Say that George chooses to do A knowledgeably provided that the beliefs on the basis of which George chooses to do A are in fact knowledge. In a paradigmatic case of deliberation (this sentence can be taken to be stipulative of what I mean by "paradigmatic case of deliberation"), the beliefs on the basis of which George chooses include counterfactual claims of the form "Were I to do A, F would happen" and "It is not the case that were I not to do A, F would happen."[note 1]
I now claim that: If George chooses to do A knowledgeably and in a paradigmatic case of deliberation, then it can happen that he failed to choose to do A.
Here is the argument:
- It is false that were George not to do A, F would happen. (Premise: by definition of "knowledgeably" and since only truths can be known)
- F actually happened. (Premise: since George does A and knows that were he to do A, F would happen)
- Whatever happens, can happen. (Premise)
- If C cannot happen but D can happen, then were C to happen, D would happen. (Premise)
- Suppose it cannot happen that George does not choose to do A. (Premise for a reductio)
- F can happen. (By (2) and (3))
- Were George not to choose to do A, F would result. (By (4) and (6))
- Thus (7) is true and false. (By (1) and (7))
- Thus, (4) is false, and so it can happen that George does not do A.
One term that has not been defined is "can happen". On any plausible reading of "can happen", all the premises will hold in a case of knowledgeable and paradigmatic deliberation, with the possible exception of (4). Thus, for any plausible reading of "can happen" that makes (4) true, we get a PAP.
In particular, on any account of counterfactuals that makes p's entailing q entail that were p to hold, q would hold as well, (4) will be verified where "can happen" expresses logical possibility. This gives us a PAP with logical possibility, though only in the case of knowledgeable and paradigmatic deliberation. Still, that's something. After all it entails that if the laws of nature are necessary and determinism holds, then knowledgeable cases of paradigmatic deliberation are impossible.
I don't know what other senses of "can happen" make (4) true.
Arguments like this provide a general template for generating relatively weak versions of PAP. Are any of the versions of PAP that the argument provides sufficiently strong to yield some kind of incompatibilist doctrine? Here is the best I can do. Say that D can happen provided that D is compatible with the laws of nature and the initial arrangement of matter in the universe. Then (4) restricted to cases where C describes a wholly non-initial arrangement of matter is not completely implausible. Given materialism (which of course I deny, but many compatibilists accept), George's not doing A will be a wholly non-initial arrangement of matter, and the argument implies that George's not doing A is compatible with the laws of nature and the initial arrangement of matter in the universe, assuming George is acting knowledgeably and paradigmatically deliberatively. So we do get an incompatibilistic conclusion, but under heavy assumptions.
What is kind of neat about the above considerations is that they do not involve freedom directly, but only deliberation.