A Puzzle Relating to Pragmatic Encroachment - DISCOVERING TRUTHS and ANNOUNCING THEM
 
The only truth I have to announce is that I'm puzzled by something. Here is a familiar case.

Smith buys a ticket in a lottery with an enormous payoff. The odds are vanishingly small that Smith's ticket is a winner. Smith knows both the odds and that the payoff is enormous. If her ticket is a loser, it's a piece of trash. Smith realizes this, so she's considering recycling it. As a matter of fact, the ticket *is* a loser, and Smith believes that it is. Smith's only evidence that it's a loser, however, is her knowledge of the odds.

One of the following must be true, but which one? (In case it isn't obvious that one of these must be true, see "PROOF" below.)

(1) Smith doesn't know that the odds are vanishingly small that the ticket is a winner.

(2) She does know that the odds are vanishingly small that the ticket is a winner, yet she's not justified in proportioning her confidence to the odds.

(3) She's justified in proportioning her confidence to the odds (and thus being virtually certain) that the ticket won't win, yet she's not justified in believing that the ticket won't win.

(4) She's justified in believing that the ticket won't win, yet she doesn’t know that the ticket won't win.

(5) She does know that the ticket won't win, yet it's not acceptable for her to reason that, since the ticket won't win, she should recycle it.

(6) It's acceptable for her to reason that, since the ticket won't win, she should recycle it, yet it's not the case that she should recycle it.

(7) She should recycle it. 

As I noted above, one of (1) through (7) must be true. But they are all hard to stomach. Very quickly, (1) through (7) have (at least) the following problems.

The scenario stipulates that (1) isn't the case, so, without reason to think the scenario is incoherent, we can't accept (1). Given that Smith wants to win the lottery, given that she paid good money for the ticket, and given that it's very little trouble for her to keep the ticket, it seems clear that (7) is false. If Smith had overriding reasons for not recycling the ticket, then (6) would seem true. But we can stipulate that she doesn't, in which case (6) looks false. And the same considerations apply to (5). It looks true if additional factors prevent Smith from reasoning that way, but we can just stipulate that they don't. Of course, Hawthorne and others would accept (4). But (4) seems unstable. Ex hypothesi, the ticket won't win and Smith isn't Gettiered with respect to her belief that it won't win. Thus, our grounds for denying that Smith knows seem to equal our grounds for denying that Smith is justified. (And speaking for myself, insofar as it's intuitive that Smith doesn't know, it's also intuitive that she's not justified.) So, it looks like anyone who wants to deny that Smith knows is pushed toward (2) or (3). But (2) seems false. Why on earth shouldn't Smith proportion her confidence to the odds? After all, ex hypothesi, she knows the odds and the odds exhaust her evidence. This leaves us with (3), which is also hard to swallow. Perhaps justified belief equals justified certainty. But then, which of our beliefs *is* justified? If we accept (3) and maintain that justified belief equals justified certainty, it will be a trick to avoid skepticism. Perhaps justified belief equals justified confidence above some threshold *below* certainty. But then, how can (3) be true while we have a significant stock of justified beliefs? Again, given that we accept (3), it will be tricky to avoid skepticism. Perhaps justified belief cannot be identified with any level of justified confidence, then. In this case, what does belief amount to, how is it related to confidence, and what explains how Smith could be justified in being virtually certain that the ticket won't win, yet not justified in *believing* that the ticket won't win?  

I'm not sure what to say about these options, except that none of them looks particularly good. What do people think? If forced to pick among them, which should we pick?

Blake

PROOF: Here’s the proof I promised above.
 
Let 'p' through 'u' name the propositions in (1) through (7) as follows.

p = Smith knows that the odds are vanishingly small that the ticket is a winner.
q = Smith is justified in proportioning her confidence to the odds.
r = Smith is justified in believing that the ticket won't win.
s = Smith knows that the ticket won't win.
t = It's acceptable for Smith to reason that, since the ticket won't win, she should recycle it.
u = Smith should recycle the ticket.

Now consider the following presentation of (1) through (7).

(1) ~p
(2) p & ~q
(3) q & ~r
(4) r & ~s
(5) s & ~t
(6) t & ~u
(7) u

Either p is true or it's false. Suppose the latter. Then (1) is true. Suppose the former. Then either (2) is true or q is. Suppose q is. Then either (3) is true or r is. Suppose r is. Then either (4) is true or s is. So suppose s is true. Then either (5) is true or t is true. Suppose it’s t. Then either (6) is true or u is. But if u is true, then so is (7). So, one of (1) through (7) must be true.

Alex Jackson
2/13/2010 07:01:12 am

yo blake,

i think you're right 3 is the one to go. (i would guess that's what hawthorne would say too, rather than ditching 4.) and i agree that means that belief is not a matter of having some high degree of belief. and i agree that leaves us with hard questions.

here's an embryonic thought. suppose an agent has two methods they can employ in decision-making. the first method is to accept some propositions, and reason on the basis of them. the second method is to assign probabilities to the relevant propositions, and do an expected utility calculation. you wouldn't expect there to be a nice mapping between believing p (as part of method 1) and giving p a very high credence (as part of method 2). that's why the agent has two different methods.

alex

Reply
Hicks
2/14/2010 01:37:10 pm

Hey Blake,

Thanks for the clarity!

I waver between 3 and 5. Since you and Alex like 3, I'll say a few words in defense of 5:

Suppose, plausibly, that some form of fallibilism about knowledge holds. Then we should have some doubts concerning our knowledge even when we in fact know.

So long as we rationally have some doubts, we will be rational to take bets against known propositions--provided the payout are sufficiently (very) high: roughly, we will be betting that we don't know (even though, by assumption, we do).

The payoffs of lotteries are so great that we should bet against our knowledge--we might be wrong! So it is rationally impermissible for us to act on our knowledge of the outcome, even when we know we'll lose.

Hicks

Reply
Mary
2/16/2010 04:07:27 am

I'd also like to defend [5]. From a certain perspective, you should act in a way that maximizes expected utility, where how to determine what maximizes expected utility is something like what's described in decision theory. this is what Alex was calling method 2.

In that case, premises like "The ticket is a loser." are irrelevant to the decision unless this is taken to be short for "There is 0 chance that this ticket is a winner."

From this perspective, using what Alex called method 1 is at best a kind of heuristic. It saves time and mental effort, but it sometimes leads us astray. The lottery problem is one example of where it goes wrong.

Viewed this way [5] seems like the best option. But the first part of [5] is irrelevant. It doesn't matter whether Smith knows or only justifiably believes that the ticket won't win. Either way, what's true is that "It's not acceptable for her to reason that, since the ticket won't win, she should recycle it."

Of course, the question still arises as to whether you need to justifiedly believe or know (or something else) what the odds and payoffs are in order to use them in your expected utility calculation.

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