## Friday, December 3, 2010

### Beta 2: a theorem

Finch and Warfield's version of the Consequence Argument for incompatibilism uses:

1. beta 2: If Np and p entails q, then Nq
Here, "Np" is: p and nobody (no human?) can ever do anything about p. The argument for incompatibilism is easy. Let P be the distant past and L the laws. Suppose p is a proposition that is determined by the distant past and the laws. Then:
1. P&L entails p. (Premise)
2. N(P&L). (Premise)
3. Therefore, Np. (1-3)
In other words, if something is determined by the distant past and the laws, nobody can ever do anything about it. In particular, if all actions are determined by the distant past and the laws, no one can do anything about any actions. And this is supposed to imply that there is no freedom.

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.
Here, "→" is the subjunctive conditional. So, Np holds if and only if p and nobody could do anything such that if she did it, we would have not-p.

Anyway, here's an interesting thing. Beta 2 is a theorem if we grant these axioms:

1. If q entails r, and pq, and p is logically possible, then pr.
2. If x can do A at t then it is logically possible that x does A at t.
Axiom (6) is really plausible. Axiom (5) is a consequence of David Lewis's account of counterfactuals. Analogues of it are going to hold on accounts that tie counterfactuals to conditional probabilities.

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.

#### 18 comments:

Alexander R Pruss said...

A paper based on this proof has just been accepted by the Canadian Journal of Philosophy.

Alexander R Pruss said...

A preprint is here (maybe only temporarily).

Mark said...

Very interesting argument.

Consider the following:

1. If God exists, then God has no control over human free choices.
2. Necessarily, if a human makes a free choice C, then God reacts in manner M.
3. If beta-2 is true, then if God exists, then God's reaction M is out of God's control. (from 2 and 3).

An example can probably be filled in for (2), but I can't think of any non-controversial ones.

Ex: necessarily, if all humans freely chose to reject Christ, then God does not create humans. (assumes Molinism it seems).

Alexander R Pruss said...

Interesting, but as you say it's hard to fill in 2.

You can probably do something with necessity modulo the past that will work just as well. God might promise that he'll do M if you do C. Given the promise, it's necessary. Probably you can generalize beta-2 to cases like that.

But that just seems right: God doesn't freely choose to keep his promises over breaking them.

Mark said...

I have a question about your definition of Np.

It seems to imply:

Np-->p.

So ~p-->~Np. And, if ~Np, then you have control over p.

So, if ~p, then you have control over p.

Worse, necessarily, Np-->p.

So, if necessarily ~p (i.e. p is impossible), then you necessarily have control over p.

Alexander R Pruss said...

Np doesn't say you have control over p. It says that p is true and someone has control over p.

Mark said...

Wait, I thought Np (generally) says that no one HAS control over p.

1. Necessarily, 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.

2. So, necessarily, if Np, then 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. (from 1, equivalence).

3. Necessarily, 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. (from 1, equivalence), then p. (simplification).

4. So, necessarily, if Np, then p. (2 and 3, hypothetical syllogism).

5. So, necessarily, if ~p, then ~p. (4, modus tollens).

6. So, necessarily ~p (i.e. p is impossible) implies necessarily ~Np. (axiom K)

But Np= someone has control over p, and conversely ~Np= someone has control over p.

So, by 6 (and the definition of ~Np), if p is impossible, then someone has control over p.

Maybe you can say this:

Np if and only if, "if p then 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".

Mark said...

My bad.

(5) is supposed to say:

So, necessarily, if ~p, then ~Np. (4, modus tollens)

Also, (minor change) I meant to say:

So, by 6 (and the definition of ~Np), if p is impossible, then necessarily, someone has control over p.

Alexander R Pruss said...

Yeah, I left out a negation: Np iff p is true AND no one has control over it.

The definition I explicitly give in the post is "p and nobody (no human?) can ever do anything about p". This is more or less van Inwagen's definition.

Np entails p, so if p is impossible, Np is impossible as well.

But ~Np does NOT say that someone has control over p. It says that either p is false OR someone has control over p.

Mark said...

Oh yes, you're right.

I can't see why that wasn't obvious to me in the first place.

Thank you.

I have a slightly unrelated question:

if a person is strongly reasons-responsive such that he always chooses A were reasons R available, would you regard him as free?

Alexander R Pruss said...

You mean, even when there are better reasons on the other side?

Mark said...

Strong reason-responsiveness (means, roughly, to me): when person P would necessarily choose A over ~A if P's reasons for A (necessarily) trump reasons for ~A.

It seems that you reject that a strong reasons-responsive agent has free will.

It's kind of like the case of God and his essentially good nature.

It seems intuitive to me that God does have free will with respect to keeping promises, for example.

Maybe the intuition comes from the idea that if the will to do A originates from the self and is directed by reasons, then that is sufficient to say that one has control with respect to doing A.

Mark said...
This comment has been removed by the author.
Mark said...

I have one more question, if you don't mind.

Don't the Frankfurt counterexamples show that your definition of control does not entail the lack of free will/ moral responsibility?

In other words, don't they show that:

~{[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]--> ~F(x,p)}.

F= free to do/choose...
Let p be the choice or action.

The counterfactual intervener (doesn't have to be an agent) ensures that there is nothing that agents can do to make it the case that they choose otherwise at a close world, yet these agents are still free persons, according to Frankfurtian cases at least.

Mark said...

Correction (for the last comment): the consequent is supposed to be ~(Ex)F(x,p).

(For clarification, the action could be merely be an effort of the will)

Alexander R Pruss said...

I don't think Frankfurt examples work. I think Widerker got this right. The examples require a prior sign of how one will choose that is not in fact available if libertarianism is true.

Mark said...

Do you think the consequence argument can be made without a transfer principle (and be at least minimally convincing)? Fischer has tried to defend such a version of the CA (in "Free Will and the Modal Principle").

Alexander R Pruss said...

Well, in the case of this paper, I don't assume any transfer principle for choices, powers or responsibilities. All I assume is one axiom for counterfactuals.