Tuesday, December 3, 2013

Another round on the lottery paradox

I've had a few rounds with Thomas Kroedel over the lottery paradox. His latest response to my latest response is coming soon to Logos & Episteme.  

As he sees it, the lottery paradox (a version of it, at least) arises because these are plausible but inconsistent: 

(1-J) For each ticket, I’m justified in believing that it will lose. 
(2-J) If, for each ticket, I’m justified in believing that it will lose, then I’m justified in believing that all the tickets will lose. 
(3-J) I’m not justified in believing that all the tickets will lose. 

His solution is to reject (2-J), denying that epistemic permissions agglomerate. 

My objections are these: 

O1: Agglomeration isn't an issue if there aren't permissions to begin with. I'd reject (1-J).
O2: There's a good reason to think that we should reject (1-J), which is that we know that we cannot know the relevant propositions.  
O3: Once we get started, so to speak, adding beliefs to the belief set, there's nothing available to him that could explain why we should 'draw the breaks'.  Once we're clear on ex post and ex ante rationality, I don't think that you can both say that it is ex ante rational to believe each of the lottery propositions and then say that it is not ex post rational to believe all of the lottery propositions.  

Anyway, in my last round, I challenged Kroedel to explain why epistemic permissions do not agglomerate. I probably wasn't as clear with my challenge as I should have been, so I'll just note a worry about his latest response.  He offers as examples that show that permissions do not agglomerate cases in which we're permitted to believe conjuncts but not conjunctions.  Here's a general worry about the principle he assumes.

He wants a case with this structure: Jp, Jq, ~J(p&q).  Here's a worry about this. Suppose you were to argue for the conjunction, p&q, from two premises, p and q.  

* Since you believe (a) the premises and (b) believe the premises entail the conclusion, you should (in some sense) believe the argument is sound. (At the very least, you know the conclusion follows and are thought to be permitted to believe the premises, so I'd think you can permissibly believe that the argument is sound.)

* Since the argument is a counterexample (allegedly) to agglomeration of permission, you should not believe the conclusion is true.
* If you do what you should, does that mean you should be like this: I believe the argument is sound, but I don't believe the argument's conclusion?

To my mind, that's a crazy way to be.  I'm pretty sure that this is similar (if not identical) to Adler's remarks concerning the lottery and the conjunction rule from Belief's Own Ethics.  

Another problem that simply doesn't arise if we simply reject (1-J).  Since I don't think there's yet been an adequate response to O3, I still don't think that the worries I've raised can be put to rest.  

Of course, there's this persisting problem.  If I ask you whether the ticket won or lost, you might say, 'I don't know'. Kroedel and I agree that you'd speak the truth. If you then added 'Of course, it will lose', I think you're being irrational.  This is a point on which we disagree, but I haven't yet seen any plausible story about how that could be a rational state of mind to be in.


Andrew Bacon said...

I think you're right that there is no p and q such that:

You're justified in believing that [you're justified in believing p, justified in believing q but not justified in believing p&q].

There's definitely something incoherent about this. But that doesn't mean that there aren't any counterexamples to agglomeration: a counterexample would just be a p and a q such that you're justified in believing p, justified in believing q but not justified in believing p&q.

The argument doesn't even establish that you're not justified in believing that there are counterexamples: being justified in believing that there are counterexamples doesn't entail that there are things you are justified in believing to be counterexamples. (Compare: I can know there are unknown truths, even though it's impossible for me to know of any particular proposition that it's both true and unknown.)

David Duffy said...

Surely 1-J is an approximation useful in some contexts, but never in that of 2-J? So neither J nor T.

Rachel McKinnon said...

Well, as you probably know, I'm all for rejecting 1-J. I don't think we're justified in believing lottery propositions, and *that's* what explains why we don't know.