During Arminius' day Calvinists used six distinctions to hold God decreed Adam's fall while denying God is the author of sin. Below are my paraphrases and summaries of the distinctions and Arminius’ responses. Here’s a link to the original text. (link)
The Act and the Sin
The first distinction is "in sin there are two things, the act and its sinfulness. God, by his own ordination, is the author of the act, not of the sinfulness in the act.
Arminius argues that the distinction works for sins of omission (i.e. giving money to the church for public praise is a good act done for the wrong reasons) but not for sins of commission, because the acts themselves are against the law. But Adam's fall was a sin of commission, so the distinction is unhelpful. Also, since God's goal was to illustrate His glory and justice, the fall was decreed in that it was evil, not just in that it was an act, since sin (not the act) is forgiven or punished.
Even if God predetermined someone to sin for some reason other than evil (i.e. someone steals because he wants money not because he hates God), God is still the author of sin. Often the devil solicits to evil based on some natural good associated with the sinful act.
Necessity and Compulsion
This distinction is between doing something willingly or by force. “If the decree of God, in which he ordained that man should fall, compelled him to sin, then God would, by that decree, become the author of sin, and man would be free from guilt: but God’s decree did not compel man. It only imposed a necessity upon him so that he could not but sin; which necessity does not take away his liberty. So since man sins freely, he is the cause of his own fall and God is free from the responsibility.
Arminius responds by accepting the distinction but rejecting the idea that the distinction resolves the difficulty. Freedom and necessity are opposites and to say an act is ultimately both necessary and free is a contradiction. So saying sin in necessary with respect to the first cause, God, and free with respect to the second cause, man, is a contradiction. A person acts freely when he has the power not to perform that act. Whoever necessitates sin is the cause of that sin. So if God’s decree necessitates sin, God is the cause and author of sin.
The Decree and its Execution
The distinction is that while God decreed from eternity to devote certain persons to death, and to accomplish this ordained the fall, yet God does not execute that decree, by their actual condemnation, until after the people themselves have become sinful by their own act, and, therefore, God is free from responsibility.
Arminius responds that “he cannot be the ordainer of the punishment, who was the ordainer of the crime.” Augustine rightly says, "God can ordain the punishment of crimes, not the crimes themselves," that is, He can ordain that they should take place.
Efficacious and Permissive Decree
Arminius states that Calvinists attempt to use a “permissive decree”. They define a permissive decree as an act of the divine will, by which God does not bestow, on a rational creature, that grace, which is necessary for the avoidance of sin.
Arminius responds that this type of permission joined with the enactment of a law, embraces in itself the whole cause of sin. If God imposes a law which a person cannot obey without grace, and denies grace that person, God is the cause of sin by the removal of the necessary hindrance.
If the efficacious vs. permissive decree distinction is rightly explained, it removes the whole difficulty and God is not the cause or author of sin; for the action of God will has reference to its own permission, not to sin. Nor is “God, in the exercise of His will, permits sin” equivalent to “God wills sin, since God wills permission not sin.
God’s Goal vs. Man’s Goal
The next distinction is “God intends, in His decree, to illustrate His own glory, but man intends to gratify his own desire.”
Arminius responds that a good end does not approve, or make good, an action which is unlawful in itself; for "we are not to do evil that good may come;" but it is evil to ordain that sin shall be committed. Also, man’s desire results from the decree of God, and, therefore, man is relieved from responsibility.
Addition of the End
The last defense is "We are accustomed to state the decree of God, not in these terms, that 'God has determined to adjudge some men to eternal death and condemnation’, but we add, 'that His justice may be illustrated to the glory of his name’ .”
Arminius responds that the addition does not deny the previous statement and that the addition, even of the best end, does not justify an action which is not in itself formally good.