Monday, 4 June 2007

Lottery Argument Against Defeasible Evidence

This post is an updated version of one I published over at the Web of Belief.

I wish to argue that it is a conceptual requirement of justification that it be factive. On this view, it is a conceptual requirement vis-à-vis some type or token reason {R}, that {R} may only justify a subject’s belief that p if {R} guarantees the truth of p. When {R} meets this stipulation, I will describe {R} as a factive reason for believing that p. I contrast having a factive reason for p with having evidence for p, in which evidence is essentially defeasible. Typically, when we describe some evidence {E} as defeasible, we mean that {E} may be evidence for p despite the fact that {E} ∪ {E*} is not evidence for p. In such a case, we would say that {E*} defeats {E}, or that {E*} is a defeater for {E}. In the discussion that follows, I will be using the expression ‘defeasible’ more broadly to refer to any evidence {E} for p, in which {E} fails to guarantee the truth of p. I take defeasible evidence to include a wide cross-section of non-truth-entailing evidence or reasoning, including inferences to the best explanation, abduction, analogical reasoning and probabilistic reasoning. What unites all these various theses is that they offer a criterion of justification that falls short of truth. On my view, it cannot be part of our concept of justification that {E} may justify a subject’s belief that p if {E} is construed as defeasible evidence for p.

For the sake of simplicity, I will be construing the notion of defeasible evidence primarily along probabilistic lines. (This is only for heuristic purposes since I do not believe our quotidian concept of evidence, even when defeasibly construed, is essentially probabilistic in nature.) Along these lines, some evidence, {E}, may be described as defeasible vis-a-vis some proposition p just in case the probability of p on {E} is less that one. I will say that p is likely when p has a greater probability than not-p or when p exhibits a probability of {> 0.5 • < class="fullpost">† will take the form of a reductio beginning with the assumption, ‘S may justifiably believe that her ticket, t1, will lose’, and concluding with the negation of the aforementioned truism. Assuming that the first premise is the least plausible of all the premises in my argument, then my argument should establish that my first premise ought to be rejected. My reductio runs as follows:
(A1) S may justifiably believe that her ticket, t1, will lose.

(A2) If S may justifiably believe that t1 will lose, then she may also justifiably believe that t2 will lose, she may justifiably believe that t3 will lose ... she may justifiably believe that ticket tn will lose.

(A3) S may justifiably believe that tickets t1, t2 ... tn will lose. [from (1) and (2)].

(A4) S may justifiably believe that either t1 will not lose or t2 will not lose ... or tn will not lose.

(A5) Propositions of the following form comprise an inconsistent set: (a) p1, p2 ... pn, either not-p1 or not-p2 ... or not-pn.

(A6) S recognises that propositions of the following form comprise an inconsistent set: (a*) t1 will lose ... tn will lose, either t1 will not lose ... or tn will not lose.

(A7) S may justifiably believe a set of inconsistent propositions that she recognises to be inconsistent. [from (3), (4), (5), and (6)].
Expressed in probabilistic terms, I take the lottery argument to show that a subject is not justified in believing that p based on evidence that makes p probable with a probability less than one. In my next post, I will respond to the objection that my lottery argument does not generalise across all cases of beliefs implicating defeasible evidence.

† See Dana Nelkin's paper “The Lottery Paradox, Knowledge and Rationality” for a discussion of the lottery paradox regarding knowledge and justifiably held belief.


Aidan said...

I tried to post a comment on the original post, but apparently it got lost or something.

Anyway, I'll try again, much more briefly this time. How exactly does (7) follow from (3), (4), (5) and (6)? I see no valid argument in the offing.


I'm not sure I see the difficulty you're hinting at. Can you explain why you think (7) does not follow from (3),(4),(5) and (6)?

Aidan said...

I'd nearly finished a reply, when my computer wiped it. Urg.

This is going to have to be much quicker than I'd originally intended it. Firstly, it looks like you need some closure principle for 'may justifiably be believed' in a couple of places. For example, you'll need something like that to infer from (3) and (4) that what S may justifiably believe about this lottery has the form of (a*). But what mandates that your opponent accept such a closure principle? If anything, it looks somewhat suspect once one allows that justification doesn't require probability 1.

More importantly, let me grant the closure step (or whatever you want to employ to get the result from (3) and (4) that what S may justifiably believe about this lottery has the form of (a*). Now let the subject S not recognize that what she may justifiably believe about this lottery has the form of (a*). That's consistent with S's recognizing that if she believed something which had the form (a*) she'd believing a set of inconsistent propositions.

The upshot is that (3) and (4) might be true, and your opponent might grant you whatever you need to conclude from that that what S may justifiably believe about this lottery is (by (5)) inconsistent. But even in the presence of (6) that's not enough for (7); it's not enough that S recognize that something of the form (a*) would be inconsistent - she's also got to realize that what she could justifiably believe about this lottery has the form (a*).

But I'm not seeing the motivation for that. The number of tickets n might be very large (and the larger n is, the more plausible (1) and (2) are). And we're discussing the structure of the beliefs S may justifiably have about this lottery, not the beliefs she in fact has. Why should we accept that S is able to recognize that the huge set of beliefs she may justifiably have has the form of (a*)?

(Sorry, the set-back earlier has meant I don't now have time to proof-read this comment).


You raise some of the very points I was hoping to address in my follow up post, so I'll save a detailed reply to your comments for then. However, since it may be some time before I actually get to post a follow-up, here's a quick preview:

I see my lottery argument as presupposing something like 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.

I find (CR) intuitively plausible. Given the plausibility of (CR) if we were presented with two competing theories and (ceteris paribus) one denied (CR) and one did not, I think we should choose the one that does not deny (CR). I eventually intend to articulate just such a thesis. Moreover, denying (CR) puts one in the unenviable position of having to claim that S may be justified so long as she does not draw logical consequences from her beliefs. (I believe this final point has some bearing on your second objection).

More to come soon.

gregates said...


I find that there is something strange in the way you set up your argument. You equate having indefeasible justification for P with believing P on the basis of evidence such that the probability of P given the evidence is 1.

This seems incorrect. Suppose I know the following:

(1) The universe contains infinitely many planets.
(2) The chances that life has come to be spontaneously on any given planet is very small but non-zero.

On the grounds of this evidence, I can justifiably believe (even by your lights) that the probability that life has come to be spontaneously on some planet in the universe is 1. I can therefore come to justifiably believe that life has come to be spontaneously on some planet in the universe.

However, this justification is not indefeasible. It could be--there is no contradiction in assuming--that life has never come to be spontaneously anywhere in the universe (perhaps it has always existed wherever it does, or in some places it was created by an intelligent designer). I might even acquire evidence that this is so, without that evidence undermining my knowledge of (1) and (2).


I agree with you that there is something 'strange' in the way my argument is presently set up, or at least, that my position needs further clarification and defence. This is something I hope to do in future posts. However, it is not immediately clear to me how your criticism bears on my argument. The lottery example presupposes a finite reference class. In fact, the lottery case is only intelligible if we assume that there are a finite number of tickets. However, your objection implicates an infinite reference class.

gregates said...

I don't suppose my point does bear directly on your lottery argument, i.e., for all I've said, your argument is valid. I meant only to suggest that it's a further step from the conclusion of the argument to the claim that justified belief must be indefeasible, since the conclusion of the argument seems only to be that justified belief requires that the proposition believed have probability 1 relative to the evidence. My example was meant to show that the two notions come apart.

Herbert Huber said...

Despite some comments I think your argument (A1) ... (A7) is perfectly sound.But your conclusion from it: A justification for a subject’s belief has to guarantee the truth of p, is too narrow. I think your conclusion isn't even right for knowledge, let alone belief.
It dismisses almost all propositions as valid for belief.

I agree with Kyburg who asks: What's wrong with (A7)? The Lottery Paradox doesn't show that knowledge or belief is nearly impossible, it shows in my opinion, that the closure for conjunction is not applicable for belief propositions. I.e. (A7) is a set of inconsistent propositions, but without conjunction of p1, p2, ... pn you wont get a contradiction. We all have a lot of inconsistent beliefs with or without knowing it. One has only to be careful to conjunct propositions with a degree of justification < 1.

Henry Kyburg put it this way: One person's modus ponens is another person's modus tollens.I think the Lottery Paradox shows to be careful with the conjunction of beliefs.

Larry Hamelin said...

The problem is in A3. You're abruptly shifting from probabilistic math to boolean math, without justification.

As you note, "justifiably believes X" means that P(X) > 0.95. Therefore, A1 means S believes (correctly) that the probability of ticket 1 losing is > 0.95.

However, A3 assumes that S believes that the probability that ticket 1 will lose is 100%, which is false. It also assumes that the probability of individual tickets winning are independent, but they are, of course, dependent: if it's known that ticket 1 is definitely a loser, then we know the probability that ticket 2 will win is higher.

The claim that believing some probability is arbitrarily high is not equivalent to believing that the probability is 1 seems trivial.