In Charles Ryrie's article "The Extent of the Atonement" he first defines the key issue. Ryrie disagrees with Berkhof that the key issue is if Christ came into the world to save all men or only the elect. Instead Ryrie frames the issue in term of did Christ make provision for the salvation of all people or not?
Next Ryrie deals with exegetical considerations. He digs into 2 Peter 2:1, 1 John 2:2, 1 Timothy 2:4-6, 4:10, Hebrews 2:9, John 3:16, and Acts 17:30. In each case Ryrie shows how the text teaches unlimited atonement and points out the problems with alternative explanations.
Finally, Ryrie deals with theological considerations. He explains that unlimited atonement is more compatible with preaching the Gospel to all. Then he explains that the objection that the value of Christ's death is not wasted even if not all believe, since God planned for the whole world to be savable, which itself has value. Last, Ryrie deals with the 'double payment' argument. His passover analogy was very helpful: An analogous question might be asked, Did the Israelite who refused to apply the Passover blood to the door of his house have his sins paid for twice! When the Passover Lamb was killed, his sins were covered. But if he did not put the blood on the door, he died. Was this a second payment for his sins? Of course not. The first and sufficient payment was simply not applied to that particular house. Death after failure to apply the blood was just retribution for not appropriating the sufficient sacrifice. The Atonement of Christ paid for the sins of the whole world, but the individual must appropriate that payment through faith. The world was reconciled to God (2 Cor. 5:19), but those reconciled people need to be reconciled to God (v. 20).