You are assuming that we have free will and asking someone to refute this hypothesis.
No, you are misrepresenting my position. I am assuming that we have determinism, and asking whether free will could exist at the same time - whether they are
compatible. (Hint: the answer is yes, they are compatible.) If they are compatible, then
every argument that goes "we can't have free will, because determinism" is
unsound.
This isn't the same as proving, or trying to prove, free will. There can be - there are - unsound arguments against untrue propositions. So it would require a very different sort of argument to establish, even as a probability, that we have free will. What sort of argument? See below.
If no contradiction exists [between determinism and free will], the compatibalism is correct.
Exactly. That's just a restatement of the definitions of compatibilism and incompatibilism.
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To establish as best we can with available evidence whether free will does or doesn't exist, first we need to understand what "free will" is supposed to mean. To do that, we should look for real life usage of the term, where "real life" means outside of philosophical or religious debates. That way, we avoid biased opinions of those who have a dog in this fight, and we also get a sense of why philosophers and religious leaders have thought the topic to be important. And a great source of real life usage of the term is the legal system.
For example,
this document on rape and sexual harassment mentions free will in its definition of consent:
c. Consent
Positive cooperation in the act or attitude pursuant to an exercise of free will. The person must act freely and voluntarily and have knowledge of the nature of the act or transaction involved. A current or previous dating relationship shall not be sufficient to constitute consent.
And this is further clarified by contrast with conditions that contradict consent, such as:
3. Where a person is prevented from resisting by any intoxicating or anesthetic substance, or any controlled substance, administered by or with the privity (knowledge) of the accused.
Another rich source of implicit and explicit definition of free will in the law is the insanity defense. For example
Julie Grachek writes:
The insanity defense addresses the policy issues inherent in the question of criminal culpability:14 the connection between responsibility and blameworthiness. The view is that in a criminal justice system based on free will,15 the acts of mentally ill persons lacking complete free will cannot fairly be judged in the same way as the acts of sane, free-willed persons. Therefore, the insanity defense reflects the “criminal justice system’s view that the conduct of individuals who lack some degree of mental capacity should not be judged according to general volitional and cognitive principles.”16 Society’s recognition of a moral difference between the acts of a sane person and a mentally ill person17 results in the insanity defense serving dual roles in the criminal justice system: 1) as a way to distinguish between offenders who are able to conform their conduct to the law as a result of punishment from those offenders who are not able to conform their conduct to the law despite punishment,18 and 2) as a method of ensuring offenders posing a threat to society are restrained.19
In this passage, free will is associated with "some degree of mental capacity", "volitional and cognitive principles", "ability to conform conduct to the law", and "ability to [take into account] punishment". I would have said, take into account any and all foreseeable consequences of action, but the general idea is similar.
OK, so now that we have a proto-definition of free will, let's look at the known universe for any beings that have: mental capacity? Check. Volition and cognition? Check. Ability to conform to the law? Check. Taking consequences, such as punishment, into account? Check!
And what kinds of beings have these qualities, in copious amounts? Are they roughly the same ones that most people regard as having free will, such as typical adult human beings without severe mental illness? By golly, gee willikers, yes they are!