Thursday, January 31, 2008

Lotteries, closure, and conjunctive beliefs

There's a nice thread developing at FQI concerning lottery propositions and conjunctive beliefs. Suppose we have a 1,000,000 ticket lottery with a guaranteed winner. We ought to grant:

(A1) It would be irrational to believe all tickets in the lottery will be losers.

Now, consider:
(1) Henry is justified in believing that his ticket (t1) will lose.
(2) If Henry is justified in believing that t1 will lose, then Henry is justified in believing that t2 will lose, … , Henry is justified in believing that t1,000,000 will lose.

Therefore,

(3) Henry is justified in believing that t1 will lose, Henry is justified in believing that t2 will lose, … , Henry is justified in believing that t1,000,000 will lose.
(4) If Henry is justified in believing that t1 will lose, Henry is justified in believing that t2 will lose, … , Henry is justified in believing that t1,000,000 will lose, then Henry is justified in believing that t1, t2, t3, … , and t1,000,000 will lose (JC).

Therefore,

(5) Henry is justified in believing that t1, t2, t3, … , and t1,000,000 will lose.
(6) Henry is not justified in believing that t1, t2, t3, … , and t1,000,000 will lose (since he knows one ticket must win).

Therefore,

(7) Both Henry is justified in believing that t1, t2, t3, … , and t1,000,000 will lose, and Henry is not justified in believing that t1, t2, t3, … , and t1,000,000 will lose.
(8) On pain of contradiction, then, either (1), (2), (4), (6), or (7) is false.

I'm all for denying (1), but JD, Tim, and Trent seem to want to deny the closure principles that commit us to the contradiction.

Consider this exchange:
Adam: Zach will be at the party.
Ben: Who else will we know at the party?
Adam: Yolanda will be at the party.
Ben: Got it.

Ben: Chris, Adam believes that Zach and Yolanda will be at the party.
Adam: Wait! I don’t believe that.
Ben: But you said…
Adam: I said Zach will be at the party …
Ben: And you believe that, yes?
Adam: Yes. I was sincere. I also said Yolanda will be at the party.
Ben: And you were sincere in saying that as well?
Adam: Yes. But, I never said I believed Zach and Yolanda would be at the party. Listen carefully. Zach will be at the party and I’m not wrong to believe that. Yolanda will be at the party and I’m not wrong to believe that. However, it would be epistemically imprudent to believe that both Yolanda and Zach will be at the party.

An argument.
(1) Adam is out of his mind.
(2) You cannot be out of your mind _and_ epistemically rational.
(3) There is no relevant difference between Adam's firm refusal to accept the conjunction of two beliefs neither of which he will give up and the refusal of someone who believes of tickets t1 - tn that each will lose without believing the collection contains nothing but losers.

Therefore,

(4) Even if we do not assume that closure holds in full generality, insisting that the lottery case is the counterexample to closure is costly. There is no apparent explanation as to why this subject seems so completely off his nut except that now is not the time for him to refuse to accept conjunctions while refusing to suspend judgment as to either conjunct.

6 comments:

Alastair Norcross said...

Hi Clayton,
I'm confused. Why isn't the obvious difference between the 2 beliefs case and the 1,000,000 beliefs case the numbers? The way you put the party case, the guy is naturally assumed to have a high degree of confidence in each of his beliefs (I can't make assumptions about his degree of justification without more details). The conjunction of just two such beliefs should then be held with a fairly high degree of confidence, even if not quite as high as that for each individual belief. If you extended the example, though, to have him asserting of each of, say, a thousand friends, that they would be at the party, he wouldn't seem nearly so nuts to refuse to assert that the conjunction of all thousand would be at the party. Am I missing something crucial here? I only do this epistemology stuff as light relief from the hard work of ethics!
Alastair

Leo Iacono said...

Picking up on Alastair Norcross's comment, I agree that Adam would not be crazy to refuse to assert the massive conjunction. In fact it seems like that assertion would be positively improper. On the other hand, shouldn't Adam also balk at asserting the many conjuncts well before he reaches 1000? It really would be crazy for him to confidently assert, of 1000 people, that they will be at the party, but then to refuse to assert the conjunction. Even though it seems okay for Adam to confidently assert that one or two people will be at the party, it does not seem okay for him to keep doing that with respect to more and more people. Is that right? If so, what's the explanation?

mvr said...

Leo & Clayton,

Doesn't it depend on the evidence plus whatever makes him cautious about conjoining. Sticking with a ten person case it seems to me that I can reasonably believe that each one will come to the party without thinking that all of them will. So, like Alastair, I tend to think that the numbers matter. But I don't see that they matter in such a way that we ought to treat the things that occur to us first as special. If I have equal evidence that each one of my ten friends is coming I can reasonably believe that each will come. And I can do that even while thinking my evidence is not good enough to conclude that they all will come.

I know Clayton doesn't agree with me here. So I don't expect to convince him. (I've said nothing new.) But given your position Leo, I think you should admit there is something funny in allowing the order in which I consider propositions with equal evidence to matter.

Leo Iacono said...

Mark--
I was making claims about what it would be proper to assert -- how assertion is connected to reasonable belief is up for debate. I stick by what I said about proper assertion, though. Do you think it could be at the same time proper to assert, flat out, that Emily1 will be coming, and proper to assert, flat out, that Emily2 will be coming, ..., and proper to assert, flat out, that Emily10 will be coming to the party, but improper to assert flat out that Emily1 & Emily2 & ... & Emily10 will all be coming? That strikes me as impossible, though how this bears on reasonable belief is unclear to me. I do agree with you that it would be weird if only the first few propositions in line, so to speak, got to be justified even though all the propositions were epistemically on a par.

mvr said...

Leo, OK, I wasn't talking about assertion in the first instance, though since I think that if something is true there is generally some context in which it is also assertable. And I suppose I think that if the negation of something is assertable (as in "They won't all come.") then that something won't be assertable. So I should refuse to assert "They'll all come," in the above example..

Suppose I go through my list and stop. Then you say, "So you think they're all coming." And I say, "No, not every last one, though I don't know which of them will make a liar out of me." That sounds OK to me. As would your reporting, of me that I believe that Emily1 is coming, or even in the case where you know Emily1 is coming that I know Emily1 is coming. If I know something it seems like it can't be nuts for me to believe it. Neither would it be wrong for you to go on to report that I know that they are not all coming, though I believe of each one that she will come.

Perhaps my ear is not delicate enough.

Leo Iacono said...

Mark,
I may well be the one with the tin ear, but I don't share those intuitions. I guess it just seems to me that when you flat out assert something, you are in some sense committing yourself to its being true, and if you commit yourself to the truth of a bunch of propositions, then you thereby have committed yourself to the truth of their conjunction. Anyhow, maybe I am taking things too far afield by dwelling on assertion--if so, sorry about that.