Valuable ignorance: delayed epistemic gratification

Abstract

A long line of epistemologists including Sosa (Epistemic explanations: a theory of telic normativity, and what it explains. Oxford University Press, Oxford, 2021), Feldman (The ethics of belief. Philos and Phenomenol Res 60:667–695, 2002), and Chisholm (Theory of knowledge, 2nd edn. Prentice-Hall, Englewood Cliffs, 2007) have argued that, at least for a certain class of questions that we take up, we should (or should aim to) close inquiry iff by closing inquiry we would meet a unique epistemic standard. I argue that no epistemic norm of this general form is true: there is not a single epistemic standard that demarcates the boundary between inquiries we are forbidden and obligated to close. In short, such norms are false because they are insensitive to the potentially ambitious epistemic goals that agents may permissibly bring to bear on an inquiry. Focusing particularly on knowledge-oriented versions of the norm, I argue that beliefless ignorance has a positive role to play in epistemic life by licensing prolonged inquiry into questions that we especially care about.

1 Introduction

I spent much of the morning searching for a runcible spoon. Searching was a physically active process: my eyes swiveled; my hands rummaged. In the end, I found what I was searching for: I then had the runcible spoon. I stopped searching.

While my eyes were swiveling and my hands rummaging, my mind was doing something too: it was wondering. I was wondering about the question “Where is my runcible spoon?” I was curious about its location and sensitive to new information about where it might be.¹ In the end, I came to know the answer to the question I was looking for. It was there, wrapped in a five-pound note. I stopped wondering.

Why did I stop searching for the runcible spoon once I had found it? It’d be awfully silly to search for something that I already had in my possession. There is a norm, it seems, governing searching and having: Don’t search for things you already have.

Once I realize that I have violated this norm—once I discover I’ve been searching and having the same thing at the same time—I could return to compliance with the norm in one of two ways: I could stop searching for the thing or I could stop having it in my possession. Here’s a funny fact: even though the norm just says, “don’t do both,” I always resolve this tension in the same way. I always choose to stop searching and never choose to stop having. I never chuck my keys into the distance so that my having them does not normatively conflict with my searching for them. Thus, the adage that you always find things in the last place you think to look.

Why the asymmetry? Because aims are asymmetrical. Because one searches for things in order to have them. Once I have something, there isn’t a reason to search for it anymore. If I’ve got a choice between searching and having, I choose having.

Plausibly, the same sorts of things that we’ve said about the physical activity of searching apply to the mental state of wondering. Just as there is a norm not to search and have things at the same time, there’s also this (epistemic) norm: don’t wonder about questions you already know the answer to. We occasionally stumble into doing this, as when we ask about a friend’s plans for the day only to recall that we’d asked before and easily would have remembered if we had bothered to reflect. But it’s epistemically embarrassing to have done so. Versions of this norm—or a similar norm prohibiting inquiring into Q while believing an answer to it—have been recently articulated by Hawthorne (2004), Fitzpatrick (2005), Whitcomb (2017), Friedman (2017, 2019b), Millson (2021), and Sapir & van Elswyk (forthcoming). If Friedman (2017, 2019b) in particular is right, the problem isn’t just that it’s incoherent to perform inquiring acts while knowing the answer to the question one is inquiring into—there is something incoherent about having an interrogative attitude toward a question while also having knowledgeably closed inquiry on the question. Notice that it’s quite odd to say either:

• 1. I know that the Yankees are winning, but are they winning? (cf. Friedman, 2017; Whitcomb, 2010).

  or

• 2. They know that the Yankees are winning, but they wonder: are the Yankees winning?

And a natural explanation for this is that such sentences indicate the violation of an epistemic norm. To wonder about Q while knowing the answer to Q is to mentally ask and answer a question at the same time—and that’s incoherent.

Relying on this tradition, I’ll assume that the following norm is true:

Ignorance Norm for Wondering: Necessarily, if one knows the complete answer to Q at t, then one ought not to wonder about Q at t.

Following Friedman (2017), the “ought” should be understood as having wide scope—the norm says: don’t both wonder about Q and know Q’s answer.

And once again, it seems that there’s an important asymmetry. One wonders about questions in order to know their answers.² Once I know the answer to a question, there isn’t a reason to wonder about it anymore. If I’ve got a choice between wondering about a question and knowing the answer (and I shouldn’t do both), I want to choose knowledge. Choosing wonder over knowledge would, it seems, be relevantly like chucking my keys into the distance upon finding them.

Here’s a principle that captures the thought that it’s defective to wonderingly inquire instead of knowing when knowledge is on offer:

The Knowledge Rule: If one is taking up the question Q, then one ought to close inquiry into Q iff one would (upon closing inquiry) thereby know the complete answer to Q.

Once again, the “ought” has wide scope. And one can always comply with The Knowledge Rule by not taking a question up. I will elaborate on the terminology of the rule in §2, but it’s worth flagging up front that I take outright believing³ an answer to a question to constitute closing inquiry into that question.

The Knowledge Rule is one avatar of a venerable tradition of biconditional norms on closing inquiry (or on outright belief). The most recent defense of a norm in that tradition belongs to Sosa, who builds a normative aim into the concept of judgmentally facing a question:

Sosa: “When one faces judgmentally a question whether p,” one aims “to affirm alethically re <p?> iff one’s alethic affirmation would be apt,” where such apt affirmation entails knowledge (Sosa, 2021: 51).

And although Chisholm and Feldman don’t talk about inquiry explicitly, they do talk about considering hypotheses or propositions. And question-directed states like wondering about a proposition or entertaining whether it is true seems to be among the prominent ways to consider a proposition.⁴ They propose these norms:

Chisholm: “We may assume that every person is subject to a purely intellectual requirement—that of trying his best to bring it about that for every proposition h that he considers, he accepts h if and only if h is true” (Chisholm, 1977: 14).

Feldman: “For any proposition p, time t, and person S, if S considers p at t, then S has a duty to have the [unique]⁵ attitude toward p that fits the evidence S has at t concerning p” (Feldman, 2002: 368).

These norms vary in several ways, most notably in the kind of epistemic aim that they take to be central to closing inquiry. Sosa focuses on apt affirmation, Chisholm truth, Feldman evidential fit. But they all share a biconditional structure: given that one is attending to a question, one ought to (or at least aims to) close inquiry iff by closing one would meet some particular epistemic threshold: if one’s belief would be apt, would fit the evidence, would be true.⁶

But these norms are all false. Or so I’ll argue. It’s not the case that there is any one epistemic standard such that, conditional on taking up a question, one is obligated to believe (or aim to believe) a complete answer just in case that belief would meet it. Expositionally, it’s useful to focus on just one such rule, and as author, I call dibs on picking which one. I will focus on The Knowledge Rule, as this is the version of such norms that seems most attractive to me. But I hope that, so long as one’s own favored rule does not require absolute certainty, it will be clear enough how what I say about The Knowledge Rule applies to variants of the rule. There will be some important (but not maximally good) threshold at which one could successfully close inquiry and a corresponding question about whether we should always close (or aim to close) those inquiries that have been taken up just in case we have the opportunity to do so successfully.

The Knowledge Rule is false, I will argue, because it is sometimes epistemically proper, from the vantage point of inquiry into Q, to be inquiringly ignorant about Q rather than to know the answer to Q. Even when knowledge is readily available. A guiding insight of the paper is that in epistemology, as in other domains, delayed gratification is a powerful strategy. One needn’t always settle for immediate knowledge: one can continue inquiry in hopes of even greater epistemic accomplishments (like knowing better, understanding, or being certain).

It turns out that ignorance has an instrumentally valuable role to play in our epistemic lives: it enables prolonged inquiry. This is surprising—to call someone ignorant is to insult them! But if this paper is right, it can be even worse to be a know-it-all.⁷

2 Terminology and assumptions

Ignorance can be valuable, I will argue, because it enables prolonged inquiry. But what is inquiry? And what is it for an inquiry to be open or closed? In this section, I clear the terminological ground.

2.1 Inquiry and wonder

To inquire involves having an inquiring state of mind that is goal-directed, question-directed, and characterized by sensitivity to new information (Friedman, 2017, 307–308). That isn’t necessarily all that inquiry is: perhaps inquiry also involves doing something with the aim of resolving the inquiry—not just having the curiosity itch but trying to scratch it.⁸ But prototypical inquiring attitudes include wondering or being curious about a question. Inquiring states are question-directed in that they take questions as objects: one wonders whether it will snow tonight or why the kids are being so quiet. One doesn’t wonder that it will snow tonight. Like inquiry, wondering is goal-directed: to have a question is to be on a quest for an answer. This paper will explore what kind of answer (or epistemic access to that answer) successfully resolves wonder-accompanied inquiry.

Friedman (2017) argues that all genuine inquiry involves an inquiring attitude. Following suit, I will limit the scope of this paper to inquiries that involve a wondering attitude on the part of the agent. Even if it turns out that “inquiry” in English picks out some activities that are not wonder-accompanied,⁹ wonder-accompanied inquiries are of special epistemic interest. And that’s because the questions that agents ask themselves are of special epistemic interest. The questions that agents ask themselves¹⁰ or wonder about stand to an agent’s beliefs roughly as QUDs stand to assertions.¹¹ As an analogue of a QUD, we might talk about an agent’s research agenda as containing the questions that they wonder about and that play a role in their belief formation.¹² Our beliefs are responsive to our wonderings—wondering motivates us to find an answer. And our wonderings are responsive to our beliefs—beliefs prompt us to ask new questions that expand or explain what we treat as known.¹³ For this reason, this paper will restrict its attention to inquiries that involve a wondering attitude on the part of the inquirer.

In the lexicon of this paper, all wonderings constitute open inquiries. If Q is on your research agenda, you have an open inquiry into Q. Inquiry into Q is closed for an agent when they believe a complete answer to Q.^{14, 15} The sort of belief that closes inquiry is full or outright belief only. It’s whatever it is that is the internal counterpart to flat-out, unhedged assertion. This concept of belief is, I suspect, a theoretical posit rather than an ordinary language unpacking of the English words “belief” (cf. Hawthorne et al., 2016, 1402).¹⁶ But it’s a concept worth positing: We want something that stands to our research agendas roughly as assertions stand to QUDs.

The Ignorance Norm says one shouldn’t inquire when one knows the relevant question’s complete answer. Hamblin (1973) thought that questions were sets of candidate answers. More recently, Groendijk and Stokhof (1984, 2011) have argued that questions impose a partition on possibility space, creating exhaustive and exclusive cells that determine what count as its answers. For instance, in a context where it’s known that Ann, Beatrice, and Chani are the only racers, the question < Who won the race? > divides worlds into three discrete cells according to which Ann wins, Beatrice wins, and Chani wins. An answer is partial, in Groenendijk’ and Stokhof’s sense, when it rules out one of the cells as containing the actual world. So, ‘Ann came in last’ is a partial answer to the question since it rules out any of the worlds in the cell in which Ann wins the race. Complete answers, in contrast, rule out all but one of the cells as the home of the actual world. ‘Beatrice won by a step’ or, more simply, ‘Beatrice won’ are complete answers to the question. The Ignorance Norm—and the Knowledge Rule too—utilize the concept of complete answerhood because it’s intuitively at least sometimes permissible to continue wondering about a question that is partially but not fully resolved: it’s fine to wonder who was at the party even if I can already rule out answers according to which my best friend was there. Complete answerhood is not the only notion of full question resolution on offer in the semantics of questions, but in keeping with the literature, we’ll use it as the default concept here.¹⁷

Here are some consequences of the definitions so far. First it isn’t definitionally ruled out that a question could be open and closed for an agent at the same time: if it is possible for an agent to wonder about Q while also believing an answer to Q, then inquiry into Q is both open and closed for them. Second, one doesn’t have to have inquired into Q in the past for inquiry into Q to be closed now. If I have always believed that Santa exists, the question whether Santa exists is closed but has never been (merely) open for me. Third, not all inquiries that were open before but are not open now are thereby closed. When I stopped wondering how Santa delivers all those presents in one night, I ended inquiry without closing it, for I did not replace my wondering with a belief in a complete answer to the corresponding question (the inquiry wasn’t closed, it was abandoned). Finally, some questions get stuck (or set) open, but their being set is not another way of being closed. Agnostics have not closed inquiry on whether God exists even if they are convinced that they are unlikely to ever form an opinion one way or the other.

I now want to introduce a new notion, that of taking up a question.¹⁸ When one believes a proposition as an answer to a particular question, one counts as taking that question up (and also resolving it). But the more interesting space, for our purposes, is when one takes up a question without having yet resolved it. It’s one thing to wonder about a question or to have it on one’s research agenda. But sometimes one aims to do more: sometimes one gives it one’s attention and tries to answer it. This expresses an active inquiry or front-of-mind wondering. But usually, one doesn’t simply try to answer a question no matter what. I could try to answer a complicated multiplication question by believing the first answer that comes to mind. But that’s not good epistemic behavior—not for forming outright beliefs. When I take up a question, I try to answer the question to a certain epistemic standard. Our question now: What is (or should) that standard (be)? Given that I’ve taken a question up and that my primary aim at the moment is to do well in my inquiry into Q, how should I decide whether to close inquiry or continue it?

As we’ve already seen, one plausible answer to this question—and the target of this paper—is The Knowledge Rule:

The Knowledge Rule: If one is taking up the question Q, then one should close inquiry into Q iff one would (upon closing inquiry) thereby know the complete answer to Q.

I find it helpful to break up the embedded biconditional in The Knowledge Rule into two directions:

LTR: If one is taking up the question Q, then: one ought to close inquiry into Q only if one would (upon closing inquiry) thereby know the complete answer to Q.

This is something like a knowledge norm on closing inquiry. Or at least on closing inquiry to an important class of questions (the questions one takes up). Although not uncontroversial, I find this norm, or something in its vicinity, quite plausible.¹⁹ Nothing that I say here will challenge the left-to-right direction. In fact, I will assume for the remainder of the argument that the LTR direction is true. Now let’s look at the other direction:

RTL: If one is taking up the question Q, then: one ought to close inquiry into Q if one would (upon closing inquiry) thereby know the complete answer to Q.

This says once you can have knowledge, you ought to take it (given that you are taking the question up). Once you’ve taken up a question, you aim to come to close inquiry if doing so meets a certain standard. The RTL direction says that knowledge is the standard one should hold inquiry to.

Since RTL will be the main target of the paper, it’s useful to be as clear as possible about its implications. RTL is a wide-scope norm. In that sense, its commitments are rather weak. It prohibits a certain combination of attitudes and states. To conform with RTL, don’t do this combination of things: (1) take up the question Q, (2) be in a situation such that were you to close inquiry on Q, you would do so knowledgeably, and yet (3) fail to close inquiry into Q. But by itself, this norm gives no advice on how to escape this tension should you find yourself in violation of the rule.

The Knowledge Rule is attractively simple and (given that we are taking up a question and that facts about whether we would know the answer to a question are fixed) cleanly partitions those cases in which we should close inquiry from those we should not. As is often the case, however, the simple answer runs into difficulty upon closer examination. And it isn’t just that The Knowledge Rule is false in some nitpicky way: it’s false in a way that gives us insight into the way that ignorance can play an epistemically useful role in our lives. In particular, ignorance that leaves inquiry open can enable us to be especially ambitious inquirers. This is the argument to which we now turn.

3 On behalf of continuing inquiry

In the remainder of this paper, I offer an argument against The Knowledge Rule. The argument has two parts. In Sect. 3.1, I lay out two kinds of thresholds that plausibly matter for determining when to close inquiry: a minimum threshold determined by the practice of inquiry and an agential threshold determined by the agent’s domain-internal goals. The Knowledge Rule, however, only has room for one threshold. If both thresholds discussed in Sect. 3.1 are normatively relevant, then there is conceptual space for cases that counterexample the Knowledge Rule. In Sect. 3.2, I argue that there are cases that fit the space—cases of agents who permissibly continue inquiry into a question even though, if they were to close inquiry, they would do so knowledgeably. And that those agents are permitted to do this because they are properly aiming at an agential threshold that requires more than mere knowledge. In Sect. 3.3, I respond to objections.

3.1 Two thresholds for closing inquiry

Elle Woods is studying for her upcoming exam at Harvard Law School. It’s late. She’s studied very hard. And she’s trying to decide whether she should close her books and fall asleep or stay awake and keep studying. She’s tired and would quite like to go to bed, but not so tired that she’d underperform on her test tomorrow if she stayed up a bit longer. Question: should she stop or continue studying?

Here’s one consideration that matters: If Elle stopped studying now, would she pass the exam? Unfortunately, Elle might not be in a position to know the answer to that question as she’s furiously taking notes on Gordon v. Steele. But if she knew whether she had already studied enough to pass, that might help her decide whether to keep studying or not. And whether or not she in fact continues studying is plausibly sensitive to how confident she is that she would pass without further study.

Then again, it might not. After all, Elle Woods might have more ambitious goals than merely passing: she might want to get an A. If so, here’s another consideration that matters: If Elle stopped studying now, would she get an A on the exam? Unfortunately, Elle might not be in a position to know the answer to that question as she’s furiously taking notes on Gordon v. Steele. But if she knew whether she had already studied enough to get an A, that might help her decide whether to keep studying or not. And whether or not she in fact continues studying is plausibly sensitive to how confident she is that she would get an A without further study.

It seems that there are (at least) two thresholds that are relevant for when Elle should stop studying. The first is a threshold that Elle is subject to merely in virtue of being an academic student: her performance on the exam only counts as good enough if she passes. The second threshold depends on Elle’s own academic goals. In general, we can think of there being two different (potential) sources for normatively relevant threshold-setting in a domain: the sort that one has in virtue of the activity and the sort that one has in virtue of one’s own domain-internal aims. Intuitively, if Elle got a B on the exam, her performance would be successful in one sense and unsuccessful in another. The “two thresholds” account of examinations makes sense of this: getting a B would easily clear the first threshold that Elle must clear as an academic student (to pass) while falling short of the threshold that Elle must clear to satisfy the academic goal that she has set for herself.

One final observation: We can imagine Elle being explicitly reflective about these two aims while she is studying. For example, she might ask herself, “Have I studied hard enough to pass?” or “Have I studied hard enough to get an A?” at regular intervals throughout the night. This is certainly possible. But I think it’s more realistic to imagine that, at a certain point, Elle simply feels like she is prepared enough to take the exam, without necessarily forming an explicit judgment about what grade she is likely to get upon sitting for it. At what point in the evening Elle feels prepared enough depends (among other things) on the grade she would be happy with, but she need not be thinking explicitly about what grade she will get as she makes the decision to continue studying or not. She can rely simply on whether she feels prepared enough.

With this background in mind, let’s consider the following academic norm for studying:

The Pass Rule for Studying: If one is taking an exam E, then one should stop studying for E iff one would thereby be positioned to pass E upon taking it without further study.

This rule can’t be right. For it is perfectly reasonable for Elle to continue studying even after she has reached the point that she would pass upon sitting without further study. After all, Elle may want to get an A. Let’s consider, then, another rule:

The ‘A’ Rule for Studying: If one is taking an exam E, then one should stop studying for E iff one would thereby be positioned to get an ‘A’ on E upon taking it without further study.

But this rule is no better. After all, Elle doesn’t have to aim for an A. If she merely wants to graduate, then it’s fine for her to stop studying well before she has studied enough to have any hope of securing an A. Indeed, there isn’t any grade for which there is a general norm that one should stop studying iff one would thereby be positioned to get that particular grade upon taking an exam. And that’s because any norm like that would be insensitive to the agent’s own academic aims for their performance. When a student should stop studying depends, in part, on what grade they want to get.

I contend that The Knowledge Rule similarly goes wrong by being insensitive to an agent’s own epistemic aims in taking up an inquiry. Deciding when to close inquiry is relevantly like deciding when to stop studying for an exam. Both decisions limit our sensitivity to future evidence. And for both, there is a minimum threshold that we must pass as students or inquirers and also an agential threshold set by our own aims within the activity.

Just as there is a “passing grade” standard that students have just in virtue of being students, so there is a standard that inquirers have just in virtue of being inquirers. In order to successfully close an inquiry, an agent must come to know the answer to the question they inquired into. This is the LTR direction of The Knowledge Rule:

LTR: If one is taking up the question Q, then: one ought to close inquiry into Q only if one would (upon closing inquiry) thereby know the complete answer to Q.

Since I am assuming the LTR direction of the Knowledge Rule, I won’t say any more about it here. Others have argued that epistemic accomplishments less than knowledge do not satisfy obligations to close an inquiry (Kelp, 2014, 2021).

But the minimum threshold is not the only threshold relevant to successful performance. Just as it matters to when a student should stop studying what grade they are aiming for, it matters to when an agent should close inquiry what their epistemic aims are. Like a student who aims not just to pass but to get an A + , it’s laudable to seek the higher epistemic goods. It’s good to seek not just knowledge but knowledge + .²⁰

What is knowledge + ? Knowledge + , as I define it, is the collection of epistemic goods that (1) entail knowledge and (2) are better than mere knowledge. Here are some examples. It’s good to know, but it’s even better to know that one knows (KK): when you know that you know, you don’t just know the answer, you know that you got it right. KK entails knowledge. So, KK is one variety of knowledge + . Alternatively, knowing requires a certain degree of justification. But one can have knowledge with even more justification than knowledge requires. Knowledge-with-extra-justification entails and is better than knowledge: knowledge-with-extra-justification is thus another variety of knowledge + . If you understand p well enough to explain how you know that p, then you also know that p. But it’s better to understand well enough to explain how you know that p than merely to know that p, so that too is a kind of knowledge + . The same goes for knowing full well or being rationally certain.

I’ve suggested that there are two thresholds that are relevant to consider when evaluating an agent’s decision to close inquiry: first, the minimum success condition (knowledge) that we inherit just in virtue of being inquirers, and second the epistemic aim that an inquirer themselves brings to inquiry (which may involve a species of knowledge +). In the very best cases, when an agent closes inquiry, they clear both thresholds. But notice that I’ve avoided saying that agents are always required to meet both thresholds. Or even that meeting the second threshold would always be a good thing. For notice that agents may set their sights either too low or too high. In order for it to be a good thing for an agent to meet the second threshold, the epistemic aim of the inquirer must be appropriate.

I don’t have a full theory about the range of epistemic aims that are appropriate for an agent to bring to bear on an inquiry. But I want to argue that agents who are aiming for knowledge + are operating in the good range—at least often enough.

Let’s reconsider Elle’s study night. Elle is a bright student, and it’s reasonable for her to aim for an A if she wants it. But there are some things it would not be reasonable for her to aim for. Or at any rate, there are some grades she could aim for such that her aims would cease to clearly influence our judgment about when she should stop studying and go to bed. Elle might set her sights too low by aiming for a 50% (a failing grade). After all, no matter what Elle’s own goals are, she won’t count as successfully taking the exam unless she passes. If Elle sets her sights too low, her own academic aim in studying for an exam won’t figure into our evaluation of when she should stop studying: other things being equal, she should keep studying at least until she can pass.

Plausibly, Elle could also set her sights too high. Imagine that no one has ever got a perfect score on one of Professor Stromwell’s exams—they are, by design, next to impossibly difficult. One might think that it isn’t really appropriate for Elle to aim for a perfect score on such an exam. After all, getting a perfect score just isn’t realistically achievable.

In aiming for knowledge + , inquirers do not (typically) set their sights too low. By definition, knowledge + entails knowledge; so, achieving knowledge + meets the minimum threshold for epistemic success for closing inquiry.

The more worrying possibility is that by aiming for knowledge + , agents sometimes set their epistemic sights too high. Suppose, for instance, that I am walking down the street and see a parked car. I am in a position to know that it is a parked car. But I could try to aim for rational certainty, a species of knowledge + . I could go kick the tires to make sure it isn’t a cleverly designed car façade or ask passersby if they also see a car there to rule out that I’m hallucinating it, or work through Al-Ghazali’s dream arguments. This sort of behavior strikes us as bizarre, epistemically fetishistic even. I might achieve a kind of knowledge + by testing additional evidence for the car’s existence, but the sort of agent who cannot resist such inquiries—like the student who can’t help but aim for a perfect score—seems destined for a bad end. Aiming for mere knowledge rather than knowledge + in our inquiries would keep us safe from this kind of epistemically self-destructive behavior.²¹

As an aside, I think it’s not totally obvious that this (admittedly) wacky behavior is always wrong epistemically. We sometimes encourage our students to experiment with setting Cartesian standards for belief, at least from the safety of the epistemology room.²² But the hard line is not required. Here is what is important for my argument: for at least some questions, there is a gap between the minimum success condition for closing inquiry (knowledge) and the highest epistemic status that an agent could reasonably aim for (some species of knowledge +). Epistemically, we should mind the gap. I’ll argue for this in §3.2 by way of example, presenting cases in which it seems permissible for agents to aim for some species of knowledge + .

But I also think it’s the intuitive verdict. Perhaps we shouldn’t aim for anything like rational certainty for perceptual beliefs, for instance, on the grounds that rational certainty isn’t a realistic goal in that domain. But we are often capable of having beliefs that are modestly more justified than knowledge requires. Compare S₁ who knows that p on the basis of some evidential set E to S₂ who knows that p on the basis of E⁺, where E⁺ includes all the evidence in E plus additional evidence that further supports p. S₁ and S₂ both have knowledge that p, but S₂ has knowledge with a greater degree of justification. So, S₂ has a kind of knowledge + . But surely we are often in a position to have the kind of knowledge + that S₂ exhibits.

Before turning to Sect. 3.2, let’s recap. We’ve found a reason to be suspicious of the Knowledge Rule: it is insensitive to the agent’s epistemic goals during an inquiry. It says that knowledge is the standard an agent should hold inquiry to but makes no mention of the (perhaps higher) individual aims an inquirer may bring to an inquiry. This isn’t to disparage those who stop at knowledge. Much of the epistemic life can be run effectively on the pass/fail model. But thinking about a second threshold for successfully closing inquiry—closing inquiry when an agent’s legitimate epistemic goals have been reached—allows us to recognize a wider range of achievements. We now have the conceptual framework required to consider cases of inquirers who aim for knowledge + .

3.2 Cases

In this section, I develop cases involving agents who are aiming for knowledge + in their inquiries. From our third-person vantage point, we’ll be able to tell that the agents are already in a position to know the answer to the question that they are inquiring into and, indeed, would come to know the answer if they closed inquiry immediately. Although the inquirers themselves do not know that this is their epistemic position. But given that their aim is knowledge + , and that they have a reasonable chance of obtaining knowledge + if they continue inquiry, it’s intuitive (I will argue) that it is permissible for the agents to continue wondering. If that’s correct, then the RTL direction of The Knowledge Rule is false. Here are the cases:

3.2.1 The detective

A detective is working through several bins of evidence to determine the innocence or guilt of a suspect. After looking through the evidence in 19 of 20 bins, the detective is extremely confident, and rightly so, that the suspect is innocent. Accordingly, she is also reasonably quite confident that the evidence in the 20th bin will also support, or at least not overturn, what the detective takes to be a compelling case for the suspect’s innocence. And indeed, it will turn out that the 20th bin does not contain any defeaters for her belief, the suspect is innocent, and there aren’t any Gettier-traps or other knowledge-defusing devices nearby. If there hadn’t been a 20th bin, she would immediately close inquiry, coming to know the complete answer to the question of whether the suspect is innocent. But there is a 20th bin. And she remains curious. After all, there is new, easily obtainable evidence that is available to her. And she won’t feel satisfied with her inquiry until she has considered the rest of the easily obtainable evidence in the final bin. After all, she wants to be thorough—she wants to form her judgment on as wide a basis of evidence as has been collected. She continues wondering whether the suspect is innocent until after she looks at the final box.

⁂

3.2.2 Coco’s cuckoo

Till and Coco see a cuckoo bird on the other side of their garden. “That’s a cuckoo,” Till tells Coco knowledgeably. That could have settled the matter for both of them. But it didn’t. Coco—who has recently but enthusiastically taken up birding—isn’t quite ready to agree. She normally takes Till’s word for things—she’s not the untrusting sort; however, she takes herself to have higher epistemic standards than Till on this particular topic. And while she doesn’t have any reason to think Till is unreliable on this question, she (rightly) thinks her interest in birding has made her a bit more reliable than Till. And so, she elects to take her own counsel. “You’re probably right,” Coco admits. “But what if it’s a young sparrowhawk instead? I’m going to get a closer look and compare it to the description in Collins Bird Guide.” Coco fetches her book and binoculars, cautiously inches partway across the garden, and meticulously studies the bird’s appearance against the entry in the guide. Eventually, Coco feels she has been thorough enough for birding standards: “Yep—that’s a cuckoo,” Coco concludes, cheerily. “You were right all along.”

⁂

3.2.3 The philosophy student

A philosophy student is studying Descartes’ cogito and trying to make sense of what the structure of the argument is. They have read the relevant sections of the Meditations several times and listened closely to their instructor’s interpretation of the argument. They are a good reader of the text, and their philosophy instructor did indeed present an accurate representation of Descartes’ argument. If on an exam they were asked the question, “What is the structure of the cogito?” they would give a knowledgeable response.²³ Nevertheless, the philosophy student is deeply interested in the way that the cogito works, and they remain curious about the argument’s structure. They are not yet satisfied with their inquiry, and this is partly explained by their (perhaps unarticulated) interest in really understanding the argument in a way that enables them to explain how they know (to themselves or to someone else). Instead of committing to a belief about the structure of the argument now, they continue wondering and start reading the SEP article on Descartes that evening.

⁂

In all three cases, it’s extremely intuitive that it’s permissible for the subject to continue wondering until they’ve gathered enough evidence to meet their lofty epistemic goals. Maybe it’s not wrong of them to stop inquiry right now. After all, their evidential situations are already quite good. But we think the ones who continue curiously inquiring have done even better, epistemically speaking. They have been more thorough, more rigorous, more epistemically ambitious. And, plausibly, part of the reason they have been more rigorous is that they were (permissibly) aiming for more than mere knowledge. They wanted to be extra justified or extra confident or to understand well enough to explain: they wanted knowledge + .

In part, these cases draw on insights from Raz (1975: 37–38) and Schroeder (2012: 471–72) that the availability of future evidence—the 20th bin of evidence, Collins Bird Guide, the SEP article—can make a difference to when inquiry is best closed.²⁴ But in the cases as I’ve presented them, the availability of future evidence plays a secondary role. What’s most important and distinctive about these cases is that the agents are aiming for an especially ambitious kind of epistemic access—something more ambitious than mere knowledge. The availability of future evidence ensures that these ambitions are not obviously going to be frustrated, and so that the agents’ ambitions are sensible given the constraints of their epistemic circumstances. They aren’t obviously in the situation of the student who tries to get a perfect score on Professor Stromwell’s exam. But it’s their epistemic aims themselves that are doing the normative work.

Notice that the availability of future evidence does not, by itself, prevent the protagonists from being in a position to know. After all, there’s almost always more evidence to be had on a given question—we can seek out more testimony, run additional tests, read one more paper, and so on. But even when we’ve been responsible enough in our evidence-gathering to know the answer to our questions, we often still face the choice of whether to continue wondering or not. That’s the choice the paper’s protagonists face.

In the next section, I’ll consider some alternative interpretations of the cases above. But let’s take stock now. The RTL direction of The Knowledge Rule reads thus:

RTL: If one is taking up the question Q, then: one ought to close inquiry into Q if one would thereby (upon closing inquiry) know the complete answer to Q.

Recall that this wide-scope norm prohibits doing the following three things simultaneously: (1) taking up the question Q, (2) being in a situation such that were one to close inquiry on Q, one would do so knowledgeably, and yet (3) failing to close inquiry into Q. All three of the protagonists meet each of those conditions at a certain point in the story. Coco, for instance, is taking up a question as she actively inquires into whether the bird is a cuckoo, pulling out her binoculars and Collins Bird Guide. Coco is also such that were she to close inquiry on that question at that time, she would do so knowledgeably. After all, were she to close inquiry then, she would knowledgeably take Till’s word for it that the bird is a cuckoo. And third, despite this, she has not yet closed inquiry into Q. Rather, she is keeping an open mind during her investigation. For the detective and the philosophy student, mutatis mutandis. If RTL were true, Coco and her fellow protagonists would be epistemically criticizable. Nevertheless, these agents, motivated by ambitious but not unachievable epistemic standards, were behaving fine epistemically. Perhaps even admirably. This judgment contradicts RTL.

In this subsection, I’ve argued that it is sometimes permissible for agents to continue inquiry even though they would know the answer to their question by closing inquiry. They can continue inquiry if they are aiming for knowledge + , at least if they are likely enough to secure knowledge + by inquiring further.

Knowledge is great. But it isn’t the best thing epistemically. One can know, but one can also know better, or know that one knows, or understand, or be rationally certain. Keeping inquiry open enables agents to aim for these higher epistemic goods. Agents may not have to raise their epistemic standards in this way, but they are permitted to.

3.3 Responding to objections: in defense of extended curiosity

I’ve suggested that each of our three protagonists above have continued wondering about their initial questions past a point at which they could have had knowledge and are praiseworthy for doing so in a distinctively epistemic way. In this subsection, I’ll consider objections to this interpretation.

Objection One: The protagonists can’t permissibly wonder; they can only permissibly gather evidence. The detective could simultaneously know that the suspect is innocent and still examine the contents of the final box of evidence. Coco the birder could know the bird is a cuckoo while still flipping through Collins Bird Guide. According to the objector, while our protagonists may indeed verify what they already know by looking at additional evidence, they are not epistemically permitted to wonder about the question they are gathering evidence for.²⁵

To be clear, I’m not arguing that it would be impermissible for the protagonists in our story to knowledgeably verify what they already know rather than to continue wondering what the answer is. I just don’t think our intuition that the protagonists are behaving permissibly depends on interpreting them as not really wondering about the questions they are gathering evidence for.

Here’s one reason I find it unconvincing that the knowledgeable verifier interpretation is the only one in which the protagonists are behaving permissibly: it’s most natural to think that the protagonists continue gathering evidence because they are genuinely curious, not despite their lack of curiosity. Perhaps the detective is just going through the motions and doing their due diligence by responsibly looking at every bin of evidence. But Coco the birder and the philosophy student continue inquiring because they are gripped by a question they continue to wonder about. Their intense curiosity manifests itself as a disposition to look for especially strong evidence. A more promising objection is to insist that, if our protagonists really are behaving permissibly, their curiosity shifts shape over the course of the stories. That’s the objection to which we now turn.

Objection Two: The protagonists are wondering about different questions at the beginning and end of the story.²⁶ Our protagonists are most naturally interpreted as manifesting genuine curiosity as they gather evidence. But one might think that the question that they are inquiring into changes over time. Initially, Coco the birder, for instance, is asking whether that is a cuckoo or not. But in the middle of the story, Coco is wondering something else, perhaps the question whether that is a cuckoo or a young sparrowhawk or the question whether I (Coco) know that is a cuckoo.

Again, I don’t want to insist that it would be wrong for Coco to proceed in that way, by asking different questions over the course of the inquiry. I just don’t think that our intuition that the protagonists are behaving well depends on stipulating that they change questions mid-inquiry.

First, not every way of changing the question would thereby allow agents to knowledgeably resolve their initial questions. Suppose, for instance, that Coco starts by wondering whether that is a cuckoo or not and then shifts to wondering whether that is a cuckoo or a young sparrowhawk. That’s fine and well, but Coco shouldn’t believe that is a cuckoo as the answer to whether that is a cuckoo or not if she is still wondering whether that is a cuckoo or a young sparrowhawk. Notice that (3).

• 3. #I know that is a cuckoo, but I wonder: is it a cuckoo?

  is not improved by switching to (4):

• 4. #I know that is a cuckoo, but I wonder: is it a cuckoo or a young sparrowhawk?

But perhaps Coco’s question shifts in a different way: perhaps she starts to wonder a question whose (known) answer would give her a variety of knowledge + with respect to (the original question of) whether that is a cuckoo. That’s what she is aiming at, after all. Imagine, for instance, that she starts by wondering whether that is a cuckoo but then starts to wonder whether I know that is a cuckoo and consults Collins Bird Guide because of her curiosity in this new question.

I think this gets the (typical) phenomenology of fascination wrong. Notice that the subjects of the two questions are importantly different. The first is about a bird, the second is about Coco and her epistemic accomplishments. But Coco a birder nor an epistemologist! Coco is interested in the question whether that is a cuckoo not whether I know that is a cuckoo. That’s why she picks up Collins Bird Guide and not Chisholm’s Theory of Knowledge. And what’s wrong, after all, if she continues to focus on what the bird is rather than what she knows? Birds, not knowers, are the object of her fascination. When we really care deeply about a topic or have ambitious standards for an inquiry, we tend to hold onto our curiosity in the initial question longer. And this seems like a virtue, not a vice. The depth of such curiosity motivates exceptional epistemic performance.

Objection Three: It is necessarily an epistemic failing that the agents wonder about a question given that they are in a position to know its answer. After all, it’d be absurd for the agents to say, “I’m in a position to know that p, but I’ll keep wondering whether p anyway.” ²⁷

Given the knowledge norm for assertion,²⁸ it would indeed be absurd for any of our protagonists to say (and so be obliged to know) that they are in a position to know that a particular answer to their question is true while still inquiring into it. That’s because p is an obvious and salient entailment of “I’m in a position to know that p.” So, anyone who knows “I’m in a position to know that p” had better know that p too: omitting this obvious entailment would be an epistemic failing, not a success story. And anyone who knows that p but wonders whether p is in violation of the Ignorance Norm for Wondering. So anyone who says “I’m in a position to know p, but I wonder: whether p?” has either failed to see an obvious and salient consequence of what they know or has violated the Ignorance Norm for Wondering.²⁹

But we shouldn’t think there is an independent norm prohibiting wondering about Q while being in a position to know its answer. I briefly sketch two reasons why not. First, there are cases (aside from the core examples of this paper) in which it seems permissible for agents to wonder despite being in a position to know the answer. Consider a sudoku puzzle. One can permissibly wonder—right up to the moment of discovery—which number goes in a particular box, all the while knowing that one already has all the information necessary to solve the puzzle. In such cases, wondering about a question or inquiring into it is a way of working out what one is already in a position to know rather than a way of getting in position to know something new.

Second, notice that while sentences of the form, “I know that p, but I wonder whether p?” preserve their awkwardness when exported into the third-person, the awkwardness of “I’m in a position to know that p, but I wonder whether p?” is restricted to the first-person. Compare:

• 5. #They know that the Yankees are winning, but they are wondering: are the Yankees winning?

• 6. ?They are in a position to know that the Yankees are winning, but they are wondering: are the Yankees winning?

Perhaps (6) still sounds a bit strained, but it can be made clearly better by denying that the subject knows what they are in a position to know:

• 7. They are in a position to know that the Yankees are winning—but they haven’t figured it out yet. So they are still wondering: are the Yankees winning?

This recovery is unavailable, however, for (5). So, the cross-personal pattern of linguistic data³⁰ that supports the Ignorance Norm for Wondering does not back a similar norm that prohibits wondering about Q while being in a (mere) position to know its answer.

Objection Four: But since the protagonists are wondering about Q, shouldn’t they be especially sensitive to whether they are in a position to know Q’s answer?

Not necessarily. They may be thinking about whether they are satisfied with the progress of their inquiry so far. What it takes to satisfy inquiry has an open texture: it depends on what the agent’s epistemic aims (explicit or implicit) in that context are. And, as we’ve noted, those aims could be more stringent than mere knowledge. It is (in part) this diversity of higher epistemic aims that allows agents to overlook the fact (even when it’s available to them) that they are in a position to successfully close inquiry with knowledge. And this is proper. After all, if the agent isn’t aiming at (mere) knowledge but at (some variety of) knowledge + , then they may be more sensitive to whether their inquiry has progressed to the point that can give knowledge + than to whether their inquiry has progressed to the point that can give mere knowledge.³¹

Recall that we noticed something similar when thinking about Elle’s study night. When it is proper for Elle to stop studying and go to sleep depends on facts about what grade she would get if she took the test without further study. But there’s no reason to insist that Elle is thinking about what grade she would get while she is studying. Rather, she wants to do well and at a certain point she feels that she is prepared enough. Of course, when she feels prepared enough will depend on what her academic goals are, but there is no reason she needs to be explicitly thinking about her academic goals in order for us to make sense of her decision about when to stop studying.

Just so, when it is proper for our protagonists to close inquiry depends on facts about what epistemic accomplishment they would achieve if they closed inquiry now. But there’s no reason to insist that the protagonists are thinking about their epistemic accomplishments while they are investigating. Rather, they want to do well in their inquiry and at a certain point they feel satisfied enough with their resulting epistemic position. Of course, when they feel satisfied will depend on what their epistemic goals are, but there is no reason they need to be explicitly thinking about their epistemic goals in order for us to make sense of their decision about when to close inquiry.

Objection Five: The protagonists are professionally permitted to continue wondering, not epistemically permitted. Sure, it’s better for detectives to be extra thorough and continue inquiry even when they could knowledgeably close inquiry, but (the objector insists) that is because the professional norms of detectives require unusually stringent standards beyond the norms of everyday epistemic agents.³²

But I don’t think the professional standards are actually interfering with our judgments of the case. For one thing, detectives, birders, and—one dares to hope—philosophers are supposed to be professionally good epistemic agents within their domains. If detectives do better to continue inquiry, it’s presumably because there is something epistemically better about continuing inquiry. It would be shocking to discover that the generic, professional norms of detectives require them to act in epistemically worse ways than in professions in which the aims are not explicitly intellectual.

That doesn’t mean there is no difference at all between the lay investigator and the detective. Perhaps the detective has a professional duty that we do not normally have—a duty to look at all the relevant, available evidence before concluding inquiry. But if so, that’s a duty for the detective to do something that would be epistemically praiseworthy (though perhaps not required) for any inquirer properly situated to do. What matters, ultimately, isn’t whether the agents are professionals but whether the agents are curious in the question in the sort of way that leads them to aim for knowledge + . And curiosity does not, by itself, smuggle in professional or otherwise non-epistemic norms.

Objection Six: The protagonists are only pragmatically permitted to continue wondering, not epistemically permitted. For example, perhaps it is pragmatically useful for the philosophy student to have a policy of wondering about questions that are likely to appear on an exam until they know the answers well enough to explain them because they are likely to get higher grades on their exams and essays if they maintain curiosity until they attain this kind of epistemic access to the answers. Similarly, the detective is likely to have a better professional track record if she continues wondering, Coco enjoys the process of getting out Collins Bird Guide, and so on.³³

Still, even when these negative practical consequences are built in, we think that the protagonists are positively evaluable in some sense. Their continued curiosity still manifests an admirable ambition for high epistemic achievement in the questions they are in fact taking up. That this kind of positive evaluation remains even when the cases are engineered so that continued wondering is pragmatically disadvantageous further suggests that our protagonists are epistemically rather than pragmatically permitted to continue inquiring.³⁴

4 Conclusion

In this paper, I’ve argued that The Knowledge Rule is false:

The Knowledge Rule: If one is taking up the question Q, then one should close inquiry into Q iff one would (upon closing inquiry) thereby know the complete answer to Q.

Indeed, although biconditional norms of this general form have been defended by a variety of epistemologists including Chisholm (1977), Feldman (2002), and Sosa (2021), no norm of this general form can be right. And that’s because The Knowledge Rule—and rules like it—are insensitive to the proper epistemic goals that inquirers can bring to an inquiry: goals that may be more ambitious than mere knowledge.

Given the Ignorance Norm for Wondering, ignorance is the price of inquiry. At least, of inquiry that is motivated by genuine wonder. One must forego knowledge if one wants to permissibly inquire into a question in that way. But the price may be reasonable if knowledge + is both the agent’s (desired enough) aim and (likely enough) reward.