Fallibilism

Philosophy Compass (2012) 7/9: 585-596 F A L L IB IL IS M Baron Reed Northwestern University [email protected] Many of the central figures in the history of epistemology have held that knowledge must be grounded in foundations that are infallible. Although these foundations have been characterized in a variety of ways, in each case it has been agreed that they are infallible in that they preclude error on the part of the person who has them.1 The doctrine of fallibilism stands in opposition to this picture. Roughly stated, the basic idea is that the subject can know something even though it could have been false. This is not the same as saying that the subject can know something that is false—it is very widely accepted by philosophers that, if a belief counts as knowledge, it is true. Rather, the claim is that a belief held with a particular epistemic grounding can be knowledge even though the subject could have held that belief with the same grounding in circumstances where the belief is false (and, of course, in those circumstances the belief would not count as knowledge).2 To put the claim a different way, the epistemic grounding that is sufficient for knowledge does not preclude the possibility of error. 1. THE MOTIVATION FOR FALLIBILISM While infallibilist views have been prominent historically, it is now fallibilism that enjoys widespread popularity. There have been two motivations leading to this change. First, one of the most common responses to skeptical arguments has been to treat them as depending on a stringent conception of knowledge.3 Because our cognitive faculties are imperfect, the thought goes, there is no way we could meet the demands of such a conception of knowledge. But our faculties are still very good; surely they allow us to achieve a more modest sort of cognitive success. Fallibilism, then, takes that modest success to be knowledge. Second, on the assumption that we do have the variety and amount of knowledge that we think we have, infallibilism does not seem to model it very well.4 For example, one of the things that I currently take 2 myself to know is the fact that the jaguar is the biggest cat in the Americas. But, if I were asked how that knowledge is grounded in such a way that it is infallible (say, by relation to the given element in experience), I would be utterly at a loss.5 Not only do I not remember the way in which I acquired the belief, it also seems to me that it could be wrong. I don’t think that it is wrong, but I also wouldn’t be terribly surprised if it were. Although there are still some defenders of infallibilism, it has largely been supplanted by fallibilism as the dominant framework in contemporary epistemology.6 As Stewart Cohen has said, “the acceptance of fallibilism in epistemology is virtually universal.”7 But there are many, widely divergent ways in which this framework has been developed—ranging from externalist views like Alvin Goldman’s reliabilism to internalist views like Earl Conee’s and Richard Feldman’s evidentialism.8 For that reason, fallibilism serves both as the doctrine that unifies contemporary epistemologists and as a primary source of the most serious problems they still face. 2. A MORE PRECISE ACCOUNT OF FALLIBILISM Fallibilism has most commonly been formulated in two ways.9 According to the first, a person S knows that p in a fallibilist way just in case S knows that p on the basis of some justification j and yet S's belief that p on the basis of j could have been false (or mistaken or in error).10 For example, I know that the Cubs beat the Dodgers the last time they played; I know it because my brother told me what happened, and he is usually reliable about this sort of thing. But, if he had misread the box score in the newspaper, I still would have believed him. In that case, my belief would have been held with the same justification, but the belief would have been false. Because this case is possible, the knowledge I actually have (when my brother did not misread the box score) is fallible.11 According to the second common formulation of fallibilism, S knows that p in a fallibilist way just in case S knows that p on the basis of some justification j and yet j does not entail (or guarantee) that p.12 In the above example, it is my brother's testimony that enables me to know that the Cubs beat the Dodgers. But my brother's saying that the Cubs beat the Dodgers does not entail that this is so. His testimony is logically consistent with the Cubs having beaten the Dodgers, but it is 3 also consistent with various other possibilities: the Dodgers may have beaten the Cubs and yet my brother misread the box score or misspoke or uncharacteristically told a lie, etc. These two formulations of fallibilism are equivalent, given the standard interpretation of entailment. To say that j entails p is to say that it is impossible for j to be true and p to be false. So, to deny that j entails p, as the second formulation does, is to say that it is possible for j to be true and yet p false. And this is just what the first formulation says: S could have justification j even in cases where it is false that p. Both of these formulations, however, fall prey to the same problem: neither is able to allow for fallibilistic knowledge of necessary truths.13 Let us suppose that it is necessarily true that p. In that case, it is impossible for S to be in a situation where her justification for this belief is true and yet it is false that p. This will be so for any justification, no matter how poorly it might otherwise seem to justify belief that p. Similarly, where it is necessarily true that p, every justification will entail that p. But this will be so simply because everything entails a necessary truth. There are two lessons to be learned from this problem. First, entailment is not itself an epistemic relation.14 Although it would valuable for a subject to know (or grasp or be acquainted with the fact) that her evidence entailed a proposition, it is not the entailment relation itself that would allow her to know the entailed proposition. Given that the entailment relation can hold even in cases where there is no epistemic link between a subject’s grounds for belief and the belief itself, this is a lesson that should be accepted by fallibilists and infallibilists alike. Second, fallibilism requires a broader formulation: S knows that p in a fallible way just in case S knows that p on the basis of some justification j and yet S's belief that p on the basis of j could have failed to be knowledge. There are two ways in which this might be so. First, the subject’s belief could have been false. This, of course, has been the point of focus for those who give the first of the common formulations above. But, second, the belief could have been true but only by accident. Suppose, for example that Julia has walked by the same clock tower every day for years and has always found it to be accurate. Today, she sees that it says the time is two o’clock. She comes to believe this, and her true belief is both justified and an instance of knowledge. 4 Nevertheless, it could have been the case that the clock had stopped exactly twelve hours earlier, in which case it would no longer be generally accurate but still would have, by chance, indicated the correct time. In that case, Julia would have believed it was two o’clock, and her belief would have been both true and justified (given her long track record of deriving knowledge from using the clock). But her belief would have failed to have been knowledge.15 Notice that the fallible nature of Julia’s knowledge about the time can be explained in two ways: it could have been accidentally true (as above), or it could have been false (if, for example, the clock had stopped ten, rather than twelve, hours before she walked past it). In the case of knowledge of necessary truths, however, this is not so—they could not have been false, no matter what the quality of the subject’s justification might be. Even so, a subject’s belief regarding a necessary proposition could have been true by accident. For example, a student may learn many mathematical truths from her teacher, who is generally a highly reliable source for testimony of this sort. But the teacher, excellent though she is, could have made two mistakes on a single problem: when she performed the calculation, and then again when she wrote the answer on the board, she transposed the numbers. In some cases, these two mistakes will cancel each other out—the number she mistakenly writes on the board turns out to be the correct answer to the problem. If this were to happen, the student would have a belief that is true (indeed, necessarily true) and justified (given the teacher’s general reliability, the student’s well-placed trust in the teacher, etc.) but not knowledge. Such cases are rare, no doubt, but their possibility means that the student’s knowledge, acquired on the basis of her teacher’s testimony, is fallible. What makes fallibilistic knowledge possible is the fact that it is grounded in something other than entailment—something more loosely connected with the truth of the subject’s belief. This looser grounding is probability, which permits both of the ways in which a subject’s justified belief can fail to be knowledge: P can make Q probable even though Q is false, and P can make Q probable even though Q is true for reasons that have nothing to do with P (and, in that sense, is accidentally true).16 This permits a second way of thinking about fallibilism, which can replace the entailment formulation above: S knows that p in a fallible way just in case S knows that p on the basis of some justification j, 5 where j makes probable that p.17 There is widespread disagreement over how best to understand probability in epistemology. Some philosophers have taken epistemic probability to be a relation holding between propositions, perhaps knowable a priori.18 Others have taken the relevant sort of probability to be grounded in relative frequencies or propensities.19 Nevertheless, the fact that these theories are recognizably grounding justification and knowledge in probability marks all of them as versions of fallibilism. 3. THE GETTIER PROBLEM If philosophers have been pushed to fallibilism by skeptical worries, it is only fair to note that fallibilism itself has presented a number of significant epistemological problems. Chief among these is the so-called Gettier problem, which has to do with accidental truth of the sort discussed in the previous section.20 In his classic paper, Edmund Gettier argued that any account of knowledge that (a) allowed the possibility of justified but false beliefs and (b) allowed justification to be transferred by deduction would be unable to account for the following sort of case. Smith has a great deal of evidence for the proposition that Jones owns a Ford (such as remembering that Jones has always driven a Ford, has spoken about having to make his car payments, etc.). At the same time, Smith has no evidence that his friend Brown is traveling, but he knows that each of the following propositions is entailed by the proposition for which he has a lot of evidence: that either Jones owns a Ford or Brown is in Boston, that either Jones owns a Ford or Brown is in Barcelona, and that either Jones owns a Ford or Brown is in Brest-Litovsk. As it happens, Jones has just sold his Ford, but Brown really is in Barcelona. So, while two of his inferred beliefs are justified and false, Smith’s belief that either Jones owns a Ford or Brown is in Barcelona is both justified and true. But, surely, it is not knowledge. So-called Gettier cases have generally been taken to have shown that justified true belief accounts of knowledge, where the justification in question is fallibilistic, cannot be correct.21 At a minimum, some other condition must be added to the account to rule out accidentally true belief. Various solutions to the Gettier problem have been proposed. They include requiring that the subject’s justification not rely on any falsehood;22 requiring that there not be any true proposition which, if the subject were to believe it, would undermine the subject’s 6 justification for the belief in question;23 and requiring that there be some causal or counterfactual relation between the belief and the fact that the belief makes true.24 There are variations on each of these strategies to responding to the Gettier problem, and objections and counterexamples have been raised for each of them. It is safe to say, I think, that none of these responses has been widely accepted.25 Perhaps as a result many philosophers have largely set it aside, apparently reasoning that, since everyone faces the Gettier problem, it doesn't weigh on them any more heavily than it does on anyone else.26 Others have become pessimistic about the prospects of finding a solution at all and have embraced instead views that, they argue, are not susceptible to problems with accidentality in the first place.27 Finally, it has been argued that the Gettier problem cannot be solved and that fallibilism will always bring with it a problematic kind of accidental success. If this is so, fallibilism may permit an escape from traditional skepticism only to fall prey to a new type of skeptical problem.28 4. THE LOTTERY PARADOX In its initial form, the lottery paradox poses a problem for rational belief.29 It depends on the following two principles: (1) If it is highly probable that p, then it is rational to believe that p.30 (2) If it is rational to believe that p and it is rational to believe that q, then it is rational to believe that p & q. Each principle is quite plausible on its own. If one’s goal is to have as many true beliefs as possible while minimizing the risk of having false beliefs, then one will obviously want to pursue the strategy of believing highly probable propositions. And, if two propositions are each rational to believe, then surely it is rational to believe their conjunction. After all, one can be certain that the conjunction is entailed by what one is already rational to believe; because one is certain of the entailment, it seems as though adding the conjunction to one’s set of beliefs does not bring with it any additional risk of error. Nevertheless, the two principles together yield a counterintuitive result. Suppose there is a fair lottery in which 1000 tickets are sold and in which only one ticket will win. For each ticket, there is thus a .999 7 probability that it will lose. Principle (1) tells us that it is rational to believe of each ticket that it will lose. So, where proposition pi is the proposition that ticket ti will lose, it is rational to believe that p1, that p2, …, that p1000. Principle (2) tells us that it is rational to believe the conjunction of all these propositions: that p1 & p2 & … & p1000. But, because we know it is a fair lottery, it is also rational for us to believe that some one of the tickets will win—i.e., it is rational for us to believe that either not-p1 or not-p2, …, or not-p1000. We know (and rationally believe) that this is equivalent to the proposition that not-(p1 & p2 & … & p1000). Using principle (2) again, it is rational to believe that p1 & p2 & … & p1000 & not-(p1 & p2 & … & p1000). But that proposition, of course, is a contradiction. The above result is a paradox (in the strict sense of the term) if one is also committed—as many philosophers are—to the view that it is never rational to believe a contradiction.31 Others have argued that the lesson we should draw from it (and from the preface paradox, as well) is that it in fact is rational to have a belief set that is inconsistent.32 Whether or not this is a satisfactory solution to the original version of the lottery paradox, we shall see that it does not work for a version of the paradox framed in terms of knowledge. Consider the following epistemic versions of the two principles underlying the lottery paradox: (1′) If it is both highly probable and true that p, then (ceteris paribus) one knows that p.33 (2′) If one knows that p and one knows that q, then one knows that p & q.34 These principles seem to be just as plausible as the principles governing rational belief above. More to the point, the fallibilist is in no position to reject (1′), if my characterization of fallibilism as probabilistic in nature is correct. When a proposition is highly probable and true, it is hard to see how the proposition could fail to be known. Such a failure could happen only in the event that what makes the proposition justified for the subject is somehow disconnected from what makes it true, as happens in Gettier cases. But, supposing (in accordance with the ceteris paribus clause) that there is no such disconnection, the subject will have fallibilist knowledge when she believes highly probable, true 8 propositions. Principle (2′) simply allows the subject to conjoin separate bits of knowledge into one.35 If she knows them separately, it seems obvious that she will still know them when she has put them together. Consider, again, a fair lottery with 1000 tickets, of which only one will be a winner. For each ticket, there is a .999 probability that it will lose. Principle (1′) says that, where it is true that a ticket ti will lose, I can know the proposition pi that it will lose. Suppose it is true that t1 will lose. My belief that p1 is then true and highly probable; moreover, its truth does not seem to be accidental in the way that the justified true beliefs in Gettier cases are.36 So, it looks as though I know that p1. By parity of reasoning, the same should hold with respect to the other tickets in the lottery, with the exception of the ticket tj that loses. In that case, my belief that pj will be justified but false, and therefore it will fail to be knowledge. Principle (2′) will allow me to conjoin all of my separate bits of knowledge, but the conjunction I know to be true will not include the proposition that pj. Hence, I cannot be said to know the contradiction that all of the tickets will lose and one of them will win. So, principles (1′) and (2′) do not lead to an epistemic version of the paradox in the same way that the paradox for rational belief arose.37 Nevertheless, there is another way of showing how the two epistemic principles above lead to a problematic result in lottery cases.38 Suppose that ticket 1000 will be the winning ticket of our fair, 1000ticket lottery. Suppose also that I consider the lottery tickets in a methodical way, by deciding for each one in turn whether it will win or lose. I recognize that t1 has a .999 probability of losing; it is also true that it will lose. So, according to principle (1′), I know that p1. In the same way, my consideration of whether t2 will lose allows me to know that p2. Principle (2′) then allows me to conclude that p1 & p2. More generally, as I continue considering the tickets one by one, principle (1′) allows me to acquire new pieces of knowledge, and principle (2′) allows me to add them to the conjunction I know to be true. Ultimately, I end up knowing that p1 & p2 & … & p999. Before considering t1000, I remember that one of the tickets will win. I know that t1 through t999 will lose, so t1000 must be the winner. The belief I end up with is true, and I can see that it is obviously entailed by things that I know, so it is surely justified as well. But it does not seem to be knowledge. Explaining what is defective about acquiring justified true beliefs that fall short of knowledge in this way will require us to reject one of 9 the two principles underlying the knowledge version of the lottery paradox. The most common response has been to deny or modify (1′). So, for example, Dana Nelkin has argued that my belief that p1 fails to be knowledge because the fact that makes it true does not bear a causal or explanatory connection to the belief.39 Adapting a strategy from Gilbert Harman, we might hold that (1′) should apply only to the subject’s entire set of knowledge rather than to individual instances of knowledge.40 Or, following John Hawthorne, we might take (1′) to fail in lottery-like situations because they make error possibilities salient in a way that undermines the subject’s ability to have knowledge.41 Each strategy will face difficulties of its own. For example, requiring a causal or explanatory connection between the belief and the fact that makes it true may mean that it is impossible for anyone to ever know anything about the future, about general facts, or about abstract objects. Applying principle (1′) only to the subject’s entire set of knowledge will make fallibilism impossible since the probability of one’s body of knowledge taken as a whole will surely be low. And, finally, it will be difficult to say why the salience of error possibilities undermines knowledge in lottery situations. If fallibilism is correct, there are error possibilities for every instance of knowledge; if the ones that matter are all and only the salient error possibilities, it will obviously be very important to have an account of salience that is not merely ad hoc. The other main option is to reject principle (2′).42 Doing so means rejecting or modifying the idea that knowing two separate propositions thereby permits one to know their conjunction. There is a clear fallibilist rationale for this strategy: if knowledge is essentially probabilistic in nature, then conjoining one’s knowledge will also compound the risk of error. In response, some philosophers have objected that abandoning (2′) leaves deductive reasoning no clear role to play in epistemology.43 Whatever response we make to the lottery paradox, then, it seems that fallibilism will require some modification of our basic assumptions governing knowledge. 5. CONCLUSION Although many philosophers have come to regard fallibilism as the only serious option in epistemology, there is still much work to be done in understanding the nature of fallibilistic knowledge. Accepting fallibilism may involve the rejection of some traditional ways of conceiving of 10 knowledge. In addition to the problem it poses for closure (discussed in the previous section), fallibilism also seems to conflict with the standard conception of epistemic possibility. On the usual way of thinking, it is epistemically possible (for S) that p just in case S does not know that not-p. Equivalently, it is epistemically impossible (for S) that not-p just in case S knows that p. But, if fallibilism is correct, it looks like a subject can know a proposition while its contradictory remains possible—in fact, the rough formulation of fallibilism says as much.44 A further problem for fallibilism arises from the fact that it seems to license problematic assertions with forms like, “I know that p, but it might be false that p” and “It is probable that p, and I know that p.” Some philosophers—e.g., David Lewis—have thought these so problematic that they have regarded them as compelling reasons to favor infallibilism.45 But the issues here are complex. There are similar forms of assertion, also licensed by fallibilism, that seem not to be (as) problematic: e.g., “I know that p, and yet it might be false that p.”46 There are also similar, problematic forms of assertion that are licensed by every epistemological theory (fallibilist or infallibilist): “It is true that p, but I know that p.” Reflection on a wide range of assertions may leave us skeptical that we can recover much of epistemological significance from attention to the ways we can and cannot talk about knowledge and epistemic possibility. Finally, and most fundamentally, the problem of accidentality remains to be solved. Without a compelling solution to the Gettier problem, the fallibilist approach to epistemology cannot ultimately be considered a success.47 REFERENCES Alston, William. 1992. “Infallibility,” in A Companion to Epistemology, J. Dancy and E. Sosa (eds.), Oxford: Blackwell, p. 206. Ayer, A.J. 1956. The Problem of Knowledge. London: Penguin. 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Rysiew, Patrick. 2001. “The Context–Sensitivity of Knowledge Attributions,” Noûs 35: 477–514. Shope, Robert. 2002. “Conditions and Analyses of Knowing,” in The Oxford Handbook of Epistemology, P. Moser (ed.), Oxford: Oxford University Press, pp. 25-70. ——. 1983. The Analysis of Knowing. Princeton: Princeton University Press. Stanley, Jason. 2005. “Fallibilism and Concessive Knowledge Attributions,” Analysis 65: 126-31. Strawson, P.F. 1992. Analysis and Metaphysics. Oxford: Oxford University Press. Unger, Peter. 1975. Ignorance: A Case for Scepticism. Oxford: Clarendon Press. Vogel, Jonathan. 1992. “Lottery Paradox,” in A Companion to 17 Epistemology, J. Dancy and E. Sosa (eds.), Oxford: Blackwell, pp. 265-7. Weatherson, Brian. 2003. “What Good Are Counterexamples?” Philosophical Studies 115: 1-31. Williams, Michael. 2001. “Contextualism, Externalism and Epistemic Standards,” Philosophical Studies 103: 1-23. ——. 1999. “Skepticism,” in The Blackwell Guide to Epistemology, J. Greco and E. Sosa (eds.), Oxford: Blackwell, pp. 35-69. Williamson, Timothy. 2000. Knowledge and Its Limits. Oxford: Oxford University Press. Zagzebski, Linda. 1994. “The Inescapability of Gettier Problems,” Philosophical Quarterly 44: 65-73. NOTES 1 Infallible foundations have been conceived in various ways—e.g., by the ancient Stoics as cognitive impressions, by Descartes as clear and distinct perceptions, and by twentieth-century philosophers like C.I. Lewis as “the given” element in experience (for more on these views, see Reed (2002a), Descartes (1984), and Lewis (1970), respectively). It should be noted that infallible foundations preclude error in a limited way. For example, the clear and distinct perception that a circle is a shape allows the subject to know, without possibility of error, that a circle is a shape. But the subject might still combine this clear and distinct perception with other beliefs that are not clearly and distinctly perceived in such a way that a false belief is the result. Indeed, it is because this is possible that the Stoics took only sages (who possess a systematic body of cognitive impressions) to be free from error; see Cicero (2006), p. 34. The rest of us are still prone to making mistakes, despite the fact that we do have cognitive impressions. Descartes also seems to be aware of the difference between limited certainty and absolute certainty; see Reed (forthcoming-b). 18 2 I am using ‘grounding’ here in a broad sense, so as to include not only support from reasons and evidence but also possession of so-called externalist properties—such as reliability—as well. Other terms that can be used in roughly the same way as ‘grounding’ in this sense are ‘justification’ and ‘warrant’. Fallibilist theories of justification, such that one can have justification for the belief that p even in cases where it is false that p, are relatively uncontroversial, given that one have justification for a belief without the belief being justified simpliciter. But it is worth noting that fallibilism about knowledge will entail commitment to fallibilism about, not only justification, but justification simpliciter. 3 This sort of fallibilistic response can be seen to follow in the wake of the first systematic presentation of skepticism in the Hellenistic world; see, e.g., Brittain’s introduction to Cicero (2006). For present-day versions of this response, see: Cohen (1988); Williams (1999), p. 54; and Feldman (2003), pp. 122-128. See Unger (1975) for a skeptical argument that relies explicitly on an infallibilist conception of knowledge. 4 For this argument, see, e.g., Strawson (1992), pp. 91-6. 5 And, even if I did know it in that sort of way when I first acquired the belief, I do not now know it through its relation to the given. So, that cannot be the explanation for my current knowledge. 6 For defenses of infallibilism—less than full-blooded, in some cases— see Fumerton (2006), ch. 2; D. Lewis (1996); and Dodd (forthcoming). 7 Cohen (1988), p. 91. Michael Williams agrees: “We are all fallibilists nowadays” (2001, p. 5). 8 See, e.g., the various papers in Goldman (1992) and in Conee and Feldman (2004). 9 Throughout this section, I rely on Reed (2002b). 10 Here and throughout the paper, I am using the term ‘justification’ as a placeholder. The reader can substitute for it evidential relations, modal relations like sensitivity or safety, reliability properties, or whatever features in her preferred account of knowledge. For variants on this formulation of fallibilism, or the analogous account of infallibilism, see, e.g., Ayer (1956), p. 54-56; BonJour (1985), p. 26; BonJour (1998), p. 16; Alston (1992); and Pritchard (2005), p. 17. See also Lehrer’s first definition of ‘incorrigibility’, which is (in my terms) a definition of 19 infallibility (1974, p. 81). 11 The relevant sort of possibility here (and in the earlier formulation, in which a belief “could have been mistaken”) is metaphysical or broadly logical. This is why Dretske’s conclusive reasons view counts as a type of fallibilism. Although he says, “If S has conclusive reasons for believing P, then it is false to say that, given these grounds for belief, and the circumstances in which these grounds served as the basis for his belief, S might be mistaken about P” (1971, p. 13), the impossibility he has in mind is physical or nomological in nature. In a similar way, David Lewis requires that, for a subject to know that P, her evidence rule out, not absolutely every possibility in which not-P, but just “every possibility in which not-P—Psst!—except for those possibilities that conflict with our proper presuppositions” (1996, p. 554). In other words, the subject’s evidence must rule out every not-P possibility except for those that she is entitled to ignore. Although Lewis presents this as a kind of infallibilism, I think the view achieves the “impossibility of error” only through a kind of creative accounting. (Compare: “Of course we turned a profit last year. We brought in more than enough revenue to cover our expenses—Psst!—except for those costs we’re ignoring.”) Because it is not the subject’s evidence (or justification or reasons or whatever provides her epistemic grounding) that eliminates all of the ways in which her belief could (in the broadest sense) be false, her evidence does not make her infallible with respect to the belief. (To be clear, this is not meant as an objection to the views of either Lewis or Dretske. Either view may be correct as an account of knowledge (or, in Lewis’s case, of knowledge attributions); my point is merely that neither of them is really a version of infallibilism.) 12 For some examples of this formulation of fallibilism, or the analogous formulation of infallibilism, see Cohen (1988), p. 91; Merricks (1995), p. 842; Jeshion (2000), pp. 334-335; Conee and Feldman (2004), ch. 12; Stanley (2005), p. 127; and Dougherty and Rysiew (2009), p. 128. 13 This is a problem that has been recognized for some time; see, e.g., Lehrer (1974), pp. 82-83; Hetherington (1999), p. 565; Merricks (1995); Reed (2002b); and Fumerton (2006), p. 60. 14 For this reason, entailment should not feature in accounts of epistemic possibility, as it does in Stanley (2005) and Dougherty and Rysiew (2009). 20 15 This case is derived from Russell (1948), p. 154. 16 See Reed (2002b), p. 151, for more on fallibilism and probability and Reed (2000) for more on accidentality. 17 There are three clarifications worth noting. First, the probability relation is not incompatible with there also being an entailment relation between the subject’s justification and the belief it justifies. In fact, there will be such an entailment when the subject’s belief is of a necessary proposition. My point here is simply that the subject’s justification will not supervene on that entailment. Second, if one’s justification makes certain one’s belief, and certainty implies probability, then it would seem that the beliefs one knows with certainty would be instances of fallibilistic knowledge. To preclude cases of this sort, we can understand fallibilistic knowledge to occur only in cases where one’s justification makes the belief in question merely probable. (On this point, I am grateful to an anonymous referee.) Third, in the relevant sense of probability, it cannot be the case that necessary truths are assigned an unconditional probability of 1; if they were, the problem with necessity would arise again, for necessary truths would then have probability 1, no matter what one’s evidence might be. For this reason, probability in the epistemic sense cannot be unconditional in nature. Beliefs—even beliefs about necessary propositions—can be epistemically probable only relative to their grounding. (On this point, I am grateful to Sherri Roush.) For more on fallibilism understood in terms of probability, see Reed (2002) and Fantl and McGrath (2009), ch. 1. 18 This sort of view is grounded in Keynes (1921). See also Fumerton (2004); Kyburg (2003) and (1971); and Chisholm (1989a, pp. 54-56 and 63-64) and (1989b). 19 For more on these interpretations of probability, see Russell (1948), part five, and Mellor (2005). Goldman’s reliabilism is one prominent example of a view that takes justification to be grounded in probability as a measure of either actual or counterfactual frequencies; see his (1979), pp. 114-115. 20 The problem is first explicitly stated in Getter (1963), but an earlier example of this sort—the clock case mentioned above—can be found in Russell (1948). See Shope (1983) for an account of the most influential early responses to Gettier’s paper. See Reed (2000) for more on the nature of epistemic accidentality. See Pritchard (2005) and Lackey 21 (2008) on the related phenomenon of epistemic luck. 21 Gettier’s specific targets are Chisholm (1957) and Ayer (1956), but virtually all epistemologists in the years since his paper was published have recognized the general significance of the problem of accidentality. For a case of justified, accidentally true belief that is perhaps in some ways different than Gettier’s original examples, see Ginet’s barn façade case in Goldman (1976). For cases of beliefs that are both accidentally true and accidentally justified, see Reed (2000). 22 See Myers and Stern (1973) and Armstrong (1973), p. 152. Harman (1970) and (1973), pp. 120-4, argues that the rules of belief acceptance cannot be probabilistic because, if they were, they would not allow this sort of response to the Gettier problem. Feldman (1974) objects that this sort of response will not work in all sorts of Gettier cases—there are some that do not rely on false premises. 23 For this, so-called defeasibility account, see, e.g., Lehrer and Paxson (1969) and Klein (1971). See Shope (1983), ch. 2, for objections. 24 See, e.g., Goldman (1967) and Nozick (1981). See Ginet’s barn façade case in Goldman (1976) for an example of the sort of case that has proven to be problematic for these externalist accounts of knowledge. 25 See Shope (1983) and (2002) for many of the proposed solutions and proposed counterexamples. 26 I should also mention that a few philosophers have tried arguing that Gettier cases are not really a serious problem. Thus, Hetherington (1999) takes the subjects in Gettier cases to have borderline instances of knowledge, and Weatherson (2003) argues that we might be better off rejecting the intuitions underlying Gettier cases in favor of the simple and intuitive theory they undermine. 27 Two examples of this strategy include Plantinga (1993), p. 48, and Williamson (2000), pp. 4, 30. However, it has been objected that both of their views do in fact face the problem of accidentality; see Greene and Balmert (1997) and Reed (2005), respectively. 28 See Reed (2007) and (2009) and Zagzebski (1994). See Hetherington (1996) for another way of linking the Gettier problem and skepticism. 29 The lottery paradox was first formulated by Henry Kyburg (1961, pp. 197-9). See also his presentation of it in Kyburg (1970). For other helpful presentations of the lottery paradox, see Vogel (1992), Nelkin (2000), and Olin (2003). 22 30 Some philosophers will prefer to talk, not about rational belief outright, but rather about rational degree of belief. It is thought that this will permit a solution to the lottery paradox because the degree of belief it is rational to have will drop as the subject conjoins propositions she individually has some higher degree of confidence in. See Foley (2009) for discussion of this option and of what he calls the “Lockean thesis,” which links rational degree of belief with rational outright belief. 31 See, for example, Lehrer (1974), pp. 202-4. 32 It should be noted, however, that having a belief set with inconsistent members is not necessarily the same as believing an outright contradiction. On this point, see Kyburg (1970), p. 56-60, where he draws a distinction between a weak and a strong principle of consistency; both principles rule out believing contradictions, but the former permits having beliefs that entail a contradiction (how this is possible is explained below; see note 41). See also Klein (1985) and Foley (1987), pp. 241-7, and (1993), pp. 162-73. The preface paradox arises when an author, who is committed to the truth of every claim in her book, nevertheless reflects that her fallibility makes it reasonable for her to believe that at least one of those claims is false; see Makinson (1965) and Olin (2003), ch. 4. 33 Could the knowledge form of the lottery paradox be avoided by moving to a theory that is focused on degrees of knowledge rather than knowledge outright? Although this strategy might plausibly work for the belief form of the paradox, matters are complicated here by the fact that there will have to be a minimal threshold below which one does not possess knowledge of any degree. Given any plausible threshold for the minimal degree of knowledge, it should be possible to formulate the lottery paradox so that the subject meets that threshold with respect to each of her beliefs about the losing tickets. 34 It may be necessary to add two clauses to this principle: (a) one believes that p & q and (b) one does so because one knows that p and one knows that q. These additions would prevent (a) cases of knowledge without belief and (b) unrelated lucky guesses that p & q from counting as knowledge. In what follows, though, I shall ignore these complications. 35 (2′) is what has become known as an epistemic closure principle (in 23 this case, governing conjunction). 36 To secure this result, we will need to build into the case that my belief has been acquired in the way specified by one’s favorite fallibilistic theory—e.g., it is the product of reliable belief-producing processes, or it is held on the basis of one’s evidence about the odds. When that is the case, it will not typically be a coincidence that the belief is both true and justified, as we saw happen in the case of Smith’s belief that either Jones owns a Ford or Brown is in Barcelona. 37 See Nelkin (2000) for an attempt to construct a knowledge version of the lottery paradox along these lines. Olin (2003), p. 204, correctly points out that it will not work, given that one of the allegedly known propositions is in fact false and thus fails to be knowledge. 38 39 See Reed (2010) and Hawthorne (2004). Nelkin (2000), p. 390. See also Dretske (1971), pp. 3-4, who disallows knowledge of lottery propositions because the probabilistic basis for them does not make it impossible for the subject to be mistaken in those circumstances. Nozick (1981) would disallow knowledge of lottery propositions because they fail his sensitivity condition, which holds that, if the proposition in question were false, one wouldn’t believe it. In the case of a lottery proposition, one would continue to believe (on merely probabilistic grounds) that the ticket will lose even in the case in which it wins. 40 See Harman (1973), p. 119. Harman’s argument is framed in terms of rational acceptance rather than knowledge. 41 Hawthorne (2004), pp. 160-2. 42 See Reed (2010) for the rejection of (2′). For the rejection of (2)—the rational belief version of the principle—see Kyburg (1970); and Foley (1987), p. 243, and (1993), pp. 162-73. As Kyburg notes, it is possible to draw a distinction between it being rational to hold inconsistent beliefs and it being rational to hold contradictory beliefs only if one rejects (2). 43 See Pollock (1983). This issue also intersects with the recent debate as to whether knowledge is deductively closed; see Hawthorne (2005) for the case to be made in favor of closure. 44 For a solution to this problem, see Reed (2010) and (forthcoming-a). 45 See Lewis (1996) and Fantl and McGrath (2009) on the “madness” of fallibilism. DeRose (1991) argues that assertions of the form, “I know 24 that p, but it might be false that p,” clash because the conjuncts are logically inconsistent. For more on the debate over what have come to be called ‘concessive knowledge attributions’, see Rysiew (2001), Stanley (2005), Dougherty and Rysiew (2009), and Dodd (2010) and (forthcoming). I am grateful to an anonymous referee for pressing this point. 46 Notice that assertions of this type have the same logical form as assertions of the first type mentioned above (“I know that p, but it might be false that p”). The fact that the assertions of the one type sound much more “clashy” than do assertions of the other type indicates that the clash in question does not derive from any logical contradiction. 47 For helpful comments on a draft of this paper, I am grateful to an anonymous referee for Philosophy Compass, Sherri Roush, and, especially, Jennifer Lackey.