SC is the view that determinism is compatible with moral responsibility, but not compatible with freedom to do otherwise.
CP is the claim that determinism is compatible with freedom to do otherwise.
I think most Calvinists who call themselves compatibilists actually are, more properly, defined as semi-compatibilists.
Classical compatibilists said, in an unrefined way, that all that is required for freedom is "ability to do what one wants to do." But, this has seen attack. For example, some people suffering from mental disorders do "what they want to do" yet we wouldn't say that they acted freely in these instances.
Contemporary compatibilists have sought to refine this idea, ruling out cases like the above. One such answer is that of Fischer. This can be called "Reasons-Responsive Compatibilism," RRC. The discussions about RRC are myriad, in depth, and detailed. I'll offer an "enough-to-get-started" quote from the Stanford Encyclopedia of Philosophy,
A reasons-responsiveness theory turns upon dispositional features of an agent's relation to reasons issuing in freely willed action. Appropriately reasons-responsive conduct is sensitive to rational considerations. The view is not merely that an agent would display herself in some counterfactual situations to be responsive to reasons, but rather that her responsiveness to reasons in some counterfactual situations is evidence that her actual conduct itself — the causes giving rise to it — is also in response to rational considerations. SOURCE
Now, there are a few things that sparked discussions leading to contemporary compatibilism. In interest of time, I'll mention one of them - Frankfurt Style Counter examples, FSC's. Names after philosopher Harry Frankfurt, the thought experiments ask us to imagine cases where a person S does an action A, and S did A of her own decision, i.e., she was not coerced, and S, in some of these experiments, was morally responsible for A. A classic example is,
Jones has resolved to shoot Smith. Black has learned of Jones' plan and wants Jones to shoot Smith. But Black would prefer that Jones shoot Smith on his own. However, concerned that Jones might waiver in his resolve to shoot Smith, Black secretly arranges things so that, if Jones should show any sign at all that he will not shoot Smith (something Black has the resources to detect), Black will be able to manipulate Jones in such a way that Jones will shoot Smith. As things transpire, Jones follows through with his plans and shoots Smith for his own reasons. No one else in any way threatened or coerced Jones, offered Jones a bribe, or even suggested that he shoot Smith. Jones shot Smith under his own steam. Black never intervened. SOURCE
Thus, for obvious reasons, many have concluded that FSC's are decisive refutations to a widely held libertarian constraint on free will called, in contemporary literature, PAP's: Principle of Alternate Possibility. Some, not willing to give up libertarianism, have rejected PAP's as a necessary feature of what is entailed by making a free choice. Theologically, we can also see how the notion of PAP isn't easily accepted by the Calvinist. God has determined the end from the beginning, and no one can thwart his plan. Many libertarians would hold that the historical past could have been exactly the same (including God's decree), yet S could have done other than S did in this world. Or, given the Calvinist notion of depravity, sinners cannot please God in their sinful nature. They do not have the ability to please God. But, they are still responsible, and Scripture says that they do what they do freely, i.e., without coercion (Acts 2 comes to mind, for one example).
Thus SC's think they have good reasons to believe that "freedom to do otherwise" is not necessary for free will or moral responsibility. S does what S wants to do, given the appropriate RRC, and these seem to satisfy requirements for freedom. So, the SC denies any "freedom to do otherwise" constraint. This does need to be specified in more detail, though. For example, there are different senses of "can" as used in "he can do otherwise." One way the compatibilist can agree that S "could" have done otherwise is by claiming that S was physically "able" to do A. Somewhat analogous, there is a sense in which Jesus' bones "could" not have been broken since Jehovah prophesied that they wouldn't be. But, there is another sense where they "could" have indeed been broken - they weren't made of steel. Jesus wasn't Wolverine. He had human bones and human bones "can" be broken. Thus, physically, an unregenerate "can" accept Christ since he can physically say the words "I trust and believe on the Lord Jesus Christ." But, in another sense, he "cannot" (cf. John 6:44). So, "can" all men accept Jesus, sic et non.
Next, contemporary compatibilism has made interesting strides in regards to how one can be morally responsible for doing something that he was determined to do.
SC denies that determinism is compatible with "ability to do otherwise," (not all contemporary compatibilists agree. So, this is why I think Calvinists should be SC's) but they say that determinism is compatible with moral responsibility. Again, this is a detailed maze of arguments and counter-arguments. A person is morally responsible when she can not only do moral right and/or wrong, but also is accountable for her actions. Thus a person is a fitting subject of praise or punishment.
Contemporary compatibilism is very entrenched in this debate. Many think it is incoherent to punish or reward someone for doing what they were determined to do. Or, to punish or reward for simply acting according to some pre-determined nature. Interestingly, some say that you shouldn't beat your dog for panting, yet they give an "atta boy" to their dogs for "being loyal." Besides my canine digression, a contemporary distinction is drawn by Fischer regarding SC's view of moral responsibility being compatible with determinism. He makes a distinction with respects to Regulative Control RC and Guidance Control GC. And agent with RC can regulate between different alternatives. An agent with GC brings about her conduct, even if there are no other alternatives to her actions. Here in GC we see the denial of PAP's which were, you will recall, allegedly (and decisively IMHO) refuted by FSC's. Furthermore, invoking views of secondary causation like those found in the Westminster Confession of Faith allow for agents to guide their actions. Moreover, that all is determined by God does not mean that agents can't guide or bring about their actions. So, that S determined that S* would A does not imply that S did A instead of S*. For example, Acts 2 tells us that it was not God, but then lawless men, who put Jesus to death. We are also told that God foreordained this. Thus we see that S foreordaining that S* would do A does not imply that S* isn't the one who does A.
Thus, for Fischer, it is only GC that is necessary for moral responsibility. RC could not be necessary for morality given FSC's. GC assumes what is called Source Control. Source Control simply maintains that the agent must be the source of his actions in some important or crucial way. Many incompatibilists have constructed an argument against Source Control theories (remember, the Confession says that there is a sense that we cause our actions, hence I find a source theory there). The key premise for the argument is:
A person acts of her own free will only if she is its ultimate source .
Obviously "ultimate source" would need to be defined, but most Christians (libertarians included!) must deny that we are the ultimate source of our actions. Furthermore, it is an "important way" to be involved in bringing about your conduct if you are a means in bringing it about. Calvinism says that God works through means. He does not accomplish ends irregardless of the means. This is fatalism. Fischer employs an actual-sequence RRC mechanism which is fully possessed by the agent (i.e. it is his own mechanism) and issues in the agent’s guidance control over his own action. Thus the agent brings about his actions in an important way, even though s/he is not the ultimate source of said actions.
Lastly, some have asked about whether our choices are “real” given determinism. I admit I don’t really understand the point. Isn’t a choice “real” if it occurs at all? Furthermore, before me I have a Ruth’s Christ steak and a plate of cheese and crackers. There are two options and, given Semi-Compatibilism, I have the ability to choose whichever one I want to. So, I choose, of course, the sizzling and mouth tantalizing steak from Ruth’s Chris (which sizzles because it is on a plate that is about 500 degrees, and is cooked in butter). This was pre-determined. Given God’s decree, I could not have actually picked the cheese and crackers. I could not because as, say, Isaiah 46:10 tells us: “Declaring the end from the beginning, And from ancient times things which have not been done, Saying, 'My purpose will be established, And I will accomplish all My good pleasure';” God declared that I would pick the steak, He cannot be wrong, hence I could not have, in a libertarian way, have chosen the cheese. But, did I not “really” chose the steak? After all, that’s what I wanted.
I think we can take our cue from Jesus. Before I explain, allow a brief digression, and apologies to those who aren’t familiar with the eschatological terms. I think one of the best lines I have ever heard was in a debate between gene Cook and Hyper-preterist H.L. James. Now, H.L. was kind of ridiculing the idea of a bodily resurrection. He pointed out that Jesus had holes in his hands, and so asked if Gene would have tattoos, scars, or things like that in heaven. Gene’s voice and response was unforgettable. In a humble sounding voice, he meekly replied: “If it’s good enough for Jesus, it’s good enough for me.” Now, I don’t think Gene think we will have said scars, but even if we did, so what. Our Lord has them. Of course he has them for theological reasons that we don’t need to get into. My point in all this is to talk about a choice Jesus made. If it’s “good enough” for Him to cal it a choice, then it’s “good enough” for me to call my choices, choices. If it’s “real enough” for Jesus, it’s “real enough” for me.
In John 6:70 Jesus tells us that he chose all 12 disciples, yet one, speaking of Judas, was the devil. In John 13:18 Jesus tells us that he chose Judas, along with the other 11, “so that the Scriptures may be fulfilled.” Now, this shows that it was pre-determined that Judas would betray Jesus. We are told in Hebrews 6 that it is “impossible” that God can lie. So, was it “possible” that Jesus could have, in a libertarian way, chosen rabbi Larry over Judas? If Jesus had libertarian free will, wouldn’t his “choice,” to be real” have to be between at least two live options, both of which were possible to actuate? If it was “possible” that Jesus could falsify Jehovah’s prophecy, God would have turned out a liar. But, this is “impossible,” therefore it seems “impossible” that Jesus could have chosen otherwise. Yet. Jesus calls it a “choice.” Hence if my choices aren’t real, neither was Jesus.’ If the libertarian doesn’t allow me to have a “real” choice, so what? If it is good enough for Jesus, it’s good enough for me.
As many can see, the debate is a detailed and rigorous one. If the contemporary Christian wants to engage in a contemporary defense of our basic beliefs about God, His sovereignty, and man's responsibility, s/he would do well to study some of the contemporary literature. We should not sit back, at least those interested, while the Arminian makes headway in the area of analytic philosophy, making many of us look sloppy. I feel it can only strengthen our extremely strong theological case if we learn to employ contemporary philosophic arguments for compatibilism (semi, for us) and moral responsibility in light of determinism. Furthermore, many Arminians simply will avoid Scripture in the debate, and it would serve well to be able to answer them on their own grounds. Lastly, the above is, as I said, a rough and ready, all too brief, discussion of contemporary compatibilism. I recognize that many can find fault with the above. My main goal was to briefly explain some of the language and ideas used in contemporary debate, and possibly spur future interest. Hopefully I at least achieved the latter.