So, there's a connection between the Calvinist iteration of Total Depravity and this compatibilist free will idea. The Calvinist understands Total Depravity more as Total Inability; that a person is completely unable to choose to pursue God and, coupled with that, God must intervene to change the nature of those He elects for salvation so that they will inevitably believe and if He does not change a person's nature, they will never believe because they are unable to, doing so would be contrary to their nature. By now you should start to see the relationship to compatibilist free will: You are only able to make choices that are consistent with your nature… any choice that's inconsistent with your nature is a choice you are not able to make. Calvinists believe that man's nature is inclined against God. And the compatibilist version of free will states that no one can make choices that are contrary to their nature. This would explain why it is that God has to somehow supernaturally change a person's nature to enable them to believe, because without that they never will.
However, that all assumes that compatibilist free will is true. And I think we've shown that compatibilist free will is not true. But what would Total Depravity look like if libertarian free will were true, and man IS actually capable of choosing that which is contrary to his nature? Hmmm… that's a real game-changer, isn't it?
Now Calvinists will frequently take the opportunity of an attack on Total Depravity to play the semi-pelagianism card. Semi-pelagianism denied that man's nature wasn't damaged enough by the fall to render him completely unable to pursue God. Well, it's interesting to notice that if compatibilist free will turns out to be false and libertarian free will turns out to be true, then we can agree that man has a totally corrupt and rebellious nature, but he is nonetheless capable of choosing that which is contrary to his nature: Belief in God.
This thinking can be seen to have implications, then, for what the Calvinist expects from his post-salvation life as well. The Calvinist understanding is that the regenerated man has a "new nature" (and I would agree) but given compatibilist free will, it should be impossible for the Calvinist to choose that which is contrary to his new nature. That is, regenerate man should no longer be drawn toward sin… he is a changed man. However, this is fraught with difficulty because every Calvinist knows that sin persists to some degree in the life of the believer in spite of his new nature. From this flows a prodigious amount of double-speak, since the Calvinist is now torn between this idea that they have a new nature, and yet they plainly see that sin remains. That this is true actually demonstrates that the compatibilist concept of free will is false.
But again, libertarian free will can be seen as a solution here. Only if one is committed to the compatibilist notion of free will can these two realities be seen to conflict. Because given libertarian free will, it is not at all unthinkable that the regenerated person is capable of choosing that which is against this new nature. And isn't that what Paul struggles with in Romans 7?
This is interesting enough, seems to me… but when you combine this with some considerations related to general revelation, it becomes easier to see what God is up to and how this all works together.
Two passages in particular can be seen to reveal God's strategy, or at least part of it. Psalm 19 and Romans 1:20. These passages make it clear that the creation around us isn't merely there to accommodate and provide sustenance for man. Rather, these passages state that there is another purpose designed into creation, and that is to testify to the presence and existence of God. Notice that this is not the Bible's role… the Bible has another role. But the role of creation, clearly, is to alert man to God's presence. And this can be seen then as an appeal to man's libertarian free will, to give us reason to choose--quite against our nature--to pursue God in precisely the same way that my wife can coax me into eating a few slices of zucchini on the grounds that my diet needs to improve.
The conclusion this brings me to is that the Calvinist's understanding of Total Depravity, and in fact much of their theology, appears to be driven more by a compatibilist view of free will than it is driven by what the Bible actually says.