Finch and Warfield's version of the Consequence Argument for incompatibilism uses:
- beta 2: If Np and p entails q, then Nq
- P&L entails p. (Premise)
- N(P&L). (Premise)
- Therefore, Np. (1-3)
Here's a cool thing I arrived at in class when teaching about the argument. Suppose we try to come up with a definition of the N operator. Here's a plausible version:
- Np if and only if p and there does not exist an action A, agent x and time t such that (a) x can do A at t; and (b) (x does A at t)→~p.
Anyway, here's an interesting thing. Beta 2 is a theorem if we grant these axioms:
- If q entails r, and p→q, and p is logically possible, then p→r.
- If x can do A at t then it is logically possible that x does A at t.
The proof of beta 2 from (5) and (6) is easy. Suppose that Np is true and p entails q. For a reductio, suppose that ~Nq. If ~Nq, then either ~q or there are A, x and t such that (a) x can do A at t; and (b) (x does A at t)→~q. Since Np is true, p is true, and hence q is true as p entails q. So the ~q option is out. So there are A, x and t such that x can do A at t, and were x to do A at t, it would be the case that ~q. But ~q entails ~p, since p entails q, so by (5) and (6) it follows that were x to do A at t, it would eb the case that ~p. And so ~Np, which contradicts the assumption that Np and completes the proof.
So it looks like the consequence argument is victorious. The one controversial premise, beta 2, is a theorem given very plausible axioms.
Unfortunately, there is a problem. With the proposed definition of N, premise (2) says that there is no action anybody can do such that were they to do it, it would be the case that ~(P&L). While this is extremely plausible, David Lewis famously denies this on his essay whether one can break the laws. I think he's wrong to deny it, but the argument in this formulation directly begs the question against him.
Note that in the definition of the N operator, we might also replace the → with a might-conditional: were x to do A at t, it might be the case that ~p. (This gives the M operator in the Finch and Warfield terminology; see also Huemer's argument.) The analogue of (5) for might-conditionals is about as plausible. So once again we get as a theorem an appropriate beta-type principle.