The following argument is valid:
- Normally, if an embodied person freely does A, then x could have done otherwise than she did. (Premise)
- It is a common occurrence that an embodied person freely does something. (Premise)
- If a general conditional holds normally, and specific cases of the antecedent are common, then it is nomically possible that the antecedent and consequent hold simultaneously. (Premise)
- Therefore, it is nomically possible that there be an embodied person x and an action A such that x does A freely and x could have not done A. (By (1)-(3))
- Metaphysically necessarily, if an embodied person x does A freely and x could have done otherwise, then determinism is false. (Premise)
- Therefore, it is nomically possible that determinism does not hold. (By (4) and (5))
- If determinism holds, then it holds of nomic necessity. (Premise)
- Therefore, determinism does not hold. (By (6) and (7))
Observe that the Principle of Alternate Possibilities in (1) is not subject to any Frankfurt-style counterexamples. I got this Principle based on an idea of David Alexander, but I don't think he endorses this version.
I think the tough question is whether (3) holds. But I think it at least holds as a probabilistic principle: if normally (if p, then q), and if cases of p are common, then probably a case of both p and q is nomically possible. In fact, a stronger probabilistic claim seems to hold: probably some case of both p and q actually holds. (When I talk about cases, I am assuming that the conditional is a quantified one: for all x, if P(x), then Q(x).) If so, then the conclusion would be that probably determinism does not hold. (Not earthshaking in light of the fact that there is some direct reason from physics to think it does not hold.) But the stronger non-probabilistic claim is also plausible. How could something be normal and yet nomically impossible?
[Edited to fix typo in argument and attribution of PAP.]