(Comments in blue by BV.)
1) Frankfurt-style examples are intended to be counterexamples to PAP.
PAP: A person S is morally responsible for intentionally doing X at t only if S can intentionally refrain from doing X at t.
BV: The following formulation better captures what Frankfurt actually says in his 1969 J. Phil. article, namely, "a person is morally responsible for what he has done only if he could have done otherwise."
PAP*: A person S is morally responsible for intentionally doing X at t only if (i) S intentionally does X at time t, and (ii) S could have intentionally refrained from doing X at t.
Assumption: I assume that intentionally refraining from doing X is identical to intentionally doing some Y, where Y is not identical to X.
BV: Do you need this assumption? The assumption appears false. If I refrain from doing one thing, it doesn't follow that I do some other thing: I could do nothing at all. If I refrain from smoking a cigarette during the next ten minutes, it does not follow that I do something else during that period. I take it that not-doing-X is not an action. It is an action-omission. If 'X' and 'Y' range over action-types, then not-doing-X is not identical to doing-some-Y.
2) A Frankfurt-style example is going to be a genuine counterexample to PAP just in case it entails the antecedent of PAP as well as the negation of its consequent: i.e.,
(I) S is morally responsible for doing X at t;
(II) It is not the case that S can intentionally refrain from doing X at t.
BV: Note that (II') corresponding to PAP* is
(II') Either it is not the case that S intentionally does X at t, or it is not the case that S could have intentionally refrained from doing X at t.
3) The following are two claims I shall prove:
Claim 1: Any Frankfurt-style example that is interpreted to entail (I) and (II) is inconsistent: i.e., it also entails the negation of either (I) or (II).
Claim 2: Any Frankfurt-style example that is interpreted as entailing (II*) instead of (II) is consistent but is not a counterexample to PAP.
(II*) It is not the case that S can behave in a manner other than X at t.
But, (II*) is not the negation of the consequent of PAP. Hence, (I) and (II*) do not refute PAP. Therefore, standard Frankfurt-style examples are either inconsistent or they are not genuine counterexamples to PAP.
4) Suppose a Frankfurt-style example (choose your favorite example) entails (I) and (II).
4.1) Then such an example includes a backup mechanism that is capable of directly causing S to do X at t in the event S intentionally refrains from doing X at t. But the very existence of such a backup mechanism entails
(III) S can intentionally refrain from doing X at t.
Because if (III) were false and S could not intentionally refrain from doing X at t, then there would be no need for a backup mechanism.
4.2) (II) and (III) are contradictories.
4.3) Therefore, the assumption stated in (4) must be false.
4.4) This proves Claim 1 above.
BV: I agree that (II) and (III) are contradictories. But (II') and (III) are not contradictories. So even if you succeed in refuting your PAP, you haven't refuted Frankfurt's PAP*. If I haven't blundered, it seems that the debate now shifts to what exactly the Principle of Alternate Possibilities is.
5) Suppose that a Frankfurt-style example entails (I) and (II*).
5.1) As before, such an example entails (III) as well. But, now, notice that as long as we maintain a sharp distinction between behavior and action, (II*) and (III) are perfectly consistent.
5.3) Therefore, the supposition stated in (5) does not lead to a contradiction. Frankfurt-style examples that entail (I), (II*), and (III) are perfectly consistent.
5.4) But notice that none of these propositions; i.e., (I), (II*), and (III) contradict the consequent of PAP. Therefore, consistent Frankfurt-style examples are not counterexamples to PAP. This proves Claim 2.
6) Since this holds for any arbitrary Frankfurt-style example, we can state the following:
(*) Every Frankfurt-style example is either inconsistent or it is not a counterexamples to PAP.
7) The only potentially vulnerable move that I can see in this argument is the claim that the existence of a backup mechanism entails (III): S can intentionally refrain from doing X at t.
7.1) But, how can the proponents of Frankfurt-style examples deny such an entailment? The very point of Frankfurt-style examples is that the existence of such a backup mechanism (however it is described) is feasible and that its sole purpose is to insure that in the event S intentionally refrains from doing X at t, then the backup mechanism induces S to do X at t. Thus, the rationale of such a backup mechanism presupposes that S can intentionally refrain from doing X at t; but this just is (III). Hence, any Frankfurt-style example entails (III).
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