The Infallibility constraint 173 We repeat the process one hundred times, and he answers likewise

each time. He is wrong in one case.³⁰

Series of additions S is given a hundred simple addition problems.

The problems involve three small numbers, between 10 and 100. S solves them using normal paper-and-pencil methods for adding, and each time checks the results by an independent arithmetic method. He is wrong in one case.

Let me add a useful third series:

Series of photographs We have a series of a hundred black and white photographs of groups of about twenty adult people, in the style of school class or worker team photographs. Their quality is variable:

some are much grainier than others. S is shown them and asked each time whether a certain good friend of his is in the photograph.

He is wrong in one case. (In that case, he mistakenly thinks that the friend is in the photograph.)

The first series is a variant on lottery cases.³¹ The common judgement is thatSdoes not know, in any case of the series, that the top card is white.

(Whether or not he indeed believes that it is.) The second and third series are ordinary cases of mathematical and perceptual knowledge. I would not say that in every case of these series where S gets the answer right, he knows. But it is hard to deny that he does so in someof the cases of the series, provided nothing unusual is going on. At the 10^th addition problem, say, Shas warmed up a bit, is still attentive enough, calculates rightly that 84+37+31 is 152 and has checked the result. In the 10^th photo, which is in a reasonably good condition, S immediately notices the familiar face of his friend in the first row, and further examination only increases his confidence. These are paradigmatic cases of what we ordinarily call knowledge.

The contrast between the lottery case and the two others creates a dilemma (Cohen,1988, 92;Greco, 2003, 112-3 ). If we require infallibility

30. The probability of the black card turning up at least once in such a series is≈0.63;

the probability of it turning at least twice up is≈0.26.

31. SeeHawthorne(2004, 1–2) for a presentation of lottery cases and references.

for knowledge, we have to deny knowledge in the ordinary series. But if we say that reliability is sufficient, we have to grant knowledge in the lot-tery series. We can sharpen the dilemma using the apparatus of chapter 3. Prima facie, all cases in one series form a natural range, and they do not appear to involve different methods. A method infallibility requirement requires avoidance of error in all cases in the range; a method reliability requirement only requires avoidance of error in most cases in the range.

The reliability requirement (in conjunction with belief and truth) cannot be sufficient, otherwiseSwould know in the lottery series. The infallibil-ity requirement cannot be necessary, otherwise Swould fail to know in the ordinary series. But there seems to be no modal requirement stronger than reliability and weaker than infallibility.

4.2.2 The structure in ordinary cases

What distinguishes the lottery series from the ordinary ones? An important point is that the ordinary series have astructurethat the lottery one lacks (Unger,1968, 162).³² Some of the photographs are more grainy than others, the faces are more recognisable in some than in others, and so on. Some addition problems are more difficult or confusing than others, and S was more attentive and less tired while doing some than others.

By contrast, the lottery series is uniform with respect to epistemically relevant properties. The black card was further from the top in some cases than in others,Swas less bored in some cases than in others, but that did not make him any closer to knowing that the top one was white. More generally: structured series contain natural ranges of cases in which errors are scarce and natural ranges of cases in which error is more frequent.

Call the firstfavourable areas,and the secondunfavourable areas. Of course we could single out an error-free set of cases in the lottery series. But that would be an arbitrary selection, not an epistemically natural range. Any non-arbitrary selection in the lottery series will include errors just as well

32. Unger does not say what he means by a “more structured” case (Unger,1968, 162), so I cannot tell if he had in mind the idea I develop here.

4.2. The Infallibility constraint 175 as any other; that is what makes the lottery random. By contrast, in the

ordinary series,favourable areasare distinguished. We can picture the idea by imagining the series of cases as disposed in a line. In the lottery cases, errors are distributed randomly along the lines, while in the structured cases, they tend to be in some segments of it. (See also the illustrations on pp.179–181.)

4.2.3 The Gettier problem for fallibilism

The introduction of structure opens the way for requirements that are both stronger than overall reliability and weaker than overall infallibility:

Favourable area requirement Sknows thatpin a case only ifS’s method is reliable over the series and the case is in a favourable enough area.

How favourable is favourable enough? Is it required that S’s method be infallible in the favourable area? Or is some high measure of reliability sufficient? The later proposal runs into two problems.

First, we can build lottery series with reliably favourable areas. Con-sider a variant of our lottery series in which the size of the deck varies, but always contains a unique black card. Sis aware of the changes of size and answers as before. Errors tend to be grouped in cases in which the deck is small; when the deck is big,Sis almost always right. If sufficient for knowledge (in conjunction with truth), the favourable requirement would entail that S knows in cases in which the deck is very big. But S does not know that the top card is white, even when the deck is very big. One way to press this judgement is to focus on the favourable area in which the deck is, say, 1000 cards. We may presume that there are a few cases in the area where S is wrong.³³ The situation in that area is the same as in the original series, namely, we have an unstructured area in which errors are randomly distributed.³⁴

33. I am assuming a suitably long series. In cases in which one plays only a few times, the relevant “series” of cases is spread over counterfactual possibilities.

34. “Randomly” is loosely used here. What is meant is that errors are not grouped in an epistemically natural sub-range of cases.

(What if the deck isextremelybig, say a billion cards? Intuitions are less clear, and I think that these sort of cases raise deep issues that I will treat separately: see sec.4.3.5. Here I can simply point out that few fallibilists would be happy to require such a high improbability of error and that the Gettier problem below stands.)

Second, if there are errors in the favourable area, one can build Gettier cases. Suppose there is an error case in a favourable area of the photo-graph series. In that case, Sjudges that some person at the front row is his friend; the photograph is quite clear but S is mistaken. Now alter the case so that one person in the back isS’s friend. S’s belief is true in the modified case; it is also based on a reliable method and made in the favourable area. But it is not knowledge. In the addition series, a Gettier case can similarly be created by introducing mutually cancelling errors.

(SeeSosa,1985, 240,Zagzebski,1994, 69,Merricks,1995, 845 for a similar argument.)

The Gettier problem arose for the simpler reliability requirement as well. Suppose that reliability across all the series is sufficient for knowl-edge (in conjunction with belief and truth). Then the Gettier cases above will be wrongly classified as knowledge; so the simple reliability require-ment is not sufficient. Now if we move into a favourable area, but still merely require reliability in the favourable area, the same problem reap-pears. No requirement of reliability (or combination thereof) over natural ranges is sufficient for knowledge.³⁵

35. AsSosa(1985, 239) puts it: “If a process is reliable but fallible, it can on occasion fail to operate properly. But that opens the possibility that a process yield a belief throughimproperoperation where by lucky accidentB[the belief] turns out to be true anyhow.” Our reasoning shows that if one requires that the process operatesproperly, where properly is again less than infallibility, the problem reappears.

Sosa does not propose a solution to the problem in that paper, though infallibility is suggested (“What might be missing? Perhaps some closer connection between the belief and its truth? Perhaps these cannot be so independent as when they come together only by lucky accident, the way they do in our example of fallible memory.” p. 240) He concludes that the problem is not so much a problem for reliabilism as a theory of justification than for the idea that knowledge is justified true belief. The implication is thus that justification is not infallible, but that some other requirement on knowledge may be so.