A Limited Atonement or a Particular Remdemption states that when Jesus Christ died on the cross, He did, in fact, effectually buy redemption for the elect of God. In better wording from The Five Points of Calvinism Defined, Defended, and Documented, it is said that "Historical or mainline Calvinism has consistently maintained that Christ's redemmin work was definite in design and accomplishment--that it was intended to render complete satisfaction for certain specified sinners, and that it actually secured salvation for these individuals and for no one else" (pg.40).
The "L" of the TULIP teaches that when Jesus died, He died for specific people in order to save them. He did not die for everyone in the same way. Classic Arminian theology teaches a General Atonement whereby Jesus died for the "sins of the world" in the same way. However, the atonement is limited in its effectiveness in Arminian theology by the exercising of faith. Though Jesus died for all the sin of all the world, people will still perish in their sins if they do not exercise faith in Jesus Christ according to classic Arminianism. (If anyone cares, we can discuss the differences of "election" and the nature of faith in both views later.) The point that I am making here is that both Arminians and Calvinists limit the effectiveness of the atonement in some type of way.
The Arminian objection to the Calvinistic view of the atonement comes from Biblical texts like these (leaving aside philosophical discussion for a moment):
"And He Himself is the propitiation for our sins, and not for ours only but also for the whole world" (1 John 2:2).
"For there is one God and one Mediator between God and men, the Man Christ Jesus, who gave Himself a ransom for all, to be testified in due time" (1 Timothy 2:6).
"For God so loved the world that He gave His only begotten Son, that whoever believes in Him should not perish but have everlasting life" (John 3:16).
From these verses, it seems apparent that Jesus Christ died for the sins of the whole world. How, faced with this evidence, can a person hold to the view that the atonement is limited only to the elect in its effectiveness?
Several legitimate answers have been offered to this question. One being that the word "world" does not necessarily mean "every individual on the planet." Rather, it often means the world in its scope. In other words, it means that Jesus did not only die for Israelies, but also for Iranians, Russians, Americans, Belgians, and other various European folk.
The Arminian answer has its own problems. The natural question is that if Jesus died for all the sins of all the world, why do people go to hell? The answer normal answer is that it is because they die in unbelief. The Calvinistic response to that is, "Well, isn't unbelief a sin? And if it is a sin, didn't Jesus die for that one as well?"
All the classical questions aside, I do believe that there is a way in which the death of Jesus Christ on the cross is effective to every individual and yet remain only salvifically effective for the elect. Here is how the understanding goes. I will quote from Deuteronomy 21:1-9:
1 “If anyone is found slain, lying in the field in the land which the LORD your God is giving you to possess, and it is not known who killed him, 2 then your elders and your judges shall go out and measure the distance from the slain man to the surrounding cities. 3 And it shall be that the elders of the city nearest to the slain man will take a heifer which has not been worked and which has not pulled with a yoke. 4 The elders of that city shall bring the heifer down to a valley with flowing water, which is neither plowed nor sown, and they shall break the heifer’s neck there in the valley. 5 Then the priests, the sons of Levi, shall come near, for the LORD your God has chosen them to minister to Him and to bless in the name of the LORD; by their word every controversy and every assault shall be settled. 6 And all the elders of that city nearest to the slain man shall wash their hands over the heifer whose neck was broken in the valley. 7 Then they shall answer and say, ‘Our hands have not shed this blood, nor have our eyes seen it. 8 Provide atonement, O LORD, for Your people Israel, whom You have redeemed, and do not lay innocent blood to the charge of Your people Israel.’ And atonement shall be provided on their behalf for the blood. 9 So you shall put away the guilt of innocent blood from among you when you do what is right in the sight of the LORD.
Look carefully at verse eight and what the elders ask God to do. They ask that God will not lay the charge of murder to the people of Israel. Verse nine indicates that when they do this, God will provide atonement for Israel for the murder incident. Here is my point: If God provides atonement for all Israel, does this mean that the sacrifice covers the Israelite who is guilty of the murder as well? Yes, in the sense that he will enjoy the benefits of an uncursed land, but no in the sense that God will still hold that individual accountable for the crime. In the same way, I believe, God's mercy is shown to unbelievers every day because of the atonement of Jesus Christ, yet at the same time they remain guilty even though atonement has been provided for.