends to us is not at the expense of His justice, which He met in Christ on the cross. If a) common graces (mercies) are extended by God to the non-elect as well as the rest of creation, and b) Christ's sufferings could have nothing to do with these mercies, we expose God to a compromise of justice for mercy since we leave no room for any satisfaction of justice in relation to creation apart from the elect.