It was alleged that the Arminians are semi-Pelagians because they believe things that the semi-Pelagians might believe, but which have nothing to do with the specific heresy of semi-Pelagianism.
To properly address your comment above, a person could write a small book. R.C. Sproul wrote a book years ago dealing with history behind the free will debate entitled: "Willing to Believe".
The term "Arminian" can and is used in a variety of ways. Most "Arminians" are not classical Arminians in the sense of holding the same beliefs as the man Arminus behind the term. There are so many shades of Arminianism that it's not even funny, but they all have something in common, they all deny "total inability". They all affirm reguardless of position on original sin, even depravity, that all humans are capable, within the will, of responding to God with faith. The semi-palagian position could be described as cooperating with the Grace of God, while the Arminian position could be described as free to believe, or visa versa. I agree with Dr. Sproul's analysis:
"Though the Pelagian controversy ended with the condemnation of Pelagius and his followers, the views of Augustine were not universally accepted in all their details. At first, opposition arose to some elements of Augustine’s thought in North Africa. Some monks from the monastery of Adrumetum in North Africa objected to Augustine’s view of predestination and to his view that fallen man is morally unable to incline himself to the grace of God. Questions arising from this debate prompted Augustine to write On Grace and Free Will and On Rebuke and Grace. These works were answered by the abbot of the monastery, Valentinus, in a cordial and respectful manner.
As discussion continued in North Africa, a more violent opposition to Augustine’s views erupted in France, particularly in the south at Massilia. Friends of Augustine, Hilary and Prosper, reported to him this opposition and urged him to write a response. Augustine did so in his final two works, On the Predestination of the Saints and On the Gift of Perseverance. In these works Augustine dealt more gently with his critics than he did with Pelagius, regarding them as brothers in the faith. This attitude anticipates the aura of future controversies. In the main, both Augustinians and semi-Pelagians tend to regard Pelagianism as a heresy so serious that it is non-Christian, while the on-going controversy between Augustinianism and semi-Pelagianism is an intramural debate among believers. Though the issues involved are deemed to be quite serious by both sides, they are not regarded to be so serious as to be essential to Christian faith." -
Sproul, R. (1997). Willing to believe: The controversy over free will (electronic ed.) (69–70). Grand Rapids, MI: Baker Books.
and further into the book...
"Repeatedly the Synod of Dort charges the Remonstrants with teaching the doctrines of Pelagianism. Is not this charge overly severe and unfair? Both Arminius and the Remonstrants sought to distance themselves from pure Pelagianism. Arminianism is often said to be semi-Pelagian, but not, strictly speaking, Pelagian. What the fathers of Dort probably had in mind is the link between semi-Pelagianism and Pelagianism that renders the semi-Pelagian unable to escape the fundamental thesis of Pelagianism." -
Sproul, R. (1997). Willing to believe: The controversy over free will (electronic ed.) (140). Grand Rapids, MI: Baker Books.
