Skepticism About a Priori Knowledge

CHAPTER FORTY-SEVEN Skepticism about A Priori Knowledge

OTÁ VIO BUENO

1 INTRODUCTION A priori knowledge is, roughly, knowledge that does not depend on experience (except perhaps for whatever experience that may be needed to acquire the relevant concepts). This is a rough formulation since, when analyzing the notion of a priori knowledge, some care is needed to characterize the independence-from-experience condition so that what is taken to be known independently of experience (such as mathematical and logical results) turns out to be known a priori (according to the proposed analysis of a priori knowledge), and what is taken to be known based on experience, and thus is dependent on experience (such as particular physical traits of the world), is not known a priori (according to the relevant account). The challenge is then to secure that the proposed analysis is extensionally adequate. Two questions then emerge: (a) Can we have a priori knowledge? (b) Can we know that—or, at least, determine whether—we have such knowledge? Corresponding to these two questions, there are two forms of skepticism regarding the a priori : (aʹ ) One can provide arguments to the effect that, despite appearances to the contrary, one doesn’t have knowledge of the a priori , such as mathematical claims and claims about the validity of arguments. This is a priori skepticism (see Beebe 2011 ). Alternatively, (b ʹ ) one can provide arguments to the effect that conceptual analyses (characterizations, deﬁ nitions) of a priori knowledge (see, for instance, Kitcher 1980, 2000; and, for an account of a priori justiﬁ cation, Casullo 2003) are ultimately problematic. This is skepticism about a priori knowledge . And if it is unclear what a priori knowledge ultimately is, it is similarly unclear how we can determine whether we have any such knowledge. In this chapter, I will consider both forms of skepticism regarding the a priori , focusing in particular on skepticism about a priori knowledge. After all, by challenging the very concept of a priori knowledge, it seems to me that this form of skepticism is more basic. Given the centrality of a priori knowledge to so many cognitive endeavors (from mathematics and logic to philosophy), the presence of

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these forms of skepticism highlights a central issue that needs to be addressed by those who claim to have the corresponding knowledge.

2 A PRIORI SKEPTICISM Most people think they know several mathematical claims, including simple arithmetical results, such as that 71 is the sum of 13 and 58. Most people think they also know whether a given conclusion follows logically from certain premises, such as that q follows logically from p and if p then q . Since such mathematical and logical results are arguably instances of what is known a priori , most people think they have a priori knowledge (although, unless they have some philosophical training, they are unlikely to describe that knowledge in these terms). A priori skepticism challenges such knowledge claims, and any others that are based on a priori considerations (Beebe 2011). The form of the argument for it is the same as the familiar brain-in-a-vat or evil-demon arguments, in which, ﬁ rst, a necessary condition for knowing something about the world is that one knows that one’s experiences are not being generated in a brain-in-a-vat scenario or by an evil demon. Support for this premise is provided by the closure principle, according to which (roughly) one knows what follows logically from things that are already known and that are established on the basis of that knowledge. Second, it turns out that one is unable to know that one is not in such skeptical scenarios. After all, one’s perceptual experiences are insensitive to such scenarios: one would believe that one is experiencing the world even if one weren’t (as the case would be if one were in such skeptical situations). The result is that one doesn’t know anything about the world. The argument for a priori skepticism is entirely analogous. In fact, it is an application of the familiar evil-demon argument to a priori knowledge claims, such as knowing the validity of modus ponens . As James Beebe notes:

(P1 ) If I know that modus ponens is correct, then I know that my belief that modus ponens is correct is not based on faux intuitive experiences induced in me by a bumbling evil demon.

(P 2 ) I don’t know that my belief that modus ponens is correct is not based on faux intuitive experiences induced in me by a bumbling evil demon. (C) Therefore, I don’t know that modus ponens is correct ( Beebe 2011 : 590; the numbering of the premises and the conclusion of this argument has been changed; of course, nothing hangs on this). Interestingly, the argument above does not depend on establishing the invalidity of modus ponens . Those who take this form of inference to be valid would insist on the impossibility of demonstrating the inference’s invalidity: on their view, any such attempt would inevitably lead one into a contradiction. Rather, the argument is meant to challenge one’s knowledge that modus ponens is valid (or correct) on the grounds that one’s belief in the validity of this inference could be based on a bumbling evil demon. Whether modus ponens is ultimately valid or not, one cannot know that it is valid.

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There is also a different route to a priori skepticism (Beebe 2011: 593–594). It emerges from the consideration of alternative logical and mathematical practices, in which established mathematical and logical truths may not hold. This route surfaces from Ludwig Wittgenstein’s work (see Wittgenstein 1956, as understood by Stroud 1965 ; for a discussion, see Beebe 2011 : 593–594). From the perspective of those who adopt the traditional view regarding logical and mathematical truths—according to which these truths are metaphysically necessary—one may not understand under what conditions mathematical and logical truths would be false. However, one’s inferential and calculation practices, which shape the relevant logical and mathematical results, could differ signifi cantly from the traditional ones. This would generate genuine alternatives to these practices, so that despite the agreement that modus ponens is valid, one would not agree that from p and if p then q , it would follow that q ; one could also follow the rule “adding 2” in such a way that after reaching 1,000, the rule continues as 1,004, 1,008, etc. It is ultimately a contingent fact that we developed the mathematical and logical concepts we currently have. Had the history of our species been different, we could have developed correspondingly different concepts. As Wittgenstein points out: I am not saying: if such-and-such facts of nature were different people would have different concepts (in the sense of a hypothesis). But: if anyone believes that certain concepts are absolutely the correct ones, and that having different ones would mean not realizing something that we realize—then let him imagine certain very general facts of nature to be different from what we are used to, and the formation of concepts different from the usual ones will become intelligible to him. ( Wittgenstein 1956 : I 141; quoted in Beebe 2011 : 594) It is possible then, in light of the contingency of the way in which mathematical and logical concepts have been developed, that these concepts could have evolved differently, thus resulting in different logical and mathematical practices (see also Stroud 1965: 513, and Beebe 2011: 594). As a result, one doesn’t know the correctness of one’s current mathematical and logical practices. It seems to me that a priori skepticism provides a signifi cant challenge to those who claim to know mathematical and logical results. But the challenge depends on the version of the challenge that is advanced. The second, contingency-based, formulation of this skeptical challenge seems to me more effective. The concern about the fi rst, brain-in-a-vat or evil-genius-based formulation, is that it invites a particular sort of Moorean response. One could easily complain that we have more reason to believe in the truth of the relevant mathematical and logical results than

we have to believe in the truth of the second premise, (P2 ), of the argument for a priori skepticism, according to which we don’t know that our relevant logical or mathematical beliefs are not based on experiences induced by an evil demon (or that emerge from a brain-in-a-vat scenario). Of course, as a principled response to skepticism, this move blatantly begs the question, since it assumes that we know precisely those items that the skeptical argument challenges. From this point of view, the response is clearly ineffective. But one can think of it as expressing a particular attitude: one that dismisses the entire skeptical argument as being so general, so

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detached from the particular mathematical and logical issues under consideration, that it verges on being irrelevant. The response reinforces the point that, due to its generality, there is very little in the fi rst argument for a priori skepticism that engages with the content of logical or mathematical beliefs per se. The fact that the argument is a particular instance of the usual brain-in-a-vat or evil-demon argument clearly illustrates this point. In the end, the Moorean response is perhaps better thought of as an expression of the diffi culty of taking such an argument very seriously. In contrast, the contingency-based formulation of a priori skepticism is more relevant. It engages with the content of the relevant mathematical and logical concepts. It is the fact that these concepts could have been developed differently that allows for the possibility that alternative logical and mathematical results could have been entertained. However, despite the advantage of this formulation of a priori skepticism, it can be made stronger. There is no need to leave the argument, as Wittgenstein, Stroud, and Beebe have left it, as a mere possibility, namely, that alternative mathematical and logical practices, shaped by different concepts, could have emerged. These possibilities are, in fact, actual. Challenges have been raised to the validity of modus ponens (McGee 1985) as well as to the correctness of classical mathematics, by intuitionists ( Dummett 2000), predicativists (Weyl 1918/1994), and paraconsistentists (Mortensen 1995 ; for a discussion, see da Costa , Krause , and Bueno 2007). Even classical arithmetical results have been challenged within these circles (Priest 1997, 2000 ). The alternative logical and mathematical practices, posited as mere possibilities in the formulation of a priori skepticism above, are, in fact, not only possible, but actual. This means that one cannot claim to know, so easily, traditional mathematical and logical results, since they have all been challenged. Of course, the challenge is more focused and targeted at particular logical and mathematical results. In contrast to the fi rst argument for a priori skepticism, this is a local rather than a global skeptical challenge. It is far more relevant, specifi c, and cannot be so easily dismissed. As an illustration, consider the challenge one can pose to modus ponens using the 2016 US presidential election as a context for the counterexample (this is adapted and updated from McGee 1985 ):

(P1 ) If a Republican wins the election, then it if is not Donald Trump who wins, it will be Ted Cruz.

(P 2 ) A Republican will win the election. (C) If it is not Donald Trump who wins, it will be Ted Cruz. The premises of the argument are clearly true: Ted Cruz is the closest Republican contender to Donald Trump, and a Republican will win (and, in fact, did win) the election. However, the conclusion is clearly false: if it is not Donald Trump who wins the election, it will be Hillary Clinton, rather than Ted Cruz, who does it. (In fact, Clinton eventually won the popular vote by about 3 million votes!) This local, substantive form of a priori skepticism is clearly something to be reckoned with.

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3 EMPIRICISM AND A PRIORI KNOWLEDGE Before considering skepticism about a priori knowledge, it’s important to dispel a concern about a priori knowledge that has often been raised. Traditionally, from Sextus Empiricus through David Hume, skepticism and empiricism have often been closely connected. And a priori knowledge (and a priori justifi cation) have often been a source of puzzlement particularly for those who are sympathetic to empiricism. How is it possible to have such knowledge (and justifi cation) independently of empirical information? Those who are unmoved by these concerns insist that the content of the relevant statements, since their truth does not depend on empirical factors, is non-empirical, too, and therefore considerations to support the statements in question need not, and in fact should not, invoke empirical reasons, that is, reasons based on empirical information. Note that empiricists, just for being empiricists, need not be worried about a priori knowledge (or justifi cation). Suppose empiricism is understood as involving the requirement that all information about the world is based on experience. There are at least two ways in which empiricism is not in tension with the a priori (a priori knowledge and justifi cation): (a) when experience is understood in a suitably broad sense or (b) when the offending statements, which are known or justifi ed a priori , are not about the world. With regard to (a), consider that among the experiences in question there are intellectual seemings (or intellectual intuitions; see Chudnoff 2013 ). In this case, one could have suitable experiences about objects of a non-empirical sort. Admittedly, it would require a correspondingly broader empiricism to accommodate such a broader notion of experience, and not every sort of empiricist will be willing to embrace this proposal. With regard to (b), one could argue that (pure) mathematical statements (those that are only about numbers, functions, sets, categories, and other mathematical objects and structures) are not about the world, since none of the objects invoked in such statements are empirical. If the truth of mathematical statements does not depend on what goes on in the empirical world, and if such statements are not about the empirical world either, there is nothing in these statements that can contravene empiricism—understood in the more restricted sense as a conception that is only concerned with information about the empirical world. This move clearly limits the scope of empiricism, which is then ultimately restricted to empirical objects alone. But that makes a priori knowledge (and justifi cation) safe for empiricists. At this point, one can be concerned about (allegedly) synthetic a priori statements, that is, statements about the world whose knowledge and/or justifi cation do not depend on empirical information about it (the world). Understandably, such statements raise concerns for empiricists. After all, if the statements in question are about the world, how can knowledge of their truth be independent of what goes on in it? Consider a statement such as “The standard meter is one meter in length.” This is clearly a statement about the world, but, the argument goes, it is known independently of experience, given that the standard meter specifi es what one meter is. In response, it can be pointed out that knowledge of this statement ultimately

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depends on traits of the world, namely, the particular length of the standard meter (whatever it turns out to be). So, it is unclear that this statement is a priori in the end: it is known on the basis of knowledge of the length of the standard meter (an empirical fact about the world). Not surprisingly, empiricists have traditionally been skeptical about the existence of synthetic a priori statements in the fi rst place. It is, of course, an issue whether mathematical statements are synthetic a priori . Kant famously argued that they are. But empiricists have very little reason to grant that, and may have concerns about the way Kant conceptualized the notions of analytic and synthetic. In fact, whether the subject is contained or not in the predicate crucially depends on how subjects and predicates are understood (for instance, are they formulated intensionally or extensionally?), and what the proper notion of containment is (e.g., should it be understood set-theoretically or in some other way?). However these notions are spelled out, arguably (pure) mathematical statements are not about the (empirical) world at all, given that they concern abstract mathematical objects, which are neither spatiotemporally located nor causally active. That every metric space is a topological space (but not vice versa ) is a fact about the relevant spaces (which are abstract if they exist at all) and not about any particular (partial) spatial instantiation of such spaces. Any such concrete (partial) instantiation depends on a suitable physical interpretation of the abstract notions of metric and topological spaces. Depending on how such interpretations are worked out, different instantiations end up being provided. Just consider the different metrics (metric functions) that characterize different metric spaces. The way in which such metrics are physically interpreted yields different instantiations of the corresponding spaces. The same metric space is compatible with a variety of distinct physical instances. These considerations illustrate that (pure) mathematical statements do not depend on concrete objects for their truth or falsity, and thus neither knowledge nor justifi cation of mathematical objects depend on knowledge or justifi cation of empirical facts. The result is that, properly understood, a priori knowledge and a priori justifi cation need not pose particular diffi culties for empiricists—assuming, of course, that there are any. We can now address the second kind of skepticism regarding the a priori : it challenges the very characterization of a priori knowledge. We start by considering an infl uential attempt at providing an analysis of this concept.

4 ANALYZING A PRIORI KNOWLEDGE A priori knowledge, we noted, is taken to be knowledge independent of experience. 1 Although the intuitive idea underlying the notion of a priori knowledge seems clear enough, it turns out to be extremely difﬁ cult to characterize properly what a priori knowledge is . In what follows, I illustrate this by arguing that a classic, and prima facie plausible, characterization of a priori knowledge—the one provided by Philip Kitcher ( 1980 , 2000 )—fails. Kitcher’s account has been criticized for being too strong (see, e.g., Harper 1986). As I argue here, however, the account turns out to be too weak , as it’s ultimately unable to distinguish empirical and a priori knowledge. In the end,

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the proposal seems unstable, since it seems to over-generate (turning clear cases of empirical knowledge into a priori knowledge) or under-generate (failing to classify clear cases of a priori knowledge as such). This raises a partial doubt, in light of the failure of this account, regarding the prospects of characterizing the notion of a priori knowledge more generally. Kitcher’s account of a priori knowledge was developed as part of an investigation— and, ultimately, a defense—of the claim that mathematics is not a priori ; thus, making it safe for empiricists. 2 So, one of the challenges the account needs to meet is not to have stacked the deck in the fi rst place. Kitcher is, of course, aware of the problem, and he explicitly tries to avoid this diffi culty. The proposed analysis is meant to capture the two crucial features of a priori knowledge: (i ) the fact that this type of knowledge doesn’t depend on experience, while acknowledging that (ii ) some of the concepts involved in a priori knowledge can only be obtained from experience. To accommodate ( ii ), one introduces the notion of experiences that are “suffi ciently rich for an agent X to entertain the proposition p .” That is, X has experiences that yield all the relevant concepts invoked in p . To accommodate (i ), the requirement of independence of experience, one introduces the idea that the same type of processes that produce, in the actual world, the belief that p in X will warrant that belief in all (epistemically) possible worlds. In this way, the justifi cation for an a priori belief does not depend on experience, given that processes of the same type will warrant that belief in different possible worlds. The defi nition is formulated in two steps. First, a priori knowledge is characterized in terms of a priori warrant . Second, a priori warrant is then formulated in terms of a process . Here is the account (which revisits and improves on an earlier account provided in Kitcher 1980 ): X knows a priori that p iff X knows that p and X ’s knowledge that p was produced by a process that is an a priori warrant for p . α is an a priori warrant for X ’s belief that p just in case α is a process such that for any sequence of experiences suffi ciently rich for X for p (a) some process of the same type could produce in X a belief that p ; (b) if a process of the same type were to produce in X a belief that p , then it would warrant X in believing that p ; (c) if a process of the same type were to produce in X a belief that p , then p . ( Kitcher 2000 : 67) As Kitcher notes, if only condition (a) is met, we obtain weak a priori knowledge . If conditions (b) and (c) are satisfi ed as well, we obtain strong a priori knowledge . Is this defi nition adequate? The responses have been unanimous in stressing that the defi nition is too strong (see, e.g., Parsons 1986; Harper 1986, Summerfi eld, Casullo, and Peacocke 1994). Talking about a particular a priori warrant, William Harper insists that it might “even approach the outrageously high standards set out by Philip Kitcher in his attempt to explicate a priori knowledge” (Harper 1986: 268 n. 8).3 As it turns out, however, I don’t think that Kitcher’s standards for apriority are too high. If anything, as will become clear below, they are not high enough .

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5 SKEPTICISM ABOUT A PRIORI KNOWLEDGE: SOME DIFFICULTIES The main problem with this account of a priori knowledge is that, ultimately, it cannot distinguish a priori and empirical knowledge, and so, it fails to deliver an extensionally adequate characterization of the a priori . To see why this is the case, let’s fi rst highlight what might be taken to be, initially, a benefi t of the view. Recall that, according to this account, the notion of a priori warrant is defi ned in terms of the notion of process . But the latter is left completely open in this proposal. No constraints are imposed on what counts as a process. (The term seems to be taken as a primitive.) It might be argued that the lack of constraints here is not without reason. It’s sensible to expect that any account of a priori knowledge should leave it open which processes yield this type of knowledge, whether they are proofs, conceptual analyses, or perhaps some sort of introspection.4 So, the openness of the notion of process might be taken as an advantage of the proposal. Moreover, the notion of process had better be left open indeed. After all, it’s not clear that a process adequate to defi ne the notion of a priori warrant could be exactly characterized without somehow presupposing the notion of the a priori : either the processes in question are extensionally equivalent to a priori warrants or they are not. If they aren’t , these processes clearly fail to characterize, in an extensionally correct way, the notion of an a priori warrant. If, however, the processes under consideration are extensionally equivalent to a priori warrants, then in virtue of what does the extensional equivalence obtain? To circumscribe correctly the class of a priori warrants, one needs to have already some sort of grasp of what counts as an a priori warrant, but this means that, ultimately, the notion of a priori has to be presupposed . To avoid this sort of circularity, it’s not by chance that the notion of process is then left completely open. But this openness comes with its price, and it’s in virtue of this openness that the account is unable to distinguish a priori and empirical knowledge. Let’s start with weak a priori knowledge. Consider the process of knowledge acquisition about quarks provided by particle accelerators, and consider a community of physicists X who, by using particle accelerators, have now formed the belief p that quarks have been detected. Of course, the community of physicists, understanding what quarks are, has had a series of experiences suffi ciently rich for p . The trouble is that particle accelerators yield a process of detection of quarks that immediately satisfi es condition (a) in the defi nition above. After all, for any sequence of experiences suffi ciently rich for the community of physicists X for the belief that p (i.e., that quarks have been detected), there is always a process of the same type—such as, a particle accelerator built in the other side of the planet—that could produce in X a belief that p . 5 In other words, according the account, particle accelerators provide us with (weak) a priori knowledge of quarks! Kitcher may grant this point, given that he himself acknowledges that the notion of weak a priori knowledge is far from adequate. But more than a simple inadequacy, this suggests that the notion of weak a priori knowledge is importantly problematic: in the end, it does not seem to be a notion of a priori knowledge. If the latter cannot

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be distinguished from empirical knowledge, there’s something deeply unsettling with the proposed account. But perhaps this is only a feature of weak a priori knowledge, and strong a priori knowledge may fare better. Alas, the same problem also plagues the latter notion. To see why this is so, let’s examine the particle accelerator case a little further. Is condition (b) in the account above satisfi ed in this case? This seems to be so. Let’s consider all epistemically possible worlds—in particular, worlds in which the community of physicists X has experiences suffi ciently rich for the belief in quarks p , and in which it is known that quarks exist. We should now consider processes of the same type as those that, in the actual world, produce in X the belief that p . But what would count as a process of the same type in this case? Certainly, at least other particle accelerators built in these epistemically accessible worlds. Clearly, any such particle accelerators also warrant the belief that quarks have been detected, given that it is a process of the same type as the one in the actual world. After all, if the processes are of the same type , presumably the same propositions will be true in each of them. (Compare: if two structures are isomorphic, the same propositions will be true in each structure.6 ) Note that this doesn’t require infallibility: a warranted belief can, of course, be false. But epistemic processes of the same type will validate the same beliefs; that is, in part, what makes them processes of the same type. In other words, epistemic processes are individuated, in part, by the beliefs they validate. Those beliefs that are true in one process will be true in other processes of the same type; those that are false in one process will be false in other processes of that type as well. As a result, in each of these worlds, the belief p , being generated by the same type of process, will be warranted. Condition (b) is met. In response, it might be argued that cases of empirical knowledge fail to satisfy condition (b). If it is merely nomologically (as opposed to metaphysically or logically) necessary that the world has a certain structure, then there will be worlds in which particle accelerators don’t give us a window into the structure of the world. Particle accelerators in those worlds will be like witch detectors in the actual world. There are no witches to be detected, and we are not justifi ed in thinking that the output of a witch detector tells us anything about the natural world. Ditto for particle accelerators in worlds where the laws of nature are different from the laws of nature in the actual world. This response, however, faces two diffi culties. First, if in some worlds particle accelerators fail to warrant X in believing that quarks exist, that means that, in these worlds, the accelerators in question end up not being processes of the same type as those that produce the corresponding beliefs in the actual world. After all, in the actual world, particle accelerators do warrant the physics community in believing in the existence of quarks. And if the processes are indeed of the same type , presumably they will warrant the same propositions. In other words, condition (b) is met—by contraposition. It may be argued that the condition to the effect that processes of the same type warrant the same propositions yields too easy a response to skepticism. If perception is a process that yields warranted beliefs in the actual world, but does not in an evil-demon world, then the process relied on in the latter is not of the same kind as

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the one invoked in the former. After all, true beliefs are generated in one case, and false ones in the other. Similarly, testimony arguably generates true beliefs, at least in the typical cases. In contrast, a testimony that conveys false information, and thus yields false beliefs, would be a different type of epistemic process, given the distinct outcomes in the two cases (true vs. false testimonies). But this would make skeptical challenges extremely easy to address, since the challenges can be set aside as engaging with a different kind of epistemic process rather than with veridical, truth-conducive procedures.7 In response, arguably one does not perceive in an evil-demon scenario, given that the beliefs that are yielded are not produced, as in the case of perception, by the objects they are about. Since this is a constitutive feature of perception, the experiences one undergoes in an evil-demon scenario are of a different kind, despite the many similarities they share with perception, particularly regarding their phenomenology. (The same point goes through in the case of hallucinations.) Similarly, a false testimony, as opposed to a true one, doesn’t typically yield true AQ: Please beliefs. Despite the similarities between the testimonies of a liar and those of a check if the truth-teller, there is a signifi cant epistemic difference between them: in one case, edits made to the sentence truth is preserved and transferred; in the other, it isn’t. It’s not unreasonable, thus, “Despite the to consider them as distinct epistemic processes. similarities Note, however, that this doesn’t provide a response to skepticism. After all, one between the...” retain typically doesn’t know when a testimony is true or when it is false, nor does one the intended know whether the scenario one experiences is of an evil demon or not. And without meaning. these pieces of knowledge in place, the skeptical challenges still stand. Returning now to the case of worlds involving quarks, a second diffi culty emerges. If in some worlds quarks don’t exist (just as witches don’t exist in the actual world), then, in these worlds, we had better not have knowledge of them. After all, how could we have knowledge of an object that doesn’t exist? So, in these worlds, we couldn’t have a priori knowledge of quarks, given that we simply don’t have knowledge of them. (Recall that, on this account, for one to know a priori that p , one needs fi rst to know that p .) This seems to be good news for the account. The trouble, however, is that in the worlds in which quarks do exist, processes of the same type as those that produce the belief in quarks in the actual world will also warrant X in believing in the existence of quarks. After all, it was the nonexistence of quarks that undermined X ’s warrant in p . In other words, in those worlds in which quarks exist, the warrant is no longer undermined. But, in this case, condition (b) will be, once again, satisfi ed. How about condition (c): is it met? Recall that we are considering all epistemically possible worlds in which the community of physicists X has experiences suffi ciently rich for the belief in quarks p . We are also considering processes of the same type as those that produce in X the belief that p . Given these assumptions, for exactly the same reason discussed above, in each of these worlds the proposition p (that quarks have been detected) will also come out true. Hence, this satisfi es condition (c). After all, is it possible that p is not true? Well, note that the fi rst requirement of the account of a priori knowledge is that X knows that p . Typically, this entails that p is true. So, p ’s truth follows from the assumption that X has knowledge that

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p . Moreover, given that the processes under consideration are of the same type (as those that yield the belief that p in X ), the same propositions will come out true in each of these processes (and as noted above, this doesn’t assume infallibility). But p is one of these propositions, and p is true in the actual world. Hence, condition (c) is also met. The result is then a counterexample to the proposed account of strong a priori knowledge. After all, it is typically assumed that knowledge of quarks is empirical rather than a priori . However, for the reasons provided above, according to the proposal under consideration, we have strong a priori knowledge of quarks! In the end, it seems that even the strong version cannot distinguish empirical and a priori knowledge.

6 SKEPTICISM ABOUT A PRIORI KNOWLEDGE: EPISTEMIC COMMUNITIES, MODALITY, AND WARRANT 6.1 Epistemic Communities on the Fringes Let me consider some ways in which one could try to resist the counterexample above. Adapting a point made in Kitcher (2000), one could suggest that, among the epistemically possible worlds, there is one in which the community of physicists X is actually on the fringes, and their methods of knowledge acquisition about quarks are taken by everyone else outside the community X to be mistaken. In this case, condition (b) of the account won’t be satisfi ed, given that X won’t be warranted in believing that p —or, at least, X won’t be taken to be warranted in believing that p . As a result, as one would expect, the account won’t entail that we have strong a priori knowledge of the detection of quarks. In response, note fi rst that it’s not clear that the proposed situation shows that condition (b) is not met. After all, to establish that (b) is not satisfi ed, we need to provide a process of the same type as the one that produces in X the belief that p , but which fails to warrant X in believing that p . Moving the community of physicists X to the fringes doesn’t establish that their methods for believing that p no longer warrant this belief. The fact that the methods are not taken to be warranted by anyone else outside X doesn’t entail that the methods aren’t warranted. Condition (b) is a claim about the warrant of the methods (or processes), not a claim about whether these methods are taken to be warranted. Second, if considering the community X to be on the fringes counts as an epistemic possibility that undermines condition (b), exactly the same move can be made in the case of mathematics. After all, consider a possible world in which a community of mathematicians Xʹ uses methods that are taken by everyone else as being mistaken, and so these methods are not taken to warrant X ʹ in believing that p ʹ (say, that inaccessible cardinals exist). This would then establish that X ʹ doesn’t have strong a priori knowledge of the existence of such mathematical entities.8 As a result, it follows from the account that in order for us not to have strong a priori knowledge of the existence of quarks (by allowing the possibility of communities of physicists on the fringes), we can’t have strong a priori knowledge of the existence of

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certain sets (given the possibility of communities of mathematicians on the fringes). In other words, for the account not to over-generate —that is, for it not to turn clear cases of empirical knowledge into a priori knowledge—it ends up under-generating — that is, it fails to classify clear cases of a priori knowledge as such. Thus, once again, a priori and empirical knowledge are not properly distinguished by the account.

6.2 Truth: Contingent and Necessary Matters But perhaps there is an important aspect of the distinction between a priori and empirical knowledge that the account above is able to capture. According to the traditional view about mathematics, mathematical objects are necessary objects: they exist in all possible worlds. And so, in particular, they exist in all epistemically possible worlds. Hence, in the case of mathematics, as one would expect, condition (c) can be met easily, given that the consequent of (c)—being necessarily true— will make (c) true. But the same doesn’t happen in the case of contingent objects, such as quarks. After all, quarks only exist in some worlds, and presumably only in some epistemically accessible worlds. So, when we deal with contingent matters, condition (c) is not immediately met. In response, note fi rst that it’s an advantage of the proposed account that it doesn’t identify a priori knowledge with knowledge of necessary truths (see Kitcher 2000 : 69). Since Kripke (1980) , it has become clear that apriority and necessity are different notions. After all, we may know a priori things that are contingent (such as our own existence), and we may fail to know—and hence fail to know a priori —necessary things (such as extraordinarily complex mathematical results). Hence, taking mathematical objects as necessary doesn’t lend any support per se to the proposed defi nition of a priori knowledge. Second, even in the case of mathematics, the fact that condition (c) might be easily met is irrelevant. After all, as noted, to have a priori knowledge we need fi rst to have knowledge (see the formulation of Kitcher’s account above). And given that we may not be able to know a particular mathematical result, for example, due to its complexity, we will not have a priori knowledge of this result, either. Exactly the same point applies to empirical knowledge.

6.3 Warrant: Contingent and Necessary Matters But let’s follow a little further the traditional account that insists that mathematical objects exist in all possible worlds. Presumably, in all epistemically accessible worlds, any process of the same type as those that produce in X a belief that p will also warrant X in believing that p . After all, no empirical constraints play any role in the warrant process, and presumably, nothing that could happen in any world would change the warrant in question. As a result, if we deal with necessary entities, condition (b) can be met without difﬁ culty. But the same doesn’t happen in the case of contingent objects, such as quarks, which presumably exist only in some epistemically accessible worlds. So, perhaps this aspect of the distinction between a priori and empirical knowledge—the fact that the latter, but not the former, depends on what goes on in each world—can be secured via this account.

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But it’s not clear that the account captures even this much. After all, consider again the possibility of a world with a community of mathematicians on the fringes, so that the techniques developed to warrant mathematical beliefs fail to warrant them. This possibility would undermine the claim that even mathematical results are known a priori . Hence, mathematical knowledge, just as empirical knowledge, turns out to be world-dependent. To overcome this difﬁ culty, one could reject the possibility of epistemic communities of mathematicians on the fringes, by insisting that they are epistemically impossible. But if this move is made, then epistemic communities of physicists in the fringes will also have to be ruled out. After all, it will simply beg the question to claim that, given that physics is not known a priori , the relevant epistemic possibilities can be entertained! But, as a result, and as we saw above, if the possibility of epistemic communities of physicists is not entertained, we will end up having a priori knowledge of physical objects! Once again, the proposed analysis is unstable: it either over-generates or under-generates.

7 CONCLUSION Can the criticism of this account of a priori knowledge be generalized to other attempts to characterize this type of knowledge? Or is this criticism restricted only to this particular characterization? It is possible that the point made here has a broader scope, although I will not try to argue for this claim here. Suffi ce to note that, in general, attempts to characterize an intuitive notion in an extensionally adequate way are typically plagued with diffi culties, as challenges to proposed conceptual analyses of knowledge (Gettier 1963 ), modality ( Shalkowski 1994 ), and logical consequence (Etchemendy 1990) clearly show. A priori knowledge is no exception in this respect. In conclusion, although the intuition that motivates one to develop an account of a priori knowledge is clear—a priori knowledge is knowledge independent of experience—it’s not so apparent that one can turn this intuition into a problem- free characterization of a priori knowledge. In particular, if we judge from the diffi culties faced by Kitcher’s account, it’s much less obvious that one can articulate an extensionally adequate account of a priori knowledge that captures the above intuition. Given this diffi culty, the problem of how to characterize a priori knowledge remains open, and thus, in the end, so does the answer to the question: just what is a priori knowledge? In light of the considerations discussed in this book, it seems that both a priori skepticism (particularly in its local form) and skepticism about a priori knowledge are alive and well.9

NOTES 1. As also noted, this is not meant to include the concepts required to understand the propositions that are known a priori . After all, we may know a priori propositions whose concepts can only be obtained from experience.

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2. See Kitcher ( 1983 ) for his defense of this view about mathematics. 3. Similar complaints can be found in, e.g., Parsons (1986 ) and Summerﬁ eld, Casullo, and Peacocke (1994 ). 4. In the end, maybe the issue regarding the “sources” of a priori knowledge shouldn’t be settled a priori ! 5. Note that condition (a) only requires the existence of a particular process that could produce in X the relevant belief that p , not a process that has actually done so. 6. In other words, isomorphic structures are elementarily equivalent. 7. My thanks to Baron Reed for pressing this point. 8. Kitcher agrees, of course, with this conclusion. But if he made this move as part of the characterization of a priori knowledge, he would be clearly stacking the deck. 9. My thanks go to Diego Machuca and Baron Reed for all their help and feedback on earlier versions of this work.

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