What are the various views on what God intended for atonement to accomplish?

The question whether Christ’s atoning work was a general provision or a particular accomplishment has been a matter of intense debate for centuries, and good men differ on the issue. Note that what is in debate here is not whether all men will be saved—all Bible-believing Christians of necessity recognize that some people will not be saved and will suffer eternal damnation. In this sense every one who denies universalism recognizes some sort of “limitation” on Christ’s atonement. The real question at issue is the design of the atonement. What did God intend for it to accomplish? There are three basic views (and a vast number of variations), all ostensibly held by people describing themselves as Calvinistic:

Some hold that atonement is universal in intention and provisional in character. The atonement is a perfect sacrifice with unlimited substitutionary potential that renders all men savable. The atonement is limited in its application by man’s free will alone. God’s eternal decree to elect is not a truly limiting factor, since this decree is based on his advance knowledge of the free acts of men. This view is consistent with the understanding of Arminianism, but is held by some who call themselves “4-point Calvinists.”

Some hold that atonement is undefined in intention (or in historical terms, hypothetically universal), and both provisional and redemptive in character.In its design, the atonement had unlimited redemptive potential, but God’s subsequent elective decree in eternity past mitigated the potential of Christ’s death, rendering it redemptive/substitutionary only for the elect. In his kenotic state, however, Christ did not know the identity of the elect, and self-consciously died in the same way for all, not knowing for whom his life and death were actually substituted, and for whom he was merely making lesser provisions. This view is known as Amyraldianism.

Some hold that atonement is particular inintentionand redemptive in character. The limitation or particularity of the atonement is established by God’s prior decree to elect in eternity past. As a result, the atonement is actually substitutionary in its eternal design and thus intended for the benefit of the elect alone. This view is that of historic or “Five Point” Calvinism.