Traditionally, Libertarians have guarded the definition of free will very closely. Libertarians don't allow compatiblists to remove essential aspects from the concept of free will, like say, reducing ability to hypothetical ability. But while we maintain constant vigilance against compatibilist reductions to free will, it seems to me we are less aware of needless and harmful additions to the concept of free will; additions that may be just as damaging as compatibilist reductions.
Both Arminians and Open Theists frequently use the term free will, but do we mean the same thing by it? Compatibilist reductions to free will are subtle; saying we are able to do otherwise, meaning we are able to do otherwise if we want to. Likewise, Open Theists may have subtle and dangerous additions to free will.
Libertarians generally agree that man must be able to choose otherwise than he actually chooses in order to be free. This is a personal ability; a power the agent has. Such a twofold possibility is normally thought of in causal terms. On this, Open Theists and Arminians agree.
However, Open Theists are not yet satisfied. In addition to the ability to cause this or that, Open Theists insist that we must be able to convert propositions from possibly true to actually true. Not only must Bob be able to choose chocolate or vanilla, Bob must also be able to make it true that "Bob chose chocolate".
Open Theists and Arminians both are willing to say, Bob can make true that the proposition "Bob chose chocolate". But disagreement remains in how they interpret the idea of Bob making propositions true. Open Theists would not be satisfied with the idea that Bob can be the basis of truth for the proposition "Bob chose chocolate". Nor would they be satisfied with the idea that Bob has the power to choose chocolate or vanilla, such that, were he to choose chocolate, "Bob chose chocolate" would be true, and were he to choose vanilla, "Bob chose vanilla" would be true. Arminians can cash out the idea that Bob makes propositions true with such ideas, but for Open Theists, that's not enough.
Nor do Open Theists hold to the deviant notions that Bob causes truth directly or that Bob logically implies truth. They understand that Bob causes actions and events and truth corresponds to those actions and events. The relation between Bob and the events is causal, but the relationship between the events and truths about those events is logical; and Open Theists aren't mixing those things.
So what exactly are Open Theists adding to libertarian free will? Open Theists hold the idea that propositions about future free will acts, in an absolute sense, cannot be true. (I say in an absolute sense, because some Open Theists reinterpret statements about the future in a relative, probabilistic sense, meaning given current factors, Bob will choose chocolate is more likely than not, but not 100% certain). If the statement, "Bob will eat chocolate" is true, then Bob is not free with respect to eating chocolate. Propositions about events become true the moment the events happen and not before. Bob himself has the power to change the proposition "Bob chose chocolate" from possibly true to actually true. This is how Open Theists cash out the idea of Bob making statements true and this is the power that Open Theists add to definition of libertarian free will.
Of course, this addition has profound implications for the foreknowledge argument, since the foreknowledge argument obviously rules out the Open Theists addition to libertarian free will, but LFW itself seems to remain untouched by the foreknowledge argument. Likewise this addition has broad implications into the relationship between time and truth. Indeed, it's hard to see if Open Theists first give man an extra ability and therefore develop their views of the relationship between time and truth or if they first take a stand on the relationship between time and truth and based on that develop their views of man's abilities. Either way these views seem symbiotic. I believe there are strong theological and logical reasons to reject the Open Theists concept of the relation between time, and truth and therefore the need for their additions to libertarian free will. God willing, I will post on the foreknowledge argument and the relationship between time and truth latter.
Why do I believe that from a historic perspective, Open Theists are adding to the notion of free will rather than Arminians subtracting from it? For one thing, the church at large has generally accepted the idea that God knows the future. For another, when theologians dealt with logical determinism and logical possibilities, many of them affirmed freedom in a divided sense, but not a compound sense and proposed it as a solution to the foreknowledge argument. As an example of the divided sense, while Bob can choose chocolate (divided sense), it cannot be that God knows Bob will not choose chocolate and Bob will choose chocolate (compound sense). Similarly, Bob can lift 100 lbs or 150 lbs, but from this we cannot conclude Bob can lift 250 lbs, since that would be a composition fallacy. Not that Open Theists commit division and composition fallacies, but rather they are opposed to the whole business of the truth about what we are going to do being logically deductible via a rational inference.
Another way of some theologians said the same thing is that they affirmed the necessity of the consequent, but not the necessity of the thing consequent, which is basically a distinction between causal possibility and logical deduction. One can find a long standing, deep rooted tradition of explaining our abilities in a way that conflicts with Open Theism.
Arminians and Open Theists agree we cannot do the logically impossible; we cannot make contradictions true. Our ability to act cannot contradict any other truth and hypothetical scenarios in which we use that ability must also be free from contradiction. But for Open Theists, statements about what will do may not be true, nor may truth about what we will be deductible from other truths. For them, this would mean something other than us determined this truth, since they are not open to the idea that our future action is the basis of truth for the present future tense proposition.
I have been picking on Open Theists a bit here and that's not entirely fair. I have often seen Calvinists trying to form reducto ad absurdem arguments against Arminianism and in the process they impute the Open Theists additions to LFW to Arminians. But detecting and avoiding either additions or subtractions from LFW should help avoid problems.