Defending the Lottery Argument (Part 1)

Over the weekend I finally had the honour of meeting Aidan McGlynn, of The Boundaries of Language fame, who was one of the delegates at the Arché Vagueness Conference here at St Andrews. Aaron Bogart, from over at Struggling to Philosophize, was also at St Andrews over the weekend and promises to visit again later this summer. Looking forward to sharing a pint and a chin-wag with you again, Aaron.

Now, down to business. In honour of Aidan’s very thoughtful comments, I’ve decided to dedicate this (and the upcoming) post to responding to his two objections to my lottery argument. Aidan's first objection is that my argument presupposes some type of closure principle. While he does not actually spell out any particular closure principle, given that my argument has to do with justification, he presumably has something akin to the following justification closure principle in mind:

(JCP) If S may justifiably believe p and p entails q, and S competently deduces q from p and accepts q as a result of this deduction, then S may justifiably believe q.

According to Aidan, the inference from (A3) and (A4) to the claim that what S may justifiably believe about this lottery has the form (a*) seems to rely on some such closure principle. However, Aidan asks, ‘what mandates that [my] opponent accept such a closure principle? If anything, it looks somewhat suspect once one allows that justification does not require probability 1.’

First, I should point out that strictly speaking, the inference from (A3) and (A4) to the claim that what S may justifiably believe about the lottery takes the form of (a*) does not rest on (JCP), or any other type of closure principle. Rather, it rests on the following conjunction rule:

(CR) If S may justifiably believe p at time t and justifiably believe q at t, then S may justifiably believe p and q at t.

However, a closure principle does seem to make an appearance elsewhere in the proof. According to the set-up of the argument, S is aware that one ticket must win and only one ticket can win. Given (JCP), it follows that S may justifiably believe that it is not the case that t1 will lose, and t2 will lose, and so on, to the millionth ticket. Given this fact, a few brief words in defence of my argument's reliance on (JCP) may still be in order.

I begin by noting that I do not share Aidan’s intuition that I have no right to expect my opponent to accept (JCP). (JCP), and closure more generally, is accepted by the vast majority of philosophers. Of course, some thinkers (most notably Nozick and Dretske) have challenged certain formulations of closure. But theirs is the minority position. (This of course does not mean that closure is true, but it seems sufficient to establish its status as the default position.) Moreover, it is worth pointing out that the version of closure implicated in the lottery argument is quite modest. If someone may justifiably believe that p, and competently deduces q from p, it seems quite reasonable to think that she would have reasons adequate to justify her belief in q. In fact, I'm not even sure Nozick and Dretske would be opposed to (JCP), since it merely implicates justified belief, rather than knowledge. (I'm not trying to be coy here, I'm honestly not sure what Nozick and Dretske would say about (JCP). Does anyone care to fill me in on what, if anything, Nozick and Dretske have to say about justification closure?)

Interestingly, (JCP) is one of the essential premises in Gettier’s original argument against JTB (see: 'Is Justified True belief Knowledge'), and rejecting (JCP) would therefore constitute a rather straightforward way of resisting Gettier's conclusion. However, I think the fact that relatively few philosophers have taken this route (I only know of one) is testimony to just how widely accepted (JCP) is and the unwillingness of philosophers in general to give it up. Again, this does not establish that (JCP) is true. However, it vouches eloquently for its status as the default position in discussions of this kind. Finally, and perhaps most importantly (depending on your philosophical commitments), (JCP) does seem to have the endorsement of common sense.

That being said, I do not believe that the lottery argument necessarily presupposes (JCP), since S has independent reason for justifiably believing that t1 might not lose, t2 might not lose and so on. Even if closure where false, S's knowledge of how a fair lottery works, would be enough to justify her belief that any given ticket might not lose.

I believe what is doing the real heavy-lifting in the lottery argument is (CR). I take (CR) to have no less intuitive support than (JCP). Thus, substantive argumentation is necessary if one is to urge the rejection of (CR). Again, I believe the burden of proof rests on my opponent in this regard.

However, a brief word of caution and clarification may be order with regards to (CR). Many erroneously conflate the conjunction rule regarding epistemic justification (which is highly plausible) with the conjunction rule regarding probabilities (which seems clearly false). This mistake is particularly common in discussions of lottery-type cases which explicitly make reference to probabilities. To see that the conjunction rule regarding probabilities is probably false one simply has to recall that (on a standard probability analysis) the probability of a conjunction is less than that of the individual conjuncts. Thus, adding enough conjuncts can render a conjunction of individually probable items improbable.

However, the same is not true of a conjunction rule regarding epistemically justified beliefs. In contradistinction from probability, justification is generally preserved, and may even be strengthen, when we add new justified beliefs to our set of beliefs. Sharon Ryan puts the point much more succinctly than I can:

Imagine that a person S is justified in beliving that following list of individual claims: I am in an airport, I see a plane taking off from the runaway, I hear a loud noise, The noise I hear is probably a plane, The noise I hear is probably not a giant snoring. Imagine S forms a conjunction of all of the individual claims listed above. It seems reasonable to think that the justification for the conjoined beliefs is no weaker than is the justification for all the individual conjuncts. If anything, justification is strengthened by conjoining this set of individual beliefs. It is not true that the more conjuncts you add, the less justified the set becomes. (Ryan, 'Epistemic Virtues of Consistency', p. 124)

If Ryan is right, and I think she is, then although a conjunction rule regarding probability may be false, the conjunction rule regarding epistemic justification is true. Moreover, it is the latter, rather than former, that is implicated in the inference from (A3) and (A4) to the claim that what S may justifiably believe about the lottery takes the form of (a*).

Finally, it should be noted that rejecting the conjunction rule regarding justification has a rather counter-intuitive consequence—namely, that a subject may justifiably believe a set of statements she recognises to be inconsistent. (This does not, however, follow from rejecting the corresponding rule regarding probability which is altogether silent on the question of epistemic justification.) While a subject who believes that 'each ticket will lose' and also believes that 'not all of the tickets will lose' is not believing a contradiction, it is still not possible for all the members of her set of beliefs to be true. Since a lottery subject may be aware of this fact, then rejecting (CR) means that a subject may justifiably believe a set of statements which she knows cannot all be true. Personally, that strikes me as odd.