The definitive argument used by the non-calvinists is that God has given man "free will." This means that all human beings can do as they please in any situation. The only restraining influences on them is law, whether it is man's law or God's law, and conscience. This belief in man's free will teaches that "sinners," those who are "dead in their trespasses and sins" (Eph.2:1), can nevertheless, choose either to receive God's gift of forgiveness and eternal life (Rom.6:23), or reject it. A sinner to the non-calvinist is like the prodigal son who "came to himself" (Lu.15:17). At first this sounds like it is indeed proof of man's free will. But upon examination of the context of the prodigal we see that in fact, it proves the very opposite.
First, remember that it was "himself" that got the prodigal into trouble in the first place (Lu.15:11-16).
Second, his coming to "himself" in the pig slop did not bring him to a realistic view of his father or his relationship with his father (17-19).
Third, the prodigal's only concern was his own welfare; he was hungry.
Fourth, the prodigal's self-conceived solution was to "work" for his father rather than to be a son. And again, this was only because the son was still selfish and hungry. He had no concern for what he had done to his father or any of his family. The prodigal's reasoning was completely at the mercy of his circumstances. He had no other option; he had wasted his inheritance and now was looking for a way of life that would feed and sustain his fleshly desires. He still wanted to do life his way.
When one comes "to himself" he hasn't come far at all. He certainly hasn't come far enough. But he has come as far as he can; back to himself. "But when he was still a great way off, his father saw him and had compassion, and ran and fell on his neck and kissed him."
The prodigal was not on his way back home, he was on his way to work as a servant, not as a son. The point of the parable is that God is merciful and gracious and loving and greatly desires the return of His children (as Jesus was telling this parable the children were the Jews). It was the father's intervention that restored the son, not the son's intentions or choice. And the father pronounced by his grace that "my son was dead and is alive."
Eph.2:13, "But now in Christ Jesus you who once were far off have been brought near by the blood of Christ."
Jesus said, "I have come to seek and to save that which was lost" (Lu.19:10). The context here is important; it targets those who are of the promise and family of Abraham, not everyone (Eph.3:1-7).
Eph.2:4, "But God, who is rich in mercy, because of His great love with which He loved us, even when we were dead in trespasses, made us alive together with Christ (by grace you have been saved)...not of works, lest anyone should boast."
If coming to ourselves could possibly save us in the sense that our spiritually dead selves could make the decision to choose to be saved, would there not be the equal possibility that we would make the wrong decision about this eternal matter which would be more in keeping with our past decisions? The dead in sin can only make dead in sin decisions.
Coming to ourselves is nothing more than that; and it is the best a lost sinner can do. Without God's intervening grace this would be the closest we could ever come to having "free will."