October 04, 2007

Lewis Conference (Day 2)

The talks today were on the semantics for conditionals. John Cantwell proposed a branching-time framework that aimed to unify our understanding of indicative and subjunctive conditionals. The variation in truth-value of corresponding indicative and subjunctive "Oswald sentences" is, on John's view, to be explained without positing a plurality of conditionals. The job can be done by tense and our understanding of open futures.

Hannes Leitgeb offered a probabilistic semantics for subjunctive conditionals. His very precise proposal (which I won't go into here) is a version of the thought that subjunctives are true just in case the consequent is sufficiently likely (in some objective sense) given the antecedent. By default Hannes rejects the strong and weak centering assumptions---respectively,

(A & B) --> (A []--> B), and

(A []--> B) --> (A --> B)

What this means is that, unlike the standard semantics, we get the desirable outcome that the truth of A and B is not sufficient to imply a counterfactual dependence between A and B, and that the truth of A and ~B is not sufficient to undermine a counterfactual dependence between A and B. The actual world can be one of the exceptional worlds where what does occur is not highly likely to occur (and where what is highly likely to occur does not occur).

Hannes replaces the centering assumptions with weaker centering-like assumptions---viz.,
(T []--> (A & B)) --> (A []--> B), and

(A []--> B) --> (T []--> (A --> B))

I believe T is meant to be a tautology, and so, the following rough paraphrase can be given: the truth of A & B does entail A []--> B, when A & B is sufficiently likely on its own, and the truth of A & ~B entails the negation of A []--> B, when A & ~B is sufficiently likely on its own. Perhaps we can put it in something like Lewisian terms. The stronger of the two says that no world is as close to the actual world as are the very likely worlds; and the weaker thesis is that no world is closer to the actual world than are the very likely worlds.



Photos:

1. Statue

2. John Cantwell

3. Hannes Leitgeb

4. Niels Bohr Mansion

4 comments:

Daniel Nolan said...

If you have cases where ~A, B, but A boxarrow B, then you'll have modus ponens failure for the counterfactual conditional. (And with that semantics you will also get modus tollens failures.)

I guess I'm not seeing why ensuring that there can be counterexamples to modus ponens is that desirable in a theory of conditionals. Especially since these will be rather straighforward syntactically: they need not rely on the sorts of embedding needed for cases like Van McGee's, for example.

Joe said...

Thanks, Daniel. I didn't want to suggest that the conjunction is desirable---i.e., that it is desirable to give up both strong and weak centering. I do think it is desirable to part with strong centering, but that doesn't force a rejection of counterfactual modus ponens. I agree that things get weird when we give up weak centering.

Why do I think it is desirable to give up strong centering? Well, there are plenty of cases where A&B, but where it might easily have been that A&~B. However, by strong centering, A&B entails A[]-->B. And if it might easily have been that A&~B, then if A were the case ~B might have been the case. Formally, A<>-->~B. But 'would' and 'might' are dual operators. So A<>-->~B entails ~(A[]-->B). Contradiction.

Robbie said...

Hey Joe,

I was wondering about this argument... It looks to me (forgive me if I'm misreading) that the arg is meant to go:

Setup:
(1) A&B. (Premise)
(2) Might(A&~B). (Premise)
(3) A&Might(~B). (from 1,2)
(4) A[]-->B. (from 1, SC)

(5), which'll be the crucial move, is,
"if A, then it might be that ~B" (from 3 by SC?)
(6) ~(A[]-->B) (from 5 by duality)
(7) contradiction (4,6).

The trouble is over 5. To invoke duality, you need to read this as Lewis's diamond-arrow.

(5*) A<>-->~B

But read like this, I don't see why one should accept it as true, or following from the setup. In particular, you don't get (5*) from (3) from SC, because what that gives is:

(5**) A[]-->Might(~B)

Which is a rival analysis of "might"-counterfactuals, not Lewis's diamond-arrow one. (Bennett thinks they're equivalent given a suitable reading of "might", but I don't believe him!). Of course, you can get to (5**) by appeal to SC from (3), but I see no reason to think that (5) entails (6) (it's not definitionally dual to the would-counterfactual, like Lewis's one).

Of course, you might think that the relevant might-counterfactuals in question are just intuitively true in lots of A&B situations, so whatever reading (5) has, it *should* be true. If that's so, I'm inclined to treat it as one more piece of evidence against analyzing English might-counterfactuals in terms of A<>-->B.

So I'm thinking what you need is some argument that might-conditionals *are* true in the circs you mention and that whatever they are, they satisfy duality. I'm comfortable denying that conjunction.

Robbie

joe said...

Thanks Robbie. You helpfully pinpoint where some work needs to be done to hone the argument from Dual and SC. Another alternative is to skip use of Dual altogether. When it might easily have been that A&~B, it intuitively follows that the counterfactual A-->B is false. But if it turns out that in fact A, B, then by SC the counterfactual A-->B is true.