#### 5.1. THE MANY LOTTERY PROBLEMS

Lotteries raise many interesting epistemological issues, but not all of them are directly relevant to reliabilism. I will present three lottery problems that are not particularly relevant to reliabilism, and a fourth one that is. Dealing with that fourth problem will be the main burden of the rest of this paper.

First, there is the issue of how to understand the relation between belief as an all-or-nothing notion on the one hand and degrees of belief on the other. Many philosophers think that there are two related doxasticphenomena: on the one hand, we can characterize our doxastic life in terms of whether we have one of three mental attitudes with respect to any proposition p – we can accept the proposition, reject it (i.e., believe its negation), or suspend judgment with respect to it (which, as a positive mental attitude which can be more or less justified, should be clearly distinguished from adopting no attitude whatsoever towards a proposition). On the other hand, we can characterize our mental life in terms of a much more fine-grained range of attitudes that we can take with respect to any proposition *p*– we can believe it to a very low degree, or to a low-ish degree, or to a high, or very high degree, etc. There is an initially attractive principle that links the ternary notion with the degree notion: *S* accepts that *p* iff *S* believes that *p* to degree *n* (for a suitably high *n*). Similar principles could be offered for linking rejection and suspension of judgment to degrees of belief. Another initially attractive principle is that if *S* accepts that *p* and *S* accepts that *q*, then *S* accepts that *p and q*. Lotteries present a problem for the conjunction of these two principles. For suppose that there is a fair lottery with a very high number of tickets. Then, I will believe to a very high degree, of each ticket, that it will lose. Therefore, given the first principle, I accept, of each ticket, that it will lose. But, given the second principle, I accept that all tickets will lose. But I certainly don't accept anything like that! A similar argument can be run in terms of rational acceptance.^{9} It is clear that this problem raised by lotteries is not peculiar to reliabilism: every epistemological theory that wants to say something about the relation between the two ways of characterizing our doxastic life must face it.

Second, lotteries can be used to raise a particularly interesting closure problem. For I know that I won't have enough money to afford an African Safari for me and twenty of my friends next summer. I also know that if I won't have enough money, then the lottery ticket that I hold will lose. But I don't know that the lottery ticket that I hold will lose.^{10} Obviously, the closure problem is not peculiar to reliabilism either.

Third, there is the problem of explaining exactly *why* we don't know that a lottery ticket will lose, when we base that belief merely on the very low probability that it has of winning.^{11} It might seem as if reliabilism is a particularly salient target of this problem. For, according to reliabilism, what determines whether a belief is justified has something to do with its probability, and we can make the probability that the ticket will lose as high as we wish, and we will still think that it doesn't amount to knowledge. But what is, exactly, the problem? The main problem cannot be that reliabilism has the consequence that our lottery beliefs are justified, because that they *are* justified is a widely-held view.^{12} Some may think that the problem lies in trying to give an account of knowledge as reliably produced, un-gettierized true belief. Lottery cases are not standard Gettier cases in that they don't involve a bad luck/good luck combination,^{13} and so such an account would have the consequence that a subject who believes that she holds a losing ticket knows that she does. This result at least goes against current trends in epistemology, and so it might seem to be a problem. But it isn't. In standard lottery cases, all of this is assumed to be true: (i) that the subject is justified in believing that she holds a losing ticket; (ii) that the subject doesn't know that she is holding a losing ticket; and (iii) that the case is not a Gettier-case. That construal of lottery cases entails that *any* account of knowledge as justified, un-gettierized true belief is incorrect. So either some of the standard assumptions about lottery cases have to be given up or any account of knowledge as justified, un-gettierized true belief is wrong. In either case, reliabilism about justification is in the clear.

Finally, a problem raised by lottery considerations that is, *prima facie* at least, a serious problem for reliabilism has to do with *comparative* assessments of justification. Jonathan Adler has recently argued against reliabilism on the basis of this case:^{14}

A company that manufactures widgets knows that exactly one out of every thousand of their products suffers a singular defect as a by-product of (an ineliminable imperfection in) their excellent – and much better than average – manufacturing system. . . . Some managers would like to reduce the percentage of defects. The plan is to introduce a special detector, well designed to read ‘OK’ just in case the widget is not defective. . . .

Smith and Jones, who both know of the one in a thousand defects, are each given a detector. . . .

As each widget comes to his station, Smith momentarily glances away from the video game he is playing to stamp it ‘OK’, expressive of his corresponding (degree of) belief, while wholly ignoring the detector. As Smith knows, out of each batch of 1000, he is guaranteed to be correct 999 times by this method (and so better than by use of the detector, as explained next).

. . . Knowing of her manager's well founded confidence in the detector, Jones applies it to each widget carefully and skillfully, and assigns it ‘OK’ (mostly) or ‘Defect’ (rarely) according to her determination. Given the complexity of operating the detector and normal human limits, the probability of an error in any evaluation is .003, though Jones is a first-rate technician.^{15}

Adler claims that three things are true about his case. First, provided that she is not mistaken, Jones *knows* that the widgets that she stamps ‘OK’ are not defective, but Smith doesn't have the corresponding knowledge. Second, even though Smith can be justified in assigning a very high credence to the proposition that a given widget is not defective, he is not justified in ‘*all out* accepting it as true.’^{16} Third, Adler thinks that all of this is true despite the fact that Smith uses a method that is more reliable than Jones's.

So Adler holds the fashionable view that a subject in a standard lottery case doesn't know that she is holding a losing ticket together with the unfashionable view that the same subject is not justified in thinking that she holds a losing ticket.^{17} I will later argue (in section 5.3.) that reliabilists have resources available to hold that Smith indeed isn't justified, if they wish to do so. But the claim that I am interested in now is the *comparative* claim that Jones is more justified than Smith when they both believe of a certain widget that it is OK.^{18} There are two questions that we must ask here. First, is it really the case that Jones is more justified than Smith? Second, does reliabilism entail that Jones is not more justified than Smith? Regarding the first question, there are two ways of understanding Adler's case, and in one what way it is indeed true that Jones is more justified than Smith, whereas in the second it isn't. I’ll say more about those two ways of understanding the case in the next section, when I examine the consequences that a probabilistic understanding of reliability has. But let me first address, in the remainder of this section, the issue of what are the consequences that a truth-ratio understanding of reliability has for the case.

For a variety of reasons, the answer to that question isn't straightforward. Notice, first, that reliabilism as I have defined it characterizes an all-or-nothing notion of justification: if the process that created the belief is sufficiently reliable, then the belief is justified, otherwise it isn't. It is natural, however, to think that it is in the spirit of reliabilism to claim that the more reliable the process, the more justified the belief is. Let us then construe reliabilism as incorporating this natural understanding of a degree-theoretic notion of justification.^{19} Our question now is, then: who uses the more reliable process, Smith or Jones?

The answer to *that* question isn't straightforward either. In order to see why this is so, we must first determine what is the evidence that Jones and Smith use to arrive at their respective beliefs. Notice that, as Adler describes the case, Smith acquires his beliefs inferentially. Smith's evidence is constituted by his (justified) belief that any given widget belongs to a class of widgets of which exactly 1 in 1000 are defective. In Jones’ case, her evidence is constituted by her justified belief (or perhaps her visual experience) that the detector reads ‘OK.’^{20}

According to both of the definitions of reliability in terms of truth-ratios, Smith's procedure has a 0.999 degree of reliability (Smith is *guaranteed* to be right *exactly* 999 out of 1000 cases). The consequence that Smith is justified in believing that a specific widget is OK can be avoided by setting *r* to a number higher than 0.999 – but it is easy to see that, no matter how high *r* is set, we can always create a case where the procedure used by a counterpart of Smith is reliable to at least that degree.

What about Jones? What is the truth-ratio of the process of believing that a widget is OK based on a justified belief that the detector says that it OK? The answer is not as easy in this case as it was with Smith. All that Adler tells us about the detector is that the ‘probability of error’ in using it is 0.003. Adler clarifies that by probability of error he means both the probability that the procedure indicates that a widget is OK when it is defective as well as the probability that the procedure indicates that a widget is defective when it is OK. Notice that these probabilities don't *guarantee* anything about the actual truth-ratio of Jones's beliefs. It is compatible with this information that Jones correctly classifies *all* of the widgets, as well as that she correctly classifies *none* of them – although this latter possibility is, of course, much less probable.

Does **reliability as high counterfactual truth-ratio** deliver a result for Jones? We might be tempted to think that it does, because we might be tempted to think that to say that a certain procedure has a 0.003 probability of error is to say that applications of that procedure will issue in a false belief in exactly 3 out of 1000 possible worlds. However, we must remember that, according to **reliability as high counterfactual truth-ratio**, not all possible worlds where the procedure is applied should be counted towards establishing the reliability of the procedure. Even if there is a direct translation from probabilities to truth in possible worlds, those 3 in 1000 worlds where the procedure results in false beliefs might count as too weird (too improbable!) to be ‘close’ to the actual world.

So, even though it is true that defining reliability in terms of truth-ratios gives us the result that Smith is justified, it doesn't give us the *comparative* result that Smith is more justified than Jones – simply because it is not clear what the truth-ratio of Jones's procedure is. This is a serious problem for any understanding of reliability in terms of truth-ratios, which must be added to the conceptual problems for that understanding presented in section 3.

#### 5.2. LOTTERIES AND RELIABILITY AS HIGH CONDITIONAL PROBABILITY

What are the consequences of *reliability as high conditional probability* with respect to Adler's case? It might already be obvious that the result, in the case of Smith, is 0.999, but it is not so obvious what the number is for Jones. The source of this obscurity is an ambiguity in the interpretation of the case. But, however the case is interpreted, *reliability as high conditional probability* is in the clear.

Let us first ask how we can compute the conditional probabilities in question (that is, the conditional probability that a given widget is OK given Smith's evidence and the conditional probability that it is OK given Jones's evidence). In the usual Kolmogorov axiomatization of the probability calculus, conditional probability is defined in terms of unconditional probabilities, as follows:

provided that Pr(B) > 0 (and otherwise the conditional probability is undefined). Bayes's theorem, in turn, states a main consequence of such definition:

from where it is easy to see that the conditional probability of *A* given *B* is a function of the *unconditional* probabilities of *A* and *B* and the inverse conditional probability of *B* given *A* (what in the Bayesian jargon is called the ‘likelihood’ of *A* given *B*). Moreover, if we know the likelihood of the hypothesis on the evidence as well as the likelihood of the *negation* of the hypothesis given the evidence, we do not even need to know the unconditional probability of *B*, if we re-write the denominator in Bayes's theorem according to the ‘total probability theorem’:

In Smith's case, we have all the information we need to calculate the conditional probabilities. We know that the probability that a particular widget belongs to a class of widgets of which exactly 1 in 1000 is defective is the same whether that widget is OK or not, something very close to 1. So, where ‘W+’ abbreviates the proposition that widget n is OK and ‘S’ abbreviates the proposition that widget n belongs to a class of widgets of which exactly 1 in 1000 is defective:

We now need to know what is the probability that a widget is OK given that Jones's detector says that it is OK. There are here two ways of understanding Adler's case. According to the first one, Jones *is* more justified than Smith, but his procedure is also more reliable. According to the second one, Jones's procedure *isn't* more reliable that Smith's, but then Jones isn't more justified than Smith either. In either case, there is no problem for reliabilism.

The two ways of understanding Adler's case arise from an unclarity in Adler's presentation. What does he mean when he says that the ‘error rate’ of the detector is 0.003? The most natural way of understanding that claim is as giving us the probability that the detector says that widget is OK (defective) when it is actually defective (OK). That is, under this first understanding of the case, we have it that (letting ‘D+’ be the proposition that the detector says that widget n is OK, ‘D-’ the proposition that the detector says that widget n is defective, ‘W−’ the proposition that widget n is defective and, as before, ‘W+’ the proposition that widget n is OK):

Under this understanding of the case, it is true that we would naturally judge that Jones is more justified than Smith in his beliefs regarding widgets. But **reliability as high conditional probability** does have this consequence:

So, no matter what *r* is, we do get the comparative result that Jones's procedure is more reliable than Smith's, and so Jones does count as justified if Smith does.^{21} Whether either of them is justified depends, of course, on the choice of *r*, but it seems plausible to suppose that Jones at least will count as justified on any reasonable choice of *r*. Notice, too, that **reliability as high conditional probability** also gives the right result for a proposition that Adler doesn't consider: would Jones be justified in believing that a certain widget is *not* OK, based on the fact that the detector says so? Well, given that the error rate of the detector is 0.003 and that only 1 in 1000 widgets are not OK, on average, 3 out of 4 widgets that the detector indicates as being defective are actually OK. Given this, Jones shouldn't trust the detector when it says that a widget is defective – that is, if she does believe that a given widget is defective based on the detector's verdict, her belief is not justified.^{22} And Pr(W−|D−) is quite low (roughly, 0.25), which means that, as desired, Jones is not justified in believing of a certain widget that it is not OK.

Let us turn now to the second understanding of Adler's case.^{23} Under this interpretation, the 0.997 figure is not the be understood as giving us a likelihood, but rather as a *stipulation* of the conditional probability that a certain widget is OK given that the detector says that it is OK. This interpretation is very hard to square with what Adler actually *says*, but let us nevertheless consider it on its own merits. The suggestion, then, is that the description of the case stipulates that:

If this is the correct understanding of the case, then there is no denying that Smith's procedure is more reliable than Jones's – at least under the definition of reliability under consideration. But probabilities cannot be stipulated in an unconstrained manner. As revealed by Bayes's theorem, conditional probabilities constrain (and are constrained by) prior probabilities (the probability that a widget is OK) and likelihoods. In Adler's case, the prior probability is fixed at 0.999. So, stipulating that the conditional probability that a widget is OK given that the detector says that it is OK is 0.997 constrains the conditional probabilities that we can assign to the detector's saying that a widget is OK given that it is OK, as well as the conditional probability that the detector says that a widget is OK given that it is not OK. There is no unique number that we *have* to assign to those probabilities, but all the possible assignments are discouraging. For instance, the following is one such assignment:

Once aware that the detector has such a high error rate, few would still judge that Jones is more justified than Smith in thinking of a particular widget that it is OK. One way to bring out the fact that we wouldn't now judge that Jones is more justified than Smith is by considering what Jones's judgments will be with respect to particular batches of 1,000 widgets.^{24} For any given batch, 999 of 1,000 widgets will be OK. Of those, the detector will correctly classify as OK approximately 350 – that is, it will misclassify approximately 650. What's even more disturbing, out of 1,000 widgets that are defective, the detector will misclassify as OK 999. That is, trusting the detector is *worse* than tossing a coin and believing that a widget is OK if and only if the coin comes up heads! Reliabilism does indeed entail that, under this interpretation, Jones is worse justified than Smith, but that, it seems, is as it should be.

Stipulating that Pr(W+|D+) = 0.997 has the consequence that the likelihood must be very low because the prior probability is 0.999. Couldn't we change the example so that the detector still has a reliability of 0.997 but the prior is much lower, so that the likelihood is also high? We could, of course. For instance, if we stipulate that fully half of the widgets are defective, then the reliability of the detector can be 0.997 while Pr(D+|W+) = 0.997, whereas Pr(D+|W−) = 0.003. But now, of course, Smith will be wrong half of the time, and so will come out as being less justified than Jones according to reliabilism.

Why do we tend to judge, with Adler, that Jones is better justified than Smith? It cannot be just because Jones uses a detector – the detector might well be disastrous (as with the second interpretation of the case). The reason for our judgment, I submit, is that we interpret the 0.997 figure as giving us the likelihood that the detector will say that a widget is OK when it is, and thus we correctly assume that the conditional probability is higher for Jones than it is for Smith. If we are explicitly told that 0.997 *is* the conditional probability for Jones, then (as soon as we realize what that entails about the detector) we no longer think that Jones is better justified than Smith. That is, there is no interpretation of the case where it is true that Jones is better justified than Smith but **reliability as high conditional probability** doesn't have this conclusion.

Now, we could, if we want, compare the justification that Smith has in the original case with Jones's justification in the modified case, where half of the widgets are defective.^{25} Smith's procedure will still have a reliability of 0.999 whereas Jones's procedure will have a reliability of 0.997, and so, according to reliabilism, Smith is more justified than Jones. Again, this is as it should be. Of course, if Smith were to use his procedure at Jones's factory, then he *would* be less justified than Jones, and if Jones were to use his procedure at Smith's factory, then he *would* be more justified than Smith – and reliabilism does have that consequence. But this shouldn't make us give up the intuitive claim that Smith's procedure *in Smith's factory* results in beliefs that are more justified than the ones produced by Jones's procedure *in Jones's factory*.

What this last case does help to bring out, however, is the fact that it is not the case that Smith will count as *un*justified on any reasonable choice of *r –* as was the case with the definitions of reliability in terms of truth-ratios, it is possible to construct a lottery case with enough tickets to guarantee that Smith's procedure is reliable to any degree that we wish (in the sense of having a conditional probability at least that high). If one thinks that Smith may well be justified, and particularly if one thinks that as the numbers get higher the more justified Smith is, then the definition of reliability as high conditional probability is the correct one.

#### 5.3. GETTING SMITH TO COME OUT AS UNJUSTIFIED

But what if one follows Adler in thinking that Smith is not even justified in believing that a given widget is OK?^{26} In that case, is reliabilism shown to be false? No: There are plausible ways of defining reliability in terms of probability that do have the consequence that Smith is unjustified.

Defining reliability in terms of conditional probability is closely related to Carnap's notion of ‘confirmation as firmness.’^{27} There is another quantity that Carnap also defined, ‘confirmation as increase in firmness,’ and which can serve as the basis for a different definition of reliability:

**Reliability as increase in probability:** a type of the form *believing that p based on e* is reliable if and only if Pr(*p*|*e*) − Pr(*p*) > *r*.^{28}

Applying **reliability as increase in probability** to Adler's case gives us the following results:

This means that, independently of the choice of *r*, Smith is not justified in his belief – the result that we were looking for. Whether Jones is justified or not depends, again, on the choice of *r*. And, even though 0.000997 seems too low to make Jones come out as justified, it should be noticed that *r* need not take the same value for all cases, and it can be a function, for instance, of Pr(*p*|*e*) – it wouldn't be far-fetched, for example, to suggest that *r* must be inversely proportional to Pr(*p*|*e*) (which is itself directly proportional to Pr(*p*)). For instance, if we set *r* = 1 − Pr(*p*|*e*), *r* will be 0.001 for Smith and 0.000003 for Jones. So, in so far as Adler's case goes, **reliability as increase in probability** gives the right results.^{29}

One feature of Adler's case is that it is constructed so that it is clear what is the unconditional probability that a given widget is OK (or defective). But, as a rule, this will not be so. The problem is that, in many cases, such unconditional probabilities will not be available. Take a paradigmatic case of belief formation: I take a glance outside my study window and, as a result of having certain characteristic experiences, come to believe that there are flowers in the yard. What is the unconditional probability that there are flowers in the yard? And the unconditional probability that I will have those experiences? The problem is not that these questions are *hard* to answer; it is, rather, that they have no answer.^{30} We can, of course, make sense of different questions: what is the probability that there are flowers in the yard, given that it is May? What is the probability that I will have certain experiences, given that I look in a certain direction? But, again, if the usual definition of conditional probability is correct, then the answer to *these* questions depends on the answer to the previous ones.^{31}**Reliability as increase in probability** may be correct as a definition of reliability when it makes sense to think that there are unconditional probabilities to be had, as is the case in Adler's example. But cases of that sort will be the exception, not the rule.

We are not forced to accept the usual definition of conditional probability in terms of unconditional probabilities. There are other axiomatizations available that take the notion of conditional probability as *primitive*. When it does make sense to think of the unconditional probabilities, then Bayes’ theorem can be seen as a *constraint* on conditional probabilities, but when it doesn't make sense we can still meaningfully talk of conditional probabilities.^{32} Nevertheless, despite the fact that talking of conditional probabilities doesn't commit us to unconditional probabilities, **reliability as increase in probability***itself* traffics in unconditional probabilities. That is reason enough to look elsewhere for a definition of reliability to use when unconditional probabilities are not available.^{33}

**Reliability as increase in probability** is one of the many measures of *confirmational relevance* that have been proposed in the literature on Bayesianism.^{34} The interesting question for us now, then, is whether there is a measure that doesn't itself involve reference to unconditional probabilities. Fortunately, the answer is ‘Yes.’ In particular, what is called the ‘likelihood ratio’ measure provides us with the basis for a definition of reliability that doesn't rely on unconditional probabilities:

**Reliability as high likelihood ratio:** a type of the form *believing that p based on e* is reliable if and only if

for some suitable *r*. When the proposition in question just doesn't have an unconditional probability, the right definition of reliability to use is not **reliability as increase in probability** but **reliability as high likelihood ratio**.

Now, what does **reliability as high likelihood ratio** say with respect to Adler's case? For the proposition that a certain widget is OK, it gives the results that we are looking for in this section:

No matter how small we set *r*, Smith will not count as justified under this definition. Intuitively, Smith's procedure has a likelihood ratio of 1 because it is *insensitive* to whether any particular widget is OK or not. Jones, by contrast, will count as justified under any plausible value for *r*. The high likelihood ratio of Jones's procedure reflects the fact that *it is* highly sensitive to whether a particular widget is OK or not.^{35}