2:16 AM
@HWalters Let me start with intuitions. That there are CFW mechanisms in our brain is a platitude, that some instinctive actions are rationalized ex post facto was recognized since before Freud, and Libet-type experiments provides plentiful evidence of confabulated intentions. Libertarians like Kane specifically restrict manifestations of FW to "torn decisions", etc. We do not need "compatibilist intuitions" to affirm CFW mechanisms, indeed we know of them acting pace the opposite intuitions.
The nontrivial claim of compatibilism is that CFW accounts for all of it. It is conceivable that AI research will reveal that LFW is physically possible and neuroscience will demonstrate that we lack it. I am not aware of anyone who takes such a possibility seriously, and it would render current debates moot. You can look at Mackie's Ethics, or Dennett's Giving Libertarians What They Say They Want, or Mele's Free Will and Luck.
Much of the debate is driven by moralizing about what we "want" or "need" FW to be for ethics, etc. But such considerations, and therefore most "intuitions" on both sides, are irrelevant to the ontological question, which I take the OP to ask. Where the FW ontology is concerned (social psychology and ethics are another story), cognitive psychology and neuroscience data, combined with semiotic and AI modeling are far more credible than culturally conditioned feels, even statistically averaged.
So I will simply deny that "it's through our intuitions that we understand what free will is and isn't", or at least that we should do so. Beyond that the chief objection to folk libertarianism is that taken at face value it is incoherent either by itself or when combined with physics. Compatibilism is a cheap way out, it amounts to saying that what LFW models strive to capture needs no capturing (the situation with consciousness is similar). And it may be right.
But solutions tend to be preferred to dissolutions if they can be had. I am not saying that the moment a viable LFW model is proposed compatibilism will go out the window, but the power of intuitions is fleeting. Paraphrasing Planck, those with the old ones die out and the next generation grows with new ones. Personally, I believe the current debate is marred by unfamiliarity of most participants on both sides with non-classical physics.
Not in the sense of specifics, but their general lines of reasoning about causality, chance, identity, etc., clearly follow Kantian, if not Aristotelian, stereotypes. And we already know that satisfactory integration of a subject into natural science along those lines is hopeless. Even if QM plays no role physically it gives a new vocabulary for the subject/object interaction, which is little utilized. So compatibilism and libertarianism are both likely to be more wrong than right.
@labreuer Intuition can be wrong, but this wrongness is not established "logically", logic is simply a wrong standard for empirical science. Empirical claims can be confirmed or infirmed, adopted or abandoned, but as Quine said "any statement can be held true come what may, if we make drastic enough adjustments elsewhere in the system". And for adopting, epistemic conditions (simplicity, fruitfulness, unification, availability of alternatives) play as much of a role as "facts" and "logic".