Epistemic closure, assumptions and topics of inquiry


According to the principle of epistemic closure, knowledge is closed under known implication. The principle is intuitive but it is problematic in some cases. Suppose you know you have hands and you know that ‘I have hands’ implies ‘I am not a brain-in-a-vat’. Does it follow that you know you are not a brain-in-a-vat? It seems not; it should not be so easy to refute skepticism. In this and similar cases, we are confronted with a puzzle: epistemic closure is an intuitive principle, but at times, it does not seem that we know by implication. In response to this puzzle, the literature has been mostly polarized between those who are willing to do away with epistemic closure and those who think we cannot live without it. But there is a third way. Here I formulate a restricted version of the principle of epistemic closure. In the standard version, the principle can range over any proposition; in the restricted version, it can only range over those propositions that are within the limits of a given epistemic inquiry and that do not constitute the underlying assumptions of the inquiry. If we adopt the restricted version, I argue, we can preserve the advantages associated with closure, while at the same time avoiding the puzzle I’ve described. My discussion also yields an insight into the nature of knowledge. I argue that knowledge is best understood as a topic-restricted notion, and that such a conception is a natural one given our limited cognitive resources.

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  1. 1.

    The expression “ruling out” is somewhat metaphorical. One way to make it explicit is to say that a possibility is “ruled out” when there is sufficiently strong evidence that the possibility in question does not obtain.

  2. 2.

    The principle of epistemic closure can be spelled out in different ways; see Kvanvig (2006). In particular, a number of authors have adopted a formulation of the principle in terms of competent inference; see, among others, Williamson (2000), Hawthorne (2004) and Barker and Adams (2010). Here is an example of such a formulation:

    Inference-based closure. Suppose \(S\) knows that \(p\); \(S\) knows that \(p\) implies \(q\); and \(S\) competently infers \(q\) from \(p\), thereby coming to believe \(q\). Then, \(S\) knows \(q\).

    The inference-based version is more explicit about the underlying psychological process that allows the epistemic agent to know a proposition through known implication. To ease exposition, I will work with the implication-based formulation, although the claim of this paper should also apply mutatis mutandis to inference-based closure.

  3. 3.

    See Nozick (1981), Dretske (1970) and Dretske (1971).

  4. 4.

    See Hawthorne (2004).

  5. 5.

    As Vogel (1990) has noted, the phenomenon generalizes to situations beyond lottery scenarios. Imagine John parked his car in a relatively crime-free neighborhood and went to eat in a nearby restaurant; unless we knew John returned to his car and found it still parked, we would be disinclined to take ourselves to know the proposition John’s car has not been stolen, despite its high probability. Further, consider this scenario: Mike is young and in good health; the proposition Mike will not suffer a sudden heart attack is highly probable, but again we would be disinclined to take ourselves to know it, unless we had more precise information about Mike.

  6. 6.

    Philosophers have advanced different accounts for why (H-1) and (not-H-2) are both true. Some have argued that our evidence is not causally or explanatorily connected with a lottery proposition, whereas it is connected with an ordinary proposition; see Achinstein (1978) and Nelkin (2000). Others have invoked modal notions such as “safety” and “tracking”, and argued that the belief in a lottery proposition is not safe nor tracking, whereas the belief in an ordinary proposition is safe or tracking; see Pritchard (2005), Williamson (2000), DeRose (1996) and Roush (2006). Still others have argued that we are not in a position to rule out the scenario in which a lottery proposition is false, but we are in such a position while considering an ordinary proposition; see Dretske (1971), Vogel (1999) and Lewis (1996).

  7. 7.

    Roush (2006) disagrees with (H-3); see note 13.

  8. 8.

    Here is a response. Let’s suppose Jack is not participating in any fair lottery. There is still a non-zero chance that he might inherit one million dollars from a distant relative. If one denies that Jack knows he will be unable to afford an expensive vacation because he is participating in a fair lottery, by parity of reasoning, one will also have to deny that Jack knows he will be unable to afford an expensive vacation because he might inherit one million dollars. But if so, there would be hardly any situation in which Jack can know he will be unable to afford an expensive vacation. This is a dangerously skeptical conclusion.

  9. 9.

    I am sympathetic with this point. I say something along these lines in Sect. 4. However, instead of construing the knowledge claim as ‘Jack knows [Jack will be unable to afford a luxury vacation assuming Jack does not win the lottery]’, I suggest that we construe the knowledge claim as ‘Jack knows [Jack will be unable to afford a luxury vacation] assuming Jack does not win the lottery’. The two construals differ in the scope of the knowledge operator; on the significance of this difference, see note 27.

  10. 10.

    Another version of the puzzle goes as follows:

    (T-1) I know that it is 3 PM (e.g. by reading my watch).

    (not-T-2) I do not know that my watch is not mistakenly reporting that it is 3 PM (or at least, I do not know that by simply reading my watch).

    (T-3) I know that ‘it is 3 PM’ implies ‘my watch is not mistakenly reporting that it is 3 PM’. (The implication holds because if it is 3 PM and my watch says that it is 3 PM, it follows that my watch is not mistakenly reporting that it is 3 PM.)

    (T-2) I know that my watch is not mistakenly reporting that it is 3 PM (by closure).

    For reasons of brevity, I do not discuss this version of the puzzle here, but I believe that the machinery offered in the paper should be able to handle it, as well.

  11. 11.

    Authors working in the framework of “relevant alternative theories of knowledge” have recently proposed restrictions for epistemic closure, e.g. Lawlor (2013) and Barker and Adams (2010).

  12. 12.

    Nozick (1981) proposed to characterize knowledge in terms of the modal notion of “tracking”. Dretske (1970) and Dretske (1971) held that knowledge that \(p\) consists in having a conclusive reason for \(p\) or in having ruled out all relevant alternatives in which \(p\) is false. On both accounts of knowledge, epistemic closure fails. Epistemologists who recently questioned epistemic closure include Sharon and Spectre (2013), Adams et al. (2012), Sherman and Harman (2011) and Lawlor (2005).

  13. 13.

    One way to solve our puzzle is by endorsing a form of skepticism or dogmatism. These solutions, however, are usually considered unattractive because they entail that we would know too little or too much. In other words, they contradict epistemic fallibilism which many epistemologists endorse. Another option consists in denying that lottery/heavyweight propositions follow from ordinary propositions. I find this difficult to accept, especially if the implication holds between ordinary and heavyweight propositions. Now, Mark has hands implies Mark is not a brain-in-a-vat, in virtue of the meaning of the words used, because a handed creature such as Mark cannot be a handless creature such as a brain. So long as we master the words ‘hands’ and ‘brain’, the implication is unobjectionable. More plausible is to think that an ordinary proposition need not imply a lottery proposition. For instance, if Jack will be unable to afford a luxury vacation, it need not follow that his lottery ticket is a loser, for even if he were to win the lottery and collect the money, he might suddenly lose the money and thus be unable to afford the vacation. All we can say, it would seem, is that if Jack wins, it is highly likely that he will be able to afford the vacation, or that if Jack wins, he will be able to afford the vacation on the assumption that he does not suddenly lose the money Roush (2006). This is a fair point, but all it shows is that sometimes a proposition follows from another only tentatively, not definitely. This conclusion fits well with the claim of this paper, i.e. that we should take into account the role that assumptions play in our epistemic inquiries (see Sect. 5). Finally, another way to tackle our puzzle is to take issue with the notion of knowledge. Some philosophers endorse probabilism. This is the view that a rational epistemic agent should assign degrees of belief to propositions in accordance with the axioms of probability theory; see Easwaran (2011). The relation between degrees of belief and knowledge is unclear, but the probabilist can argue that if an epistemic agent assigns the same degree of belief to two propositions, she cannot exhibit any form of epistemic asymmetry relative to these propositions. The probabilist, then, can argue that since ordinary, lottery, and heavyweight propositions are (roughly) equally probable, an epistemic agent cannot be in a position to know some and not others. The probabilist can try to solve our puzzle by resisting our intuitions about knowledge. Instead, I try to provide a solution that is more faithful to our intuitions.

  14. 14.

    See Feldman (1995).

  15. 15.

    Tucker (2010) makes a roughly similar point. The Warrant Transmission argument can also be seen with probabilities. Let \(Pr(p)=k\) and let \(Pr(p\rightarrow q)=1\). It follows that \(Pr(q)\ge k\), because \(Pr(p) \le Pr(q)\). This suggests that the (probabilistic) strength of the warrant does not decrease across an implication. Importantly, this holds provided \(Pr(p\rightarrow q)=1\).

  16. 16.

    See Wright (2004) and Pryor (2012). Similarly, Jack has warrant for believing that he will not be able to afford an expensive vacation, because, say, he has a modest salary and his bank account contains little money. Yet, Jack does not have warrant for believing that he will not win the lottery just because of his modest salary and the little money in his bank account.

  17. 17.

    For a more detailed discussion, see Hawthorne (2004) and Holliday (2014).

  18. 18.

    The proposition Mark has hands and Mark has hands and Mark is not a handless brain-in-a-vat are equivalent, and Mark should realize, on reflection, that they are equivalent. Yet, it is rather problematic to say that if Mark knows that he has hands, then Mark knows that he has hands and that he is not a brain-in-a-vat. This is at least as problematic as saying that Mark knows he is not a brain-in-a-vat. So, the equivalence principle (in its unrestricted form) seems suspect.

  19. 19.

    See Lewis (1996).

  20. 20.

    The view expounded here is that it is epistemic agents (inquirers or potential knowers) who rely on assumptions. This view should be contrasted with a possible alternative view, i.e. one on which knowledge attributors rely on assumptions when they utter sentences of the form ‘\(S\) knows that \(p\)’. I do not develop this alternative view here. See Sect. 8 for some remarks on this score.

  21. 21.

    See e.g. Harman and Sherman (2004), Sherman and Harman (2011) and Wright (2004).

  22. 22.

    Some might think that, in particular circumstances, it is possible to know anti-skeptical propositions such as ‘I am not dreaming right now’. I do not have a compelling response here. I maintain that, in most if not all circumstances, we do not know anti-skeptical propositions and that this is a starting point for the puzzle raised by Nozick and Dretske. If we thought we knew anti-skeptical propositions, this would presumably solve the puzzle, but it would solve it in a way that dismisses it at the outset.

  23. 23.

    Propositions that are deeply entrenched are similar to Wittgenstein’s bedrock propositions from On Certainty.

  24. 24.

    According to condition (d), a proposition such as Mark has hands AND Mark is not a brain-in-a-vat would not count as an assumption that underlines Mark’s knowledge that he has hands, because Mark has hands AND Mark is not a brain-in-a-vat alone guarantees the truth of Mark has hands.

  25. 25.

    On the role of assumptions in scientific inquiries, see chapter 3 of Longino (1990).

  26. 26.

    See Dretske (1971) and Lewis (1996).

  27. 27.

    I have in mind a narrow scope construal of the knowledge operator. Alternatively, ‘\(S\) knows that \(p\) on the basis of evidence \(e\)’ can be made explicit by using a wide scope construal, namely:

    S knows [\(p\) assuming \(\alpha _1, \alpha _2\), etc.] on the basis of evidence \(e\).

    I suspect that the wide scope analysis will turn out to be problematic. For suppose Jack claims that he will not be able to afford a luxury vacation. In response, suppose his interlocutor points out that Jack will receive one million dollars from a distant relative, as shown in a notarized letter Jack just received. At this point, Jack will have to correct himself in some way; he will have to admit he misspoke, or at least, he will have to clarify his earlier claim. It seems that Jack’s knowledge claim has been contradicted. But if the true content of Jack’s knowledge claim were simply that he will be unable to afford a luxury vacation assuming e.g. he will not win the lottery or assuming e.g. he will not inherit one million dollars, then Jack would not need to correct himself. It would be as though his interlocutor did not really contradict him. This seems odd, and that’s why I prefer the narrow scope construal.

  28. 28.

    I am here assuming that \(S\)’s knowledge of \(p\) rests on assumptions \(\alpha _1, \alpha _2, \dots \), while \(S\)’s knowledge of \(p\rightarrow q\) rest on no assumptions. This is a simplification. If the knowledge of \(p\rightarrow q\) also rests on assumptions, say, \(\beta _1, \beta _2, \dots \), the resulting knowledge of \(q\) will rest on assumptions \(\alpha _1, \alpha _2, \dots \) as well as \(\beta _1, \beta _2, \dots \)

  29. 29.

    This would violate, for example, clauses (b) and clause (d) from the previous section.

  30. 30.

    Some authors deny just that. For instance, Stine (1976) writes that ‘one does know what one takes for grated in normal circumstances’ (or assume, in my terminology). Although I am unable to offer an argument against this view, I can say that, on this view, we would know, by implication, that we are not brains-in-a-vat or we would know other anti-skeptical propositions insofar as they are part of what we take for granted. But claiming that we know anti-skeptical propositions strikes me as prima facie implausible; see also footnote 22.

  31. 31.

    On this score, some authors believe that knowledge is a form of cognitive achievement; it is the result of an epistemic effort. See, among others, Heller (1999), Vogel (1999), Greco (2010) and Holliday (Forthcoming).

  32. 32.

    Along similar lines, some recent work in linguistics and formal semantics attempts to offer an account of meaning in terms of what is at issue. For instance, Groenendijk (1999) distinguishes between indicative sentences, which provide the data, and interrogative sentences, which raise the issues in a conversation. The meaning of a sentence is thus understood as its context change potential, where a context encodes the data and what is at issue in a conversation. In other words, the meaning of a sentence amounts to how the sentence can change the data given what is at issue in the conversation.

  33. 33.

    The notion of a topic is related to other notions such as subject matter, aboutness, domain, framework. On the notion of aboutness and subject matter, see Yablo (2014). In order to define a topic (or a domain, subject matter, etc.), we can begin by defining a vocabulary for all the objects, entities, concepts we want to refer to (or use) in the course of our inquiry. The topic set, then, will contain at least the atomic propositions which can be constructed from our vocabulary. An interesting question is whether all complex propositions which can be constructed from the atomic ones will be in the topic set. To answer this, we should not forget that we, as epistemic inquirers, have bounded resources (see Sect. 7 for more clarifications).

  34. 34.

    An anonymous reviewer rightly points out that the notion of a topic of inquiry might be too restrictive, in the sense that we sometimes acquire new knowledge even without having a topic of inquiry explicitly in focus. One example of this is the airplane crash example mentioned in the text. Another example—which I owe to the reviewer—is this. Suppose I look outside and see the neighbor’s orange cat. Indeed, after looking outside I come to know that there’s an orange cat outside. This scenario suggests that we can gain new items of knowledge with no inquiry whatsoever, and if there is no inquiry, it’s difficult to say what one’s topic of inquiry should be. I wish to say two things in response to this objection. First, when we are looking outside the window, we are probably looking for something that is outside the window. Thus, the presence or absence of a cat of a certain color seems to be part of a (suitably defined) topic of inquiry, namely the topic of inquiry that answers the question ‘what is outside the window?’ If we were not looking for anything whatsoever outside the window, we would hardly notice the cat, and thus we would fail to learn that there’s an orange cat outside. The second point I want to make is that the notion of a topic of inquiry is not as rigid as it might appear at first. As emphasized in the text, the available evidence—whether the inquirers actively seek it or not—does affect what the topic of inquiry is going to be. At times, the evidence in support of a proposition is so compelling, forceful, and unambiguous that we do automatically come to know the proposition, almost with no effort whatsoever. What the orange cat example suggests is that, at times, it is the evidence we encounter which selects (parts of) our topic of inquiry. When this occurs, we are not fully in control of our topic of inquiry. This is unavoidable: our topic of inquiry is shaped by the interactions between the world and us as epistemic inquirers.

  35. 35.

    This formulation echoes Stephen Yablo’s proposal that epistemic closure applies so long as there is no change in subject matter (or topic of inquiry, in my terminology) (Yablo 2014). But Yablo’s conception of subject matter, I think, is not the same as my notion of topic of inquiry. Further, Yablo does not explicitly draw the connection between epistemic assumptions, a topic-restricted principle of closure, and our resource-bounded nature (see, in particular, Sects. 5 and 7 of this paper).

  36. 36.

    Here one might object that we have no guarantee that the Nozick–Dretske puzzle would not arise even for the truly ideal, unbounded agent who is endowed with infinite intellectual and empirical capabilities. Here is a brief, tentative response. The Nozick–Dretske puzzle arises provided we adopt a fallibilist epistemology according to which an epistemic agent can know things without ruling out all the possibilities of error and deception. Now, I do not think that a fallibilist epistemology is adequate to theorize about the ideal agent who is endowed with infinite intellectual and empirical capabilities. The agent with infinite empirical and intellectual capabilities should be able to rule out all possibilities in which a given proposition is false. But if we admit that the ideal agent can rule out all possibilities of error, the Nozick–Dretske puzzle would no longer arise.

  37. 37.

    This should make clear that I am referring to so-called attributor contextualism, according to which the truth-values of knowledge attributions depend on the attributor’s purposes, goals, expectations, practical interests, etc. See, among others, Cohen (1986) and DeRose (2002).

  38. 38.

    The agent’s evidence, or lack thereof, affects the agent’s justification, which, in turn, affects whether the agent is in a position to know or not.

  39. 39.

    One could say that insofar as knowledge attributions can be true or false, contextualism is also, though indirectly, a view about knowledge. I do not enter into this difficult question here.

  40. 40.

    This does not mean that the agent will fail to know simpliciter the propositions that are off topic. Consider two different topics, \(T1\) and \(T2\). It might well be that the epistemic agent can know certain propositions relative to \(T1\), while she fails to know them relative to \(T2\) because they are off topic.

  41. 41.

    For a similar, proposition-sensitive view of topics of inquiry, see Chapter 4 of Lawlor (2013).

  42. 42.

    To be sure, there is a way to resist this conclusion, namely by saying that there cannot be a topic of inquiry that includes whether we are brains-in-a-vat. A topic of inquiry—some might argue—is largely determined by the evidence that the inquirers reasonably expect to obtain. If so, it would be impractical for a group of epistemic inquirers to investigate propositions about which they have no reasonable hope to gather suitable evidence. I do not know if this line of argument is a defensible one, and I leave it as an open question whether it is or not.


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I would like to thank Krista Lawlor, Rahul Chaudhri, Carlos Nunez, Mark Crimmins, Samuel Asarnow, Peter Hawke, and two anonymous reviewers for helpful comments.

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Di Bello, M. Epistemic closure, assumptions and topics of inquiry. Synthese 191, 3977–4002 (2014). https://doi.org/10.1007/s11229-014-0522-2

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  • Assumption
  • Bounded rationality
  • Contextualism
  • Epistemic closure
  • Knowledge
  • Topic of inquiry