Of all the so-called "doctrines of grace", Limited Atonement is the least defensible, both in theory, and indeed in the admission of many Calvinists. Indeed, most 4 point Calvinists reject Limited Atonement. There are no scriptures that present Limited Atonement, at least not explicitly. Thus, when arguing for Limited Atonement, they do so using deductive reasoning.
Now, obviously, deductive reasoning is not a bad thing. Christians do it all the time, and indeed have to do it all the time. Certainly, we do have some explicit statements. Creationists, for example, have Exodus 20:11, explicitly telling us that God created the heavens and the earth in 6 days. The only way God could have been more explicit than this is to be more explicit than even your average Creationist is, saying "in six literal, 24 hour, earth days". But other times, we need deductive reasoning. God never says "I am a Trinity". Yet, we can make a compelling case from other explicit statements that He is. Just as we do not need to say the word "tiger" to make a reader think of tigers, we do not need the Bible to say "Trinity" to know that God is a Trinity.
The problem for Calvinists is that while we don't have any explicit statements in favor of Limited Atonement, we have many explicit statements against it. What Calvinist hasn't heard John 3:16 cited at them? God so loved the world that He gave His only Son. 1 John 2:2 likewise says "And he is the propitiation for our sins: and not for ours only, but also for the sins of the whole world." (emphasis mine). This same John records John the Baptist's reaction to seeing Christ: "Behold the Lamb of God, which taketh away the sin of the world." (John 1:29b).
"That's all well and good", says the Calvinist, "but world does not always mean the planet or all people within". And they're not wrong. There are several examples within scripture where the word "world" means some locality. In Luke 2:1, for example, Caesar Augustus did not seek to register the entire planet, only those within his jurisdiction. But there are obviously other places where the word does mean world. For example, Romans 5:12 tells us "Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned:" Was Australia untouched by sin? Even the aforementioned 1 John 2:2 specifies the whole world.
So, the fact that the word "world" does not exclusively refer to all people on Earth is not sufficient to rescue Calvinism. All it does is render the verses that state it temporarily ambiguous, validating the Calvinist interpretation, but not solidifying it. In order to solidify it, Calvinists must not only demonstrate that their interpretation fits, but that it is the best fit.
Now, I don't have as much experience debating Calvinists as I have debating, say, Evolutionists. Nevertheless, I have never seen a compelling reason the Calvinist interpretation that "world" means only the elect is the best fit for any verses that tell us Christ atoned for the world. In fact, the irony is, the only way any definition of the word "world" would exclude, at least in its normal sense, those who perish, is if it also excludes many Calvinists. If "world" is the same world Caesar Augustus registered, for example, then no Calvinist outside of the Roman Empire can be saved. Augustus didn't register anyone in Australia, so did Christ not atone for anyone in Australia?
We could even go a step further and return to Romans 5:12. If sin entered the world through Adam, then Christ would need to atone for the sins of Australians. Yet, in 1 Corinthians 15:22, Paul tells us "For as in Adam all die, even so in Christ all shall be made alive." As in Adam, so in Christ. That obviously can't mean all descendants of Christ are saved, for He never had a wife, or kids. It can't be promoting Universalism, because the condition for salvation is faith. The only way this verse makes sense is blood. Adam's blood flows through us, killing us in our sin, Christ's blood flows over us, covering our sin. But if Adam's curse is global, so is Christ's atonement. The entire reason Christ's atonement covers anyone is because it covers everyone.
But it gets still worse for the 5 point Calvinist. Just as there is no definition of the word "world" that could cover all Calvinists without also covering all people, so also is there no definition of any word that would explain why scripture repeatedly refers to unbelievers whom God "bought", and willed to be baptised, yet "resisted His will" etc. In 2 Peter 2:1, for example, we read "But there were false prophets also among the people, even as there shall be false teachers among you, who privily shall bring in damnable heresies, even denying the Lord that bought them, and bring upon themselves swift destruction."
Now, who is God buying the false teachers from? According to God Himself, "everything under Heaven is mine" (Job 41:11b). Are false teachers under Heaven? Of course. So they were already God's, as if doubt could be placed upon that. Yet, God is said to have "bought" them, just as in Acts 20:28, God "purchased" the Church with His blood. But these people Peter talks about are not of the Church, by definition. They bring damnable heresies, they deny the Lord who bought them, they bring upon themselves swift destruction.
But what if they should repent and be baptised? Another instance in scripture confirms that this is the will of God, and that we are actually able to resist that will. "But who can resist the will of God?" the Calvinist replies, seemingly in step with scripture, for Romans 9:18-19 seems to indicate this is impossible. But who can resist God's will? Sinners. Sinners, by definition, frequently resist God's will. God wills us to be holy as He is Holy, and yet, we are not. We violate His will by violating His explicit commands. But furthermore, in Luke 7:30, we read "But the Pharisees and lawyers rejected the will of God for themselves, not having been baptized by him." God willed for the Pharisees and lawyers to be baptised by John, but they are explicitly stated, by Luke, under inspiration of God, to have rejected God's will.
This is where Calvinists tend to get bogged down in man made theology to try to oppose scripture itself. Luke 7:30, in their minds, cannot mean the Pharisees and lawyers rejected God's will because that would make them stronger than God. Right? Not necessarily. There are two main problems. The first, as I'm sure every Calvinist will agree, is that you can't trump scripture. If scripture says X, you can't say "but it can't say X, because X would mean Y, and Y can't be true, therefore verse X is wrong".
The second problem is that this apparent contradiction is easily resolved when you keep God's character in mind. For lack of better terms, God has different "levels" of will, and He can sacrifice the "lower" levels for the "greater". We see this in the garden of Gethsemane, where Jesus prays "O My Father, if it is possible, let this cup pass from Me; nevertheless, not as I will, but as You will." (Matthew 26:39). We see, then, that Jesus willed to not be crucified. Can you blame Him? It was an awful experience for Him, and He didn't even deserve it. So here we see God having two different wills (unless, of course, you deny the deity of Christ). On the one hand, He willed not to be crucified. On the other hand, He willed to save His people. The result of this conflict? Glory to God, we are redeemed by His precious blood!
In the same way, God has two wills, a lesser and a greater, when it comes to our salvation. His first will: Our salvation. His second will: Our freedom. Without that freedom, our salvation is actually less meaningful. We become robots, not humans. Does a robot bear God's image? Can a robot ever bear God's image? Ask the robots who bear our image. Those things are creepy, just look at them. I don't know about you, but I'd prefer a wife to even the most realistic looking human robot. I'd prefer a child to a robot baby. Robots do not compare to human beings, in our eyes, or in God's. The trouble is, free will is what separates us from these robots. They merely imitate consciousness. We legitimately have it, and because we have it, we can freely choose to accept God's will for our lives, or, as the lawyers and Pharisees explicitly did, reject it.
So did God fail? If God atoned for the sins of the whole world, why is the whole world not saved? Return to John 3:16: "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." Whoever believes in Him has everlasting life. If someone rejects God's will to be saved, that's not His failure, that's their loss. There is not a single human being on this Earth who, right now, cannot repent and be saved. That's not a failure. That's perfect success according to the mission statement.
And so we see that, by the explicit statements of scripture, Limited Atonement just doesn't work. Christ made propitiation for the sins of the world, and there is no context in which the word "world" could cover all the saved while excluding the unsaved. Even further explicit statements show that the atonement covers those who were not saved, for those whom God willed to repent often refused to do so, and those whom the Lord bought preached damnable heresies, to their own destruction. Thus, Limited Atonement is by far the least defensible of Calvinism's 5 points, and it should be discarded as swiftly as any other falsehood.