The Arminian says that Christ died for every man. But every man will not be saved. But how does this work light of the substitutionary power of the atonement (which most Arminians would affirm)? Is Christ’s blood powerful (by cleansing those for whom He sheds it) or powerless (by only cleansing some of the sins for those who will receive it)?
Some Arminians see the logical discrepancy in the popular view of the atonement, and, as a result, affirm the idea that Christ didn’t die for anyone, but died just as an example: to “show how serious God is about sin”. As J. Kenneth Grider, a proponent of this idea, wrote,
“A spillover from Calvinism into Arminianism has occurred in recent decades. Thus many Arminians whose theology is not very precise say that Christ paid the penalty for our sins. Yet such a view is foreign to Arminianism, which teaches instead that Christ suffered for us. Arminians teach that what Christ did he did for every person; therefore what he did could not have been to pay the penalty, since no one would then ever go into eternal perdition. Arminianism teaches that Christ suffered for everyone so that the Father could forgive the ones who repent and believe; his death is such that all will see that forgiveness is costly and will strive to cease from anarchy in the world God governs. This view is called the governmental theory of the atonement.” 
The problems with this are obvious and myriad. If Christ died simply to make it “possible” for people to be saved, without being the absolute substitute for anybody’s sin, every man would go to hell. “For all have sinned and fallen short of the glory of God”, “there is none that doeth good, no not one” and “God is of purer eyes than to behold evil, and cannot not look on iniquity”, and thus, every sin that we committed we would have to give an account for, without any substitution, before a holy God.
But, on the other hand, if Christ died to cleanse the sins of all men, than everybody would go to heaven. As Paul writes,
“Much more then, being now justified by His blood, we shall be saved from wrath through Him.” (Romans 5:9)
“Moreover whom He did predestinate, them He also called: and whom He called, them He also justified: and whom He justified, them He also glorified.” (Romans 8:30)
So we see that if Christ’s blood is shed for a man, that man will be justified and ultimately glorified. Thus, to say Christ died for every man is to say every man will be glorified.
Having established that Christ’s blood is efficacious we see that if He did die for everybody, than everybody would be counted sinless before God (which is unitarian heresy). Therefore we see it must be a particular atonement which is efficacious, which will, in fact, cleanse all of the sins of the people for whom Christ shed it. As theologian John Owen put it,
God imposed His wrath due unto, and the Son underwent the pains of hell for, either: all the sins of all men, all the sins of some men, or some sins of all men. In which case it may be said: If the last, some sins of all men, then have all men some sins to answer for, and so shall no man be saved. If the second, that is it which we affirm, that Christ in their stead and room suffered for all the sins of all the elect in the world. But if the first be true, why are not all men free from the punishment due unto their sins? You answer, “Because of their unbelief.” I ask, “Is this unbelief a sin, or is it not? If it is, then Christ suffered the punishment due unto it, or He did not. If He did, why must that hinder them more than their other sins for which He died? If He did not, He did not die for all their sins!”
Consider the words of Christ,
“I am the good shepherd: the good shepherd giveth his life for the sheep…. Ye [Pharisees] believe not because ye are not of my sheep.” (John 10)
Notice, Christ didn’t say, “Ye are not of my sheep because ye believe not” but He says, “Ye believe not because ye are not of my sheep.” This implies that the Pharisees did not believe, (and thus, were going to go to hell) because they weren’t His sheep, i.e., because Christ’s blood was not shed for them.
Thankfully, most Arminians are illogical in their thinking about the subject of the atonement and haven’t fallen into either of the two gross heresies that result from deducing this doctrine of universal atonement to its logical conclusions (universalism and the denial of penal substitution).
– Evan Murch
 J. Kenneth Grider, “Arminianism” in Evangelical Dictionary of Theology, Walter Elwell, ed., (Baker, 2001), p. 97.
 John Owen, The Death of Death in the Death of Christ, Book 1, Ch. 3