I think it's a good thing to discuss theological views without putting words in the mouth of others with different views or making assumptions.
One of those assumptions Calvinists, especially, make is that they assume Arminianism is semi-pelagian.
I get why they do this. I thought this was true for many years, and so (somewhat grudgingly) was a Calvinist because I thought there really wasn't any other option besides Calvinist or Semi-Pelagian.
But, as time went by, I gradually began to shed each letter from TULIP (it started with Limited Atonement) and by the time I got to something like a 2-point Calvinist, I realised that I could never call myself a Calvinist if I only accepted like 2 points of its 5 point doctrine.
When I actually studied Arminian thought, I was surprised.
So without further ado, let me lay my case down as briefly as possible and let the fiery discussion begin! (I'm sorry I cannot explain some of these terms in detail - for those who don't know what some of these terms mean, just ask!).
A. There are four options when it comes to your soteriology:
1. Augustinian (Calvinist)
2. Semi-Augustinian (Arminian)
3. Semi-Pelagian
4. Pelagian
I won't venture to guess who sits at (3) and (4) at this point, perhaps that'll come up under the discussion.
Calvinists, especially, tend to never consider (2) above. They think it consists of Augustinian and then Semi-Pelagian, and then full-blown Pelagianism. But there is another option.
B. Arminianism is semi-Augustinian because it does not claim a person has the ability to put their faith in Christ.
Classical Arminianism (I'm talking Arminius himself and Wesley here) teaches that a person is totally depraved and incapable of having faith in Christ by themselves. What is required is the Holy Spirit to convict and give someone faith. However, the difference between a Calvinist here and an Arminian is that an Arminian believes that the grace given by God to a person (it is in this grace that faith is given) can be rejected, whereas a Calvinist believes it can't.
The point to note is that both the Calvinist and the Arminian believe that a person does not have anything in them to have faith in Christ. Both believe in Total Depravity. Faith must be given by the Holy Spirit. So there is a point of similarity / unity here.
The difference also, of course, revolves around the 'who' is given grace. The Calvinist bases the grace ultimately on the hidden will of God (predestination). God chooses before time who he will elect and who he passes over. This is 'unconditional election'. The Arminian believes that everyone is given grace at some point in their lives. God's revelation of Himself is given to all at some stage by the Holy Spirit. Everyone has the ability at some stage to accept or reject Christ. This is not a once-off moment but it may be gradually over their lives.
One can think of this as a 'grace' that 'comes before' salvation. It's a grace where God is drawing a person in, wooing them. This is called 'prevenient (comes before) grace'. If a person responds to this grace they are eventually given faith (most especially through the preaching of the Word - Romans 10:17). This grace can be resisted, however.
The important thing to note in this, however, is that God is the initiatior. God is in charge of when revelation comes, how it comes, at what rate it comes, and so on. God prepares the heart for faith and subsequent salvation. This is a work of God and not of man.
Modern day stories abound of Muslims given dreams etc. of Christ. I myself have met one. But scriptural support comes in many forms. I'll simply paste one here and refer to another:
Acts 17:
26 From one man he made every nation of the human race to inhabit the entire earth, determining their set times and the fixed limits of the places where they would live, 27 so that they would search for God and perhaps grope around for him and find him, though he is not far from each one of us.
Romans 1, 2 and 3 is also a very interesting read - note how even though the Gentiles do not have the Law, they still have an understanding of their obligations to God.
Okay, I'm trying to be brief.
This is why an Arminian (in the pure, classical sense) is not Semi-Pelagian, because an Arminian does not believe that mankind still possesses something to come to God or to find God - some sort of inherent goodness that leads a person to salvation based on their own nature.
Therefore, Arminianism is semi-Augustinian.
C. Arminianism does believe in predestination - but it's also different
Lastly, let's talk predestination.
Two schools of thought essentially are at work here with Arminians, and sometimes they work together.
The first thing to note is that predestination in Ephesians 1 is 'in Christ'. This is an important point. For the Calvinist, predestination is found in God's hidden will - we can't really know who he elects and who he doesn't, and we can't even know if we're elected until we get to the end. This point is difficult for me when it comes to assurance because I find it hard to place my assurance on something that I cannot know.
To put it simply, the Calvinist then bases salvation on God's hidden will.
For the Arminian, salvation is conditioned on faith, not predestination. I realise a Calvinist does not say salvation isn't dependent on faith, but I think that ultimately the Calvinist conditions salvation on predestination.
Faith is not a work. That's why the Arminian can claim that salvation is not by work. Faith is given by God and to not resist this gift is not a work. Faith is always faith.
Okay, so the point is to note that Arminians see predestination as:
1. God predestines those he saw, ahead of time, would put their faith in Christ.
2. God predestines an elect 'group' (the church) and anyone who puts their faith in Christ is put in this group.
Briefly, predestination is primarily of God's anointed one (Christ). God predestined Christ. And he predestined a Church - a people to be found "in Christ". Anyone found 'in Christ' is therefore predestined. This pretty much sums up my view right now. It is true that God has predestined a call for every single person in this world, saved or unsaved, but if they come into Christ they get to walk in that call.
As a final point, one might note that the Arminian's position is an effort to make salvation more relational (it's more influence and response) and less mechanical (cause and effect).
If I've been a bit biased in this, please forgive me. But I do want to simply discuss why Arminianism is not semi-pelagian. Even some well known Calvinists seem to insist it is, which is disappointing, because it is intellectually dishonest to claim so.
Oh, one more point. I'm aware that some Arminians are actually semi-pelagian but they still call themselves "Arminians". Obviously this sort of discrepancy can be found in just about every form of belief, so I would consider that point irrelevant in this discussion - although worth noting.